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Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea: The Case of Papua New Guinea 303090993X, 9783030909932

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Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea: The Case of Papua New Guinea
303090993X, 9783030909932

Author / Uploaded
Patricia Paraide
Kay Owens
Charly Muke
Philip Clarkson
Christopher Owens

Table of contents :
Foreword
Series Preface: The Place of this Book in Springer’s History of Mathematics Education Series
Contents
Abbreviations
Chapter 1: Introduction to the History of Mathematics Education in Papua New Guinea
Beginning This Book
Geography
Cultures
Overview of Recent History
Major Changes in Education Since Independence
The Purposes of This Book
Brief Historical Overview of Western Education in PNG
Changes in Language Policies With Respect to Education
The Processes Adopted for Developing This Book
The Researchers and Authors
Processes for Developing Foundational Cultural Mathematics Education
Processes for Developing the History of Mathematics Education Since Colonial Times
Overview of the Book
Moving Forward
References
Chapter 2: Foundational Mathematical Knowledges: From Times Past to the Present—Technology
Introduction
Cameo from Charly Muke: Personal Story of Learning and Teaching Cultural Knowledge
Family and Cultural Background
Cultural Learning Before School
During School—Primary and Secondary Education
Tertiary Education and Employed Life
Cultural Techniques of Teaching and Learning
Mathematics Concepts Embedded within Cultural Activities
Conceptual Analysis
Mathematics Education Before European Influence
Oral Histories and Current Practices
First Contact with Europeans
Archaeological and Linguistic Studies
Evidence of Technology and Mathematics
String, Binding, Bilums and Tapa
Extraction of Minerals and Colors
Food Capture
Fish capture
Animal capture—nets and traps
Fighting and Animal Capture
Bows and Arrows
Spears
Shields
Design and Types of Shields
Color and Painting of Shields
Agriculture
Spades
Longevity and Necessity of Agriculture
Making Drains
Trenches
Stone Implements
Design
Tortoise-Shell Designs
Pottery and Pot Design
Antiquity of Pottery
Three-Dimensional Art
Drums
Musical Instruments
Food Implements
House Building
Recent Adaptations
Cooking, Drying and Storing Nuts, Ground Cooking and Baskets
Bridges
Cameo from Kay Owens
Concluding Comments
References
Chapter 3: Foundational Mathematical Konwledges: From Times Past to the Present—Trade and Intergenerational Knowledge Sharing
Introduction
Trade
Coast-to-Mountain Trade
Trading Routes
Island and Coastal Trade
North Coast Sailing
South Coast Sailing
Kula Ring
Shell
Obsidian Stone and Tools
The Impact of Colonialism on Trade
Wayfinding, Sailing, Canoes, Sails and Paddles
Sails
Large Canoes
Cameo from Kay and Chris Owens
Fast-River Canoes
Cameo on Canoe Racing and Mathematics
Racing of the Gogodola
Mathematics in the Traditional Racing of Gogodala
Time
Counting
Diversity of Counting Systems
Counting Objects, Shell Money and Classifiers
Intergenerational Knowledge Sharing
Pottery
String Art
Processes for Sharing Knowledge
Moieties and Clans
Observing, Discussing, and Communal Valuing
Sharing of Foundational Knowledge
Learning Foundational Medicinal Knowledge
Cameo by Patricia Paraide on Her Father’s Mathematical Activities and Education
Comment on this Tolai Foundational Learning
Moving Forward
Chapter 4: Mathematics Education from the Early Colonial Period, Before and After Both World Wars, Until the Early 1960s
Introduction
Early Contact and Colonial Times
New Guinea Pre-World War I: German Territory
Papua Pre-World War 1: British, Queensland, Then Australian Territory
Post World War I
Mandated Territory of New Guinea
Territory of Papua
The War Years
Cameo from Nagong Gejammec
Other Mission Schools During the War
Papua and New Guinea Territory Post World War II
The 1950s and 1960s
Moving Forward
References
Chapter 5: Before and After Independence: Community Schools, Secondary Schools and Tertiary Education, and Making Curricula Our Way
An Overview
Cameo from Roy Kirkby
Growth in Primary School Enrolments
Cameo from Patricia Paraide
Teachers
Cameo from Daniel and Carrie Luke and Others in the Yarning Circle in Sydney
Mid-1960s to Independence
Changes in the Education System
Growth in School Numbers and Gender Issues
The Developing Mathematics Curriculum
Teacher Education
Teachers’ Seminar
Cameo of Trevor Freestone
Cameo of Paulias Matane
Change Came with Independence, But Not Easily
Australian Policy Makers’ Perspectives on Change
Differences between Tololo’s and McKinnon’s Policies
Moving Forward
References
Chapter 6: Key Policies and Adjustments in the Decade After Independence
Education for Papua New Guinea
International Agency Schools
Cameo on International Schools by Kay Owens
Textbooks and Teacher Professional Development
Cameo from Kay Owens
Concerns About Standards of Education
Equity and Education
Gender Equity
Decentralization, and Provincial Equity
Moving Forward
References
Chapter 7: Higher Education for Mathematicsa and Mathematics Education: Research and Teachingb
Introduction
The University of Papua New Guinea (UPNG)
General Introduction
Mathematics Teaching
Cameo from Deane Arganbright
Faculty of Education
Papua New Guinea University of Technology (Unitech)
General Introduction
Mathematics at the PNG University of Technology
Mathematics Teaching
Mathematics Education Centre
Mathematics and Mathematics Education Research at Unitech
Gender Studies
Other Departments at the PNG University of Technology
University of Goroka
General Introduction
Cameo from David Shield
The Next Period of Development at the University of Goroka
Mathematics and Mathematics Education Teaching
The Glen Lean Ethomathematics Centre
Divine Word University
General Introduction
Mathematics Teaching
Cameo from Deane Arganbright
Mathematics Education Teaching
Pacific Adventist University
PNG University of Natural Resources and Environment
Western Pacific University
Primary Teachers Colleges
General Introduction
History of Teacher Education
The Teaching of Mathematics and Mathematics Education
Technical Colleges and Institutes of Technology, and Vocational Centers
Distance Education and Flexible Learning Institutes
Concluding Comments
References
Chapter 8: The Reform Period: Major Changes and Issues in Practice
The Birth of Education Reform
Research Studies
Numbers of Qualified Mathematics Teachers
National and International Committees
Structural Changes for the Reform of Education
Impact of the Reform
Implementing Change
Financial Issues for Teachers and the Impact of Fees or No Fees
No Fees
History of Curriculum Changes
Education for Rural Living
Inclusive Education
Teacher Education
Concerns about Teacher Education
Primary and Secondary Teacher Education Project (PASTEP)
Elementary and Lower Primary Mathematics Curricula
Language of Instruction in Formal Schooling
Ethnomathematics
Formal Integration of Indigenous and Western Knowledge
Moving Forward
References
Chapter 9: Revising the Reform: Standards Based Education
Introduction
Disquiet and Politics Create a Change to Curriculum and Language of Instruction
Cameo from Kay Owens
The Continuing Debate on Vernacular Languages for Instruction and Cultural Content
Cameo from Patricia Paraide
The Binary Divide
Task Force Report for the Review of Outcomes-Based Education (OBE)
Standards Based Education
Equity and Social Justice
Teacher Quality and Ways of Overcoming Difficulties
Cameo from Kay Owens
Revised Structure of Education
Funding and Change
Tuition Fee-Free Education and Accountability
Moving Forward
References
Chapter 10: Mathematics Education and Language*
Introduction
Politics of Language in Papua New Guinea
Students and Learning Mathematics in Papua New Guinea
Cameo from Patricia Paraide
Cameo from Charly Muke
Cameo from Philip Clarkson
Yarning in Sydney with PNG Nationals
Results from East New Britain
Some Theoretical Underpinnings
Teachers and Teaching in Papua New Guinea
Cameo from Kay Owens
Cameo from Charly Muke
Cameo from Patricia Paraide
Other Anecdotal Data
Indigenous Mathematics and Language
An Earlier Exception: The Tok Ples Skul Movement
Valuing Vernacular Languages in Education
Teachers Use of Indigenous Language When Teaching Mathematics
Mathematical Classroom Discourse
Non-Mathematical Classroom Discourse
Teachers’ Reasons for Using Code-switching
Summary Comments
Teacher Education in Papua New Guinea
Cameo from Philip Clarkson
Introducing Innovations
Professional Learning for Teachers
Preservice Education
Concluding Comments
Moving Forward
References
Chapter 11: Visuospatial Reasoning, Calculators and Computers
Introduction
Highlights of Foundational Mathematics Visuospatial Reasoning
Studies Conducted in the 1970s which Focused on Papua New Guinea Students’ Learning
Visualisation and Spatial Abilities Research
Later Research in PNG on Visuospatial Reasoning
Language for Location and Direction
Technologies and Mathematics in Papua New Guinea
Research on Calculators
Computers in Mathematics
Computers in Schools
References
Chapter 12: The Impact of Globalization, Colonialism and Neocolonialism on Education in Papua New Guine
Introduction
Globalization
Decentralization
Curriculum
Examinations
Local Focus for Education
Colonialism and Neocolonialism
Language as a Tool in Colonial and Neocolonial Approaches to Education
Reform Period and Neocolonialism
Revising the Curriculum and Structure of Education
The Role of Tertiary Education in Neocolonialism
Hegemony of Neocolonialism
Teaching Perspectives
“Look North”: To Neocolonialism Asian Styles
Moving Forward
References
Chapter 13: Moving Forward: Overcoming Neocolonialism in Education in Papua New Guinea
Introduction
Comparative Studies on Ethnomathematics in Schools
Northern Australia
Autonomous Region of Bougainville
Other South Pacific Countries
Solomon Islands
West Papua
Indonesia
Other Colonies
Nepal
First Nations in America
Privilege and Equity
Re-Examining Dialogue on Attitudes and Purpose
Concern for Identity and Valuing Identity
Funding Crisis and More Neocolonialism
Financing Education
Overcoming Colonialism/Neocolonialism Through Vernacular Languages
Cameo from Kay Owens
Recognizing the Cognitive Advantage of Bilingual Education
Equity
Gender Equity
Colonialism, Globalization and Gender Issues
A New Approach to Curriculum
Learning and Teaching
Translocal
Glocalization
The Value of Ethnomathematics and Ethnomodelling
A Warning to Those Intending to Develop a Translocal, Glocalized Curriculum
Conclusions
References
Appendix 1: Brief View of History of Early Contact
Early Settlers
Mainland Patrols
Detailed Accounts
Appendix 2: Teachers Colleges
Appendix 3: University Materials
PNG University of Technology –
MA245
Appendix 4: Selection of Pages and Information from School Curriculum
The IEA Curriculum: Mathematics 1997
The Papua New Guinea Context
Background Information
Creating a PNG Context for Mathematics
Ensuring Equity
The Mathematics Curriculum
Curriculum Outcomes
Mathematics and the IEA Key Outcomes
Reform Elementary School Cultural Mathematics
Materials for Schools
Elementary Teachers Guide 2003
Ministerial Policy Statement No. 01/2013 Dated 28/1/2013
Elementary Syllabus 2015
Curriculum Principles
Guiding Principles
Content Overview
Assessment
Recording
Reporting
Evaluation
Junior Primary Syllabus Grades 3-5
Rationale
Aims
Overarching National Benchmark
Level Benchmark
Grade Benchmarks
Curriculum Principles
Teaching
Learning
Equity
Curriculum
Assessment
Technology
Draft Teachers Guide for Elementary Teachers 2015
Lower Secondary Syllabus 2009
Combined References List
Index

Citation preview

History of Mathematics Education

Patricia Paraide · Kay Owens · Charly Muke Philip Clarkson · Christopher Owens

Mathematics Education in a Neocolonial Country The Case of Papua New Guinea

History of Mathematics Education Series Editors Nerida F. Ellerton, Illinois State University

Normal, IL, USA M. A. (Ken) Clements, Illinois State University Normal, IL, USA

History of Mathematics Education aims to make available to scholars and interested persons throughout the world the fruits of outstanding research into the history of mathematics education; provide historical syntheses of comparative research on important themes in mathematics education; and establish greater interest in the history of mathematics education. More information about this series at http://link.springer.com/series/13545

Patricia Paraide • Kay Owens • Charly Muke Philip Clarkson • Christopher Owens

Mathematics Education in a Neocolonial Country The Case of Papua New Guinea

Patricia Paraide Education Researcher and Consultant Port Moresby, Papua New Guinea Charly Muke Former Provincial Education Adviser Jiwaka Province, Papua New Guinea Christopher Owens Retired, now deceased Sydney, NSW, Australia

Kay Owens Adjunct Associate Professor School of Education Faculty of Arts and Education - Dubbo Campus Charles Sturt University Sydney, NSW, Australia Philip Clarkson Emeritus Professor Faculty of Education - Melbourne Campus Australian Catholic University Fitzroy, VIC, Australia

ISSN 2509-9736 ISSN 2509-9744 (electronic) History of Mathematics Education ISBN 978-3-030-90993-2 ISBN 978-3-030-90994-9 (eBook) https://doi.org/10.1007/978-3-030-90994-9 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

We cannot build a nation simply from technology; we cannot build a nation purely [based on] the wheel and … the steam engine. We must build this country; we must build our civilization on values, which have been passed on to us from generation to generation. And I say this: that if we do not, … if it is not now the basis and the stem upon which we nurture and grow our children, then I say there will be no future for this country. Narakobi, 1991. This book, a collaboration between Patricia Paraide, Kay Owens, Charly Muke, Philip Clarkson, and Chris Owens, provides a significant milestone for mathematical education in Papua New Guinea (PNG). The quote by Bernard Narakobi is an apt way to describe what this book aims to provide, namely a history of teaching and learning of mathematics in the home, the schools, and tertiary institutions in PNG. It is a gallant effort in that the authors firstly identify that thinking and processes of logic and making meaning mathematically existed within PNG among the numerous distinctive cultural groups. Secondly, the introduction of western mathematical knowledge and its processes through colonization (schools and schooling) provisioned a deliberate and gradual annihilation of Indigenous processes of logic and meaning making. What is even more audacious is the enunciation by the authors to identify that Papua New Guineans educated in schools have and continue to contribute to the demise of what were original and ancient mathematical ways of knowing and making meaning. Thirdly, the authors provide an opportunity to seek redress in the current context of the neocolonial situation to deny one’s own culture in favour of the colonizer’s ways. Each chapter, and there are 13 in total, is given a title followed by a short abstract and keywords of the chapter for the reader. What is unusual for such a publication but appropriate for this study is that for each chapter there is one or more sections that are called “a cameo from …” one of the authors or others. This is a rather ingenious way to engage with the reader through a personal encounter of one of the authors or others involved with mathematics education in PNG. Experiences within PNG by each author is shared and is in the context of the chapter. The ideas of logic and mathematical thinking and processes are humanized by this strategy—sharing of experiences through stories is a Papua New Guinean way and is a real bonus to this book. Some of the key progenitors to Indigenous mathematics in PNG, including Glen Lean, are included and referenced. Together with these pathfinders, experiences in relation to changing colonial systems to include Indigenous ways of mathematical thinking and sense-making in other cultures are also shared—especially with our nearest neighbour, and PNG’s nemesis, Australia, in its loss of Indigenous ways of mathematical logic.

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Foreword

This book provides hope, and a commencement to the realization of wisdom from the pantheon of PNG’s finest, Bernard Narakobi; we must rebuild and give hope to our future. Michael A. Mel Mbu Noman Mt Hagen, Papua New Guinea Reference Narakobi, B. (1991). Issues in education and development. In B. Avalos & L. Neuendorf (Eds.), Teaching in Papua New Guinea: A perspective for the nineties (pp. 19-28). Port Moresby, Papua New Guinea: University of PNG Press.

Series Preface: The Place of this Book in Springer’s History of Mathematics Education Series

This is a particularly important book in Springer’s “History of Mathematics Education” series. The authors, Patricia Paraide, Kay Owens, Charly Muke, Philip Clarkson, and Chris Owens, raise two questions which one might think would have clear answers—those questions are: (a) “What is mathematics?” and (b) “What factors need to be taken into account to assist students to learn mathematics well?” The questions, with possible answers, are writ large on almost every page of this book, which looks at mathematics and mathematics education dimensions for the nation of Papua New Guinea (PNG), from cultural, historical, and mathematical vantage points. The authors argue, persuasively, that different forms of mathematics have been developing, for thousands of years in that part of the world which is now called Papua New Guinea. Remarkably, there are almost 1000 different languages spoken within Papua New Guinea, and each has its own counting system, ways of thinking about, and expressing, numerical, visuospatial, measurement, and forms of reasoning. The authors’ overwhelming and clear message is that ethnomathematical considerations need to be taken seriously by all teachers of mathematics. This book is particularly strong in its treatment of how language and cultural factors should frame the teaching and learning of mathematics because those factors inevitably mould how students think, how they learn, how they live, and how they communicate with others. There are sections in the book which will be valuable to teachers who work with students for whom the language of instruction is not their first language. The view that whenever possible a student will learn best when he or she understands the language which the teacher is using is argued strongly (and, probably, controversially, for some readers). Impressive historical analyses of mathematics education documents from Papua New Guinea over the past 150 years are provided. Oral histories and lived experiences support the analyses. Books in Springer’s series on the history of mathematics education comprise scholarly works on a wide variety of themes, prepared by authors from around the world. We expect that authors contributing to the series will go beyond top-down approaches to mathematics and history, so that emphasis will be placed on the learning, teaching, assessment, and wider cultural and societal issues associated with schools (at all levels), with adults and, more generally, with the roles of mathematics within various societies. In addition to generating texts on the history of mathematics education written by authors in various nations, an important aim of the series is to develop and report syntheses of historical research that has already been carried out in different parts of the world with respect to important themes in mathematics education—for example, “Historical Perspectives on how Language Factors Influence Mathematics Teaching and Learning,” and “Historically Important Theories Which Have Influenced the Learning and Teaching of Mathematics.” The mission for the series can be summarized as:

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Series Preface: The Place of this Book in Springer’s History of Mathematics Education Series

• To make available to scholars and interested persons around the world the fruits of outstanding research into the history of mathematics education • To provide historical syntheses of comparative research on important themes in mathematics education • To establish greater interest in the history of mathematics education In this book, the authors tell a story of the history of mathematics education in PNG, paying special attention to cultural and language factors. Of particular interest are the profound effects these factors have had, and continue to have, on fundamental issues which affect mathematics in PNG schools today: “What should be the intended mathematics curriculum in a school?” Should it be the same curriculum as in any other school? What about outcomes-based education” Standards-based curricula? Learning progressions? “Should the intended curricula be the same for all learners?” And, “Who should be responsible for bringing about changes to school mathematics?” Although the context for the book is mathematics education in PNG, it provides an excellent model for future books in this series—studies which address critical periods in the historical evolution of mathematics education in different cultures and different parts of the world. We hope that the series will continue to provide a multi-layered canvas portraying rich details of mathematics education from the past, while at the same time presenting historical insights that can support the future. This is a canvas which can never be complete, for today’s mathematics education becomes history for tomorrow. A single snapshot of mathematics education today is, by contrast with this canvas, flat and unidimensional—a mere pixel in a detailed image. We encourage readers both to explore and contribute to the detailed image which is beginning to take shape on the canvas for this series. Any scholar contemplating the preparation of a book for the series is invited to contact Nerida Ellerton ([email protected]), in the Department of Mathematics at Illinois State University or Melissa James, at Springer’s New York office. Nerida Ellerton M. A. (Ken) Clements Illinois State University, IL, USA

Contents

1 I ntroduction to the History of Mathematics Education in Papua New Guinea �� 1 Beginning This Book�� 1 Geography�� 2 Cultures�� 2 Overview of Recent History�� 3 Major Changes in Education Since Independence �� 5 The Purposes of This Book�� 6 Brief Historical Overview of Western Education in PNG�� 7 Changes in Language Policies With Respect to Education�� 8 The Processes Adopted for Developing This Book�� 9 The Researchers and Authors �� 11 Processes for Developing Foundational Cultural Mathematics Education�� 14 Processes for Developing the History of Mathematics Education Since Colonial Times �� 14 Overview of the Book�� 16 Moving Forward�� 17 References�� 17 2 F oundational Mathematical Knowledges: From Times Past to the Present—Technology �� 21 Introduction�� 21 Cameo from Charly Muke: Personal Story of Learning and Teaching Cultural Knowledge �� 22 Family and Cultural Background �� 22 Cultural Learning Before School�� 23 During School—Primary and Secondary Education�� 26 Tertiary Education and Employed Life�� 27 Cultural Techniques of Teaching and Learning�� 27 Mathematics Concepts Embedded within Cultural Activities�� 28 Conceptual Analysis �� 29 Mathematics Education Before European Influence�� 29 Oral Histories and Current Practices�� 29 First Contact with Europeans �� 30 Archaeological and Linguistic Studies�� 31 Evidence of Technology and Mathematics�� 31 String, Binding, Bilums and Tapa�� 32 Extraction of Minerals and Colors �� 34 Food Capture�� 35 Fish capture�� 35 ix

x

Contents

Animal capture—nets and traps �� 36 Fighting and Animal Capture �� 37 Bows and Arrows �� 37 Spears �� 39 Shields�� 39 Design and Types of Shields�� 41 Color and Painting of Shields�� 42 Agriculture �� 42 Spades�� 42 Longevity and Necessity of Agriculture�� 43 Making Drains�� 44 Trenches�� 44 Stone Implements�� 45 Design�� 46 Tortoise-Shell Designs�� 47 Pottery and Pot Design�� 47 Antiquity of Pottery�� 51 Three-Dimensional Art�� 52 Drums �� 52 Musical Instruments �� 53 Food Implements�� 54 House Building�� 55 Recent Adaptations�� 57 Cooking, Drying and Storing Nuts, Ground Cooking and Baskets�� 58 Bridges �� 59 Cameo from Kay Owens�� 61 Concluding Comments�� 62 References�� 62 3 F oundational Mathematical Konwledges: From Times Past to the Present—Trade and Intergenerational Knowledge Sharing �� 67 Introduction�� 67 Trade�� 67 Coast-to-Mountain Trade�� 68 Trading Routes �� 69 Island and Coastal Trade�� 70 North Coast Sailing�� 70 South Coast Sailing�� 70 Kula Ring�� 71 Shell�� 72 Obsidian Stone and Tools �� 72 The Impact of Colonialism on Trade�� 73 Wayfinding, Sailing, Canoes, Sails and Paddles�� 74 Sails�� 75 Large Canoes�� 75 Cameo from Kay and Chris Owens�� 75 Fast-River Canoes�� 78 Cameo on Canoe Racing and Mathematics�� 78 Racing of the Gogodola�� 79 Mathematics in the Traditional Racing of Gogodala�� 80 Time�� 81

Contents

xi

Counting �� 81 Diversity of Counting Systems�� 82 Counting Objects, Shell Money and Classifiers �� 84 Intergenerational Knowledge Sharing�� 84 Pottery�� 85 String Art�� 85 Processes for Sharing Knowledge�� 86 Moieties and Clans �� 86 Observing, Discussing, and Communal Valuing�� 86 Sharing of Foundational Knowledge�� 88 Learning Foundational Medicinal Knowledge�� 89 Cameo by Patricia Paraide on Her Father’s Mathematical Activities and Education �� 90 Comment on this Tolai Foundational Learning�� 91 Moving Forward�� 91 4 M athematics Education from the Early Colonial Period, Before and After Both World Wars, Until the Early 1960s�� 97 Introduction�� 98 Early Contact and Colonial Times �� 98 New Guinea Pre-World War I: German Territory�� 99 Papua Pre-World War 1: British, Queensland, Then Australian Territory�� 101 Post World War I�� 103 Mandated Territory of New Guinea�� 103 Territory of Papua�� 105 The War Years�� 108 Cameo from Nagong Gejammec�� 109 Other Mission Schools During the War�� 109 Papua and New Guinea Territory Post World War II �� 109 The 1950s and 1960s�� 112 Moving Forward�� 115 References�� 116 5 B efore and After Independence: Community Schools, Secondary Schools and Tertiary Education, and Making Curricula Our Way�� 119 An Overview�� 119 Cameo from Roy Kirkby�� 120 Growth in Primary School Enrolments�� 121 Cameo from Patricia Paraide�� 123 Teachers�� 124 Cameo from Daniel and Carrie Luke and Others in the Yarning Circle in Sydney �� 125 Mid-1960s to Independence �� 126 Changes in the Education System�� 127 Growth in School Numbers and Gender Issues�� 128 The Developing Mathematics Curriculum�� 130 Teacher Education�� 132 Teachers’ Seminar�� 134 Cameo of Trevor Freestone�� 135 Cameo of Paulias Matane�� 136 Change Came with Independence, But Not Easily�� 139 Australian Policy Makers’ Perspectives on Change�� 139

xii

Contents

Differences between Tololo’s and McKinnon’s Policies�� 141 Moving Forward�� 146 References�� 146 6 K ey Policies and Adjustments in the Decade After Independence�� 149 Education for Papua New Guinea�� 149 International Agency Schools�� 151 Cameo on International Schools by Kay Owens�� 151 Textbooks and Teacher Professional Development�� 152 Cameo from Kay Owens�� 153 Concerns About Standards of Education�� 156 Equity and Education �� 157 Gender Equity�� 159 Decentralization, and Provincial Equity�� 160 Moving Forward�� 160 References�� 161 7 H igher Education for Mathematicsa and Mathematics Education: Research and Teachingb�� 163 Introduction�� 164 The University of Papua New Guinea (UPNG) �� 165 General Introduction�� 165 Mathematics Teaching�� 165 Cameo from Deane Arganbright�� 165 Faculty of Education�� 169 Papua New Guinea University of Technology (Unitech)�� 170 General Introduction�� 170 Mathematics at the PNG University of Technology�� 173 Mathematics Teaching�� 174 Mathematics Education Centre�� 177 Mathematics and Mathematics Education Research at Unitech�� 179 Gender Studies�� 179 Other Departments at the PNG University of Technology�� 180 University of Goroka�� 181 General Introduction�� 181 Cameo from David Shield�� 181 The Next Period of Development at the University of Goroka�� 182 Mathematics and Mathematics Education Teaching�� 182 The Glen Lean Ethomathematics Centre�� 183 Divine Word University �� 185 General Introduction�� 185 Mathematics Teaching�� 186 Cameo from Deane Arganbright�� 186 Mathematics Education Teaching�� 187 Pacific Adventist University�� 187 PNG University of Natural Resources and Environment�� 188 Western Pacific University �� 188 Primary Teachers Colleges �� 188 General Introduction�� 188 History of Teacher Education �� 190 The Teaching of Mathematics and Mathematics Education�� 191 Technical Colleges and Institutes of Technology, and Vocational Centers�� 194

Contents

xiii

Distance Education and Flexible Learning Institutes�� 194 Concluding Comments�� 195 References�� 196 8 T he Reform Period: Major Changes and Issues in Practice�� 201 The Birth of Education Reform�� 202 Research Studies�� 203 Numbers of Qualified Mathematics Teachers�� 205 National and International Committees�� 205 Structural Changes for the Reform of Education �� 206 Impact of the Reform �� 207 Implementing Change�� 207 Financial Issues for Teachers and the Impact of Fees or No Fees�� 209 No Fees�� 209 History of Curriculum Changes �� 210 Education for Rural Living�� 212 Inclusive Education�� 212 Teacher Education�� 213 Concerns about Teacher Education�� 214 Primary and Secondary Teacher Education Project (PASTEP)�� 215 Elementary and Lower Primary Mathematics Curricula�� 216 Language of Instruction in Formal Schooling�� 218 Ethnomathematics�� 218 Formal Integration of Indigenous and Western Knowledge�� 219 Moving Forward�� 222 References�� 222 9 R evising the Reform: Standards Based Education �� 229 Introduction�� 230 Disquiet and Politics Create a Change to Curriculum and Language of Instruction�� 230 Cameo from Kay Owens�� 231 The Continuing Debate on Vernacular Languages for Instruction and Cultural Content�� 232 Cameo from Patricia Paraide�� 233 The Binary Divide�� 233 Task Force Report for the Review of Outcomes-Based Education (OBE)�� 234 Standards Based Education�� 235 Equity and Social Justice�� 238 Teacher Quality and Ways of Overcoming Difficulties�� 239 Cameo from Kay Owens�� 240 Revised Structure of Education�� 240 Funding and Change�� 244 Tuition Fee-Free Education and Accountability�� 244 Moving Forward�� 245 References�� 246 10 M athematics Education and Language*�� 249 Introduction�� 249 Politics of Language in Papua New Guinea�� 252 Students and Learning Mathematics in Papua New Guinea�� 253 Cameo from Patricia Paraide �� 253 Cameo from Charly Muke�� 254

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Contents

Cameo from Philip Clarkson�� 255 Yarning in Sydney with PNG Nationals�� 256 Results from East New Britain �� 257 Some Theoretical Underpinnings �� 257 Teachers and Teaching in Papua New Guinea �� 262 Cameo from Kay Owens�� 262 Cameo from Charly Muke�� 263 Cameo from Patricia Paraide�� 264 Other Anecdotal Data �� 264 Indigenous Mathematics and Language�� 266 An Earlier Exception: The Tok Ples Skul Movement�� 268 Valuing Vernacular Languages in Education�� 269 Teachers Use of Indigenous Language When Teaching Mathematics �� 269 Mathematical Classroom Discourse �� 270 Non-Mathematical Classroom Discourse�� 272 Teachers’ Reasons for Using Code-switching�� 272 Summary Comments�� 273 Teacher Education in Papua New Guinea�� 273 Cameo from Philip Clarkson�� 274 Introducing Innovations�� 275 Professional Learning for Teachers�� 275 Preservice Education�� 277 Concluding Comments�� 278 Moving Forward�� 283 References�� 283 11 V isuospatial Reasoning, Calculators and Computers�� 289 Introduction�� 290 Highlights of Foundational Mathematics Visuospatial Reasoning�� 290 Studies Conducted in the 1970s which Focused on Papua New Guinea Students’ Learning�� 290 Visualisation and Spatial Abilities Research�� 293 Later Research in PNG on Visuospatial Reasoning �� 296 Language for Location and Direction�� 298 Technologies and Mathematics in Papua New Guinea�� 299 Research on Calculators �� 300 Computers in Mathematics�� 301 Computers in Schools�� 303 References�� 305 12 T he Impact of Globalization, Colonialism and Neocolonialism on Education in Papua New Guine�� 311 Introduction�� 311 Globalization�� 314 Decentralization�� 315 Curriculum�� 315 Examinations�� 316 Local Focus for Education�� 317 Colonialism and Neocolonialism �� 318 Language as a Tool in Colonial and Neocolonial Approaches to Education �� 320 Reform Period and Neocolonialism �� 323 Revising the Curriculum and Structure of Education�� 325

Contents

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The Role of Tertiary Education in Neocolonialism�� 326 Hegemony of Neocolonialism�� 330 Teaching Perspectives�� 333 “Look North”: To Neocolonialism Asian Styles�� 335 Moving Forward�� 337 References�� 338 13 M oving Forward: Overcoming Neocolonialism in Education in Papua New Guinea�� 345 Introduction�� 346 Comparative Studies on Ethnomathematics in Schools�� 346 Northern Australia�� 346 Autonomous Region of Bougainville�� 347 Other South Pacific Countries�� 348 Solomon Islands �� 349 West Papua �� 350 Indonesia�� 351 Other Colonies�� 353 Nepal�� 353 First Nations in America�� 353 Privilege and Equity �� 354 Re-Examining Dialogue on Attitudes and Purpose�� 354 Concern for Identity and Valuing Identity�� 356 Funding Crisis and More Neocolonialism �� 357 Financing Education�� 357 Overcoming Colonialism/Neocolonialism Through Vernacular Languages�� 358 Cameo from Kay Owens�� 360 Recognizing the Cognitive Advantage of Bilingual Education�� 361 Equity �� 362 Gender Equity�� 362 Colonialism, Globalization and Gender Issues�� 362 A New Approach to Curriculum�� 364 Learning and Teaching �� 365 Translocal �� 367 Glocalization�� 369 The Value of Ethnomathematics and Ethnomodelling �� 369 A Warning to Those Intending to Develop a Translocal, Glocalized Curriculum�� 372 Conclusions�� 372 References�� 374 Appendix 1: Brief View of History of Early Contact�� 381 Early Settlers�� 381 Mainland Patrols�� 381 Detailed Accounts�� 382 Appendix 2: Teachers Colleges �� 395 Affiliation of Teachers Colleges, Their Inception and Reduction�� 395 Balob Teachers College, 1998�� 397 Madang Teachers College, 1998�� 399

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Appendix 3: University Materials�� 407 PNG University of Technology�� 407 Appendix 4: Selection of Pages and Information from School Curriculum�� 415 The IEA Curriculum: Mathematics 1997�� 415 The Papua New Guinea Context �� 415 Background Information�� 415 Creating a PNG Context for Mathematics �� 416 Ensuring Equity �� 416 The Mathematics Curriculum �� 418 Curriculum Outcomes�� 418 Mathematics and the IEA Key Outcomes�� 418 Reform Elementary School Cultural Mathematics�� 424 Materials for Schools �� 426 Elementary Teachers Guide 2003 �� 427 Ministerial Policy Statement No. 01/2012 Dated 28/1/2013�� 429 Elementary Syllabus 2015�� 430 Curriculum Principles�� 430 Guiding Principles �� 430 Content Overview�� 431 Assessment�� 432 Recording�� 432 Reporting �� 432 Evaluation�� 432 Junior Primary Syllabus Grades 3-5 �� 434 Rationale�� 434 Aims�� 434 Overarching National Benchmark�� 434 Level Benchmark �� 435 Grade Benchmarks �� 435 Curriculum Principles�� 435 Teaching�� 437 Learning�� 437 Equity�� 437 Curriculum�� 437 Assessment�� 438 Technology�� 438 Draft Teachers Guide for Elementary Teachers 2015�� 441 Standards Based Teachers Guide�� 443 Lower Secondary Syllabus 2009�� 451 Combined References List�� 457 Index�� 491

Abbreviations

ATE AusAID BSc CHE DOE GEEP GESP ICT IDCE LMS MAB MEd NDOE NEB NES NHS NSW OBE PhD PNG PRIDE QLD SA SBE UN UNDP Unitech USA UOG UPNG Vic

Association of Teacher Education Australian Aid, now part of Department of Foreign Affairs and Trade Bachelor of Science Committee for Higher Education Department of Education, Papua New Guinea Gender Equity in Education Policy Gender Equity Strategic Plan Information and Communication Technologies Institute of Distance and Continuing Education (UPNG) London Missionary Society Multibase Arithmetic Blocks (Dienes Blocks) including 2, 4, 5, 6, and 10 bases Master of Education National Department of Education. It is also covering Ministry of Education and Department of Education National Education Board National Education System National High Schools (schools for Years 11 and 12 only) New South Wales, State of Australia Outcomes-Based Education Doctor of Philosophy Papua New Guinea Pacific Regional Initiative for Delivery of Education Queensland, State of Australia South Australia, State of Australia Standards Based Education United Nations United Nations Development Programme PNG University of Technology United States of America (note abbreviations to States in the publication list refer to the States of USA if not stipulated) University of Goroka University of Papua New Guinea Victoria, State of Australia

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Chapter 1 Introduction to the History of Mathematics Education in Papua New Guinea

Abstract: Papua New Guinea, with its ancient cultures and recent colonization, has a unique history. It provides a revealing, and fascinating, case study of how mathematics and mathematics education have changed over a long period of time. We present reasons for claiming that Indigenous forms of mathematics and mathematics education were present well before colonization. PNG’s ecology and multiple diverse languages and cultures provide the backdrop for a discussion on practices, policies, and politics which affect education, including mathematics education. Discussion will focus on specific areas of mathematics education that have been influenced by policies, research, circumstances, and other factors. The methodology for developing the book involves personal reflections and oral histories, as well as diverse records from first contact, archaeology, anthropology, official documents, and journal articles. Recent history of mathematics education in Papua New Guinea has been influenced by the nation’s short but complex colonization, its achievement of independence, reform policies, aid agencies, and Government ambitions. This chapter introduces the country and outlines the chapters and the structure of each chapter which intertwine documentary research with cameos of reflection and oral history.

Key Words: Colonial impact · Ethnomathematics · History of Papua New Guinea · Language of instruction There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation. Charles Hermite, nd1

Beginning This Book

Papua New Guinea is a nation and we, the authors, have all proudly contributed to the recent mathematics education of its people—which we acknowledge actually began thousands of generations ago. We are all fascinated by the history of mathematics education before and since European contact. The history of the nation and its education has twists and turns that astound us. From this history, key issues emerge which have been faced, and continue being faced, by many developing nations, especially those with multicultural communities. We expect that the details of our historical research will carry readers through the issues and provide food for thought on From http://www-history.mcs.st-and.ac.uk/Quotations/Hermite.html

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ways of improving mathematics education for future generations both within and outside of Papua New Guinea. First, we need to set the scene for education in this incredible country, which has been variously called a “lost paradise,” “the land of the unexpected,” and “the place of a million different journeys” (tourism slogans over the years). Its diversity is second to none in the world. The land itself covers tropical islands, coral reefs, swamps, high steep mountains, and wide lowland and highland valleys. There are rich soils and poor soils for agriculture, and there are earths full of mineral deposits. There is plentiful water, too much water, and not enough water. Its forests, corals, and unique wildlife still abound. The cultures and languages of PNG are multiple, deep, and colorful. Its riches and diversity are extraordinary. The possibilities for subsistence living continue for most. Adaptations and innovations abound. The cultures are among the longest surviving in the world. So, we start by providing a brief overview of the geography, cultures, and recent history of Papua New Guinea. Geography Papua New Guinea consists of the eastern half the island of New Guinea and several large islands such as New Britain, New Ireland, and the Autonomous Region of Bougainville (ARoB), together with thousands of smaller islands from the size of Manus to only a few hectares. It lies close to the north of Australia, just south of the equator. It experiences heavy rainfall in most areas, creating lush rainforests, fast flowing rivers, and swamps. It lies on the moving plates of the earth’s crust and has high mountain ranges, upland valleys, and numerous active and inactive volcanoes. It shares a border on the mainland with West Papua,2 which is under Indonesian occupation. The main island and many others formed part of Sahul with Australia in times past with a short straight of water dotted with islands to the west (south-east Asia) and another extended land mass joining Bougainville Island with the Solomon Islands further to the south. It is known that, since time immemorial, people have moved between these places in this area. To do so required not only some form of water-craft but also the ability to adapt to walking long distances in different and difficult terrains. Furthermore, through the millennia, inhabitants adapted to the end of the ice age and the following mini-ice age. Occupation sites have been found across this country dating from 40 000 years ago, and further south in Sahul to 60 000 years ago. Cultures Currently, PNG has a population of approximately nine million people which has doubled since Independence from Australia was achieved in 1975. It is one of the world’s most linguistically diverse nations, with more than 850 vernacular languages and cultures (National Statistical Office, 2014). Indigenous languages are used, with much pride, as a form of identity and solidarity in all Indigenous communities. Indigenous languages also provide a bond for Indigenous groups in urban areas throughout PNG. Languages are classified into two broad groups. There are a large number of Papuan languages, also called “Non-Austronesian,” and these languages are in several phyla, the largest and most recent one being the Trans New Guinea Phylum. However, the diversity of these languages is astounding and even a glimpse of their structures, and associated counting systems and dances,

We use this term as the one most commonly used by the Indigenous peoples of the western half of the island and surrounding islands. They are Melanesians as are Papua New Guineans, and there are similarities of culture and languages within their diversity. 2

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reveal this diversity. Linguists have determined that a Proto Trans New Guinea language existed at least 10 000 years ago. The remaining languages are classified as “Austronesian Oceanic.” These developed in the New Britain area around 6 000 years ago and spread through New Ireland to Manus, to the northern coastline of New Guinea, and south around the Papuan coast, and across to Bougainville, the rest of Near Melanesia or Island Melanesia, then out as far east as Fiji. Around Tonga, with other influences on Proto Oceania’s daughters, the Polynesian languages developed and spread around the other Pacific Islands and in some cases returned to Island Melanesia. These Oceanic migrations are often associated with the material culture of pottery with some distinct styles (called “Lapita,” which is the name of the place where it was first found) and obsidian, a hard dark-colored rock. Surprisingly, unlike the Papuan languages, these Austronesian Oceanic languages have similarities despite the thousands of kilometres of ocean which separate them. There are connections between some of the languages, and in some ways to other Austronesian and Pacific languages in Polynesia and Micronesia. Finally, it is known that some of the Papuan non-Austronesian languages have influenced the Austronesian languages, and vice versa, especially through trade and migration, and these changes point to further influence and use of mathematics over the millennia and in more recent history.

Overview of Recent History

Don Jorge de Meneses, a Portuguese explorer, is said to have discovered the principal island of Papua New Guinea in 1526 or 1527 CE. Subsequently, three European countries colonized New Guinea. These were the Netherlands followed by Indonesia, in the western half of the island; Germany in the north of the eastern half and the neighboring islands (from 1884); and, later, England in the south of the eastern half and adjacent islands. This southern section was once known as British New Guinea, and administered through Queensland, Australia. In 1905, after Australia had become an independent Federated nation in 1901, it became the Australian Territory of Papua. The colonies started well before 1900 with missionaries and traders. Visits were also made by researchers, such as the British Expedition to the South Seas. Governments provided law and order, which was mainly applied to the traders. Missionaries formalized the early, pre-existing localized schemes of education—the London Missionary Society (LMS), for example, began schools in Papua, and the Methodists in areas of New Britain which were to come under German control. Lutherans set up on the nearby mainland and on islands off these shores, and the Catholics moved into some of the village areas. England, through Queensland (which was later to become a State of Australia), ran the Papuan Territory using education as a means for protecting the “natives”3 from the exploiters, plantation owners and others. These three European countries—The Netherlands, Germany, and England, brought different approaches to education but they also had intertwined histories as far as Papua and New Guinea were concerned, due to the two World Wars. During the First World War (1914–1917), Australian troops captured Rabaul and took control of German New Guinea. Further educational developments occurred during Australian colonization of the Mandated New Guinea Territory and the Papuan Trust Territory, including a period after the Second World War when they were jointly administered, and during the preparation for Independence from Australia. The two territories became the Territory of Papua and New Guinea with the passing of the Papua and New Guinea Act by the Australian Parliament in 1949. None of the authors would use this term when referring to Papua New Guineans because, during our life times, this was a derogatory term. It is sometimes used during the earlier part of this book to reflect the language of the expatriates and colonial government. We prefer to refer to someone by their cultural group, or as “Papua New Guinean,” or as “local.” 3

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The country was named Papua New Guinea in July 1972, in preparation for self-government and then independence which was finally achieved in 1975. As colonies around the world were gaining independence, the Australian government began to consider what should happen with Papua and New Guinea. Initially issues revolved around whether the Territories should become part of Australia as a constituent territory or state, with the same rights and privileges as the other Australian states, or whether an independent nation should be legally established. People began to favor a gradual approach for working toward the latter. Policies and funding needed to be worked out first. Education, administration, leadership in terms of an elected Assembly, and associated infrastructures, were all on the agenda. Australia saw the Westminster style of democracy with an independent judiciary, a governor general, and two houses of parliament, as an appropriate method of government for this only recently colonized, but nevertheless soon-to-be independent, nation. An independent set of government departments headed by professionals would be needed. Towards that end, in the 1960s some Papua New Guineans were trained in law and in higher education. The “Bully Beef Club” and the Pangu Pati were set up to establish the idea of party politics and to decide how complex rules of debate should be introduced and enforced. The House of Assembly would have elected members with an increasing number of Papua New Guinean representatives, albeit that some were illiterate in English and in languages other than their own vernaculars (Hastings, 1971). It was argued that there would be enough Papua New Guineans who had been well-educated and privileged by earlier colonizers, to meet immediate needs. As the 1960s progressed there was increasing pressure from the United Nations for colonizers throughout the world to give independence to their colonized territories (Rannells & Matatier, 2005) and this helped to generate increasing talk of independence for PNG within Australia. In the early 1970s, political leadership in Australia changed for the first time in over two decades, and the new Whitlam Government stipulated that the time for New Guinea’s independence would be the mid-1970s. Decisions were made which were aimed at preparing the new nation for the inevitable administration and leadership responsibilities. These had an impact on education policies, with recognition that the country would need schools, technical and administrative colleges, and universities. Teacher education, for both primary school teachers and teachers in the beginning secondary schools, would also need upgrading. In 1973, the country attained self-government and on 16th September 1975 the independent nation of Papua New Guinea came into existence. It was still supported by Australian aid and many second-level administrative positions were still held by Australians and others from overseas. The Australian “kiaps” who had maintained administration and lawfulness in Districts were phased out over a period of several years after Independence. In preparation for Independence, the original 19 provinces and National Capital District (areas 1 to 20 in Figure 1.1) were formed. These provinces, grouped into four regions for some administrative purposes, are shown in Figure 1.1, with the superscript numbers in the following list referring to the areas indicated on the map: • Southern Region: Central1, Gulf7, Milne Bay10, Western16, and Oro13 (Northern); • Highlands Region: Southern Highlands15, Western Highlands17, Simbu2 (Chimbu), Enga6, and Eastern Highlands3. • Mamose Region: Morobe11, Madang8, East Sepik5, and Sandaun19 (West Sepik). • New Guinea Islands Region: East New Britain4, West New Britain18, New Ireland12, Manus9, and North Solomons14 now known as the Autonomous Region of Bougainville4. At the time of writing a vote for Independence from PNG had been taken with a large majority in favor, although the PNG Government has yet to endorse it. 4

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Source. Wikipedia Note.1 Central, 2 Simbu (formerly Chimbu), 3 Eastern Highlands, 4 East New Britain, 5 East Sepik, 6 Enga, 7 Gulf, 8 Madang, 9 Manus, 10 Milne Bay, 11 Morobe, 12 New Ireland, 13 Oro (formerly Northern), 14 Bougainville (formerly North Solomons), 15 Southern Highlands, 16 Western, 17 Western Highlands, 18 West New Britain, 19 Saundaun (formerly West Sepik), 20 National Capital Territory, 21 Hela, 22 Jiwaka. Figure 1.1. Provinces of Papua New Guinea.

In 2012, Southern Highlands became two provinces, Southern Highlands and Hela21, and Western Highlands became Western Highlands and Jiwaka22. It should be noted that author Patricia Paraide comes from East New Britain and has travelled extensively throughout the country; author Charly Muke comes from Jiwaka Province and has also travelled; Kay and Chris Owens lived in Morobe Province for 15 years and have had a further 15 return visits for various research projects with colleagues. Kay has stayed in 15 provinces; and Phillip Clarkson lived for a shorter period in Morobe Province but has travelled to many parts of the country. Major Changes in Education Since Independence Since the 1960s—i.e., even before Independence was granted—certain reports and policies were prepared for the purpose of shaping the future of education in the new nation. The most significant of this early planning was a report by a committee established in 1974 and chaired by Alkan Tololo. It recommended a recognition of community in education and that education should prepare students for their life after school. The Tololo Report was not, however, adopted and a more conservative—one might say elitist—approach to education for the new nation was put in place according to a 1976 plan. More will be presented on the Tololo Report and the effect of the adopted plan later in this book. In the 1980s came the Matane Report A Philosophy of Education for Papua New Guinea, • Self-reliance which was to be achieved through a community-based education program; • Forms 1 and 2 (to be renamed Grades 7 and 8) were to be transferred from high schools to community schools with the aim of accelerating the process of universal community education;

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• The language of instruction in the first three grades was to be in the functional language (i.e., the Tok Ples) of the local community; • Opportunities for education were to be extended to all Papua New Guineans including girls, children from isolated areas, and disadvantaged children; and • Intended, implemented and attained curricula that were “relevant to the life that students will have to live after school” should be developed. A World Bank report followed. These reports led to structural and curricular reforms at the end of the 20th century. Further policy and implementation changes occurred after 2013—which, later in this book, we will refer to as the “revised reform” or “standards-based education period.”

The Purposes of This Book

This book expands on the historical narrative to explore important issues relevant to future policymaking and implementation plans for mathematics education for PNG and other countries with colonial pasts or developing multicultural societies. Indeed, the authors of this book offer a case-study of changes in mathematics education in a very multicultural, colonized but now independent nation whose people are nearly all Indigenous. Over its period of Independence, since 1975, PNG has had very few non-Indigenous members of parliament or heads of Government departments. However, there have been significant impacts from colonization, aid, globalization, politics, lack of infrastructure, and a small but fast-growing population. A number of themes with various strengths and weaknesses arose during each period. Although this case study could be compared and contrasted with what transpired in other nations, the colonial period in Papua New Guinea was relatively brief (around 100 years) and the achievement of independence was bloodless and seemingly rapid. The Indigenous cultures are very old and relatively continuous. For these reasons, offering a longer history is relevant, and something which could motivate change in other nations. However, the main purpose for this book is to ensure that current and future Papua New Guinean educators, politicians, government decision-makers, and the general public and families have access to relevant historical narratives based on research as they make decisions for the future. There is an old adage that history repeats itself and it is hoped that, knowing their past history, people will learn from the past, follow up on strengths of the nation’s diversity, history, and culture, avoid mistakes, and make wiser decisions. We believe that generating an understanding of the nation’s history is vital for the future of the country because that can help ensure people’s cultural heritage is recognized, maintained and strengthened for the common good. It will also provide background for studying the mathematics education of the nation and the range of cultures within Papua New Guinea. In this book, the following issues will be recurrent: • What is education and what are the purposes of education in Papua New Guinea? • How can Indigenous forms of language, culture and mathematics be documented and used to advantage within the nation? • What should be the languages of instruction in PNG’s education institutions? • Who should be educated (e.g., an elite, or every child) and for how long? • What needs to be done to ensure social justice and the development and maintenance of human rights—particularly for gender equity, for persons living in remote locations (especially with groups with a small number of speakers of a language, or limited resources, or major cultural differences from others)? • What needs to be done to ensure the development of a harmonious multicultural nation?

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• How can the demands of global forms of education in PNG be reconciled with those of local forms of education? • Who should administer and pay for education? • What values and perspectives on education should be present with decision- makers, especially in relation to their views on the purposes of education and mathematics education? Some of these issues will be discussed in almost every chapter. There will be a specific chapter on language.

Brief Historical Overview of Western Education in PNG

Geoffrey Smith (1975) pointed out differences between Indigenous and Western education. First, he described Indigenous education in the following way: Knowledge was imparted when the learners needed to use it … Learning was casual and unplanned as young children had few duties and were often left to amuse themselves in the care of an older child or with their own age group. They learned how to fight or swim through play and picked up skills of canoeing or keeping garden through observing and imitating their parents when they happened to accompany them. As the children grew older, they were increasingly involved in these activities and taught the rituals which would bring success to their tasks. (pp. 1–2) Smith (1975) also described modern or Western education in the following statement: The modern school system is far more selective in determining which children will be educated and to what levels, but far less selective in ensuring that the knowledge transmitted is relevant to their situation. (p. 4) Before Independence, some Papua New Guinean people did not think much of mission education and did not encourage their children to attend mission schools. Instead, they provided their own education which prepared their children for adult lives in their communities. They did not have to learn Western knowledge and skills to be able to function well as intelligent human beings in their own ecocultural environments. They gained a wealth of knowledge and mastered appropriate life-skills, primarily from observation, trial and error, and working alongside their fathers, mothers, other relatives, and peers in their communities. During the period of colonization colonial authorities began to provide health and educational services. Unlike the missionaries—who were usually happy to use the local Tok Ples, or something similar to that, when teaching—the language of instruction was less an issue for the colonizers, so debates inevitably arose over what should be the most suitable language of instruction in formal education settings (Barrington-Thomas, 1976; Smith, 1975). The aim of the missionaries, by and large, had been to educate the people, primarily about Christianity, but a wider view of the curriculum would be needed after Independence. Many years earlier, Albert Hahl, the German Imperial Governor, had proposed to the heads of missions the payment of subsidies to help spread the German language in German New Guinea, which is now known as the Mamose, Highlands and New Guinea Islands Regions. In Papua, which is now known as the Southern Region, Governor Murray initiated a scheme a few years later to promote literacy in English. The German scheme was frustrated by the First World War, but most missions in German New Guinea cooperated with the language policy. The British and later Australian colonial authorities wanted English as the formal language of instruction. In Papua, English was taught in the upper grades where European staff were posted (Smith, 1975)— although vernacular languages and lingua franca were often, perhaps unofficially, used in lower grades by local PNG or Pacific Islands teachers.

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From a postcolonial perspective, teaching in vernacular languages or a lingua franca like Tok Pisin would have disrupted the colonial authority’s power over the Indigenous people because they could not communicate with them in those languages as well as the missionaries did. In fact, most Australian kiaps and police could use the lingua franca of Tok Pisin or Hiri Motu depending on where they were placed in remote areas. Records of the British and Australian governments show that McGregor, Murray, and others in charge, requested that local languages and associated counting systems be recorded (Owens, Lean with Paraide and Muke, 2018). It could be argued that authorities who perceived that their influence over the people was weak, categorized people, and used English as the language of power (Fellingham, 1993; Foucault, 1978, 1987). Many of the early missionaries and others, such as traders in the north east of the main island, spoke German, and some others spoke other European languages or other South Pacific languages. Most of the Indigenous Papua New Guinean people spoke only their own languages. Often, the expatriates expressed their opposition when English was first proposed as the formal language of administration, and for instruction in schools. Missionaries tended to favor having education taught through local languages or local lingua franca, arguing that that would be more meaningful to the people—especially for spreading the gospel but also for giving the people power over exploitive and, in some cases, cruel traders. Changes in Language Policies With Respect to Education From the onset of colonization, the language of instruction in formal education became an issue. Around Port Moresby in the late 18th century, the missions used Hiri Motu. In German New Guinea the missionaries used local lingua franca and German. In 1907, a Royal Commission of Enquiry endorsed Administrator Macgregor’s policy, which was “that the teaching of English be made compulsory in mission schools, and ‘native’ children be compelled to attend schools at which English is taught” (Smith, 1975). Much debate concerning the language of instruction for schools continued during the colonial and pre-independence era (Barrington-Thomas, 1976; Smith, 1975). An administration-mission conference in 1927 regarding native education resulted in a review of education for the natives, in 1929, by B. J. McKenna (the Queensland Director of Education). Data for this review were gathered mainly from the Rabaul area and the findings were questioned by Johann Flierl, a German Lutheran missionary, because McKenna did not include the Lutheran Missions and Catholic Divine Word Mission schools on the New Guinea mainland in the review (Ralph 1978). Ralph (1978) argued that “McKenna’s aim was to have the government schools in Rabaul, with only white teachers, using exclusively the English language as the medium of instruction” (p. 308). Despite these criticisms, McKenna’s Report was accepted, and “stands as a landmark in the development of secular, government education in the Mandated Territory” (Ralph, 1978, p. 309). Despite all these criticisms, the pressure to use English as the language of instruction increased after the Second World War. In 1952, the Education Ordinance was endorsed. “English was to be the language of instruction, with teaching in vernacular limited to infant classes” (Ralph, 1978, p. 309). However, this policy was shelved and a policy that all grades—even with infant classes— use English as the language of instruction was implemented (Barrington-Thomas, 1976; Ralph, 1978; Smith, 1975). Before, during and after the time that the colonizers were deciding on an “appropriate education for the natives,” and “the most appropriate language of instruction for formal learning” (Dickson, 1976, p. 23), Papua New Guineans continued their informal village ways of education, using their own languages, and teaching about mathematics in context through the various village technologies, trades, and activities required for life in their sociocultural environments.

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After Independence, there continued to be external influences on education in PNG. The following chapters provide an overview of the strengths and negative results of these influences. Overall, PNG remains an education system in which administrators, policy makers, governments and teachers struggle to provide the best for the children and students of the nation. In 2015, only 63% of children attended school (UNESCO, 2015). However, the complexity of the national languages and cultures as well as the richness of foundational mathematical knowledge and school education provides for valuable historical analyses into mathematics education in PNG.

The Processes Adopted for Developing This Book

A research book is expected to critique published information across various fields. However, this can only be done for written knowledge. Reviewing Indigenous knowledge is different because such knowledge comprises complex sets of technologies which have been developed and sustained by the Indigenous civilization. Much of the information is passed on orally through the generations. As Marie Battiste (2002) stated, “knowledge is transmitted through the structure of Indigenous languages passed on to the next generation, through modelling, practice, and animation, rather than through the written word” (p. 2). Battiste identified the limitation of some research information, and argued that Western research cannot reveal an understanding of Indigenous knowledge: It is a knowledge system in its own right with its own internal consistency and ways of knowing, and there are limits to how far it can be comprehended from a Eurocentric point of view (p. 2) In recent times, the use of Indigenous knowledge and skills has been encouraged through the school curriculum in many fields of study in many communities. Indigenous knowledge is a growing field of inquiry, both nationally and internationally, particularly for those interested in educational innovation (Battiste, 2002). During colonial times, Indigenous knowledge systems were usually classified as “primitive,” and were considered a hindrance to Western progress and development (Barrington- Thomas, 1976; Ralph, 1978; Smith, 1975). Now, however, Indigenous knowledge areas, including mathematics, are acknowledged as important in defining intended curricula (Beach, 2003; Bishop & Seah, 2003; de Abreu, Bishop, & Presmeg, 2002; Kaleva, 1998; Rosa & Orey, 2020). For example, Lean (1992) carried out a detailed study on Tolai and other PNG and Pacific Indigenous mathematics. He referred to reports from Codrington (1885), Ray (1891), and Parkinson (1907) concerning the complex counting system used by the Tolai people. This counting knowledge is confirmed by author Paraide, a Tolai. Some of PNG’s counting systems are discussed in Chapter 3 of this book, and in Owens, Lean, with Paraide and Muke (2018). There is now a wider recognition of the value of Indigenous knowledge, but it is not universally accepted as yet. Myer (1998) and Nakata (2004) argue that Westerners’ growing interest in Indigenous knowledge contributes to the elevation of its status. Their interest is largely driven by research into sustainable development practices in developing countries, and the scientific community’s concern about the loss of biodiversity of species and ecosystems and future implications for the planet Earth. However, there is also a darker side. Many Western pharmaceutical companies wish to mine Indigenous knowledge so they can research and exploit potentially worthwhile, in monetary terms, flora and fauna that point to new medicines that can be profitably produced in the West. Other overseas companies wish to establish other exploitative practices like mining but are expected to negotiate access to the land and people’s rights. However, even when the emphasis is on sustainable development, there is clearly too little emphasis on culture, cultural survival, languages, or social and cultural diversity. Nakata (2004) has further asserted that, in humanitarian and scientific areas, scientists recognize that Indigenous knowledge needs

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to be recorded and validated if it is to be incorporated into scientific bodies and used. Also, agencies that are working in developing countries show an interest because they realize the importance of local knowledge in solving problems at the local level. Battiste (2002) offered another dimension to Nakata’s (2004) discussions on Indigenous knowledge: The recognition and intellectual activation of Indigenous knowledge today is an act of empowerment by Indigenous people. The task for Indigenous academics has been to affirm and activate the holistic paradigm of Indigenous knowledge to reveal the wealth and richness of Indigenous languages, world views, teaching, and experiences, all of which have been systematically excluded from contemporary institutions and from Eurocentric knowledge systems. (Battiste, 2002, p. 4) Battiste (2002) further stated that Westerners generally think that only they “can progress, and that Indigenous peoples are frozen in time, guided by knowledge systems that reinforce the past, and do not look towards the future” (p. 4). Battiste also argued that Westerners have not acknowledged the fact that Western knowledge consists of some knowledge that is of non-European origin. For example, the Greek alphabet is largely of Syrian/Lebanese origin, and some Western mathematical knowledge is of Mayan, Hindu, and Arabic origin. Battiste (2002), Nakata (2004), and other Indigenous academics, have supported the preservation of Indigenous knowledge because it is of interest and is valuable in its own right to the Indigenous people, irrespective of what worth it has to Westerners. Indigenous knowledge was often been viewed negatively in the past, which resulted in the non-use and loss of much valuable Indigenous knowledge, especially when it belonged to people with an oral history, such as Indigenous Papua New Guineans. Battiste (2002) s uggested that one way of preserving culture in her homeland, in Canada, is through integration with the school curriculum. This strategy is now being implemented in Canada. However, that too is not without its problems. Lyn Carter (2011) among others has argued forcefully that this can be viewed as a continuation of colonialist thinking, whether those involved, with the best intentions in the world, realize this or not. Authors, including Carter, have warned that much care is needed in this process (Dei, 2011). The need to preserve culture has been embraced in curriculum reform in PNG and in other parts of the world. Indigenous knowledge is discussed here because it is valuable to the people of Papua New Guinea and how they use such knowledge in their everyday lives. From the authors’ perspective, the recognition and acceptance of the value of Indigenous knowledge and practices, and their inclusion in the PNG school curriculum is a positive step toward the validation of the various bodies of Indigenous knowledge. Students who acquire Indigenous knowledge may then be able to easily relate it to Western and other knowledge bodies that are learned in formal education, and be able to expand on and explore that collective body of knowledge further. They may compare knowledges from different Indigenous societies within PNG. Care needs to be taken, nevertheless, with respect to how Indigenous knowledge is incorporated into the schooling of students. It cannot be devalued in comparison with Western school knowledge, as can be the case when Indigenous knowledge is only used as just an introduction to a Western concept, and then forgotten about as the Western concept is fully analyzed and incorporated into the growing network of ideas of the students. The Indigenous knowledge needs to be fully worked through either in the school context or in the village. Village school authorities, especially teachers, need to recognize this and value what is happening outside the classroom walls. This can be a difficult process, one which is often under-valued in the preparation of intended, implemented, and attained curricula. Battiste (2002) also commented on this dilemma when she stressed that caution should prevail when integrating Indigenous and Western knowledge. Although there are similarities that

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can be easily accommodated for educational purposes, there are also differences that are unique to each culture. These differences cannot be separated or compared with Western cultures but should be recognized as vital components in their own particular cultures. Mathematics is a body of knowledge that is present in all cultures and yet it too, although often not recognized by Western experts, is culture-dependent (Bishop, 1988). The Researchers and Authors Patricia Paraide. The Papua New Guinean authors have been involved for many years in mathematics education in practice and research. The first author, Patricia Paraide, worked as a teacher, a curriculum adviser and a member of the PNG National Research Institute before joining Divine Word University as Associate Professor. She retired at the end of 2019 and now acts as an educational consultant. She speaks Tolai or Tinatatuna fluently. It is an Austronesian Oceanic language and culture. Her doctoral thesis raised the importance of culture and language in the development of mathematical skills, and the kinds of skills teachers require to teach mathematics well in bilingual situations. However, a critical lens on the power of domination over the less powerful, as discussed by Smith (2012), Barrington-Thomas (1976), Fellingham (1993), and Said (1993), has been a major background consideration for this historical study of mathematics education in PNG. Thus, her background strengthens the discussion on the key area of language in education, and of culture in mathematics education. Significantly, she brings a postcolonialist critique to the development of mathematics education in PNG. Several chapters, in particular those on language, reform, and Chapter 12 on neocolonialism, draw on her thesis (Paraide, 2010). As we prepared this book, she summarized her point of view:

My people journeyed through the changes in education before and during the colonial era, and following Independence. My people, like other PNG and Indigenous peoples, had their own system of education before the colonizers arrived. As Tololo (1976) discussed earlier, our young people learned their life-skills through observation, imitation, and participation. The skills that they learned growing up were useful and appropriate for their adult lives. Geoffrey Smith (1975, p. 3) made similar observations regarding the teaching strategies used by PNG’s Indigenous people to educate their young people. Tololo and Smith acknowledge that everyone in the community contributed to the education of the young Indigenous people. Peers working together and working alongside experienced adults on specific tasks were encouraged through Indigenous education. However, this changed with the arrival of the missionaries. (Paraide, personal communication, November, 2018) Charly Muke. After his University of Papua New Guinea Diploma in Education, Charly Muke was a high school mathematics teacher, completed further studies and lectured at the University of Goroka. For his Master of Education from the University of Waikato, in New Zealand, he studied indepth the counting systems of his own people raising several important and previously undocumented aspects of those counting systems. He followed this with his doctoral study at Australian Catholic University on how teachers used language and Indigenous cultural practices when teaching mathematics as they moved from using many Tok Ples (local languages) to introducing English at Year 3 of school. Subsequently he has been involved in elementary school mathematics and technology professional development for teachers. He taught mathematics in Australia for 15 years teaching mainly Australian Indigenous students—that experience provided him with a comparative understanding of mathematics education for Indigenous students. Charly’s first language is Mid-Wahgi or Yu Wooi language, which is a language of the Papuan Trans New Guinea Phylum, East New Guinea Highlands

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Stock, Central Family, Wahgi Sub-Family. Recently (in 2020), Charly was appointed to the position of Provincial Education Advisor in Jiwaka Province. Kay Owens. The preparation of this book seemed timely while the three authors who are expatriate to Papua New Guinea were able to initiate and support the development of such a history while maintaining regular connections to the country. Kay and Chris Owens lived, worked and raised their family in Papua New Guinea for 15 years and have both returned to work with their PNG colleagues to do further research and promote educational advancements. Kay taught mathematics at the PNG University of Technology, and mathematics education, health education, and education studies at Balob Teachers College. She taught in a community school, supervised practicums in village and town schools. She planned and led many professional development sessions for teachers in various parts of PNG. It was through teaching health education and her awareness of cultural impacts on health and values that raised her interest in culture-based mathematics education. During their 15 years in PNG both Kay and Chris enjoyed many bushwalks to villages all over Morobe Province. They also made trips to Buna and Popondetta in Oro Province, the Trobriand Islands in Milne Bay, Sepik River villages by motorized dugout canoes and Wosera by PMV, to Yombu Selden’s village out of Tufi in Oro Province by sailing canoe. They also visited Mt Wilhelm and Mt Michael in the Highland provinces, Lake Kutubu in the Southern Highlands, and Simbine village in Madang. Kay wrote booklets and ran numerous inservices on mathematics for nurses in many hospitals from Sandaun to Oro Provinces. She wrote Healthy Lifestyle for Tertiary Students. Both the nursing mathematics booklets and the booklet for tertiary students were used for many years.

During her doctoral studies, Kay studied visuospatial reasoning of primary school students in Australia and in a PNG community school; the PNG sample provided significant insights into the grounded-theory qualitative analysis she undertook. This study helped to establish the importance of responsiveness during problem solving. She spearheaded an elementary teachers’ professional development project with a number of her PNG colleagues, including Charly Muke. Her other research in PNG over the last two decades has included a study of architecture students’ problem solving and reasoning in designing and making a paper-cardboard sculpture (1997). This happened while she was lecturing at Balob Teachers College and living at PNG University of Technology. A study on measurement and space conducted with Wilfred Kaleva, between 2006 to 2008, demonstrated the importance of cultural identity and visuospatial reasoning for self-regulation and mathematical thinking. With Rex Matang and others between 2000 and 2003, she developed a website for the Glen Lean Ethnomathematics Centre, University of Goroka, through a USA National Science Foundation grant to the Pacific Resources for Education and Learning (PREL), Hawaii, for assisting with digitizing ethnomathematics studies in the Pacific. She worked on data collected by the late Glendon Lean, prepared an electronic database, and extended his thesis on the counting systems of Papua New Guinea, establishing a significant history of number with Patricia Paraide and Charly Muke (Owens, Lean, with Paraide, & Muke, 2018). From 1998 to 2003, she worked with former PNG colleagues, Wilfred Kaleva and Theresa Hamadi, with Philip Clarkson and Ron Toomey on the evaluation of the Primary and Secondary Teacher Education Project collecting and preparing data for two of the teachers colleges, and teachers in the field, analyzing the data and writing up sections of the reports on these colleges. In 2003, she set up a Master of Education program with Wilfred Kaleva, Api Maha and other lecturers from University of Goroka’s (UOG) Unigor, and taught an “Introduction to Research” course to the 40 Teachers College lecturers in the Virtual Colombo Plan program run by Charles Sturt University, UOG, and TAFEglobal as an AusAid program. Between 2006 and 2008, Kay, working with Wilfred Kaleva, investigated and documented the measurement systems of PNG cultures and languages. This involved her in making several village field trips, and conducting numerous interviews with students at UOG who had com-

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pleted a questionnaire on their measurement practices. She also analyzed written statements— including documents prepared by SIL and other linguists, and research reports from UOG students—on relationships between PNG cultural (Indigenous) mathematical practices and school mathematics (Owens, 2008, 2013; Owens & Kaleva, 2008a, 2008b). Among her research colleagues have been the late Rex Matang, Wilfred Kaleva, Martin Imong, and Chris Owens on PNG’s number systems, the Glen Lean Ethnomathematics Virtual Library, and measurement and spatial knowledge Indigenous to PNG. She worked with Vagi Bino, Cris Edmonds-Wathen (an Australian), Susie Daino, and Geori Kravia and others on the elementary school mathematics education project, and with Serongke Sondo and Charly Muke on both the measurement and elementary schools projects. Patricia Paraide provided insights on culture in mathematics education. Kay also worked with the elementary curriculum division, especially on mathematics education, and delivered a number of inservice programs for teachers college lecturers. She gave public lectures at the University of Goroka in 1992, 1997, 1999, 2000, 2001, 2003, 2008, 2014, 2015, and 2016. Her recent research projects involved visits to villages in Jiwaka, Central, Morobe, and Madang. This long connection with PNG education has provided a strong background for the planning and writing of this book. Chris Owens. As a Chemistry Lecturer at the PNG University of Technology Chris found that mathematical understanding was a key for success in chemistry. His love of history, the environment, and politics has provided the team with different insights in terms of both applied mathematics and historical connections. Chris was on the National High Schools Syllabus Advisory Committee for Science and this committee acted as the Year 12 examiners to set and mark papers for Year 12 examinations. He was on the Aiyura NHS Governing Council, and was an Executive member of the Institute of Mining and Metallurgy PNG Branch and PNG Institute of Chemistry. In 2006, he was external advisor to the Syllabus Committee revising the Years 11 and 12 Science Curriculum for PNG. Philip Clarkson. From 1980 to early 1985, Philip was Director of the Mathematics Education Centre (MEC), located at the PNG University of Technology (in Lae). During that time he led a number of research projects, edited the two series of MEC reports (MEC Research Reports series, and the MEC Technical Research Reports series). These were being referred to long after he left—as was reported to him in 1990 on a return visit to Port Moresby by a USA Aid worker in the Curriculum Branch, and later still in 2003 by staff at the University of Goroka and Mount Hagen Teachers College when he visited PNG again. Phil also edited PRIME, a professional journal for PNG mathematics teachers. An annual two-day MEC research conference in mathematics education was established in 1981, with each conference being accompanied by published proceedings. Conferees not only came from all over PNG, but from various Pacific Island countries and from Australia. During this time, he was a member of the various Mathematics Syllabus Committees of the National Department of Education, and was the external examiner for all the Year 11 and Year 12 mathematics subjects. That role involved him in regular visits to all National High Schools throughout the country, for consultation purposes. He often conducted professional development sessions for the staff in schools. He contributed to the writing and publication of various Mathematics textbooks and accompanying books for teachers of Years 4 through to 10 during this time (and for some years after 1985), and later in the early 1990s to the first Oxford Publications mathematics textbook for Year 1. He led many professional development sessions for local teachers, and gave many invited sessions throughout the country, including at all teachers colleges at one time or another, at the University of Goroka and the University of Papua New Guinea (on multiple occasions), mainly dealing with mathematics teaching but also on leadership and other aspects of education. He gave papers at the Waigani Seminars and various other conferences run by the Department.

Influential publications from this time, ranged from issues related to students’ affect both in higher education and schools (Clarkson, 1984; Clarkson & Leder, 1984), broader educational

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issues (Clarkson, 1986), the role language played in students’ learning mathematics in PNG (Clarkson 1992; Clarkson & Galbraith, 1992), the impact of students’ language competence on examination scores (Clarkson & Clarkson, 1993), and innovative ideas on methodologies for teaching mathematics in PNG (Clarkson & Kaleva, 1993). Clarkson’s PhD thesis was based on data from PNG schools, and he found that students’ competence in both their home language, as well as in the language of instruction, had significant impact on their mathematical performance in school. This research generated data which supported the use of student’s local tok ples languages in all teaching, including the teaching of mathematics in PNG and in other countries. His two major subsequent visits to PNG included preliminary visits with Oxford University Press in preparing to develop their PNG mathematics textbooks==although he did not participate in that process beyond the first book. Later, from 1998 through to 2003 he led the evaluation project of PASTEP (Primary and Secondary Teacher Education Project); Kay Owens and Wilfred Kaleva were on this team, with Charly Muke a respondent at one of the Teachers Colleges. PASTEP was a major project that helped develop a nationwide curriculum and teaching approach for all of PNG’s teachers colleges and provided professional development for staff of UOG, and set up computing facilities at each site (Clarkson, Hamadi, Kaleva, Owens & Toomey, 2004). Processes for Developing Foundational Cultural Mathematics Education In this book we use archaeological, documentary, and linguistic evidence to establish the antiquity and diversity of the people of Papua New Guinea. We will use archaeological findings to provide information on the kind of mathematics that has been used in PNG for tens of thousands of years. The study will use the cultural practices evident in first contact and even today to develop understandings of these foundational cultural forms of mathematics. Much of the education that occurred were with cultural practices such as those learnt in the men’s and women’s houses and in everyday or special activities. Although it will not be possible to provide all systematic, patterned ways of thinking related to these practices the diversity and depth of mathematics education should become evident. To illustrate the recent nature of first contact, Appendix 1 provides an indication of some of the first patrols into the interior of the nation and dates. It should be noted that MacGregor’s travel was across a high mountain range, the same range but a different area from that which the Japanese and Australian soldiers fought upon in the Second World War. The patrols discussed by Gammage (1998) as well as Leahy’s 1994) and Mikloucho-Maclay’s (1975) accounts have provided further evidence of technologies and forms of mathematics already in existence at first contact. The next two chapters present an account of findings from early explorers, scientists and others together with shared oral histories. We also describe some of our own observations and knowledge of these cultures as they continue today. Processes for Developing the History of Mathematics Education Since Colonial Times Each of the authors is passionate about mathematics education in PNG and has lived and breathed much of what is recorded. Considerable use was been made of key records, in particular those referred to in two books—one by Peter Smith and the other by John Cleverley and Christobel Wescombe—which provide bibliographies and commentary on the history of education in PNG. We highly recommend those two documentary sources for their recording and analyses of education during colonial times. Other documents were sourced and analyzed, and are referenced throughout the book—these include the Matane Report and several evaluations of projects and elementary school literacy. Carol Leo Abiri from St Benedict’s College of Divine Word University

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also supplied a number of documents that she had sourced for her own doctoral research on bilingual reading education. In addition, we sent out a questionnaire to friends and others who responded to a call in the Sydney Morning Herald for comments on mathematics education in Papua New Guinea. We received 22 responses from Papua New Guineans who commented on their own schooling, their children’s and/or grandchildren’s or parents’ education together with their knowledge of the system. As it transpired, most had worked as teachers, teacher educators or in other related professional positions. We also received 25 from expatriates, one of whom had resided in PNG in 1946; most of the others had been in PNG in the 1970s, 1980s, and 1990s, and one had been there in 2019. These responses provided anecdotes which confirmed our own recollections of mathematics education in PNG. In particular, we would like to acknowledge the sharing of information by Professor Deane Arganbright, and a number of expatriate Papua New Guineans permanently living in Australia, in particular Yombu Selden and husband Rod, and Dan and Carrie Luke. Michael Mel, whom Kay worked with when he was Pro Vice Chancellor of UOG and was curator of the Pacific Collection at the Australian Museum in Sydney, kindly provided much valued insight and advice while we were writing this book. However, many of the chapters draw on oral histories and on our own knowledge. For example, Paraide (2010) recalls: Our clan leaders, ToLitur, ToBola, ToUraria, Tale, ToKamai and Taragau, accepted the Catholic missionaries. Brown (1908, p. 78) [a protestant missionary] also labelled our people as “fierce, ferocious natives and cannibals” because they retaliated when the white people invaded our land. Brown’s writings also referred to our land as “unhealthy” because the early missionaries assumed that, as they had encountered a plague in Fiji at that time, our land would be the same. It is not known why the elders accepted the missionaries at that point in time, given their history of resistance towards invaders. Mennis (1972, p. 13) explained that my elders had previous contact with European traders and that, at that particular time, they had a preference for Catholic missionaries. Consequently, they accepted them when they landed on our shores. Apparently, they had heard some positive experiences about the Methodist missionaries from the Matalau people who lived in the next village. The Matalau villagers had accepted the Methodist missionaries in the late 1800s (Brown, 1908). These missionaries taught the Matalau people to read and write, treated them with Western medicine, which they claimed to be more effective than our own, and wore Western clothes. The Matalau people became peaceful and ceased to participate in warfare with the neighboring villages and invaders. The people must have viewed non-aggression as a better way of life, and desired a change. Their aggressive attitude towards invaders had changed before the Catholic missionaries arrived. Mennis (1972, p. 34) documented that strong winds had blown the Catholic missionaries off course, as they were heading towards Matupit Island. As a result, they arrived in my village. The missionaries’ stay in our village was brief, before they moved on to Kiniguan in the Kokopo area. The move was prompted by insufficient land for mission expansion in our village. From the stories that my Elders told and from the writings of Murphy and Moynihan (1936), Mennis (1972), Ralph (1978), and Jonduo (1993), the missionaries taught our people to read and write in our own language, Tinatatuna. Brown (1908) and Mennis (1972), when discussing the spread of Christianity, stated that the early missionaries used vernacular literacy mainly as a tool to spread the Gospel. It was not used to teach Indigenous knowledge because their main focus of education was the spread of Christianity. The missionaries also provided Western health and education

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services for the Indigenous people (Dickson, 1976; Smith, 1975), although the people already had their own form of education and medical practices. Stories told to me as a child showed that the missionaries often tried to prove that their form of education and medicine was better than the Indigenous ones, which may be true in some cases. They also forbade certain practices in the people’s culture, which interfered with the spreading of the “good news.” Additionally, they made the people wear clothes as they regarded the nakedness of the people as sinful (Brown, 1908). (Paraide, 2010, pp. 10-11)

Overview of the Book

Interestingly, there are a number of recurring themes during each epoch of education. These include policies, language, gender, teacher education, who were educated and why (e.g. elites for development), research influences, funding, and overseas influences or globalization. These themes will re-occur and develop throughout the book. • This chapter has introduced the country and the purpose of the book. It has outlined the processes for the development of this book and the overall content of this book; • The main thrust of the second and third chapters has been influenced by research and records, personal observations and interviews with Papua New Guineans and others from the colonial period, and oral histories provided by Papua New Guineans. The two chapters document the foundational mathematics of the activities of the many Papua New Guinea cultures and how new generations were educated within these cultures; • The fourth chapter covers the early colonial period in both the German and EnglishAustralian territories, and the impact of the First World War (WWI) on the colonial management and power that influenced education. It also covers the impact of World War II (WWII) and subsequent Australian-Papua and New Guinea relations; • The fifth chapter discusses the years leading up to Independence—from the 1960s to a year after Independence, 1976—and in particular, the report of goals and principles for education spearheaded by Tololo. Primary and secondary education and the rise of tertiary education are discussed; • The sixth chapter considers the issues for education after Independence until the 1990s and in particular the report of the committee headed by Matane; • The seventh chapter offers a summary of the history of tertiary mathematics in Papua New Guinea, and in particular it focuses on how mathematicians and mathematics educators at the universities and teachers colleges thought about the kind of mathematics education which was needed in a developing country like PNG. It looks at early mathematics education research and practices in PNG, and at some of the unique ideas of expatriate academics welcomed to this country, as well as at the ideas of early PNG academics. It also discusses changes in PNG’s teacher education since Independence; • The eighth chapter discusses a major reform which occurred during the 1990s and through to 2014. In particular, it covers associated issues relating to elementary schools, language, and the desire for universal education, as well as issues associated with outcomes-based education (OBE); • The ninth chapter discusses the revision of the reform, Standards Based Education, and the impact on mathematics education, and glocalization5; • The tenth chapter discusses issues associated with language and diversity of population. In this multicultural country of Indigenous communities and self- Glocalization has become a word that represents the current global approaches as these are interpreted locally.

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determination, there are ongoing issues associated with funding, teacher education, and administration, especially in terms of national and provincial administration; • The eleventh chapter considers the significance of research on visuospatial reasoning carried out from the 1970s through to the 2020s, and the role of ICTs in mathematics education beginning with calculators, early employment of mainframe computers in tertiary education, Japanese technological aid, up to the provision of solar-powered computers in rural areas; • In the twelfth chapter it is argued that PNG is a neocolonial country. Several recurring themes and the influences on and nature of education over the many stages of development are summarized. Implications of significant themes that arose across the various periods of mathematics education in PNG, including decision-making; funding; who to educate and why; language of instruction; level of education provision; teacher education and mathematics; attitudes towards mathematics including level, purpose and cultural connections; female education which became a significant issue throughout the post-Independence era and a factor heavily discussed in aid projects. The place of women in PNG societies is seen as one that requires significant educational influences and is seen as an issue for the teaching profession. The loss of reciprocity and the impact of neocolonialist approaches need to be addressed; • The thirteenth chapter argues that the way forward will be to change attitudes toward local cultural mathematics, and to find ways of learning that are appropriate for schools, combining the local with the global, recognizing a way forward through language and cultural identities, accepting differences, developing equity, and reducing dualities associated with culture, schooling, and mathematics. It makes some comparisons with neighboring colonized island nations.

Moving Forward

The chapters of this book offer a unique history of mathematics education in PNG. There is much to learn from the historical record that is pertinent if one wishes to understand educational issues within the country, but also within other newly developing countries. The book also has something to say about transcultural education in general. The arguments on decision-making in education, on issues that affect mathematics education, and on the strength of the mathematics associated with tens of thousands of years of cultural development and adaptation, are pertinent to future educational policy-making within and beyond the country. This book provides a case study of transcultural decision-making in our global and multicultural world. References Barrington-Thomas, E. (1976). Problems of educational provision in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 3–16). Melbourne, Australia: Oxford University Press. Battiste, M. (2002). Indigenous knowledge and pedagogy in First Nations education: A literature review with recommendations (Working paper). Ottawa, Canada: Apamuwek Institute. Beach, D. (2003). Mathematics goes to market. In D. Beach, T. Gordon, & E. Lahelma (Eds.), Democratic education ethnographic challenges (pp. 99–122). London, England: Tufnell Press. Bishop, A. (1988). Mathematical enculturation: A cultural perspective on mathematics education. Dordrecht, The Netherlands: Kluwer.

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Bishop, A., & Seah, W. T. (2003). Values in mathematics: Teaching the hidden persuaders? In A. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Second international handbook of mathematics education (pp. 717–765). New York, NY: Springer. Brown, G. (1908). Pioneer-missionary and explorer: An autobiography. London, England: Hodder and Stoughton. Carter, L. (2011) The challenges of science education and Indigenous knowledge. In G J. S. Dei (Ed.), Indigenous philosophies and critical education: A reader (pp. 312–336). New York, NY: Peter Lang. Chinnery, E. (1924). Natives of the Waria, Williams, and Bialola Watershed. Anthropological Report, 4. Chinnery, E. (1926). Territory of New Guinea Anthropological Report No. 1. Papers of Ernest William Pearson Chinnery. Australian National Library. Canberra, Australia, MS 766. Clarkson, P. (1984). Papua New Guinea students' perceptions of mathematics lecturers. Journal of Educational Psychology, 76(6), 1386–1395. Clarkson, P. (1986). Towards the upgrading of mathematics education in higher education. Papua New Guinea Journal of Education, 22(2), 129–140. Clarkson, P. (1992). Language and mathematics: A comparison of bi and monolingual students of mathematics. Educational Studies in Mathematics, 23, 417–429. Clarkson, P., & Clarkson, R. (1993). The effects of bilingualism on examination scores: A different setting. RELC Journal, 24(1), 109–117. Clarkson, P., & Galbraith, P. (1992). Bilingualism and mathematics: Another perspective. Journal for Research in Mathematics Education, 23(1), 34–44. Clarkson, P., Hamadi, T., Kaleva, W., Owens, K. & Toomey, R. (2004). Findings and future directions: Results and recommendations from the Baseline Survey of the Primary and Secondary Teacher Education Project—Final Report. Report to Australian Agency for International Development: Australian Government and Papua New Guinea National Department of Education. Clarkson, P., & Kaleva, W. (1993) Mathematics in schools in Papua New Guinea. In G. Bell (Ed.), Asian perspectives on mathematics education (pp.111–120). Lismore, NSW, Australia: University of New England and Northern Rivers Mathematical Association. Clarkson, P., & Leder, G. (1984). Causal attributions for success and failure in mathematics: A cross cultural perspective. Educational Studies in Mathematics, 15(4), 413–422. Cleverley, J., & Wescombe, C. (1979). Papua New Guinea: Guide to sources in education. Sydney, Australia: Sydney University Press. Codrington, R. (1885). The Melanesian languages. Oxford, UK: Clarendon Press. D'Entrecasteaux, B. (2001). Voyage to Australia and the Pacific 1791–1793 (E. Duyker & M. Duyker (Eds.), Trans. E. Duyker & M. Duyker. Melbourne, Australia: Melbourne University Press. de Abreu, G., Bishop, A., & Presmeg, N. (Eds.). (2002). Transitions between contexts of mathematical practices. Dordrecht, The Netherlands: Kluwer. Dei, G. Sefa (Ed.) (2011). Indigenous philosophies and critical education: A reader. New York, NY: Peter Lang. Dickson, D. (1976). Murray and education: Policy in Papua, 1906–1941. In E. Barrington- Thomas (Ed.), Papua New Guinea education (pp. 21–45). Melbourne, Australia: Oxford University Press. Fellingham, L. A. (1993). Foucault for beginners. New York, NY: Writers and Readers Publishing Inc. Foucault, M. (1978). The history of sexuality: An introduction (Vol. 1). New York, NY: Vintage Books.

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Foucault, M. (1987). The use of pleasure: The history of sexuality (Vol. 2). London, England: Penguin Books. Gammage, B. (1998). The sky travellers: Journeys in New Guinea 1938–1939. Melbourne, Australia: Miegunyah Press, Melbourne University Press. Goodman, J. (2005). The Rattlesnake: A voyage of discovery to the Coral Sea. London, England: Faber and Faber. Hastings, P. (Ed.) (1971). Papua/New Guinea: Prospero's other island. Sydney, Australia: Angus and Robertson. Hilder, B. (1980). The voyage of Torres. Brisbane, Australia: University of Queensland Press. Jack-Hinton, C. (1972). Discovery. In P. Ryan (Ed.), Encyclopedia of Papua and New Guinea (Vol. 1, pp. 246–257). Melbourne, Australia: Melbourne University Press and University of Papua and New Guinea. Jinks, B., Biskup, P., & Nelson, H. (Eds.). (1973). Readings in New Guinea history. Sydney, Australia: Angus and Robertson. Jonduo, W. (1993). Effectiveness of Tok Ples skul tisa training: Initial findings. Port Moresby, PNG: Department of Education. Kaleva, W. (1998). The cultural dimension of mathematics curriculum in Papua New Guinea: Teacher beliefs and practices. Doctoral thesis, Monash University, Melbourne, Australia. Leahy, M. (1994). Explorations into highland New Guinea 1930–1935. Bathurst, Australia: Crawford House Press. Lean, G. (1992). Counting systems of Papua New Guinea and Oceania. Doctoral thesis, PNG University of Technology, Lae, Papua New Guinea. Mennis, M. (1972). They came to Matupit. Vunapope, Rabaul, East New Britain Province: Mission Press. Mikloucho-Maclay, N. (1975). New Guinea diaries 1871–1883 (C. L. Sentinella, Trans.). Madang, Papua New Guinea: Kristen Press. Murphy, J. M., & Moynihan, F. (1936). The National Eucharistic Congress. Melbourne, Australia: The Advocate Press. Myer, L. (1998). Biodiversity conservation and Indigenous knowledge: Rethinking the role of anthropology. London, England: Intermediate Publications. Nakata, M. (2004). Indigenous knowledge and the cultural interface: Underlying issues at the intersection of knowledge and information systems. In A. Hickling-Hudson, J. Mathews, & A. Woods (Eds.), Disrupting preconceptions: Postcolonialism and education (pp. 21–35). Maryborough, UK: Smartprint Solutions. National Department of Education Papua New Guinea (NDOE). (1986). A philosophy of education for Papua New Guinea (chairperson, P. Matane). Waigani, Port Moresby, Papua New Guinea: Government Printer. National Statistical Office. (2014). Papua New Guinea 2011 National Report. https://png-data. sprep.org/dataset/2011-census-report Owens, K. (2008). Culturality in mathematics education: A comparative study. Nordic Studies in Mathematics Education, 13(4), 7–28. Owens, K. (2013). Diversifying our perspectives on mathematics about space and geometry: An ecocultural approach. International Journal for Science and Mathematics Education. http:// www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10763-013-9441-9 Owens, K. (2015). Visuospatial reasoning: An ecocultural perspective for space, geometry and measurement education. New York, NY: Springer. Owens, K., & Kaleva, W. (2008a). Case studies of mathematical thinking about area in Papua New Guinea. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Annual conference of the International Group for the Psychology of Mathematics Education

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(PME) and North America Chapter of PME, PME32 - PMENAXXX (Vol. 4, pp. 73–80). Morelia, Mexico: PME. Owens, K., & Kaleva, W. (2008b). Indigenous Papua New Guinea knowledges related to volume and mass. Paper presented at the International Congress on Mathematics Education ICME 11, Discussion Group 11 on The Role of Ethnomathematics in Mathematics Education, Monterrey, Mexico. https://researchoutput.csu.edu.au/en/publications/ indigenous-papua-new-guinea-knowledges-related-to-volume-and-mass Owens, K., Lean, G. A., with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Paraide, P. (2010). Integrating Indigenous and Western mathematical knowledge in PNG early schooling. Doctoral thesis, Deakin University, Geelong, Australia. Parkinson, R. (1907). Dreissig jahre in der Südsee. Stuttgart, Germany: Strecker und Schröder. Ralph, R. (1978). Education in Papua and New Guinea to 1950. Unpublished book. Available at National Library of Australia, Canberra, Australia. Rannells, J., & Matatier, E. (2005). PNG fact book. Melbourne, Australia: Oxford University Press. Ray, S. (1891). Note on the people of New Ireland and Admiralty Islands. Journal of the Anthropological Institute, 21, 3–13. Rosa, M., & Orey, D. (Eds.). (2020). International perspectives on ethnomathematics: From research to practices, Revemop, 2. Said, E. (1993). Culture and imperialism. London, England: Chatto & Windus. Smith, G. (1975). Education in Papua New Guinea. Melbourne, Australia: Melbourne University Press. Smith, L. T. T. R. (2012). Decolonizing methodologies: Research and indigenous peoples. New York, NY: Zed Books. Tololo, A. (1976). A consideration of some likely future trends in education in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea Education (pp. 221–225). Melbourne, Australia: Oxford University Press. United Nations Educational Scientific and Cultural Organization (UNESCO). (2015). Pacific education for all 2015 review. Apia, Samoa: UNESCO, Office for the Pacific States.

Chapter 2 Foundational Mathematical Knowledges: From Times Past to the Present—Technology

Abstract: Foundational mathematical forms of knowledge are evident in many of the cultural activities which were in place before contact with Europeans, and are still reflected in today’s activities. These activities indicate the foundational mathematics of the many Papua New Guinean cultures and their varied ecoculturally-influenced activities which cover understandings of pattern, space, measurement and number. The mathematical knowledge itself is complex, different, and diverse, with relational knowledge being particularly noticeable. This chapter focuses on technology evident in material culture such as string bags, nets, traps, bows and arrows, and canoes, and on how the construction and use of these involved mathematical knowledge. Further items and relations are discussed in Chapter 3 under trade. Much of what is said in these two chapters has been developed from archaeological, anthropological, and first-contact evidence as well as from reflections on sustained mathematical practices of today. Our two Papua New Guinea authors have contributed their own knowledge, one from a Highlands community and one from a coastal island community.

Key Words: Design and mathematics · Ethnomathematics · First Contact · Foundational knowledge · Indigenous mathematical knowledges; · Indigenous technology; · Papua New Guinea culture; · Traditional bridges; · Traditional house building; · Traditional pottery. Recent research, mainly carried on by anthropologists, shows evidence of practices which are typically mathematical, such as counting, ordering, sorting, measuring and weighing, done in radically different ways than those which are commonly taught in the school system. This has encouraged a few studies on the evolution of the concepts of mathematics in a cultural and anthropological framework. But we consider this direction to have been pursued only to a very limited and - we might say-timid extent. D’Ambrosio, 1985, p. 44 Introduction Foundational knowledge is that which a society has used for generations, often over millennia of years. It is the foundation for thinking, acting and relating. In this chapter we will argue that the multiple societies of Papua New Guinea (PNG) had strong foundational mathematical knowledge as part of their activities and ways of relating to their environments and their societies. Smith (1975) argued that: Before Western contact, the peoples of Papua New Guinea lived within the limits of subsistence economy afforded by their fertile land…Conditions of affluent subsis© Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9_2

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tence for a relatively stable society influenced both the knowledge which they required and the way in which that knowledge was distributed. (p. 1) One might question the wisdom of Smith’s use of the word “affluent,” for it is known that there are times of food shortage, places with poor soil and low rainfall, and periods of climate change and disasters (siesmic activity, landslides, ice ages and now global warming). However, adaptation has become part of the thinking. Survival over time required best uses of materials and the environments involving technological developments and mathematical processes of comparisons, reflections, evaluations, and recordings. Through discussions the effects of trials were reflected on and further decisions made. These logical developments were often communal. Certain people had specific knowledge that would be shared in reciprocal relationships. Others observed and learnt following the knowledgeable persons’ practices and discussions. Thus mathematical knowledge was incorporated into the activities and passed down through the generations. Education for Indigenous children was mostly informal and undertaken by those who came into contact with them in their homes and community environments. Often persons with specific relationship to the child passed on certain knowledge. This tended to be more formal as the child grew, especially during adolescence and early adulthood. A cameo from Charly Muke provides an example of this. Charly was immersed in his culture and is now a person that many turn to as a leader with his higher education and doctoral status. These discussions are about community enhancement with decision-making still embedded in discussion among Elders and others. The cameo also draws attention to how current cultural knowledge is important in relation to our understanding of the technological and mathematical knowledge from long ago. Cameo from Charly Muke: Personal Story of Learning and Teaching Cultural Knowledge I believe I was shaped by two forms of education that played an interwoven role in my life. They were cultural education that occurred outside of school in my home village and education at school. In this section, I will aim to describe cultural ways of learning and teaching cultural mathematical knowledge, and attempt to make connections to school mathematics. The traditional methods of learning and teaching involved watching, listening, imitating and trying out under adult supervision. The cultural knowledge including cultural mathematics that I was taught was found within cultural practices that assisted in living in the environment; both physical and social (cultural). My story will be categorized into three parts of my life; before schooling, during my schooling and during tertiary education and later employment. I will conclude by reflecting on these aspects of my story and identify teaching/learning practices and mathematical knowledge which make connections to school mathematics. I begin with some background to my cultural life. Family and Cultural Background My dad was a young man when the first whiteman came in 1936 into the Wahgi valley of Jiwaka Province. He volunteered to learn oral Tok Pisin, with limited writing skills. That enabled him to earn the job of a Tanim Tok (translator) of the first district administrator established at the nearby township of Minj in Jiwaka Province. He would travel with the district administrator to villages and translate messages for the locals. He eventually became the first councillor in 1960 when the first local level government structure was established. During that term as councilor, he negotiated provincial boundaries in the Chimbu Province and that is where he met my mum, a very young girl. I was born around 1970. I do not know my exact birth date because my mum or dad did not know how to write and keep birth dates. Both my mum and dad did not complete formal education. Therefore, they could not read or write in English. They spoke Kuman, Yu Wooi and Tok Pisin. I learnt to speak all their languages and at school, I learnt English, which was also used at times in the village.

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Their daily living was supported by numerous cultural activities that I was expected to learn. My parents were engaged in these activities and taught me during these activities so I could continue them because our living depended on them. Activities in which my parents were engaged and in which I was expected to learn are: • • • • • • • • • • • • • •

Gardening Building houses Making fences Hunting small animals and birds (boys only) Making bows and arrows Swimming Men’s and family house reflection sessions Cultural festivals Food exchange Pig killing Bride price Compensation Tribal war Legends and poems

The young girls in the village also learned to make bilums (i.e., string bags) and join in the reflection sessions that accompanied these gatherings where they sat and made bilums or met in the women’s house. The adults ensured that the practices, skills and knowledge related to these cultural activities were passed on to their young because that was the way to survive and allow generations to continue with this knowledge. Therefore, they initiated learning and teaching practices appropriate for each age bracket. This process was what my parents experienced, and so it was for me. At the same time, schools were established and my parents trusted the schooling system to give me a good life to avoid the harsh life of the village. Therefore, they sent me to school. In the following section, I will describe my cultural ways of learning during the following three age-related periods of my life: before school, during school and during tertiary education and employed life. Cultural Learning Before School I am able to recall much of what I did from about age 3 before going to school at the age of 7, so I had a period of 3-4 years of being a village child. I remember that I was expected to do two things at this age. First, I was expected to learn cultural knowledge and skills, and second, I was allowed to play cultural games with children in my age group. The method of learning and teaching appropriate for my age was mainly observing and contributing by participating in small jobs related to the main cultural activities. To illustrate, I will describe the cultural ways of gardening. Gardening began by clearing bush for a garden, preparing the land, planting, looking after the garden when crops were growing (e.g. weeding and drainage), and harvesting. It ended with a feast for harvesting. I was involved in all those parts of gardening. To illustrate the extent of my participation, I will describe how we prepared the land for planting. Gardens were normally made in a distant place far away from home. Therefore, we would travel with tools, food for lunch, and brought domesticated animals which would dig for worms in the nearby bush. To assist parents with things related to the main activity at that time, I would help chase and keep domesticated animals on the path to the garden, carry tools, and carry cooking utensils to the garden. At the garden, I would fetch water and prepare the fire area for cooking our lunch. I was expected, and happily it was also my interest, to watch and

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participate in small ways with the preparation of the land. That involved making drainage, collecting rubbish to be burnt in the center of the plot for its ash, and digging the land for planting. I would give tools and carry uprooted plants to a heap to be burnt. I would make the fire and burn grass that had been heaped on each plot of prepared land; this, as I learnt afterwards, added carbon and other nutrients to the soil before planting began. I would help dad in making drains by holding the line rope for the drains so they were in a straight line. While dad was digging the drain, I would help pick up the earth dropping from the spade to clear the drain for an easy flow of water. When the sun came up and warmed the area, I would run to the nearby creek to fetch water for mum and dad. I helped mum and dad to cook lunch while sitting in the shade. I would also help mum by moving the pigs and tying them to a shady tree. In the afternoon, after lunch, we would continue working until late before we all returned home. If we had not finished, the same routine would continue the next day. I was allowed at this early age to play if I was not helping my parents. I recall that there were two types of play in which I took part with other children in my age group. The first involved imitating and modeling adult activities and tools. At this age, I was not old enough to participate in activities that were performed by adults, but I was expected to watch and listen. However, since our parents valued such activity, we were interested in them. To fulfil our desire to participate, we imitated and modeled the activities during our play time when we were away from the adults. We imitated making gardens, building house and acting out the roles of dad and mum. Boys, imitating their dads, would build a house for the family. The girls would imitate their mums by pretending to plant in the gardens, do house duties or look after the family baby. We made cultural artefacts that were important and exciting to the children. We used mud to make axes, cars, paint our faces like the warriors, spirits, etc. We also tried to make fences, roads, bridges, and bows and arrows. The second type of play involved children’s games. We played a variety of children’s games such as the following: • Winning bottle top within rings; • Winning marbles on rings; • Aiming objects using: –– stones –– bows and arrows; • Climbing trees; • Rolling small wheels made of round meat cans; • Making dust from branches with leaves (imitating dust made by cars); and • Singing. While many of these indicate Western influence, in the past children used round and flat seeds, made spinning tops, and imitated other exciting events like dust from singsings. To illustrate that a number of these games involved mathematics, I will describe what was involved in playing bottle top. This game involved children aiming to win bottle tops from each other. The children started by drawing on the ground a circle and two parallel lines. The circle was more than a metre in diameter. One of the lines was close to the circle and the other was between three and six metres away (see Figure 2.1). Before my time I have been told that light seed pods were used for this game instead of bottle tops. The circle was called “ring” by the children and the first step in the game was to put some of your bottle tops into the ring. This was your “bet.” The children would talk to each other to negotiate how many bottle tops each would throw into the ring—these were their bets—ready to be knockout of the ring during the game. For instance, the children could bet two each in the first game, three each in the second and four each in the third game, and so on. To win, the children had to aim at getting the bottle tops out of the ring with a flat stone which they called “spear.” The

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Figure 2.1. Bets of bottle top and spear stones positioned from line 2. second step in the game, after the bet, was for the children to determine the order to be played, and this involved the two lines the children drew. They first took turns to stand behind the line close to the ring and throw their spear at the second line. The order was determined by the position of the spear stone landing close to the second line. The illustration in Figure 2.1 is from the first game; the closeness of the spears to the line determined the order for playing the game. This only happened for the first round of aiming at winning the bottle tops from the ring. The reason was that all the bottle tops were spread close together in the ring and the person who threw first had a better chance to knock them out of the ring to win than those who came later. When all the students had thrown their spear stones at the second line, they all gathered around the line, standing next to their stones and comparing to each other with reference to the line to determine their placings. Figure 2.1 shows the decision that the children made. The stone closest to the line was determined as first player followed by the second, third and fourth, etc. The third step involved trying to get the bottle tops out of the ring. In this order, the children threw their spears from behind the second line aiming at the bottle tops in the ring. If any player knocked a bottle top out of the ring, he/she would continue until he/she missed. If he/she managed to knock out a bottle top, then he/she could continue aiming and throwing at the bottle tops in the ring. The next throw occurred from the location where the spear landed. This person stopped when he/she missed a throw at the bottle tops in the ring. Then the next person in order would take his/her turn from the distant line. If all the children standing at the line completed throwing from the line but all had missed, then the second round of the game began. The first child started the second round. This time, he or she started from the location where his/her spear stone was located after he/she had missed. This continued into the third round and so on until all the bottle tops in the ring were all knocked out of the ring. The child who won more than his/her bet was regarded as the winner, the child who won less than the bet regarded as the loser, and those who won back their bets were regarded as “lucky” as they did not win or lose. The children started the next two games by betting 3 and 4 each going through the above steps again to determine their wins or losses (or neither). All the above activities were performed close to home and not far away from parents because of our age. Each helped me learn cultural knowledge, and in each case there was mathematics embedded in both the activities and games. I will make explicit connections to mathematics later, but for now I turn to the activities that I did during my primary and secondary school years.

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During School—Primary and Secondary Education During primary school years, I participated in the same cultural activities at home as those I did before schooling. However, nothing that I did at school was related to what I had been exposed to at home. The school used a different language (English), different knowledge, and the activities were never related to my home. So there was nothing I could describe that related to my home. When the school day ended and during weekends, I returned to my home and carried on with what my family perceived important culturally and played the children’s games. When I was between the age of 13 and 18 years, I left home to attend boarding high school to complete Years 7–10. Again, learning was in a different language from the one used in the village and knowledge was also completely different from that in my culture. The only thing that crossed over to my home culture, and was actively promoted by the school, was during cultural days and graduation when we would be allowed to dress in our cultural dress and sing in our own language. Everything else had nothing to do with my culture. Therefore, the only time I could speak my language and engage in my cultural activities was when I went home for school holidays. I could not go home during weekends because it was too far from home. It was during the holiday breaks that I learned to do the age-related important activities of my culture. At this age, I was big enough to be involved in the actual activity with careful supervision by adults including parents and experts who had knowledge of certain activities. Some of the important cultural activities I participated in included gardening, making houses/fences, domesticating animals and in cultural festivals. No longer was this play-acting or imitating, as it had been when I was younger, this was real participation. The tasks were planned and initiated by my parents or other adults and I helped. At this age, I was expected to live in the men’s house and listen to Elders talking, advising and reflecting on history, situations and processes related to likely future happenings. One example from this time is gardening. I will describe preparing gardens for planting. It involved digging drains, and then digging the earth to make mounds for the planting of sweet potatoes. Figure 2.2 shows the plan for drainage, plots and the mounds made in each plot. This is one part of the gardening process, but the spatial and counting aspects of mathematics are clear to see.

a. Garden plots at Vule village taken for elementary school project by Susie Daino. b. Muke’s diagram of his village plots. Figure 2.2. Drainage with mounds on each plot for planting sweet potato, usually three stalks per mound.

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Tertiary Education and Employed Life After completing Year 10 in 1987, I was admitted to a two-year teacher-education program at Kaindi Teachers College in Wewak, in the East Sepik Province. The teacher-education program included teaching school mathematics. Such teaching drew very little on the real and cultural practices with which I was familiar and with which the children would be familiar. However, my own education in cultural knowledge and skills continued when I returned to the village during holidays. As I began full-time teaching, my participation in cultural activities relating to village life declined. My living was assured through paid employment as a teacher, and this took my focus away from cultural activities like gardening, fencing and building houses, although I did these on a small scale beside my house at my teaching location. However, I did continue to contribute to cultural festivals and participate in them during school holidays. Most cultural activities could be timetabled to take place during school holidays so that all members of the village who were in schools (students and teachers) could return and participate. My story differs little from the stories of many of my peers. More and more children of my age left school for employment. Hence we were unable to pass on the knowledge of villagerelated cultural activities to our children in the same ways in which our own parents had done for us. A sad trend that is emerging is that our children who have gone to school outside the village find it hard to fit into the cultural ways of life of their village if they do not succeed in schooling and hence find paid employment. There is a need for cultural practices and knowledge to be embedded in school including in the mathematics that is taught. The way forward is to dig more deeply into understanding mathematical ideas and concepts found within cultural activities. If teachers and parents could be made aware of the connections between cultural mathematics and school mathematics, they would be more ready and able to value and accept the practices to be used for teaching and learning at school. Therefore, the following two sections aim to make such connections. Before I draw out some of the mathematical practices from cultural activities, and make connections to mathematical concepts, I will describe cultural ways of teaching and learning which will be important for this discussion. Cultural Techniques of Teaching and Learning In most cases, parents, and later Elders as well, taught their children while performing the cultural activities. The knowledge and skills related to cultural activities were taught by people actually engaged in those activities. The parents would clearly explain as they performed and reminded the children that they would do this when they grew up. Hence the teaching involved demonstration, explaining and emphasizing the importance of the job for future life. They also taught by questioning; did the children understand the reason for doing and using certain practices? When they were old enough (13–18 years), they were asked to take part in the activity under supervision. The scale of involvement increased as the children got older. Hence children’s learning took place while the cultural activities took place in real time, as well as the games they played. When examining the learning process, there were specific learning techniques and methods promoted. Briefly, for cultural skills and knowledge related to lifedetermining cultural activities, children learned by being close to their parents to observe, imitate and take part in real life-determining activities. During games, children learnt by talking to each other, explaining in their own languages, playing games collaboratively, and learning the skills of the game while taking part. In addition, children were also engaged in a reflection process in the men’s house to predict future occurrence. The learning process involved here was listening, asking questions, and engaging in active listening in order to become familiar with the flow of the story.

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Mathematics Concepts Embedded within Cultural Activities Connections can be made between some cultural mathematical knowledge embedded in cultural activities and school mathematics. For this to occur both teachers and the general public need to become aware of possibilities and generally approve of the approach. There is a need for a much deeper understanding of the conceptual and cognitive frameworks embedded in cultural practices that could provide support for the learning of school mathematics. If such understanding is explored and made explicit, both parents and teachers can better promote cultural practices in the schooling process. From international studies, it has been identified that most cultures in the world, including my own1, have engaged in what seem to be universal activities that are fundamental to the notions of doing mathematics. One such study is that by Bishop (1988) who identified six such mathematical activities: counting, locating, measuring, designing, playing and explaining. All these fundamental mathematical concepts can be seen in the cultural games and activities I have described above in which I was engaged when growing up, and they also were evident in the school mathematics I learnt. To illustrate some of the connections more clearly, I will continue to examine the two activities I described above: the bottle-top game, and the preparation of land for planting. From the bottle top game, children were engaged in various aspects of mathematical knowledge. During the bottle-top game, obviously we children were involved in playing but also designing, counting, measuring, locating and explaining. Some examples are: Designing: Designed and marked the place for playing the game; Counting: Counted bottle tops placed as a bet in the ring but also doing arithmetic to decide how many bottle tops were placed as bets from all players, how much they won or lost by, and how to decide who won; Measuring: Measured the circle’s radius (and hence observe the area), the distance between lines, the difference in distances that the spear stones were from the line to determine the order in which children played; Locating: Located the best place to draw the ring, and comparing the positions of spear stones to decide the order of play; Explaining: Communicated with each other about how many bottle tops to bet, deciding the order children would play, and finally deciding the winner of the game. Reflecting: Reflected on garden making—especially for preparing the plot for planting sweet potatoes—mathematics was involved in this activity. Designing: Designed the drainage so that water could flow easily. How many mounds needed to be made per plot. Interestingly my parents did not use paper, but this designing was carried out in their minds and then discussed between the two of them. Of course, I was listening. Locating: Located good land for gardening, determining the best location to make drains to flow easily to other water channels, locating where to build mounds to plant sweet potatoes. Measuring: Decided what should be the lengths of drains, size of the plots, height and size of mounds. Counting: Estimated and counted number of drains needed, plots, mounds, expected number of plant stalks for each mound. Explaining: Discussed why choose this land and not that, why dig drains in this or that location, how to dig drains, and how to make mounds.

i.e., Charly Muke’s Yu Wooi, Mid-Wahgi culture

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Conceptual Analysis Although the village gardeners may not use terms used in school mathematics, they are very aware of “implicit” mathematics. The following bullet points detail the activity with connections to school mathematics: • Gradients, or slopes and the angles formed by the drains to the imagined horizontal; • The best angles to use to keep the water running but without being so fast that the drains would be eroded away; • The parallel edges of the drains are formed by digging and keeping the sides of the drain equidistant as they move along the drain—albeit assisted by the string marking the edge of the drain. This is done by sight with the string being used as a guideline; • Often the spade is used to indicate the depth and width of the channel so the base of the drain is smooth for the water to flow easily. For example, they depth of the spade in the holes was used as a check for the depth of the drain; • Dividing the space between drains is also considered; • A choice is made to make two, three or four mounds spaced across between the drains. Interestingly in some areas they also stagger the mounds so they are forming a pattern of equilateral triangles rather than squares. This assists with water flow and provides the best use of the land per mound. Usually, the plots are larger and absorb the rain well. • Although counting per se may not occur, the visual representation of the number of mounds and hence vines—usually three per mound—is visually represented. In terms of linking to primary school mathematics, rows of mounds can be skip or group counted (e.g., 3, 6, 9, …), assisting an understanding of multiplication as accumulating equal groups. Comparisons of numbers for a sense of size is also evident, especially when there are several garden areas.

Mathematics Education Before European Influence

There are five key sources of evidence for discussing mathematics education before contact with European people. These sources inform each other and intersect in the arguments to be presented in this chapter. The first source is the oral histories of those who have shared their stories with their children or grandchildren and who now share these stories. These become significant for establishing the archaeological importance of certain places in PNG (Ballard, 2003; Mennis, 2014). The second source is the intergenerational maintenance of knowledge through which children learn through stories and participation in the culture of their time and in mathematical activities of their forebears. Charly’s previous cameo and Patricia’s in chapter 3 provide evidence of these two ways of learning. The third source of evidence is from descriptions given by Europeans on first contact. The fourth source is language and in particular linguistic archaeology, which informs discussions of longevity, connectivity, and migration. The fifth source is that of archaeology. Artefacts and their use and the impact of humans on the environment indicate mathematical activities. Each of these is now discussed in more detail. The first two sources are discussed in the next section, then evidence from first contacts is given, and finally the last two sources of evidence are considered. Oral Histories and Current Practices Foundational knowledge for a community includes current knowledge. Much of the foundational knowledge is expressed in oral histories both recent and from the distant part. It plays an

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important role for spiritual purposes within the community. Ecocultural knowledge and interpersonal relationships are always embedded in community activities. Mennis (2014) summed up the activities for the Bel people of Madang—this represents only one of at least 850 Papua New Guinean societies, each of which divides up activities in different ways: The men did the hard work: cutting down trees; building the houses and canoes; making the garden fences; hollowing out new drums; making sago in the bush; making the yam mounds; and preparing the gardens for taro. They also hunted with their bows and arrows and killed and butchered the pigs. In preparing for a feast, the men grated the coconut and rolled the sago patties in it and then they grated the bananas and taro. If it was a special feast, the men did the serving when there were honoured guests. They also made a special grace calling for protection from the spirits over all those present at the meal. And, of course, they made the long trading trips and bartered the pots made by the women for other trade items. On the other hand, the women planted the taro, did the weeding, carried the food, water and firewood as well as doing the cooking, looking after the children and feeding the pigs. They also fished in the sea and the rivers both with nets, fish traps and hook and line. In the evening, the women would make bilums and pots or weave fish-traps with various vines. (p. 16) Each cultural group maintained its relationships and its practices in terms of mathematical activities. In many cases, many people were needed to carry out activities such as building houses, especially long houses (Crawford, 1981). Needless-to-say, with European colonization and a breakdown in some foundational activities, relationships and beliefs, some activities were no longer practiced. However, in the case of Gogodola house-building skills and islander sailing skills, these were only recently being passed on to new generations, including the mathematics within the activity. First Contact with Europeans Europeans initially sailed into the waters of New Guinea and the Melanesian islands to the east from the 1500s. The first Europeans were Portuguese, Spanish and Dutch; then came the French and later the Germans and English. A list of contacts, dates and notes is provided in Appendix 1. An astounding number of sailors, explorers, government-directed expeditions, naturalists, and business men visited and described their contacts, usually in terms of exchange of goods and quality of contact such as peaceful, timid, threatening, or impossible (Gash & Whittaker, 1975; Souter, 1963; Whittaker, Gash, Hookey & Lacey, 1975). In fact, the many reefs and shoals that these ships had to avoid reduced contact but the need for fresh food and water necessitated landings from time to time. For example, Bougainville zigzagged south of the Milne Bay Islands and was somewhat confused about whether the lands he saw were the Solomon Islands. Eventually he sailed up the east coast of Bougainville and New Ireland. Malaria was a factor which discouraged settlement. D'Entrecasteaux (2001, translation of his diaries) made two trips in 1792 and 1793 looking, without success, for the lost La Perouse ship. His account was especially important as he provided a positive perspective of the local peoples’ skills—this from a particularly competent cartographer and navigator. During his first visit he sailed north from New Caledonia and turned north-west at New Georgia (Solomon Islands), skirting around many shoals especially off southwest Bougainville. Then travelling north, he thought Buka and Bougainville were connected. He went through St George Channel (between East New Britain and New Ireland) as he realized the current and depth of water suggested a passage that was missed earlier by Dampier. He visited

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some of the Duke of York Islands and sailed along the west coast of New Ireland to one of the Manus islands east of the main Manus Island, and then travelled west to Ambon. In 1793, he came from the east to the south of San Cristobal, south of Guadalcanal, moving back and forth with the shoals, tides and winds until he headed south. He then sailed west to Rossel Island, travelling north of many islands of Milne Bay including the Trobriands before following the mainland out to sea until he reached the Huon Gulf. He commented that the people had not had previous contact as they did not want iron such as axes or nails. That suggested to him that they did not recognize the advantages of these tools. He then passed through the Dampier Strait between Umboi Island (Siassi, Morobe Province) and New Britain, along the north coast of New Britain, east of New Ireland and then west. Sailors like D'Entrecasteaux (2001) recognized the amazing knowledge and skills of the “natives,” which is significant, given their own sea-faring skills (see comments about the canoes later). The visitors also recognized the considerable differences between people they met in terms of the quantity and quality of the knowledge of the “natives.” Some described physical characteristics of the people and their demeanor, and a few described some of their techniques such as for sailing or bartering, but few actually observed village life. Village life was explored by Mikloucho-Maclay (1975) in Madang, and along the north coast of the mainland; by D’Albertis in West Papua, and in Western Province and Yule Island, Central Province as well as in the south (Souter, 1963) of PNG. Meanwhile, missionaries such as Lawes (Papua), Brown (New Ireland and New Britain), and Frierl (Madang) began settlements and training centers and/or schools (Gash & Whittaker, 1975; Whittaker et al., 1975). Archaeological and Linguistic Studies Archaeological studies and archeological linguistics also provide significant input into our understanding of foundational mathematical knowledge. First contact explorers and government workers recorded people’s languages as best they could. The diversity of types of language did not make this easy, although the coastal languages were mainly Austronesian Oceanic and had some similarities. Linguistic studies have informed us of the longevity of the languages, including counting systems, the possible spread, the innovations, and the modifications (Owens, Lean, with Paraide, & Muke, 2018). Understanding space and geometries is also expressed in language providing evidence of foundational spatial knowledge (Owens, 2015). In partocular, archaeology provides the interpretations of how societies may have created and used objects and managed within their environments. There are numerous archaeological sites across Papua New Guinea mainly run by the University of Papua New Guinea (UPNG) supported by Australian, European, and other universities. John Muke, Charly’s brother, has completed many archaeological studies, usually with Jo Manggi, another Mid-Wahgi man. The pottery of the Lapita trade was of special relevance in terms of archaeological linguistics.

Evidence of Technology and Mathematics

Archaeological records show changes in the environments that point to there having been large populations in some places—like the coast of Central Province as well as the upland valleys around Mt Hagen in the Western Highlands. In these areas, there is evidence of developed technologies to maintain these large populations—such as in agriculture and the management of the landscape, and in the extraction, modification, and crafting of materials from the environment.

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Both agriculture and the extraction and development of flora, fauna, stone, and mineral resources require certain technologies, each involving mathematical processes.

String, Binding, Bilums and Tapa

Mikloucho-Maclay (1975) in his 1872 diary noted that the bark of thin trees2 was carefully removed in one piece and beaten on a flat stone with a stout, short stick. This is the way both tapa (used as the equivalent of cloth) and string began. The tapa provides coverings of various kinds. Cultural groups in Papua New Guinea do not weave (weft and warp) cloth from string although many use leaves to weave baskets, walls, sleeping mats and other items. When the bark is beaten into fibres, these are then rubbed on the thigh to make string. The string is used in continuous looping techniques and made into continuous string bags called bilum, or other items, some of which are coverings. There are several different kinds of loops in these objects. The figure-ofeight looping (Figure 2.3) was mainly used and this allowed for stretch especially in the more open form such as those made over a strip of pandanus leaf. The looping techniques varied in both the loops that are formed and in their tightness, but looping forms an interesting network in non-Euclidean geometry perspective. The making and giving of these objects had significance in terms of relationships between people (Mackenzie, 1991). Mackenzie (1991), for just one language group Telefol, provided names, uses, and diagrams of six open-looping techniques and four tight-looping techniques used in bilum-making, called men in Telefol, four binding or joining techniques, and four construction processes such as to form a base or tighten an opening. There are 26 different names given to different bilums depending on their size and purpose, decoration or containment. Many of the bilum types were associated with the relationship between the maker and the recipient. Design and technique required considerable pattern recognition of the different techniques, visuospatial memory (visual and kinaesthetic imagery), trialing of techniques, learning of techniques with all the associated corrections, and associated relationship knowledge. While bilums are the main carrying device, held mostly on the forehead by the band and hanging down the back (men often wore theirs around their neck and hanging in front), other uses of bilums in different places are for decoration, often associated with dogs teeth, pigs tusks, shells, feathers and possum fur, front covering skirts for the men usually quite long for dancing warriors, mourning veils and mourning ropes, fishing nets, hair covers, amulets, armbands, bases for singsing feathers, and small threedimensional objects like dolls or pigs or for head objects. Designs are created in the bilums by both changes of looping technique or changes of color, the knowledge of which may be used to identify the places of origin of a bilum. Initial patterns based on “diamonds” or horizontal squares have been modified to create new designs, and these at times spread across country and language groups. Women combine shapes such as two trapezia for a hexagon in the “soccer-ball” or “50 toea” designs. They narrow shapes, produce lines, and make concentric shapes, repeated and reflected shapes, letters, and pictures. They think and work along oblique lines, count or subitize the number of “stitches” in the pattern, and continually check that they are making a neat, repeated pattern. They visualize the overall product and the small shapes as they make each one. Traditionally, as noted above, string is made from a variety of inner barks especially the tulip tree (also used for tapa). More recently introduced products like sisal, and later trade-store string, plastic string, and wool are now also used. Using the traditional technique, two lengths of different colored trade-store wool rubbed on the leg is at times employed to make string of a diversity of color by women in creating new designs. A woman might work with up to 20 different threads of different colors (some repeated) to create a design. These designs are shared generally for relationship building just as the bilum is given as a gift to mark a special relationship. These were probably tulip trees

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a. Basic bilum loop b. Headdress Simbu c. Headdress Western Highlands d. Bena Bena, Eastern Highlands e. Headdress Eastern Highlands Figure 2.3. PNG dress and decoration- bilum usages, basic bilum loop.

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Bilums are now also sold for money. The diversity of new designs is extraordinary, not to mention the mathematical thinking and visuospatial memory behind the pattern making (Owens, 2012, 2015).

Extraction of Minerals and Colors

Lime was extracted from limestone rocks, coral and shell fish. Around 1985, at Labu, women gathered many long shell fish from the lake to make lime. Owens observed them heating the shells on limbom palm racks stacked up over a fire on the beach where the wind would take the fire gradually towards the back of the racks. The heated shells fell down and were collected. From the feel of the shells, they knew if they were ready to become powder or whether they needed further heating—in which case they were returned to the fire. After placing the shells on flywire, any sand was shaken off the shells (other material may have been used foundationally). The shells were then placed in sacks which were pounded to make lime powder. This technology involved recognizing the amount of heat and wind needed for the processing, building the racks with wood at right angles alternately in one direction and then the next, and knowing the amount of time needed for the various parts of the process. Lime is chewed with the areca betel nut, but also used for white coloring especially in the Oceanic societies around the coast and throughout the East Sepik and Sandaun areas. It was used on both wooden carved images, painted on panels and door boards, and on pots. In Papua and other places, it was used on large shields. White ochres were also used for these purposes and for skin decoration together with yellow, brown, black, and red ochres. Various colored clays were found in specific areas of ground. The colors were highly prized for decoration. White, yellow, brown, and red ochres were fairly common. However, other sources of minerals provided blacks, blues and greens and were traded. Discussing pigments in the highlands, Hughes (1977) says: Green pigments are rare. …one was colored by cholorite. The other was brightly colored by a basic sulphate of copper and aluminium called spangolite. Difficult to get to and laborious to gather, this color nevertheless was gathered in sufficient quantity to trade to villages 30 km away. Blue colors were traded more widely still. Within that same large area, I know of only four sources, one of which is a modern discovery. All consist of vivianite of a beautiful pale but well saturated blue…in central Chimbu it was traded over an area of 600 sq km. The most remarkable pigment, most valuable and most widely traded, was a glistening shiny black mineral … which is spectacular hematite. Special techniques were used to enhance its lustre. The trade area of this paint was expanding in pre-contact days and, uniquely, it has continued to grow. (p. 31) With respect to the Gogodola, Crawford (1981) wrote: The artist works always in four basic pigments: black, white, red and yellow, which are made in the following manner: Black (idi)—a white or slate-colored clay of the same name mixed with a juice called idede, extracted by chewing the back of the idede tree (Glochidion sp.); the reaction of the idede on the clay is such that it turns pitch black; White (abilo)—a white clay of the same name, mixed with water; Red (wasa)—a red clay of the same name, also mixed with the idede juice, and very often supplemented by adding the juice of crushed Bixa Orellana (ikalawasa) seeds, either with or without the idede; Yellow (mala)—a yellow clay of the same name, mixed with the juice extracted by grating the yellow bulb of a wild ginger called komela. (p. 202)

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These are water resistant, except for abilo, so it is mixed with white tree resin from the Canarium tree. The amounts of water and pigment required for a good mix and to cover a particular area or string became shared intergenerational knowledge. The mineral colors supplement the multiple vegetable dyes used for bilums and tapa painting. Brown, purple, pink, yellow and green dyes were used. Most were obtained by soaking leaves, barks, roots and flowers of various plants to make the paints3. The colored seeds were smashed up and if necessary strained through coconut fibre to make the body, face, and wood paints. Different plants were used in different places. Marita or arang is a red pandanus that grows in colder parts and used for making red color and also processed to extract the rich red flavored soup. Onggi (2005) discussed the Telefol door boards and in particular the ratios involved in making the various colored paints. These foundational knowledges continue to today in many areas. Feathers, especially from the various Birds of Paradise, but also from other birds like cockatoos, were collected for headdresses. Beetles were also crushed for their coloring. Significantly these are carefully positioned to make unique and culturally identifiable headdresses. The mathematics of the designs relied on the shapes, quantities, colors and creativity of the maker (see Figure 2.3). Symmetry was common in these designs. Natural resources had special spiritual significance for people and although technology and mathematics may have been used for extracting, processing or crafting, this special relationship of the natural object should not be forgotten. Gammage (1998) stated that, when Black and his supporting line reached the top of the cliff above the Strickland River and looked at the oil seepage, people then regarded it as contaminated. Archaeologists have noted that, if there were no traded stones in a place, that did not mean that there was no trade in the past. It may have been that special old stones had been buried in gardens and under houses for special spiritual benefits. Trade of objects was also significant in terms of the building up of relationships, often linked to spiritual positioning (see Chapter 3 for more details).

Food Capture

Although upland valleys, sago swamps and gardens provide food, people in PNG rely on capturing food sources available in their environment. For example, in highland communities people will own land for food gardens, bush areas for plants providing nuts and materials for housing, and other areas, often higher on the mountain, for catching bandicoot or other animals. Captured food includes fish, birds, pigs, bandicoot, tree kangaroo and cuscus, lizards, snakes, rats, flying fox, and insects. Feathers, skins, teeth, tusks, and bones are also used for various other purposes. For example, the skins of tree kangaroos are highly prized for symbolic position and for warmth, while the skins of lizards and snakes are used for drums. Fish capture The sea, rivers and creeks provide fish, crustaceans, and shell-fish. There are a variety of fish traps, usually allowing fish to enter but not easily return out of the entrance because of the internal funnel. Mikloucho-Maclay (1975) was one of the first Europeans to live in a village and record everyday practices. He indicated that the large fish traps were “sturdy, meticulously made from bamboo, worked by three men, and had quite a complicated form” (p. 161). Paraide (2010) noted that people setting traps in the sea knew about ocean swells and waves and learnt how to use weights to keep the traps in place to counter the swells. The mathematics includes knowledge of mass, seasons, design of traps, and skills for spearing. An interesting adaptation occurred in the 1960s when schools printed with colored stencil sheets and methylated spirits. The used sheets with surprisingly similar colors, became popular for getting colors. 3

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Most crustaceans are speared with a three-pronged spear. Fish are also speared, but for these, often catapults of springy material are used to send the spear more quickly. Men generally fish at night knowing where the fish might be found around the reefs. Lights are used to attract the fish but often men have to dive for the fish. Rods, lines and hooks are also used. Nets were also made and even today some are handmade by villagers. In one Morobean village around 1982, Owens watched as the people all just went silently to the shore together. As a community and as if one, they had noticed a run of mullet. The net was taken out by a couple of canoes and then dragged around to the shore by many people thus capturing the fish. In Malalamai (2006), the women informed us that they made about 30 balls of string from the inner bark of the tulip tree to make a net. These were then knotted with a special row of knots at the top, with floats made like fish—the particular fish represented by the knots were the type and size of fish expected to be captured for a particular net. Shells were used as weights at the bottom of the net. When the net was full, this middle float would sink below the surface or turn vertically. Other nets, such as those made at Bongu on the Rai Coast in Madang Province used two strong canes for two sides of a triangle with the net between them. Near the join of the overlapping canes was a cross beam to strengthen the triangle and assist with carrying the net (Biro, 1899, State Library of Queensland, cited in Mennis, 2014). Animal capture—nets and traps A picture taken in Wanigela in Oro Province in 1910 shows men making a boar’s net, presumably to catch a wild pig (Percy Money’s album held in the Australian Museum). Traps vary depending on the village culture but also the type of animal which was to be captured. Traps included the lasso type with a slip knot and a heavy counter weight used especially to capture wild pigs. Other traps are covered holes into which the pig falls. For bandicoot and small animals, a platform is made that goes down with the weight of the animal triggering the noose to tighten or a log to knock the animal unconscious or dead. In 1997 Architecture students at the PNG University of Technology modelled over 15 types of traps, some being slight variations of others. To construct the traps requires technological and mathematical knowledge involving the idea of mass, counter balance and knots, as indicated by the students’ models and explanations but also demonstrated in the following report: In Nega culture (Lufa District, Eastern Highlands), whoever is constructing the trap usually adhered to the following sequential order of construction; i. Firstly, prepare all the necessary materials necessary for the construction of the trap… ii. Secondly, drive two or three sticks firmly in the ground which will create something similar to the door. Then, bend a split bamboo that will form a semi-circle (1800) with respect to the horizontal ground level. This is called We’a in Nega language. iii. Tie the driven poles that create a door with the bend [sic.] sticks with a bush vines (known as Fimita kweda) firmly. iv. Next, drive a long piece of twig in the ground approximately about a metre away from the We’a. Bend the twig at about 1800 with respect to the ground. (Selectively, experts choose a branch of a special tree that can be able [sic.] to withstand bending and avoid fracture. According to my grandfather’s advice, a branch from a K’tapa tree is normally used). This piece of twig is known as K’mewa yawa v. Tie a rope that is known as Klge’vu kweda onto the bending twig (k’mewa yawa). Prior to tying the rope, that rope must be tied in the double loop form. where one is small while the other must be big. (Both loops must be approximately in a circular manner, that is, at an angle of 3600) [sic.] vi. Place a stick that bisects the circle forming a chord. Then a small piece of stick of about 5 to 8 cm long should be placed in such a way that it bisects the sticks (Kamo kewa) perpendicularly to the horizontal stick attached.

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vii. Finally, block all other alternative routes to ensure that the bandicoot travels only by that particular path. This can be done with sticks and grasses. As a result, the bandicoot travels through the hole, where it will be trapped easily. viii. Also, attach a ripe banana on to the trap. This can be done by carefully studying the direction through which the bandicoot normally travels. (Hunters are very innovative and put ripe bananas because bandicoots are fond of eating them. So, in the struggle of eating the bananas, they can get trapped very easily.) (Aurawe, 2007) Another technology required for food capture are the bows and arrows. Since they are also used for fighting along with spears and shields, we will discuss these separately. Fighting and Animal Capture Bows and Arrows Leahy (1994) commented on the opinion of the men, assisting him during his early patrols, regarding the lengths of bows from different regions indicating that the bows were designed, mathematically, for specific people and used in skirmishes. These skirmishes were sometimes quite devastating, required alliance building, and in some cases led to whole groups becoming refugees displaced to other places.

a. Fish trap. b. Fishing net (Martin, 2007) c. Bandicoot trap (Aurawe, 2007) Figure 2.4. Traps and nets.

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The making of bows and arrows relies on measuring the height of the man who is to use them. The bow is made to a height that, when taut, would be at shoulder height and easy for a man to run with it. The bow relied on “laminating,” that is having three sections of the “string” made from cane to give it strength for the knot at each end. Arrows are designed for straightness with the choice of pitpit (Gohok in Wahgi) and heat. The pitpit would be heated over the fire and when it is soft, the men would bend it at the part that is not straight, and then they would confirm its straightness by levelling it with their eyes. Pipit is used for its soft and hollow inner part so that sharp points could easily be attached. The sharp points are mainly made from sharpened bark of palm tree (Bune in Wahgi language) wood or bamboo. These sharp tips are firmly attached to the end of the pitpit by rope. The rope is then covered with a glue-type black material extracted from honey made by insects. The honey is heated to melt so it spreads evenly and smoothly over the rope. The sealing of honey is used to protect the rope and make it waterproof so that the arrow lasts. The tips of the arrows vary according to purpose. They use two different types of arrows for warfare. The first has serrated ends made of wood so it is not easily pulled out of the flesh. The second type is smooth and long so that it penetrates deeply into the heart to kill. These tips are made from bark of palm trees (Bune). There are also variations for hunting and killing ani-

a. Bow and arrows b. Dagger with teeth shows prestige c. Band of warriors (Thomson, 1892) Figure 2.5. Arrows used in fighting and hunting

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mals. For killing a captured pig, a sharp point arrow was used and a flat spearhead for cutting through the animal’s flesh. These are mainly made from bamboo (Figure 2.5). For killing birds, a three-headed arrow-tip was prepared. The three heads are smaller versions of the spearhead bamboo made for pig killing. These arrows are used by bird hunters while sheltered near a water hole mainly found at the lower end of big trees. The hunter would have a hollow fern tree attached to the shelter at chest height to the waterhole where the arrow was placed. At the other end of the waterhole, a cuisine was placed, made of the bark of the fern tree. Using the bow, the arrow would be used to aim at the bird drinking water from the waterhole and the three headed arrow would be used to pin the bird against the cuisine for the hunter to collect, mostly alive4. To this day, in the Highlands there is periodic fighting and the need for negotiations. Tribal fighting was the oldest and ultimate traditional way of settling tribal disputes. Therefore, the knowledge to make weapons was as old as making tools for gardens in the highlands. John Muke (1993) studied tribal warfare for his PhD degree. He made comparisons between tribal warfare with football games in England or rugby league in Australia. It meant that it had rules, spiritual obligations and with special designed fighting weapons that displayed specific religious and social messages. J. Muke (1993) identified specific historical weapons including wooden shields, spears, bows and arrows, and analyzed their designs and paintings. Some specifics will be described in the section that follows. On the coast, Malalamai people also talk of the highlanders, Papuan speakers, coming down not long ago and fighting a bloody battle and they continue to maintain the skills of making their bows and arrows and keeping them handy. Traversing the Madang foothills also brings stories of recent fighting between the people from Henganofi and the villages in the foothills (stories told in 2008 to the Owens). Spears Finsch (1888) noted that in the Sepik: The spears were 66 inches to 94 inches long, with wooden points and barbed hooks. These were too light to throw by hand and too heavy for a bow. They had a special implement, a two-foot-long bamboo stick, to throw the spears with. This method we had not seen in New Guinea before. (pp. 292–293, reproduced in Whittaker et al., 1975, p. 305, translated by Christa Hustwayte) This is an interesting description as it makes one think of the woomera for throwing spears used by First Nations in Australia. The shapes and diversity of materials for making the spears in different areas and by different language groups would suggest experiments with different materials, again requiring mathematical thinking, generally memorizing the various trials and differences. Leahy (1994) also noted the shape of the long spears made by the Mogai in the Highlands together with bows and arrows. The spears had two large side parts with points for penetration. Shields Across PNG, shields with various designs and sizes are found. At Bili Bili, Madang, in 1872, Mikloucho-Maclay (1975) commented: Many huts had shields hung, newly decorated with white and red paint, … made from one piece of wood, circular with a diameter of 70 centimetres to one metre, and about two centimetres thick. On the front side near the edge two concentric circles are engraved. The figures in the middle of the shield vary considerably. These shields are painted only on exceptional occasions. … The shield covered the head and chest and could very well shield them from an arrow or a spear. (p. 201) The information was provided by interviews in Malalamai village, Rai Coast, Madang Province and supported by Muke from Mid-Wahgi, Jiwaka Province in the highlands. The Tok Ples words used in the description are in Mid-Wahgi. 4

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Leahy (1994) referred to shields in his first contact trips in the Highlands in the 1930s. Beautiful shield designs are now displayed in the National Museum of PNG and were at the Pacific Lodge Goroka. Some of these wooden shields are found in the Wahgi Valley of Jiwaka and were examined in detail by Muke (1993). As Muke (1993) indicated, perhaps more than any other object, the wooden shields embodied the symbolic and religious dimension of Wahgi warfare. The popular preference for the red color also had its religious associations (O'Hanlon, 1989). Being sacred objects, they appeared in public only during formally organized tribal wars. The design, painting and the way wooden shields were used, supposedly, to communicate a number of messages. They were often seen as an extension of the warriors, and a representation of a shadow image of the fighting force. The various parts of the shield were seen to represent the body parts of the warrior. The shields were also used as a screen to communicate aspects of Wahgi war talk (opo yu). They were meant to reveal the underlying moral conditions of the performing groups, and the metaphors used by the spectators to describe the symbolic functions of the shield recognized the relationship between the gods and the warriors (Muke, 1993). The role of the shields was to deflect arrows and allow carriers to fight at close quarters with spears. A combination of these items allowed warriors to defend, attack and protect themselves from their opponents. The shields were made with specific knowledge of its design and materials. According to Muke (1993): The raw materials for shields were (a) two types of timbers (Tapi and Kinjip), (b) rattan canes, (c) ropes and (d) cassowary feathers. Items added to the exterior surface of the board for non-functional purposes include designs and paints. The main qualities of the chosen timbers included (a) softwood grain texture, (b) easy to work with, (c) light-weight when successfully dried, and (d) able to absorb massive blows from projectiles. (p. 202) The designs on the shield were separated into independent units. This is an analytical convenience and did not account for the complex religious meanings that they were meant to express. All the shields had more than a single design. According to Muke (1993), shapes such as trian-

Figure 2.6. Types of wooden shield in the Wahgi Valley—adopted from Muke (1993, p. 218).

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gles, circles and rectangles were preferred according to wider cultural meanings. In general, the order of appearance of the shapes on the different parts of the body of the shields related to such cultural ideas as the importance of the 'head, 'navel' and 'leg' parts of the human body. In some cases, the head would be seen as more important than the navel or the leg. The following is the parts, design and color of wooden shield as described by Muke (1993). Design and Types of Shields Kekep (Shield Type 1). This category signifies part of the final stages of the shield production, and it was a technological category. All new shields that are decorated but without designs and paintings are often regarded as cut or split (Kekep) ones. It implies that the shields are recently cut and made, but have not reached the final stages of painting the cultural designs. Nginimb (Shield Type 2). This is another typology which refers to the initial stages of painting. The production of the shield reaches a Kekep stage and if the warriors were faced with a situation where they were forced to use the new shields, the immediate response was to paint the leg section with black color. It formed a rectangular shape across the shield. The appearance of a rectangle shape at the navel and head of the shields is defined as Nginimb. The term Nginimb also meant black in color but in this case, it is used to refer to a shield with the bottom rectangle painted in black ready to be used for fighting. Olge kamp mon (Shield Type 3). The criteria for classifying a shield as Olge is the appearance of a circle in the middle part of the shield. The term olge has two meanings. It referred to full moon and also it was used to describe a spider's (Olge kamp) web. In most cases, the olge (circle) design is created purposefully to cover the rope tied at the navel (dombel) part, where the handle of the shield is firmly attached for the fighter to carry. It has to be painted and was intended to hide the rope that became the handle of the shield. It is the important part where the fighter was attached to the shield as protection, and it must not be easy to be seen and cut by enemies. Since they appear in the mid-section of the shield, the top and bottom space are often filled with other designs. In particular, the Nginimp (rectangle) design appear at the bottom and Kelnge kambim (triangle) appear at the top. Mangak tandkanem (Shield Type 4). A cross on the entire wooden board without a circle is referred to as bamboo leaf design (Mangak tandkanem). It is often described as the bamboo leaf with the tip end of the leaf pointing to the navel part of the shield. The triangular shape also relates to a fighting technique known as kugang mangake. Kugang is the name used for a spear and mangke refers to the fighting formation where enemies are in close range, lined up behind wooden shields, enabling both sides to use spears to attack each other in close ranged. Kelnge Kambim (Shield Type 5). A closely spaced series of triangles drawn across the head (pipin) of the shield is classified as “the thing that lived in the Kelnge tree” (Muke, 1993). Kelgne is the name of a tree that produces particular white gluey substances and a particular insect with a triangular-shaped wings when not in flight is seen on this tree feeding on these substances. Therefore, Wahgi people have decided to refer to triangle as Kelgne kambim. The Kelnge kambim shield designs are mostly part of the Olge designs. Often the detailed motifs on the head of the shield is taken as an example of how decorations appear on the head of the people. The triangles in the form of zigzags run along the sides of the long axis of the shields. Jerepere (Shield Type 6). When the circles are split in halves and appear as semi circles repeatedly, the shields are regarded as “walking side-ways” (Jerepere). The term is also used for shields that have more than a single line defining the three shapes and at the same time that they have a combination of semi-circles or circles and smaller repeated triangles within the major shapes.

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Kukumb (Shield Type 7). When triangles, circles and rectangles appear as exclusive elements on the pipin, dombil and sipin respectively, they are described as Kukumb mon. This is a combination of Type 2, Type 3 and Type 4 but without a repetition of the triangle at the leg section similar to shields with bamboo leaf design. Whether the term refers to a type of spider or the Kukump tree remains to be clarified. Some claim that Kukumb refers to a spider web but it is likely that the pattern may have some similarities with the leaves of the Kukump tree (see also Strathern & Strathern, 1971, p. 103).

Korkanjip (Shield Type 8). Shields which have star designs have been labelled as the “lazy bright object” (Korkanjip). The term for star in Wahgi is korkanjip. Color and Painting of Shields The traditional names, like the classification of terminologies associated with the shield designs, refer to a wider framework of cultural meanings. Some names are derived from plants that produce the color (balau, singamanga, numb and bukinjs). Others are named after elements which have natural color such as the clouds (kuru), clay (tol tup), blue sky (maingek), the green leaves (ond aukem) and the brown (ondom or nganjim) stems or barks of trees and so forth. Color symbolism in the Wahgi Valley has been analyzed by a number of anthropologists in relation to body decoration (Layton, 1977; O'Hanlon, 1989; Reay 1975; Strathern & Strathern, 1971). As Muke (1993) stated, the application of colors on the designs is referred to as “shield paint details” kumbo tol mon). The term tol refers to color and it can be used as a prefix to describe the different color names. The traditional color combinations are given in Table 2.1. It should be noted that the diversity of color names is greater than in English and may be linked to its source. In an earlier section, we discussed extraction of minerals and colors, and this list provides an example for one particular culture. Agriculture Spades Paddle-like spades were found in several areas including the Sepik and in the Enga region dated well before Europeans arrived, as Swadling (2010) described: Table 2.1 Classification of Colors in Yu Wooi • Red - red—tol bang; - bright red —tol gono - dark red—kong maiam tol;

• Yellow - tol balau; - terbe, - singamanga;

• Black - black – nginimb; refers to charcoal, - dye black – jipik

• Orange - bukinjs, - kiskok; • Pink - numb,

• White - white—kuru; cloud white - tol tup—clay used as white

• Blue—maingek • Green—ond aukem • Brown—ond nganjim.

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One of the oldest-looking assemblages [Pleiestone] found, but still undated, consists of hundreds of heavily patinated, unifacially flaked pebble tools, including some stemmed ones, found by Paul Gorecki eroding out of a terrace formed by the Yuat River at Yerem … in the foothills of the central highlands (Gorecki, 1989). The only other artifact found which may date to the Pleistocene is a waisted blade which was found at the depth of 2.5 m in alluvial gravel downstream of Yerakai village, near Bimba lagoon …, in the Ambunti region …. The well-drained alluvials inland of the Sepik delta in the Ambunti region would explain why prehistoric stone digging implements and traditional wooden spades are found in this region. Comparable tools are known from the highlands, including Kuk near Mount Hagen … which has agricultural drain features dating from 9000 BP (Golson, 1977). A prehistoric tanged stone spade was found by a woman fishing in the lagoon some 15km west of Ambunti [Middle Sepik river]. (pp. 5–6) Similar implements were found in the Highlands at Wanlek (Bulmer, 1982), Wahgi valley (Allen, 1970), and Wurup south-east of Mount Hagen. A wooden digging stick with a paddle-like blade found at Tambul in the Kaugel valley, Western Highlands was dated at 4 000 BP (Golson, 1996). These tools were used across the highlands for making trenches in both dryland and wetland agriculture (Golson & Streenberg, 1985). Leahy (1994) noted the presence of such spades in the Wahgi valley during his first contact explorations. They are also still used by the Manambu people of the Ambunti region in the Sepik to dig holes for yams (see plate 85 in Swadling et al. 1988). The lack of marine shellfish in the Kowekau rock shelter, located inland of Timbunke in the Middle Sepik, which has basal dates of 14,000 BP (Swadling et al., 1988, p.18) supports the observation that the western part of the inland sea had a low salinity. (Swadling, 2010, pp. 5–6) Needless to say, wooden implements such as the ubiquitous digging stick which is still used across the country do not last long in the tropics with termites, so wooden implements are rarely found to provide archaeological evidence of specific activities. Longevity and Necessity of Agriculture In 1849, Stanley noted the small Joannet Island off Sudest was “intensively cultivated and supported about thirty to forty people in two villages” (Goodman, 2005, p. 230). The sailors had already bartered for a pig. These people also had clubs and spears with which they later attacked the group. Later the sailors came across an uninhabited island on which a family from another island were gathering food. From first contact, it was noted that the PNG people had developed agricultural methods. Numerous authors have suggested agriculture may have begun in Papua New Guinea some 12 000 years ago, where population numbers in the upland valleys were sufficiently large that plants were not only encouraged in the bush but people planted out gardens. Nuts such as pandanus and karuka nuts, tubers such as yam, taro and lesser-known tubers, and sago were planted and conserved in the bush. Banana and casuarina were exploited and cultivated before the arrival of the pig. It should be noted that Australian Aboriginal people whose heritage is older also planted root and yam plants and cultivated and cared for various nut and fruit plants (Pascoe, 2014). Population increases were assisted by the introduction of sweet potato and increases in pig populations that also needed feeding. Some gardens were made on grasslands or river flats but the archaeological evidence suggests the environment was substantially altered by cutting down rainforests, planting gardens and then leaving the area fallow for regrowth but after some time the regeneration was not as fast as the need for replanting in an area. This led to large areas of

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land no longer maintaining forests. In many of these areas, gardens were designated by squares with drains in between (Golson, 1977; Owens & Kaleva, 2008). Another mathematical aspect of gardening is time, for example how long it takes a crop to bear fruit and when the rains would come. Discussions, involving sometimes 40 or 50 men assisted planning for how much food was needed for the family, ceremonies, and large exchanges. Often the garden work was communal and shared in specific relational ways. For example, women who married into a family might garden adjacent plots or husbands would cut down the bush and make the drains, pile up the dirt and add grass for compost, and then the wife would plant sweet potato runners, water if necessary and weed. In the small mounds, there were usually three runners positioned to maximize sun impact but in the larger mounds, many more were planted. New plants and techniques were developed to increase exchange and today, money, from sales of crops. Each of these required mathematical reasoning and assessments to make the best decisions building on prior knowledge of feasts, weather, people, and exchange or money flows as well as risk-taking and problem solving. All these events were often associated with spiritual relationships and prestige. Making Drains Drains are used across PNG for gardening, directing water, and/or for land demarcation. Drain making indicates the skills of developing parallel-sided drains, with correct slope and depth for good drainage. The pattern of drains is particularly well constructed in Highland areas. At a UNESCO World Heritage Site, the Kuk valley in the Upper Wahgi Valley of the Highlands in the center of the PNG mainland, there is archaeological “evidence of plant exploitation and agriculture dating back over 10 000 years and continuing (i.e., ongoing traditional cultivation)” (Muke, Denham & Genorupa, 2007, p. 326). At first, drains were less regular in design suggesting the development of the drainage system over some thousands of years, perhaps suggesting fewer people needed to be fed in that particular area with some areas still left as forest (dated 12 000 years ago). The old drains ran for two kilometres and were up to three metres deep draining into a river. Other drainage sites were more seasonal such as at Arona in the Eastern Highlands on the shores of old lakes to retain water in ditches and in the Yeni swamp in the Sepik using small basins for water and raised mounds for the crops (dated 5 000 BP) (White, 1993). Gardening techniques have also reflected the environment. In colder areas the plants are placed in mounds so the temperature is steady during the cold months. The mounds are larger in higher areas than lower highlands. Area sizes marked into units by the drains are carefully considered to decide on the number of plants they can support. Gardens and sometimes drains are regularly reworked. Enriching the soil by burning or adding grass was common. Trenches Like drains, trenches were carefully constructed for demarcating land; trenches were also used by warriors when they needed to move from one area to another. Hela Province, in particular, has trenches that date back a long way and the first patrols under John Black and Jim Taylor found the maze of trenches remarkable and daunting (Gammage, 1998): From Tabali the country was cut by great trenches up to five yards deep and twelve wide, bordered by earthworks. They chequered the valley, a maze of corners and dead ends, and the track followed them for days. Only a big population could have built them. They kept pigs from gardens but were much deeper than needed for that. Their purpose was military, obliging attackers to tread cautiously forward, deep in mud, while defenders who knew the maze chose when to attack overland, when to ambush from a side ditch, when to block retreat … . They brought opponents face to face before they could see each other, negating the reach of rifles just as forest did. The

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line spent hours bridging them and hated to walk in them, but for mile on mile had no choice. (Gammage, 1998, pp. 74–75) Trenches are still used today for marking the land and for warfare. The mental mapping involved in making and using these trenches reflects exceptional mathematical processes as well as impressive technological and organizational skills.

Stone Implements

The heavy rainfall and humidity made it hard for things to remain for thousands of years, but stone artefacts have remained in various places especially in occupation caves. The stones were often traded from long distances away. For example, Moresby in 1873 (Moresby, 1876) noted at Hayter Island in Milne Bay Province: The stone axes we found here were the most perfect specimens I have ever seen, and had been clipped into shape, and polished with a skill that must have been the result of practice for ages. The stone used was a kind of green-stone, hard, close-grained, and susceptible to high polish, but liable to chip off in irregular scales. The blades were some as large as seven inches, and they tapered away in a beautiful curve to a sharp edge; they were set into a cleft in a handle, which described two sides of a triangle, and secured by strips of rattan. The axe was carried over the shoulder. (p. 203, reproduced in Whittaker et al., 1975, p. 302) These green stones were probably traded from the Highlands. Leahy (1994) noted one place where many axe-heads had been made, with evidence of the stone remains from fashioning a rough axe head and the use of the sandstone for sharpening them. One particular quarrying area was a set of quarries along the Tunam River in the Wahgi Valley which seemingly produced large axe heads for ceremonial purposes as well as more functional tools. Burton who researched this area during early contact suggested about 200 men would work on a site and some would prepare a finished product but many pieces of stone were traded for the new buyer to sharpen (Figure 2.7a). Mikloucho-Maclay (1975) on the Madang coast noted: All the carved decorations have to be done with a stone ground to the form of an axe, or with bones sharpened with splinters of shell or flint and one can only marvel that with the aid of such primitive instruments they are able to construct such good huts and pirogues (canoes) and these not without some fairly handsome ornamentation. (p. 87) Obsidian was also used in the Madang area for scarification. These implements not only show exchange routes were extensive but also that they were used for creating designs which represent mathematical knowledge. Allen (1977) noted the changes, over time, in how stone implements were used in the Yule Island and Moresby areas. For example, in the Moresby area, the earlier obsidian rock from Ferguson Island was not used in later times, presumably because the local people began to extract and sharpen the local chert rock. In 1973, Kay Owens watched men making dugout canoes using a stone adze for shaping the inside of the canoe in Tamigedu, Huon Peninsula Morobe, and a lady scraping sago with one on a path leading to Lake Kutubu in the Southern Highlands in 1986 and in Tufi in the mid 1980s. In 1973, Gonjon from the hinterland of Lae, Morobe, commented that cutting a tree with the stone axe used to take him a long time as he would lose the stone as he used it and then have to find it. The stones shown in Figure 2.7 were given to the Owens between 1973 and 1984. Besides cutting instruments, there are a few stone carvings probably associated with spiritual purposes, and used as mortars and clubs of various kinds. From the Angan areas of Morobe, the Eastern Highlands, and Simbu, clubs included stars with four points (see Figure 2.7c), “pine-

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a. Long flat greenstone axe head, probably ceremonial (Mt Hagen, 1973) with a polished axe head (Watabung, 1984); b. Axe and adze, stone with rattan and cane binding respectively (Bena Bena, 1973.); c. Angan (Kukukuka) tapa cloth and club (Morobe/Eastern Highlands border); d. Stone carvings (Mt Hagen, 1984). Figure 2.7. Stone axe heads and other stone technology. apple” and round circular or annular clubs. Stone implements from antiquity have been found in the mountains of Central Province, through the Papuan and Morobe ranges to New Britain, Madang Rai Coast, and Sepik Dongan cave (East Sepik) (Swadling, 2010). The other carving (see Figure 2.7d) were two of several sold in 1984 by an artist from Mt Hagen who used foundational skills. Design Design is a particularly important aspect of manufactured goods and creativity in technology. The development of the skills to make stone implements, wooden shields, and other objects (described in this chapter and chapter 3), and the shapes and lines of beauty were all parts of investigations requiring comparisons of results of various trials. Much of the detail was stored in the mind although groups did discuss activities involving systematic mathematical thinking. Designs were shared and traded, helping to build relationships between people.

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Figure 2.8. Leadership symbol of shell and tortoise shell. Tortoise-Shell Designs Tortoise-shell designs held considerable meaning for people of New Ireland and other Austronesian Oceanic groups (Were, 2003, 2010). The tortoise-shell and hard shell kapkap were used as a mark of leadership. They were valuable objects in themselves. However, Were found that the designs and not just the objects were valued and traded. The creation of designs is mathematical but particularly reliant on visual imagery. People often worked around the circle but were good at making sure the designs were symmetrical in the end and divisions of the whole were equal. The designs on circular implements such as kapkap and pots were divided into quarters, sixths or 12ths of the circle at different radiuses from the center. The gifts from Waima village, Central Province, to Kay Owens for leading an elementary school teacher workshop in 2016 is shown in Figure 2.8. Four of these kapkap were in the elaborate headdress for high-ranking men collected in 1923 by Hurley and McCulloch for the Australian Museum, indicating that knowledge was passed down for many generations as well as across societies. See also the young dance leaders in the Trobriands wearing similar symbols in 1975 (Owens, 2015, Figure 5.2, p. 146). Pottery and Pot Design The making of pots and the studies of pots both as archaeological records and in terms of their diversity and beauty, and as both art and as utility, indicate a long tradition in this technology in many centers including in the Provinces of Madang, Sepik, Morobe, and Central near Port Moresby and Mailu. Pots are still made and used in villages today. “The food tastes best cooked in an earthen pot” is commonly mentioned where pots were made or traded. Various methods were used for similar end results, from having a ball and stabbing the fist into the center to start the pot; coils and spirals which were smoothed out at different stages of the pot-making, slabs; and combinations of these methods. The art of making various clays, especially using fine sands, was developed in different areas and others had dyes made from colored clay material, such as the red-slip pottery (May & Tuckson, 2000; Mennis, 2014). Pots were fired in open fires, with great care taken to keep the heat even on all sides. This, like all developed technologies required considerable visual mathematical thinking, remembering and improving trials of products, planning, and implementing designs. The diversity of pot designs is illustrated in Figure 2.9. Some have open mouths of various widths but others are almost closed, one with three spouts for carrying water. However, there were a number of distinct designs such as the taller/longer tapered vessels usually used for cooking (and thus suiting the type of fire that was made) or in funeral rites. In the Iatmul village of Aibom, on the Chambri Lakes off the Middle Sepik River, the pots for sago and the “stoves”

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Figure 2.9. Sketches of pots illustrating diverse styles and decorations. Drawn by L. Burn from sketches by K. Owens gugumbe (third row on right of Figure 2.9, see Figure 2.10) were made using circular coils that were patted into a smooth curved shape with the additions of pottered three-dimensional faces and thin stripes and additional decorations5. Stoves were used for flat pancake plates or for keeping sticks burning to keep away mosquitoes. Plates were made for cooking, for example sago pancakes, or for standing pots on, whereas many eating and serving bowls had wide openings. Visit by Owens in 1983.

5

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a. Aibom pots, Chambri Lake, East Sepik. b. Sawos pot East Sepik c. Wosera pot, East Sepik d. Bilbil, Madang Province pot makinge. East Sepik pots (Source for d. and e ... Egloff, 1977). Figure 2.10. Various pots from the Momase Region. Pots also carried motifs in various designs. Some designs were extensive and others were on the rim. Coils were also part of some designs (e.g., the Bau people, Madang) including the break finish at the top. Two-pronged bamboo scrapers produced parallel lines, usually curved and the one shown on the bottom right has six lines (3 pairs). Early Papuan pots on Yule Island, Central Province, had shell pressed motifs. On Yule island, the pottery was strong in earlier times and although there was a hiatus, it did return with some modifications but with continuing high quality (Allen, 1977). In many places, sections were raised with the thumbs and fingers in various ways. Some used sticks to make repeated indents, often between the applied designs like the eating pot kamana from Sawos East Sepik, Figure 2.10. These high quality pots are still made today. Most colors were painted on the fired pots to complete or highlight some of the designs, especially the white lime such as the Aibom sago storage pot noranggau on the lower right of Figure 2.9. The Wosera ceremonial pot kwam, bottom middle of Figure 2.10c is almost ballshape and has faces lightly carved and painted on the surface similar to their wood lintel carvings. The technique of a white section moving into the next section is found also on boards of haus tambaram (Hauser-Schäublin, 1996; Owens, 2016). However, red slip pottery were covered with thin red clay before firing.

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The Adzera pots of the Markham were a distinct open cooking pot, some decorated around the upper section with the animal to be cooked in it. Figure 2.11b illustrates a woman preparing food for the family in Binemap, a Markham village in 2006, using a Zumin pot. The pot for cooking snakes was made about 1984 in Zumin (Figure 2.11c) by the expert potter Imat Rungan (Figure 2.11a) who also made kundu drums as well as many other pots despite his failing eyesight at the time. In the Madang area, at Bilbil and Yabob, the clay is prepared and left wrapped in balls under the house. Clay and black sand temper are mixed to prepare the balls. Typically the hand and fingers open up the mouth, stones with wooden boards help to smooth the pot, and the hand to make them beautifully round and symmetrical pots. Two small sticks incise a design and a red liquid clay is applied. The pot is fired and sago-paste used to make it shine. There are two cooking pots, a larger one for feasts and one for storing water (See Gaffney & Summerhayes, 2017; May & Tuckson, 2000; Mennis, 2014 for more details of potters at work and descriptions of the pottery making). While designs require mathematical application in practice, the use of materials also requires scientific inquiry with a sense of volume and feel applied to make the product viable. Mikloucho-Maclay (1975) reported in the 1870s that, at BilBil, that there were many pots on verandahs and under houses and the making of pots was carried out by women. The women used flat boards and smooth, relatively flat stones. At first, with the aid of the small piece of flat wood, the upper rim is made from clay, which then left to dry in the sun. When it has hardened somewhat, the rest of the sides of the pot are added bit by bit and smoothed out. The correct shape is given to the pot by holding it on the knees, the woman inserting her left hand with a round or flat stone in the pot, holding it against the internal surface of the wall and with the right hand

a. Zumin master potter Imat Rungan (Post Courier story, 1984) b. Cooking pot c. Snake bowl d. Zumin pot making (Johnston, 1972, in Egloff & Aura, 1977) e. Kundu (Egloff, 1977). Figure 2.11. Adzera, Zumin pottery, Morobe Province.

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striking on the corresponding place on the exterior with a flat piece of wood, evening out at the same time the surface and the thickness of the pot. When the pot is ready, it is first dried in the sun and then baked on a layer of brushwood covered with leaves and then sticks, etc. After staking the pots in several rows one on top of the other, and covering the whole pile with light brush, it is set alight. All the pots have approximately the one form, although of varying size. There is very little ornament on them, occasionally a row of points round the neck or a kind of star. Sometimes these ornamentations are made with the fingernails. (Mikloucho-Maclay, 1975, p. 131) Clay has also been used in other areas, such as in the Asaro area of the Eastern Highlands, for making the head gear of the mudmen (Owens, 2015, chapter 5, pp. 163, 165, 253). In the Sina Sina area of Simbu Province and elsewhere clay is used to make whistles, as illustrated in Figure 2.12. The mathematics behind the finished clay vessels included the strength of design, clay and making techniques, as well as the use of fire on circular 3D objects for even firing. Although certain people were seen as experts in the making of pots, there would often be communal activities for sharing such knowledge. Antiquity of Pottery Archaeologists have studied pottery in various diggings. It was significant in establishing that Oceanic groups moved from New Britain to various islands south and eventually across the Pacific due to the recognized Lapita pottery. The Lapita pottery had distinctive designs of crosshatching. However, the initial pottery and later pottery, still considered to be Lapita varied with later pottery having much less decoration. As Allen (1996) said: (when) a series of island village sites 2000 to 3000 years old, containing a distinctively decorated pottery, shell tools and jewelry, and a marine-adapted economy offering strong presumptions of an agricultural subsistence-base was found stretching

Figure 2.12. Flutes from Sina Sina.

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down the Melanesian island chain from New Guinea to Tonga, the argument was effectively complete: the ultimate settlement of the remote Pacific, eastern Melanesia and Polynesia, during the last 3 000 or so years was seen as a distinct and separate event from the history of those groups who had settled the southwestern fringes of the Pacific at an earlier period. (Allen, 1996, p. 11) Meyer in 1909 had found some pottery at Watom Island near Rabaul, East New Britain, dated to 500 BCE and later excavations in the Province have suggested a much earlier date for the beginning of this Oceanic language group. However, the widespread findings suggest that there was considerable interaction back and forth across this vast ocean region (Addison & Matisoo-Smith, 2010; Allen, 1996).

Three-Dimensional Art

Besides pottery, various cultures made a range of three-dimensional objects for creative and ceremonial purposes as well as for utilitarian purposes such as houses for shelter. Canoe building, house building, carvings, traps, and basket ware are discussed by Owens (2015). An example from one valley was studied for the representation of religious and cultural beliefs about the origin of their communities, food production and other cultural practices (Schmitz, 1963). These 3D representations were indicative of these skills. The Wantoat River valley is situated in the Morobe Province, on the southern flanks of the Finisterre-Sarawaged Range where the valley widens before narrowing to flow into the Leron River and hence into the Markham River. The influence of the mountain Papuan beliefs has been modified, perhaps with the Markham valley peoples’ Papuan and Oceanic worldviews. The figures and poles that are created and carried in ceremonies require a strong understanding of material properties: • • • •

When and how to dry bamboo. How inner bark becomes paper cloth. How to make dyes to last over time. And The means of joining one item to another by weaving or binding.

Additionally, they also show how mathematical thinking is used for stylized representations of objects and actions. As in many places, they use bamboo strips which are treated so they spring up and down when the person is dancing. Large curved structures with a near triangular side view are made and carried on shoulders or head. Tall poles are decorated and some structures are placed on the top representing, for example, a bird and bird’s nest or life-size woven cane models of “people” decorated with red tufts. Each of these are held into positions, some on platforms with stays, others by binding. However, the means by which the young dancer has the tall pole strapped to him so he can dance is remarkable and also shows embodied weight management. Drums There are two main types of drums used in PNG (Figure 2.13). Both are cleverly designed. The garamut is used for sending messages between islands and villages. These are made from large logs with a slit, hollow, and pounded on the side. For example, when a Hungarian, Biróó, soon after first contact was collecting birds and artefacts from around Madang, an incident arose and he was nearly killed on Kranket island. The Siar knew there was a war cry even against a European from the drumming (Mennis, 2014) and saved him. A smaller drum is hit with two sticks and used in many dances. Manus islanders, famous for their fast moving dance steps, are accompanied by these drums but they have been seen in Oro Province and other coastal provinces.

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The kundu drum is shaped like a long hour glass timer and has snake or lizard skin. These are tuned with gum from a tree so that the men’s drums are all in tune. These always have a handle and some are carved. In recent years, people have made similar drums from drain pipes. The overlap of mathematical shapes and sound production is evident in all these drums. Musical Instruments Besides the drums and pottery flutes shown in Figure 2.12, many other musical instruments were found at first contact. The bamboo jaws harp, for example, with a slit pulled with a string to make a sound with the mouth, was seen in several areas (Leahy, 1994; Moresby, 1876) Several drums and pipes are shown in Figure 2.13. Pan pipes were common and single pipes usually for just a few notes were available. Mikloucho-Maclay (1975) noted that, in 1872, “a Papuan flute, consisting of a simple bamboo tube 25 millimeters in diameter with both ends closed but with two apertures at the side, above and below” (p. 162). Some of the pipes were quite large. Mikloucho-Maclay (1975) in his 1872 diaries note: A few, probably inveterate lovers of Papuan music, raising their bamboo tubes (more than two metres long) high in the air above their heads or leaning them against trees, were uttering load drawn –out wailing sounds. Others were blowing on rather long

a. Kundu drum from Sepik Province b. Kundu drum from Morobe Province c. Slit drum d. Pan pipes (Thomson, 1892; Owens owned some in 1970s) e. Large pipes (Thomas, 1892) f. Bamboo jaws harp. Figure 2.13. Examples of musical instruments.

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shells of nuts perforated on the top and sides, which produced a very sharp whistling sound. … Breakfast … was announced by the increased howling of the bamboo tubes and it was now that I experienced all the harshness of these ear-lacerating sounds … the internal partitions removed. One end of the long tube is held in the mouth, considerably stretching the lips, and by blowing or more correctly crying into the bamboo it gives out a penetrating, protracted sound something like a dog howling. The sound of this instrument can be heard half a mile away. (p. 123) The description of the long pipes by Mikloucho-Maclay reminds one of Australian Indigenous didgeridoos. The men at Bongu on the coast of Madang encouraged Mikloucho- Maclay not to show any musical instrument to women and children. Conch shells were also used for calling across the sea to sharks and to other people. These shells could be used for producing musical scores when missionaries introduced hymns. Highlanders also used large bamboo pipes for making music or for smoking. Other instruments included some rattles with wooden handles attached to strings with shells or opened nuts that were shaken. Similar shells were often tied around their legs and arms by the Kiwai dancers and other coastal groups. Food Implements Mikloucho-Maclay (1975), in 1872, also noted that different instruments were used for eating different kinds of food and that men ate the large grubs found in rotten wood with some gusto. Several large serving implements are shown in Figure 2.14. Bamboo was also split to act

a. Marriage bowl, Siassi Islands, Morobe Province, ~1981 b. Model of marriage bowl, ~2014 c. Plates, Tami Island, Morobe Province, ~1977 d. Tami containers made from coconuts, ~1977 Figure 2.14. Several carved food implements.

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as tongs, a very important tool for moving stones when making a ground pit mumu. The technology involved was evident in the use of the natural features of the wood, such as the springy headdress decorations, its lightness and resistance to fire and water. When pots were unavailable, food like green leaves were generally cooked in bamboo tubes, often with a little pork fat or coconut milk. House Building One of the most prevalent structures requiring mathematical thinking were houses (Figure 2.15). At first contact, a range of house styles and other structures were noted, and they generally varied with the environment, especially climate, but available material and individual design and creativity also influenced these structures. D’Entrecasteaux (2001) commented on Nauna (Verdola) that “their dwelling places seemed to be numerous, large and quite stylish: therefore the number of inhabitants on that island is considerable” (p. 80) and they “wore ornaments of white shells and dark red belts” (p. 81) but they saw no weapons. During Stanley’s voyage on the Rattlesnake, Card noted that the houses on Rossel Island in Milne Bay (Louisiade Archipelago) “were larger and looked more compactly built than any we had seen before” (Goodman, 2005, p. 220). At Coral Haven, where the Austronesian Oceanic people had noses and ears pierced and decorated, no tattoos, and long combs in their fuzzy hair, the houses were built on piles and thatched (presumably with kunai grass). They found water on Sudest Island. In Leahy’s (1994) first contact diary from 1930 to 1934, he noted the use of morata roofing made from sago palms for the lowland houses in the Sepik Province rivers. He compared them with the Highland houses and noted that in the Wahgi some were rectangular with curved ends much as they are today. He photographed some with two and three central poles. Many houses in the Highlands were round and the size varied, often with status and age as an older person might like a smaller warmer house. Another innovation for the feast for a dead person in one area around Ialibu in the Southern Highlands was a very cylindrical tall house off the ground with a conical top. Coastal houses were generally quite different. Many bush material houses exist today, some with modifications adapting to new ideas from different people and areas or new skills and materials. In 1872, in the lower mountains at Tengum-mana, Madang, the houses were smaller than on the coast. The base was oval and the house consisted almost entirely of a roof, as the walls on the side were hardly visible. In front of the small doorway there is a semi-circular space, which is under the same kind of roof and supported on two posts. (Mikloucho-Maclay, 1975, p. 148) Interestingly, in this area, trees had carved figures as he had seen on canoes or drawings in charcoal or color that he termed hieroglyphs. On many other occasions, he noted carvings on trees and on house posts, often of some antiquity. Mennis (2014) also discussed the houses that were built around Madang noting that the roofs curved down to the platforms, with a family verandah space for much of the women’s meeting time and preparation time such as rope- and bilum-making. Entrances to houses required a person to get down on their haunches. The roof style has been largely replaced by woven walls and a door, with more recent designs having a triangular piece front and back which I was told in 1992 had come from the Pacific Island missionaries. In New Britain, Gash and Whittaker (1975) displayed a picture from George Brown of a house around 1900 with the thick leaf roof coming low to the ground and entrances that were not as tall as a man. Another photograph from Whittaker et al. (1975) shows similar houses but with a high pinnacle section on platforms over water. Changes occurred and by 1930 the roofs and walls were of separate materials (Paraide, 2018). Later changes occurred with many homes built on posts off the ground, rectangular bases and woven walls with Western commercial materials being more used in recent times. In one area of the Sepik, a doorway was covered by a decorated door board with a circle, wide enough for the biggest man in the village requiring the making of a rough circle and also

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56 Eastern Highlands, Simbu, Enga

Sleeping plaorms, storage, drying

a.

Large, 4m diameter needs central pole

b.

Western Highlands, Jiwaka, Southern Highlands

Kaveve village, 2008

c. Coastal e.g. Madang and Morobe

d. Enga, Hela

Charly’s village, Jiwaka, 2008

Malalamai village, Madang, 2008

Sandaun, Hela, Inland Sepik May be on slts May be further elongated for family long house

May have flaer or curves roof especially for long house

e.

Figure 2.15. Houses around PNG

Kanganaman Long House, Middle Sepik River, 1983

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f. Tree house (Thomson, 1892, p. 29

i.

g. Mee ng house (Thomson, 1892)

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h. House similar to some found in Hela and East Sepik (Thomson, 1892)

Morobe village, 1973 where Kay stayed in 1983 for prac cum supervision

Figure 2.15 (continued) the colors to paint it using set ratios for the ingredients (Onggi, 2005). In TepTep, in the mountains bordering Madang and Morobe, the people called their kunai houses “stone” because they looked like a huge rocky outcrop from a distance—although one author called them “haystack.” These houses had fireplaces on raised ground inside. There was considerable mathematics involved in setting up the floor, the dirt piled under the fire, the curves and angles of the rafters, the amount of kunai needed for a roof, and how big the door-way needed for a crouched entry. In other cold areas, hand-hewn planks provided warmth, often further insulated with dried leaves hanging around the edges. The overall size of the house was kept as small as possible for the number of residents so the fire inside could keep it warm. Recent Adaptations On the Morobe coast, at Lababia village, houses showed numerous architectural modifications with towers, L-shaped floor plans and other extensions from the traditional small rectangular house plans with a verandah either at one end or in front. The houses have adapted to use of sawmill timbers (including those made by a “walkabout” sawmill machine (a portable large machine that has been developed and become popular in PNG) and a tape measure for deciding

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lengths but there are still many aspects completed by “eye” or body, stick or rope measurements (see Owens, 2015 for further descriptions and images of houses in PNG). In Kaveve village, the villages designed a guest house with a large square section with a high tower roof, and on two sides were long areas with bedrooms off the passageways. There was a typical haus win for cooking, with seats all around the walls and open at one end and a fireplace in the middle. In Enga, where there were round houses, the villagers learnt to cut the central pole to provide extra space inside using the rafters to hold up the roof. Then, in one village, they began to add a rectangular area on to their circular houses to extend them. This required the joining of two separate 3D shapes (personal communication, Atete Henry, 1997; see Figure 2.15). Cooking, Drying and Storing Nuts, Ground Cooking and Baskets While people in a few areas dry fish, another item that is dried are karuka nuts so they can be traded to the coast. This requires the technology of cooking and storing and the making of baskets suitable for carrying along the paths to the coast. Families in Kaveve village had their own holes for cooking, often round, but there were also rectangular-shaped pits which were especially used for celebrations often made for the number of pigs to be included as well as root crops and greens. Thus, volume of the pit was associated with the number of large nuts (often in layers) with each nut being like a volume unit despite some gaps. Similarly, for the volume of the rectangular pits, the number of pigs were like units. In addition, when the stones are heated and the nuts were added and covered with leaves that were then covered by soil, one leaf was removed so that water could be added. The number of pots of water was estimated from knowledge of the size and heat of the stones and the required amount of steam that would come out of the hole when water was poured on the stones. Thus, for volume, a pot was a unit (previously tubes of bamboo with three or four nodes removed, a composite unit). These nuts were then stored in the huts with the fire smoking each whole nut. (Each nut was a cluster of layers of many actual nuts about 5 cm long). In coastal villages, in particular, root vegetables and bananas were placed in banana leaves with greens, ginger and coconut milk so that the number of bundles would be like a unit for volume. The second feature was the clever manipulation of a piece of wide bamboo being split but leaving a section of about 20 cm whole at the end to give strength. Then other cane is woven between the split slats and the whole cylinder is widened to carry the bundles of nuts after drying (Figure 2.16d). Many different kinds of baskets are woven in Papua New Guinea (Owens, 2018). Some are made from dried pandanus leaves, usually with thicker inner lining and thin strips outside for making beautiful woven designs. Other baskets are practical for carrying food from the garden or for sale like the one in Figure 2.16c made from limbom palm leaves. When women go to gardens, often at some distance from the village and into the bush, they are likely to gather numerous berries, fruits, nuts and salt-making materials that they would carry in the bag. They can also carry garden produce back to the village. Some balls of fish or other food might be wrapped in leaves to make packets and laid on top of each other in the bags. Another bag carried by men in coastal areas is shown in Figure 2.16a. Men learn to make these and other objects in the men’s house. A coconut frond can be used for weaving a mat to sit on or a basket to carry food to market (Figure 2.16b). Other baskets like these can contain food for exchange. In many places like Buka and the Southern Highlands, people have been weaving baskets from reeds. These are often adapted for different purposes such as a handbag with a lid, a tray, a mat for hot food, a baby’s bed, or a bowl. Often dark and light reeds are used for making patterns. The diversity of patterns are still being created and shared among friends in a similar way to the bilum designs shared among women.

a. Man’s carrying bag b. Market greens for market or exchange c. Limbom palm from Wosera d. Bamboo split and woven for carrying large nuts Figure 2.16. Baskets for carrying food. Bridges Long before contact with Western society the people of what is now Papua New Guinea had indigenous technologies relating to their own environment. Of these technologies, one of the most impressive is that of bridge building. The terrain of the inland areas of Papua New Guinea is extremely rugged with deep gorges and rushing rivers. The traditional ‘civil engineers’ spanned these rivers with footbridges up to 90 metres long, showing a remarkable knowledge of structural principles and making the best possible use of the materials available. (Siegel, 1982, p. 4) Siegel described the suspension bridges (which usually used vines), cantilever bridges, combined types, and girder bridges. Siegel (1982) provided detailed descriptions of many bridges indicating some of the engineering, and hence mathematical principles behind their construc-

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a. Gumbum bridge (Siegel, 1982) b. Vanapa river bridge (Thomson, 1892)

Figure 2.17. Bridges in Papua New Guinea.

tions. Yambi (2004, cited in Owens, 2014, p. 199) illustrated how the bridge structure could be related to school mathematics and provided some idea of how the bridge might have been constructed in terms of the position of different parts, what they needed to look like, and how the young men actually climbed up to create the structure and looked down at a father watching a son build the structure. The casuarina (yar in Tok Pisin) was often used. Trees planted on the side of the river could act as high points for the suspension sections. Different vines and canes were used, in at least Imongu (Ialibu, Southern Highlands), having 10 different names for these and each with a different weight limit, some remarkably strong. Vines were used mainly for the suspended foot and hand rails. For cantilever, or partially cantilever bridges casuarina formed the generally arched

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walkway. A suspended section could be at one or both ends or in the middle. In different places, they may have had cantilevers held by a couple of solid posts or a series of smaller posts pushed into the ground. Stones and logs were used as cantilevers and when trees were not available as weights for holding the suspension ropes. Siegel illustrated and detailed the engineering for 21 bridges in Morobe, Enga, Southern Highlands, Western, East Sepik and Sandaun, and a couple in other highland provinces. Although there were variations, mostly due to available resources and the type of river or the frequency and reason for its use (e.g. daily to gardens or for trade between neighbors, clans or language groups), there was remarkable similarity within these bridge types. During first contact encounters by outside explorers in New Guinea coastal valleys and higher tributaries, a variety of bridges were encountered. Bridges were described in British New Guinea from the 1890s in Central and Oro Provinces (Chinnery, 1926; Siegel, 1982). Bridges in the Morobe Province were also described. These bridges were still being built by villages in the 1980s and probably later. Gammage (1998) mentioned that the first European patrol under John Black crossed from the Hela Province to the Sandaun Province over the torrential Strickland River. They then had to climb a steep cliff. It took the men much courage and several attempts to build a bridge across the river. The only vine bridge they had seen had been cut down by the people in Sandaun to prevent them using it. This had happened in other places until people realized that the patrol would move on if they left the bridge and that the white men would trade. The bridges were a necessity for trade that had occurred prior to Europeans. Leahy (1994) discussed a number of bridges that they crossed in various places and that the Hagen men from the wide valleys were at first unsure, unlike the Waria men, about using the bridges. He also noted that they often had to strengthen the bridges. Most of the bridges, and Leahy’s photographs, show the suspension bridges made from vines with a narrow walking track. The track is held up by vines hanging from the suspension ropes or the hand rails. The high pole structure, with its cross beams, is firmly embedded in the ground and takes the suspension lines and the hand rails as they continue beyond the poles to anchor points. However, Leahy noted that one particularly wide bridge consisted of two cantilever sections joined. There were piles of rocks at each end to balance the cantilever. Cameo from Kay Owens The authors, Kay and Chris Owens, crossed both these kinds of bridges and a third basic girder bridge on several occasions in the mountains not far from Lae during the 1970s and 1980s. Some were replacement bridges indicating that the skills for building these bridges were being passed down to younger generations, but some were also in need of repair. On one occasion, all the people from a village were waiting at the side of a fast-flowing flooded creek and we watched as the young men built the temporary bridge. They sized up the height of several saplings that they knew would reach across the torrential waters, and cut them down. We were fascinated with how they could perceive the tree height needed for the horizontal distance to bridge the torrent— a skill that most Westerners would not have. They cut down vines for ropes. On a slightly wider section of the flooded creek, a couple of strong young men bounced their way across to the other side to catch and secure the long poles placed from one side to the next. In less than an hour, they had constructed a bridge with a hand-rail, wide enough for even the most clumsy of Westerners to cross and sturdy enough for all to cross including the two halves of the water tank which the villagers had proudly purchased in town and would carry up the mountains for the rest of the day. Ten years later I walked into this village, but the tank had developed a hole—metal on concrete— and again the villagers had no running water and could not spare us any as the water was down the mountain.

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Concluding Comments

This chapter and Chapter 3 provide evidence of mathematics and mathematics education within the technological knowledge of Papua New Guinean societies prior to colonization. Adapting to their environment and changes in their environment has been a strength of PNG societies. There is evidence that these are long-standing communities with continuing occupation of the lands albeit migration, warfare and climate having affected their situations. The technologies involve the use of a range of materials and processes. Comparison could be made to several scholarly and well researched books about Australian Indigenous peoples (Burarrwanga et al., 2013; Gammage, 2011; Pascoe, 2014, 2017). In these books and in these chapters is evidence of longevity, ownership, mathematical ways of thinking, and technological adaptations to their ecocultural environments. We began this chapter with a cameo by Charly Muke which illustrates the importance of having foundational mathematical knowledges that have been in the past and help to inform the present. In the next chapter we continue detailing foundational mathematics and conclude the chapter with evidence of how these mathematical knowledges and their diversity have been passed down through the generations. A cameo from Patricia Paraide will be presented. We note in both chapters the importance of how there should be continuity between home, community and school education and the important role of Elders in these activities, pride of home identity, and the importance of establishing a synergy between community mathematics and school mathematics. The rest of the book tells the history of PNG’s attempts to grapple with this most important aspect of mathematics education. References Addison, D. J., & Matisoo-Smith, E. (2010). Rethinking Polynesians origins: A West-Polynesia Triple-I Model. Archaeology in Oceania, 45(1), 1–12. Allen, J. (1970). Prehistoric agricultural systems in the Wahgi valley: A further note. Mankind, 7, 177–183. Allen, J. (1977). Management of resources in prehistoric coastal Papua. In J. Winslow (Ed.), The Melanesian environment. Canberra, Australia: Australian National University. Allen, J. (1996). The Pre-Austronesian settlement of Island Melanesia: Implications for Lapita archaeology. Transactions of the American Philosophical Society, 86(5), 11–27. https://doi. org/10.2307/1006618 Aurawe, J. B. (2007). Traditional bandicoot trap Unpublished report. Goroka, PNG: University of Goroka. Ballard, C. (2003). Writing (pre)history: Narrative and archaeological explanation in the New Guinea Highlands [Paper in: Perspectives on Prehistoric Agriculture in The New Guinea Highlands: A Tribute to Jack Golson, Denham, Tim and Ballard, Chris (eds.)]. Archaeology in Oceania, 38(3), 135–148. https://doi.org/10.1002/j.1834-4453.2003.tb00540.x Biró, L. (1899). Nemet-Uj-Guineai.(Berlinhafen). Neprajzi.Gyujteseinek Leiro Jegyzeke. Beschreibender Catalag der Enthrographischen. Budapest, Hungary: National Museum. Bishop, A. J. (1988). Mathematical enculturation: A cultural perspective on mathematics education. Dordrecht, The Netherlands: Kluwer. Bulmer, S. (1977). Between the mountain and plain: prehistoric settlement and environment in the Kaironk valley. In J. Winslow (Ed.), The Melanesian environment (pp.61–73). Canberra, Australia: ANU Press. Bulmer, S. (1982). Human ecology and cultural variation in prehistoric New Guinea. Monographiae Biologicae, 42, 169–206.

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Burarrwanga, L., Ganambarr, R., Merrkiwayawuy Ganambarr-Stubbs, B. G., Barr, D., Maymuru, D., Wright, S., Suchet-Pearson, S., & Lloyd, K. (2013). Welcome to my country. Sydney, Australia: Allen & Unwin. Chinnery, E. (1926). Territory of New Guinea Anthropological Report No. 1. Papers of Ernest William Pearson Chinnery. Australian National Library. Canberra, Australia, MS 766. Crawford, A. L. (1981). Aida: Life and ceremony of the Gogodala. Bathurst, NSW, Australia: Robert Brown and National Cultural Council of Papua New Guinea. D’Ambrosio, U (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44–48. D'Entrecasteaux, B. (2001). Voyage to Australia and the Pacific 1791–1793. Edited by E. Duyker & M. Duyker, Trans. E. Duyker & M. Duyker. Melbourne, Australia: Melbourne University Press. Egloff, B. (Senior editor), & Aura, G. (photographer). (1977). Pottery of Papua New Guinea: National collection. Port Moresby, PNG: Trustees, PNG National Museum and Art Gallery. Finsch, O. (1888). Samoafahrten.Reizen in Kaizer Wilhelmsland und English-Neu-Guinea in den Jahren 1884, 1885, an bord des Deutschen Dampfers “Samoa.” Leipzig, Germany: Ferdinand Hirt & Sohn. Gaffney, D., & Summerhayes, G. (2017). An archaeology of Madang Papua New Guinea Working Papers in Anthropology, 5. http://hdl.handle.net/10523/7294 Gammage, B. (1998). The sky travellers: Journeys in New Guinea 1938–1939. Melbourne, Australia: Miegunyah Press, Melbourne University Press. Gammage, B. (2011). The biggest estate on earth how Aborigines made Australia. Crows Nest, Australia: Allen & Unwin. Gash, N., & Whittaker, J. (1975). A pictorial history of New Guinea. Melbourne, Australia: Melbourne University Press. Golson, J. (1977). The making of the New Guinea highlands. In J. Winslow (Ed.), The Melanesian environment (pp. 45–56). Canberra, Australia: Australian National University. Golson, J. (1996). The Tambul spade. In H. Levine & A. Ploeg (eds.) Work in progress:Essays in New Guinea Highlands ethnography in honour of Paula Brown Glick (pp.142–171). New York, NY: Peter Lang. Golson, J. & Steensberg, A. (1985). The tools of agricultural intensification in the New Guinea highlands. In Prehistoric intensive agriculture in the tropics (Vol 1, pp. 347–384). Oxford, England: British Archaeological Reports, International Series 232. Goodman, J. (2005). The Rattlesnake: A voyage of discovery to the Coral Sea. London, England: Faber and Faber. Gorecki, P. (1989). Prehistory of the Jimi Valley. In P. Gorecki & D. Gillieson (Eds.), A crack in the spine: Prehistory of the Jimi-Yuat Valley of Papua New Guinea (pp. 130–187). Townsville, Australia: James Cook University. Hauser-Schäublin, B. (1996). The thrill of the line, the string, and the frond, or why the Abelam are a non-cloth culture. Oceania, 67(2), 81–106. Hughes, I. (1977). The use of resources in traditional Melanesia. In J. Winslow (Ed.), The Melanesian environment (pp. 28–34). Canberra, Australia: Australian National University. Layton, R. (1981). The anthropology of art. London, England: Grenada. Leahy, M. (1994). Explorations into highland New Guinea 1930-1935. Bathurst, Australia: Crawford House Press. Mackenzie, M. A. (1991). Androgynous objects: String bags and gender in central New Guinea. Chur, Switzerland: Harwood Academic Publishers. Martin, A. (2007). Making fishing nets in Magi. Unpublished paper. University of Goroka, PNG. May, P., & Tuckson, M. (2000). The traditional pottery of Papua New Guinea (rev. ed.). Adelaide, Australia: Crawford House Publishing.

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Mennis, M. (2014). Sailing for survival. University of Otago Working Papers in Anthropology, 2. http://hdl.handle.net/10523/5935 Mikloucho-Maclay, N. (1975). New Guinea diaries 1871–1883 (C. L. Sentinella, Trans.). Madang, Papua New Guinea: Kristen Press. Moresby, J. (1876). Discoveries in New Guinea. London, England: Government Report. Muke, J. (1993). The Wahgi opo kumbo: An account of warfare in the central highlands of New Guinea. Doctoral thesis, University of Cambridge, England. Muke, J., Denham, T., & Genorupa, V. (2007). Nominating and managing a World Heritage Site in the highlands of Papua New Guinea. World Archaeology, 39(3), 324–338. https://doi. org/10.1080/00438240701464947 O'Hanlon, M. (1989). Reading the skin: Adornment, display and society among the Wahgi. London, England: Trustees of the British Museum by British Museum Publications. Onggi, C. (2005). Deriving academic mathematics from Telefol traditional door board designs. Unpublished report. University of Goroka. PNG. Owens, K. (2012). Mathematics and Papua New Guinea bilum (string bags): Design and making. Paper presented at the International Congress on Mathematics Education, Seoul, Korea. Owens, K. (2014). The impact of a teacher education culture-based project on identity as a mathematics learner. Asia-Pacific Journal of Teacher Education, 42(2), 186–207. https://doi.org /10.1080/1359866X.2014.892568 Owens, K. (2015). Visuospatial reasoning: An ecocultural perspective for space, geometry and measurement education. New York, NY: Springer. Owens, K. (2016). The line and the number are not naked in Papua New Guinea. International Journal for Research in Mathematics Education. Special issue: Ethnomathematics: Walking the mystical path with practical feet, 6(1), 244–260. Owens, K., & Kaleva, W. (2008). Case studies of mathematical thinking about area in Papua New Guinea. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Annual Conference of the International Group for the Psychology of Mathematics Education (PME) and North America Chapter of PME, PME32-PMENAXXX (Vol. 4, pp. 73–80). Morelia, Mexico: PME. Owens, K., Lean, G. A., with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Paraide, P. (2010). Integrating Indigenous and Western mathematical knowledge in PNG early schooling. Doctoral thesis, Deakin University, Geelong, Australia. Paraide, P. (2018). Chapter 11: Indigenous and Western knowledge. In K. Owens, G. Lean with P. Paraide, & C. Muke (Ed.), History of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Pascoe, B. (2014). Dark Emu: Black seeds agriculture or accident? Sydney, Australia: Magabala Books. Pascoe, B. (2017). Dark Emu: A fresh insight into Aboriginal land-use in Australia. Geography Bulletin, 49(2), 15–16. Reay, M. (1975). Seeing things in their true colors. In R. Dobson (Ed.) Australian voices (pp. 7-–15). Canberra, Australia: Canberra Fellowship of Australian Writers & ANU Press Schmitz, C. (1963). Wantoat (G. van Baaren-Pape, Trans.). The Hague, The Netherlands: Moulton & Co. Siegel, J. (1982). Traditional bridges of Papua New Guinea. Lae, PNG: Appropriate Technology Development Institute, PNG University of Technology. Smith, G. (1975). Education in Papua New Guinea. Melbourne, Australia: Melbourne University Press. Souter, G. (1963). New Guinea The last unknown. Sydney, Australia: Angus & Robertson.

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Strathern, A., & Strathern, A. M. (1971). Self decoration in Mt. Hagen. London, England: Duckworth. Swadling, P. (2010). The impact of a dynamic environmental past on trade routes and language distributions in the lower-middle Sepik. In J. Bowden, N. Himmelmann, & M. Ross (Eds.), A journey through Austronesian and Papuan linguistic and cultural space: Papers in honour of Andrew Pawley (Vol. 615, pp. 141–159). Canberra, Australia: Pacific Linguistics, Australian National University. Swadling, P., Hauser Schaublin, B., Gorecki, P., & Tiesler, F. (1988). The Sepik-Ramu: An introduction. Boroko: PNG National Museum. Thomson, J. (1892). British New Guinea. London, England: George Philip & Son. Were, G. (2003). Objects of learning: An anthropological approach to mathematics education. Journal of Material Culture, 8(25), 25–44. https://doi.org/10.1177/ 1359183503008001761 Were, G. (2010). Lines that connect: Rethinking pattern and mind in the Pacific. Honolulu, HA: University of Hawai'i Press. White, P. (1993). Pacific explorers: Highlanders and islanders. In G. Burenhult, P. Rowley- Conwy, D. Hurst Thomas, W. Schiefenhovel, & P. White (Eds.), People of the stone age: Hunter-gatherers and early farmers (pp. 145–161). Brisbane, Australia: University of Queensland Press. Whittaker, J., Gash, N., Hookey, J., & Lacey, R. (1975). Documents and readings in New Guinea History: Prehistory to 1889. Milton, Australia: Jacaranda Press.

Chapter 3 Foundational Mathematical Konwledges: From Times Past to the Present—Trade and Intergenerational Knowledge Sharing

Abstract: This chapter considers the foundational mathematical knowledge related to trade and associated activities. Crucial to trade are spatial, numerical and measurement knowledge from across the diverse language groups. Anthropological records, oral histories, and personal accounts are used to identify the mathematical practices and to discuss the diverse ways of intergenerational knowledge sharing. The commentaries of today’s Papua New Guineans are recognized as significant in establishing this chapter, from their own and their parents’ experiences. The chapter concludes with comments on the importance of recognizing foundational mathematics education and mathematical language.

Key Words: Canoes in Papua New Guinea · Intergenerational knowledge sharing · Reciprocity and exchange · Sailing in the Pacific Ocean · Trade routes in Papua New Guinea What we need to know in order to provide any learner with learning environments conducive to expression, sharing and negotiation of meanings, still seems to be an open question. … It is necessary to get insights into the dynamics of mathematics learning of individuals who might behave and apprehend meanings in situated ways, but who certainly move across the different practices and institutions of societies, that are themselves continually in the process of change. de Abreu, Bishop & Presmeg, 2002, pp. 8–9 Introduction This chapter continues our discussion of foundational mathematical knowledge held and passed on from early time, before colonization and continuing in societies today, in many cases with large or small modifications. Although records are used to discuss the main topics of trade, canoes, and medicine, several oral histories from today will illustrate a few of the mathematical skills and knowledges in practical activities. We will also discuss some of the abstract mathematical ideas that have been recorded or discussed with us personally and have some links to school mathematics. Particularly important in this chapter is how cultural mathematical knowledge is passed on to the next generation. Trade The economy of Papua New Guinea (PNG) involves “the set of arrangements by which people, and entities established by people, are able to create, acquire and exchange things that they value” (Saul Eslake interviewed by Rivett, 2020, April 25–26, Sydney Morning Herald, © Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9_3

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p. 8). The term “economy” would reflect PNG foundational practices better than the word “trade” because there is always a group of people arranging with others about commodities of value for exchange—such as hard stone, food, canoes, and cooking pots as well as fighting or building support, designs and other knowledges. Although each group seemingly remained as an independently organized group, often with their own language and culture, nevertheless various places had shortages such as west of Port Moresby where they had neither the richer Laloki river plains nor the swamps with sago. Trade was inevitable if there was to be enough food in such places in times of shortages (taim hungry). There are many places where people go hungry in certain seasons. Items of trade usually required some form of technology such as in the making of pots, pigments, stone axes and adzes, modifications of shells for various implements, and in the capturing of fish, land animals and birds. For example, various types of clays are more widespread than the centers for pottery, and hence pots were traded. Wallabies, flying foxes, fish and other marine animals were traded from where they were abundant such as inland forests or on the coast. Salt, oils (both black oil from above the Strickland River and plant oils) carried usually in bamboo tubes, and ochres and other minerals for body painting for ceremonies and other art, were traded. Shells were traded from coastal areas into the Highlands and this cemented longstanding relationships between groups. Shells were important in marking wealth and status. Trade has been important for the people of Papua New Guinea and neighbors for tens of thousands of years (e.g., Golitko, Schauer, & Terrell, 2012). Trade of obsidian from its relatively few sources in PNG, and other objects, have provided archaeologists the opportunity to verify movements of people, relationships and reasons for these, the spread and independent development of languages (e.g., Foley, 2005, 2010; Owens, Lean, with Paraide, & Muke, 2018; Ross, Pawley, & Osmond, 2003), and past colonizations as in the Ivane Valley (Ford, 2011). For example, several Oceanic migrations along the Papuan southern coast in particular are marked especially by changes in pottery and a variety of items (from food to shell and bones) in the archaeological records. Mathematics was required in the travel and trade between coast and Highlands, and across the many valleys and communities in the Highlands, and around the coast. Positional knowledge required knowledge of rocks, mountain slopes, plants, water courses, bird and animal behaviour and, for sailing, winds, ocean currents and swells. Trade required negotiation of exchanges and remembering past relational behaviours. Coast-to-Mountain Trade Trading was evident across the Highlands regions during the first contact of the Leahys and others in the 1930s (Leahy, 1994). In particular, greenstone and other stones, sometimes fashioned into an axe head were traded (see Figure 2.7). There is evidence that gold-lip shells, baller shells, cowrie shells and other objects were traded from the coast and then traded on further inland. Leahy noted that at one point a Kukakuka axe head, which must have originated in the foothills of the Highlands, was found well up into the Highlands, and even an axe with a wellworn metal head had been traded up from the Papuan coast (near Port Moresby) for some distance into the Western Highlands of PNG. Local exchanges also provided for payments (compensation and marriage in particular) and relationship building. On two occasions Leahy noted pigs tethered to stakes; the pigs were generally in pairs in two long rows, matching for equality of size and counting occurred1 (Strathern, 1977). The bamboo strips in the chest piece worn by Highlands men represented 10 gold-lip shells. Interestingly, Leahy traded mostly in these and other kinds of shells for food and other items, and for his men’s crimes, such as rape. There were many times that he mentioned that villages wanted Counting systems in Western Highlands where Strathern gathered his data involved (4,8) cycle systems with an overlayed 10 system especially for the display of counting pigs (Owens et al., 2018) 1

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them to stay longer to exchange food for these goods. Europeans were interested in finding sources of gold and deliberately avoided attaching themselves to one group which was in conflict with another, and in that sense they were not interfering with relationships. However, the huge amounts of shell being brought in and used for payments was leading to a massive change or disruption in trade relationships. He said that it was important not to pay compensation in such a way that misdemeanours would be encouraged. Leahy’s photographs from first contact show axes and axe heads similar to those in Figure 2.7. Karuka nuts (see Figure 2.15a) from the Highlands, particularly the Goroka valley, were dried and carried in baskets (Figure 2.16d) to the coast for trade. Green snail shells and tree oil especially from the Foi in the Southern Highlands (Strathern & Stewart, 2000) were traded. Red marita pandanus and bird feathers were traded across the Highlands, and further to the west the headdresses became spectacular often containing opened wings of birds, iridescent beetles, and the colorful feathers of parrots and various birds of paradise. Even today, headdresses (as shown in Figure 2.3) are elaborate and may be rented from a man whose role is to bring together these collections of feathers. The women take many hours to prepare for an exhibition singsing for a price or a prize to support their women’s group (Mel, personal communication, 2014). Trading Routes As Leahy (19940 and those who traversed the land with him moved around the Highlands and the rivers flowing to the coast on either side of the mountain ranges, they were often walking along mountain tracks used for trade and facilitating both friendly and less friendly meetings. There was evidence of singsing grounds where large gatherings of people from several villages came together. In general, certain men were known to be able to speak several languages and to communicate with groups further along the trade route. Intermediate villages and languages continued the trade until the objects had been traded along the entire route with appropriate payments (Swadling, 2010). Strathern and Stewart (2000) provide oral histories which suggest that the Melpa moka, and exchange ceremony with many pigs, only developed into this elaborate ceremony of pigs and shell and other objects of exchange for relationship building in recent history only, a few generations before first contact. The trade of large amounts of gold-lipped shell when Leahy and others entered the Highland regions led to further modifications. The introduction of commercial crops and external monetary income and other valued goods (such as beer) continues to modify the role of exchanges although the basic essence of relationship building remains. Strathern and Stewart (2000) also noted the importance of main traders in the second half of the 20th century and provided an example of a man who discussed the importance of respect for trade partners. They recorded the story of another who noted that he had a long tally of wealth himself. I had partners in distant places. … The Hagen men brought me cassowary headdresses and packs of salt, of the round or long kind. The Jimmi Valley people brought plumes of the white bird of paradise, eagle feathers, parrot feathers, woven decorations for bark belts, cassowaries, packs of sago. The Eastern Valley men brought plumes of the red bird of paradise, white marsupial furs, parrot wings. Even Enga men came to see me, with big marsupials and necklaces of cowrie shells. They brought all these things and I acted as a middleman, switching things round between them, giving the things from the Jimmi people to the Hagen men and vice versa, and so on, so that people said, “He is in the middle of us all and he is generous.” From all sides they came to me, and I also used what they brought for my own moka making and bridewealth payments. (Trans. in 1979 by A. Strathern, cited in Strathern & Stewart, 2000, p. 35) Thus this extensive trade was dependent on the memory of gifts, comparisons of gifts in a taken-as-shared or valorized mathematical account.

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Island and Coastal Trade

Three particular sailing traditions and associated activities will be discussed based on Mennis’s (2014) study of the Bel people (Madang Province) in the north who have been voyaging from at least 1500 years ago, the Motu (Central Province) in the south from at least 2800 years ago, and then the Kula Ring (Milne Bay) in the south east. North Coast Sailing The Bel community have lived on several islands around Madang harbour and neighboring coastline for 500 years having originated from Yomba, an island in the Bismarck Sea that was sunk with an underwater earthquake and tsunamai (now Hancow Reef) (archaeological seismology and oral history stories recorded in Mennis, 2014; Gaffney & Summerhayes, 2017). However, they brought, from the Bismarck area of New Britain, their pottery-making skills learnt from their Oceanic ancestors. Miklouho-Maclay, a Russian scientist who lived in the area during three long visits— each for two to three years—between 1869 and 1883, documented his Madang travels and in so doing described many cultural practices that captured his interest. He lived with the Bongu people and learnt their language. This first-contact data provided strong evidence for foundational mathematics practices (Mikloucho-Maclay, 1975). The Taksia of Karkar Island (the other half of the island is inhabited by Waskia, who speak a non-Austronesian language) made canoes and received pots from Bilbil, the number of pots covered the length of the canoe (Mennis, 2014). The Bel people made two main types of canoes, large outrigger canoes balangut and lalong, and traded pots and objects along the coast usually from May to July using the winds for that time of year. They became quite wealthy (Mennis, 2014). In 1904, Siar, Bilbil and Bel areas drank the red juice of betel nut shared among the villagers who rose up against the Germans and they were moved off their islands. They sought refuge with trading partners or occupied areas on the mainland where they also obtained clay. The i ntervention of the Germans clearly disrupted their trading systems, and this continued with colonization and World War II. Although the pottery making, intermarriages, and recognition of relationships has continued, the larger festivals and large canoe-making have died. A recent canoe was built in 1995 with funding for the Independence celebration. At the time, the older men taught the younger men many skills. However, trade was also affected by valued inter-tribal skills and fights. For example, Bilbil knew several languages and had won the right to negotiate directly with the mountain people behind Rai coast. Their sailing skills took them to Sio in Morobe Province. The trade routes from Siassi and Umboi Islands, Morobe Province in the east, and those from Karkar Island in the west, met around Malalamai on the Rai coast of Madang Province. Even to this day, Malalamai is a center for exchange and negotiation for the coastal people and the mountain hinterland tribes. Copra is processed there and coffee purchased to sell to the coffee distributors in the towns. Nevertheless, from time to time in the trading areas, conflict arose. Mindiri people took some trade from others and then were killed so they continued to trade only locally (May & Tuckson, 2000). The Bel people also exchanged pots for stone, pig tusks, dog teeth necklaces, and wooden bowls (Gaffney & Summerhayes, 2017). Karkar Island specialized in large wooden pestles and mortars and kundu drums; Rai coast made large wooden bowls (similar to those shown in Figure 2.14) and garamut drums, bark cloth and string; Gogol Valley made black pots; Sio and Gitua another type of pot similar to the Bel pots; inland villages made the bow and arrow. South Coast Sailing Large lagatoi canoes (see Figure 3.1b) were made with either one hull and an outrigger or two hulls (like a catamaran) or one hull and two outriggers more like Malay canoes suggesting that there had been at least sporadic trading with the Malaccas along the south coast. (The

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Malaccas were also known to trade with the Kimberley Aboriginal tribes in Australia.) Oral history suggests that the Motuans came from the West but the archaeological and linguistic evidence is that the Papuan Tip Oceanic Austronesian speakers arrived from the east, settling where they could. Perhaps the first settled further west and then travelled towards the east to Central Province. Hence there are pockets of Oceanic speakers amongst the original Papuan speakers. The Koita became closely linked to the Motuans and acquired a number of their skills; the men could speak Motu and the women learnt pottery making. They provided additional support against their neighboring Koiara. In the Roro-speaking area, the Motuans both clashed with and helped some of the tribal groups especially on Yule Island which was at times a stopping place for the Motuans travelling to Gulf. Roro language is also classified as Papuan Tip Cluster of Oceanic Austronesian. The counting is of the Motu-type (paired) (Owens et al., 2018; SIL, nd). The famous trading cycle on the south coast is called the Hiri trade. Hiri trade by the Motuans to Gulf Province was also about selling pots and gaining food. Gulf Province provided much needed sago. Mennis noted: Many of the characteristics of the two trading systems the dadeng (trading) and waing (sailing) of the Madang area, and the hiri of the Motu, are comparable: the need to trade because of infertile soil; the clay pots that the women made; the position of the women in the trading system; the mythology and origin myths of the trading system; the belief in magic to protect the traders and enhance the weather; the use of geographic points; the winds; and the stars to aid navigation. Their similar ancestral origins and culture was adapted to the environmental conditions in which the people found themselves. Although there were cultural and social reasons for the trading trips, the primary reason was economic—the men were Sailing for Survival. (Mennis, 2014, abstract) During the taim hungri in Central Province, men were away in the Gulf Province waiting for the change of winds and so they were able to eat and work with that community and reduce the number of mouths to be fed in Central Province. Chalmers (1887) recorded that he encouraged fish trade with an upland tribe that was starving and he encouraged pot and fish exchanges which brought some peace to the region (Seligmann, 1910). The Motuans also traded, especially the women, with neighbors and the Mekeo while their men were away in Gulf Province. Trade and relationship building require systematic, mental records. Establishing equivalence was important and was kept in memory. Similar memory records are needed for marriages across clan groups during marriage exchanges, and this continues today. Kula Ring This exchange ring occurred between the islands mainly of Milne Bay requiring sailing between islands (Hallinan, 1985). It is the best known of exchange rings in PNG and Australia. White shells armbands travel one way and red shell necklaces the other way. Partners give in their home and receive on visits. It provides status and relationships. Malinowski (1920, 1922) became aware of these sailing exploits from his time in the Trobriand Islands in the first two decades of the 1900s. The main importance of this exchange has been in terms of social organizational theory; bringing peace between groups, establishing positional power, privilege and expectations. Magic, ritual and mythology were all important in this process, in particular the beautification of the canoes helped in wooing partners and the spirits. Kula art is also valued and meaningful (Campbell, 2002; see Owens, 2022 for a mathematical analysis of the art designs). The sailing was between neighboring islands of the Trobriands, Amphelletts, Dobu- speaking parts of the D’Entrecasteaux Islands, Tubetube, Wari, Misima, and Panayati, Woodlark, Marshall Bennett and some other islands. Inevitably many of these islands were over the horizon

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from where sailing began and, in some cases, several days of sailing away, requiring detailed knowledge of winds, tides, currents, swells, birds, and stars and sun positions. Shell Around the northern coast, especially in the New Guinea islands, shell money was well established before first contact with Europeans. The shells were joined in strips and a fathom was the main unit. This could be divided up into small equal lengths of around a cubit in length. The fathoms were also bundled together and often displayed as circular amounts, often of different sizes as shown in a picture taken in New Britain in 1883 (reproduced in Whittaker, Gash, Hookey, & Lacey, 1975; see also Paraide (2010, 2018). Reports of large amounts of shell money were reported in Buin on first contact by Thurnwald in 1908 (Thurnwald, 1936). In this area, trade and relationship exchanges occurred for women, pigs, status, and accruing money for future uses. They also traded with concerns about sorcery and would pay for assistance for protection and good health. Feasts usually meant the purchase of pigs: Any gift of friendship is described by the same name (as) a surplus payment over the price agreed … mamoko. Totakai is the excess payment of a kitere to his mumira for ensuring his good will and his willingness to credit him with abuta (shell-money) on another occasion. Dakai designates a payment for reconciliation or reparation between men of equal position. The price to be paid for a pig sold for a feast is the dacinke. A drum signal is interpreted as saying this word. Currency is also paid for weregild (reparation) 100 fathoms for a kitere, 200 for a mumira. … The rain maker, doctormagician (mekai), sorcerer and dealer in special ‘poisons’ (mara) and medicines are recompensed with 10, 20 or 40 fathoms of abuta according to circumstances. … red onu … is worth 20 fathoms of abuta … white onu is equated to only 10 fathoms of abuta. Abuta is arranged in bundles of 10 fathom lengths, which are arranged in six strings with one quarter of a string overhanging, and terminated with a small shell of an inland snail. This embodies a curious combination of the decimal system with a sexagesimal system, as evinced in the terms for the numbers in their language. … The men are prone to give something additional a totokai, when they settle their debts. … Although it would be wrong to exaggerate this profit making …, it should be noted that it existed even in the pre-European epoch. (Thurnwald, 1936, pp. 135–136, 138) In many other parts of the islands, early contacts reported the use of a complex trade system using shell money (Armstrong, 1924; Parkinson, 1907). Armstrong reported shell money business in Rossell Island, some distance from New Britain. It is used in Roviana, in the Solomon Islands. This practice remains strong in many areas especially in New Britain. Purchases covered many items and the people of Buin even purchased the red onu shell disks from Roviana, Solomon Islands, for 200 fathoms as well as stone axe heads, bilums (string bags), tortoise shell disks, fish nets, plaited bags, armlets, and pots. There were four sizes of pots for different prices. Usually these were bought or exchanged at feasts; that is there was no market per se for selling. (Gatherings for the purpose of selling seemed to begin with the European explorations such as Leahy’s). Paraide (2018) provides illustrative examples and the equivalence of different amounts of shell money which is still used today for exchanges, marriages and for other purposes. Obsidian Stone and Tools Another important trade item was the obsidian stone tool. Obsidian stone tools indicate trade connections with the Bismarck Archipelago. Obsidian stone sources are geographically limited and so there are only a small number of locations to obtain this very sharp cutting rock.

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It is likely that obsidian was traded into the Vitiaz Strait (at Long Island or Umboi Island) and then exchanged to the Rai Coast or to the Bel traders directly. Other stone tools of lesser quality, such as chert and argillite, were also traded into Madang and this likely derives from the hills behind the Rai Coast. Shell artefacts include tools such as adzes, made from giant clams, which could be used for wood-working and processing sago, and net weights which were tied to the edge of fishing nets. These artefacts also include a variety of shell ornaments such as incised shell armbands, shell and coral beads, and shell breastplates. There is also evidence for the local manufacture of many of these ornaments on-site. (Gaffney & Summerhayes, 2017, p. 28) Obsidian was found with shell and red slipper pottery dating back to 500 to 600 years ago in excavations on Bilbil island off the coast of Madang (Gaffney & Summerhayes, 2017). Such pottery was exchanged or traded in relationship building across many generations. Pia villagers told Kay Owens in 2006 that they exchanged large baskets of food with special foods on top and the quality was more important than the quantity but usually it was eight baskets for a pot (Owens & Kaleva, 2008b). Today, they said, there was no clay for coastal pots for exchange. At this point, it should be noted that an overseas volunteer in Yabob, Madang, encouraged pots to be made in a kiln for the tourist trade but this resulted in the clay supplies being severely depleted. As a consequence, the making of pots and relationship exchanges were seriously affected. In the 1870s, Mikloucho-Maclay (1975) wrote in his diaries that the baskets made from coconut fronds were similar to those he had seen in Polynesia. He also noted that people took great care of them because it often took a couple of men half an hour to cut down a tree 14 cm thick with the sharpened stone axes (see Figure 2.16a, the man told Kay Owens that he learnt how to make these in the men’s house and it took a lot of skill). On one occasion, MiklouchoMaclay commented that the leader’s son standing on the platform of the outrigger canoe was impressively dressed with flowers, leaves, woven arm bands and belts rubbed with red ochre as he came to trade.

The Impact of Colonialism on Trade

In the 1830s, Papua New Guinea began to be exploited. The established trade systems were built upon, but also colonized, by traders from other countries. Coconuts traditionally were traded along the coast to produce oil, but this was not favored for the industrialized products of Europe. In the long run the PNG rudimentary technology could not hold its own. Traders took the copra that was processed locally from the coconuts. Trade then was of commodities to be used overseas to make products elsewhere. Even the neighboring Australians were mere middle-men for economies further afield in Europe, America, and Japan. This situation has continued during colonial and postcolonial times. Meanwhile, the commodity of sugar brought profits to Australia and in the process, “blackbirding” occurred; that is to say, many men were taken from the Bismarck and Madang region in particular to be exploited on plantations in north east Australia (Queensland) from the middle of the 19th century (Amarshi, Good, & Mortimer, 1979). Hence local people were not only trading, but tragically also being traded. Exploitation of people took other forms as well. At first contact, Europeans at times encouraged men from a newly contacted region to travel with them and “experience the outside world,” as the Leahy brothers did. (Such a practice occurred throughout history often with the Europeans hoping that they would gain easier access or better communication with “new” peoples if they had “natives2” in their group.) Unfortunately, the desire to see other places and apparently to make money in those places This is not a term that the authors would use as in our life time it was derogatory in PNG. We would use the person’s cultural group or the word “local” or “Papua New Guinean.” 2

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encouraged men to go onboard ships and be taken to other parts of the Pacific from where they often never returned, although they had been promised a return passage. It was a serious vice with which the Australian Administrators of the Territories, especially the New Guinea Trust Territory had to deal. This exploitation of the country did not establish a good beginning to colonization. It was an issue that required the external governments to consider protecting the natives (sic.) but we will leave that to the next chapter of this book when we discuss its impact on education. Buying land for large plantations was also an issue (see Appendix 1). As trade was carried on between the myriad of groups within Papua New Guinea, clearly mathematical concepts were involved at many points of the process. In the next sections we draw attention to the activities of sailing where spatial ideas, measurement ideas of time, and of course numerical ideas were crucial before considering how such ideas were transmitted between generations.

Wayfinding, Sailing, Canoes, Sails and Paddles

One reason for the Oceanic spread was the development of the single-outrigger canoe with planks to raise the sides and platform houses for long voyages. The sailors’ knowledge of local currents, swells, winds, stars and evidence of land or other sea markers (day travel distances) was extraordinary. D'Entrecasteaux (2001) made several comments on their sea skills and noted the differences in the water craft. At Buka, he noted the skill of people using the dugout canoe without an outrigger. At Nauna Island off Manus where people seemed particularly content and non-fighting, he commented that their sailing canoes could move faster than his own ships with their huge sails—such was their shape and facility with the sail. In Milne Bay, he observed canoes with two sails (see Figure 3.1). Stanley in 1849, noted at Coral Haven, on the Papuan coast, that the outrigger canoes “were propelled by paddles, shaped ‘like the ace of spades with a long handle’ and a single large sail.” (Goodman, 2005, p. 226) Interestingly, the outrigger on some canoes was quite large compared to outriggers from other areas (see Figure 3.1b. The model was made at Tubusereia, Central Province in 2014). Around the coast and on the islands, most sea-going canoes had long planks sewn on top of the sides of the dugout trunk and made waterproof from the waves. The image of the Trobriands canoe from 1973 indicates that people were still making use of these canoes in some areas. Finsch (1888), a German, first noted the large canoes on Siar Island, Madang, in the early 1880s and he noted the differences between those and the ones of the Bilbil traders who made use of the base of canoes from Riwo and Siar for their island trade, often accompanied by people from those two areas (all are Bel speakers) (Mennis, 2014). In Siar there were several different clans. There were specialists from the clans: a war-chief, a weather magician, and a master builder of canoes. Each required specialist mathematical reasoning. The houses on Siar were also spacious, using limbom palm for walls and floors, half enclosed, on nine posts, with the middle roof ridge section high above the sides with a cantilevered front section (see Finsch’s 1880 image, in Mennis, 2014). Siars travelled to Karkar Island to the north and as far as the Rai coast to the east. In Milne Bay in 1873, Moresby (1876) commented: The cove we had entered was semi-circular, and fringed all round by graceful cocoanut palms, the blue water rippling up to their roots. Pretty native houses were scattered amongst the trees, every one of which seemed to have sent forth its inmates to gaze at us. There was no unfriendliness; canoes of all sizes, and catamarans darted about us, bringing fine pigs and vegetables, which were gladly exchanged for our hoop-iron. (p. 214, reproduced in Whittaker, Gash, Hookey, & Lacey 1975, p. 302)

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Sails Mikloucho-Maclay (1975) wrote in his 1872 diary about the making of sails. Strings were tied between two poles at regular intervals and then the pandanus leaf, with its thorny edge and spine removed, was woven between the strings. Numerous old photographs show the sails of boats from early contact. Sails were made from a variety of materials like purchased materials, tapa, and woven mats. Lineham (1983) provided details of the Tami islanders making a sea-going canoe as part of their intergenerational knowledge sharing (Figure 3.1d). Nowell and Gilmore (2002) also illustrated a similar occasion for the Kalasinga voyage by sailing canoe. Large Canoes Finsch (1888) travelled to the Sepik and noted that there were beautiful skilfully carved daggers from cassowary bones. The carvings were of crocodile figures, with part bird or human head and eyes inset with mother-of-pearl or other substance. They also had long hollowed tree trunks for canoes to carry up to 18 people. Large canoes like this are found on lakes (see Figure 3.1e) and slower rivers although the Sepik in flood nowadays requires an outboard motor to travel upstream. Cameo from Kay and Chris Owens While living in PNG from 1973 to 1988, we travelled on several kinds of crafts. There were still the old chugging boats from colonial days into the 1980s from Lae town up and down the coast—for villages you just jumped off and swam to shore. When we visited coastal villages we might go on a canoe to cross a river or into the lake behind Labu where the women collected seashells for lime. The mangroves were magnificent there. When crossing a river, one took the advice of locals. They knew the places where it was safest from crocodiles, firm under foot, and not too deep but they still carried their rifle just in case. We crossed river mouths by walking out to the sand bank or by sailing canoe. Another spot where knowledge of the sea was needed was at a cliff face. Despite timing it for low tide, a young man with bandido head band and torn shorts appeared from nowhere and grabbed the two girls to carry them through the crashing waves. He remembered Kay Owens’s name—she had taught him at Taraka Community School some 10 years earlier. The very fast flowing Markham River was crossed by sailing canoe on our earlier visits but a later visit it was in a fibreglass boat with a hole in it on which I had to hold a plastic bag with my foot. So long as the boat was going fast enough, the water stayed out. On that visit, two log canoes were being carved. Sails would be hoisted on canoes when needed. One memorable trip was the crossing of the fast-flowing Markham River near Lake Wanum 30 kilometres inland. The men would sail to the islands in the middle of the river and the canoe would be brought back to the tip in the lee of the island. Then we clambered back on for the next part of the crossing which was the main turbulent river—if a local swam across, it would always be on a log. (The coast-watcher Peter Ryan during the war had to do the same.) Another memorable trip was sailing from Yombu Selden’s village to Tufi with the family when the men awoke early enough to make the three-hour trip into Tufi to meet a plane. The sailors need to read the currents, swells and winds around the points and to row hard to assist the sail and ride the waves. On large lakes like Lake Kutubu, in Hela Province or on rivers like the Sepik, very large dugout canoes with motors are used but for shorter distances paddling or poling smaller canoes. We met a teacher, a former student of Kay Owens in 1985, at Lake Kutubu. This teacher showed us through their long house—both he and the children were still very familiar with how to make large structures like this. They survived by fishing, subsistence farming, and collecting food from the bush.

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The Owens family in 1983 travelled in one of these canoes with Adam from Ambunti to Pagwi, Kanganaman, Palambi, Chambri Lakes and back to Pagwi staying in the village platform rest house. On our return to Pagwi we sat in the back of a utility (PMV) holding our precious pots to Willy Biru’s village. We visited a small, decorated men’s house and noticed the malnutrition of many children, where families had poor access to food gardens. Then we went by PMV to Maprik, left to Wosera and left to Betty Barnabas’ village where we stayed a few nights before returning by PMVs to Wewak. Both were Balob student teachers, Betty was an avid reader. Later she taught at Labu Tali, out of Lae, where the school was trying to preserve the turtle eggs for future hatching rather than have them sold at the Lae market. In 1975 a student from New Ireland said that all their canoe trees had been logged or clear felled and now they needed to use “banana boats.” These fibre-glass dinghies with outboard motors are ubiquitous around the coast nowadays but the canoe drivers (skipa) have transferred their canoe skills to these boats, riding the waves and currents similarly. In the villages, people still use dugout outrigger canoes for fishing or travelling to small centers or across wide rivers. This is still occurring to this day, but the banana boats provided me with a very bumpy trip between Madang and Rai coast or Lae to Salamaua and beyond. It was in Tamigedu, half-way between Lae and Finschhafen that I first watched a canoe being built with a stone adze. I saw the same methods and tools that had been used prior to 2006. The Owens by chance saw the Tami canoe in Lae and later read the story of this canoe. Many had told of the treacherous waters around Tami. We would see smaller log canoes nearly every time we visited a beach or walked this coastline or south of Lae; planks had been added to some for travel out to sea, along the Huon Peninsula coast. On occasions, we watched people in towns, from Gulf and Central Province, make and “sail” small canoes about 0.75m long racing them across harbours making use of their knowledge of winds and canoe structures to construct a winning model (this was confirmed by Vagi Bino’s personal communication, 2016, in Tubusereai, and Sarah Chinnery’s 1923 photograph of such a model). At Malalamai village, on the Rai coast, the people discussed the making of canoes with Kay Owens and with Sondo Serongke (from the village). They drew attention to the selection of a tree and the number of different-sized canoes that could be made from the tree. Men knew the trees on their land and when it would be sensible to cut a particular tree to make two or three canoes. The hollowing of the tree relied on the feel when knocked to decide the thickness of the side of the canoe based on the sound. The relationship between sound and the thickness were recalled in making decisions. Experiments were also carried out. The height of the X-shaped supports for the beams to the outrigger were modified by someone sitting in the canoe to test for buoyancy and stability (personal communication, Malalamai village, 2006)3 . The Hiri Moala festival still celebrates the Hiri trade along the Papuan coast and boats race from Tubuserea around the Loloata islands and return. The mathematical activities and prowess of sailing are still celebrated today and in memory of the trading voyages of the past that took the men away for months at a time as they relied on the changing of the winds to return (personal communications in Tubuserea4 village, 2014).

Thanks to Serongke Sondo and family for organizing visits, providing accommodation and sharing many mathematical activities. 4 Thanks to Vagi Bino for organizing several visits and sharing many mathematical activities. 3

a. Canoe Teste Island, Lindt, 1886, State Library of Victoria b. Model gift from Tubusereia village. c.Trobriand seagoing canoe, 1973 d. Tami island canoe – intergenerational knowledge sharing (Lineham, 1983) e. Family fishing canoes for the large lake and river canoes, Lake Kutubu, 1985 (f) Gathering at the canoes, village at Lake Kutubu, 1985 (g) Sailing across the Markham, 1985 (h) Canoes (Reynolds & Coxhead, ~1940) Figure 3.1. Canoes including sailing canoes.

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Fast-River Canoes In 1875, at the mouth of the Fly River, d’Albertis saw two canoes each with about 10 men. These men gathered “six large canoes, crammed full of men, armed and decorated for war, and were seen coming up to us. … Canoe was made of a large tree trunk, nearly hollowed, without a paddle, and was about sixty foot long” (d’Albertis, 1880). In fact, d’Albertis shot at the boats and took this canoe and used it for fuel killing more men the following morning. The men’s canoes and racing canoes of the Gogodala were also intricately carved. An implement with the cutting section at right angles to the shaft was used as a plane (Figure 3.2d). The cutting implement supplemented the adze which was often stone (Figure 2.7b). Canoes are beautifully illustrated and described by Crawford (1981) whose book covers many daily activities which involve mathematical thinking from designing to making: By virtue of the terrain, the sole means of transportation is the canoe, of which there are three types: a woman’s fishing or work canoe which lacks any form of prow decoration; a man’s canoe which in earlier times was always intricately carved and painted as it was not simply a vehicle, but also an image of his totem; and the mighty racing canoes, which in appearance are similar to a man’s canoe, but far more ornate, up to three times the length (30 metres), and belong to a clan, not an individual. (Crawford, 1981, p. 111) The racing canoe is carefully made and decorated to please the spirits so it might win (Figure 3.2a). There is considerable planning to ensure the canoe is ready for the big race. The sight of several highly decorated prows in a village is exciting. Small model canoes were also made in stylized forms and used (perhaps inverted) as part of a headdress or as food bowls (summarizing villagers’ comments, Crawford, 1981).

Cameo on Canoe Racing and Mathematics

Miwasa Galligali (2001), a student at the University of Goroka, proudly described the Gogodala races, discussing distances, angles, speeds, and other mathematics linked to school mathematics. In particular she recalled how they would run along the bank watching the canoes and estimating their speeds and final positions in the race. They had poles as markers on the opposite bank. In the Balimo area, this activity of canoe racing, duda:gawa, is still strong today. Previously it was a sign of relationship between the tribal groups at the time of initiation but today it is performed usually around Christmas and New Year’s Day and Independence Day or on the opening of a new church, school, or important buildings like trade stores. Traditionally, races have been held to establish friendship and relationships between villages or clans within the tribe. They also symbolize the period in which ancestors were brought to this land by a mysteriously designed, single dugout canoe, called Suliki, across the ocean from the land unknown to many people today. The fossil of the mysterious canoe still exists, but has been buried under the ground. Canoes are designed in such a way that in water there is less friction force acting on the canoe. Because everyone wants his or her canoe to win the race there must be one or two skilled men or sakema, from the village who guide and supervize the men preparing the canoe. The length of the canoe also determines how many paddlers would be needed to paddle the canoe, and hence the speed of the canoe. Before a selected tree can be cut down and formed into canoe, the Elders carefully plan the events, and preparations are carried out. Almost all the men from the village help. After a selected tree has been felled, it is roughly hollowed out with axes and then weeks later, the canoe is pulled through the bush over rollers of logs to the lagoon’s edge. Still in its very rough form it is paddled

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a. Decorating the canoe (Crawford, 1981, p. 286) b. Canoe (Lyons, 1914) c. Canoe racing (Crawford, 1981, p. 300) d. Planing tool (Crawford, 1981, p. 115) Figure 3.2. Plane and canoes, Gogodala.

to the edge of the village where finishing touches will be done. For the next four to six months men will work under the guidance of sakema to complete the canoe in time for the coming event. When it is completed designs are painted on both external sides of the canoe and it is decorated with feathers from the birds. On the completion of all carving, painting and decorating the canoe is launched and for a number of weeks prior to the race there are frequent trials allowing the paddlers to practice both singing, and paddling in unison. The trial races between canoes from different clans in the village are also held in order to test the speeds of the canoes. Racing of the Gogodola It is a thrilling sight to witness up to ten or more Gogodala clan canoes in a race along a straight stretch of the Aramia River, or Kabili lagoon. There are variations in the size and length,

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and the number of paddlers of each canoe, and therefore, they are grouped into three divisions; A, B, and C grades. In C grade division the canoes are about 15 to 20 metres in length with 20 to 25 paddlers. B grade canoes are 20 to 25 metres long and can accommodate about 25 to 35 paddlers, and A grade division canoes are big, about 25 to more than 30 metres in length and hold up to 50 crew members. On the day preceding the race, all canoes from the various villages that are taking part are paddled to the pre-selected finishing mark. On arrival they are carried up the bank by the crews and positioned so that the prow protrudes over the bank or ridge, looking out over the river. By nightfall all participants and spectators set up camps near the canoes, and the night continues with little feasting. All night the various crews sit and sing appropriate canoe songs. Singing continues until just before the sun starts to creep over the horizon. In the early morning following the night of preparation and ceremony, as though on the given signal, the crews march to various canoes. Singing continues as the canoes are lifted bodily and carried to the water’s edge. One by one they are launched, and with crews in each canoe they are slowly pulled about four kilometres up the river to the starting mark. Before the sun could rise from the horizon the race starts, with C-grade canoes first followed by the B-grade canoes, and finally the A grade canoes. In the process of the race big waves are produced and the canoes are likely to over-turn, but the crews, while paddling are able to balance the canoes with their legs. When the race is started the crews, with the count from the first crew in front, strike their paddles in the water simultaneously at a constant rate and phase of the paddle cycle. Within the first 10 minutes of the race the fastest canoe is well ahead of the others. It normally depends on the amount of force applied by the crews using their paddles to accelerate the canoe. In the last 30 to 80 minutes of the race the canoes line up forming one lane with fastest in the front followed by the second fastest and so on depending on the speed. Even though, the crews are so tired they continue paddling until the canoe reach the finishing mark. The gaps between the canoes as they approach the finishing mark determine how fast the canoes are. Mathematics in the Traditional Racing of Gogodala By looking at the traditional canoe racing of Gogodala, one can become aware of hidden mathematical concepts that can be of great help when teaching related concepts in class. Concepts or topics that can be identified from the activity, that are included in school curricula include: 1. Area of a circle and concentric circles found in the canoe designs; 2. Ratio and proportion, in particular comparing lengths of canoes, number of men in the canoes, the ratio of the numbers of men to length, and the speeds of different canoes; 3. Rates and speed, particularly of the canoes given their time for the race of the course of 4 km. For example, the times for the six A-grade canoes, Iyobo, A:imala, Kanaba, Somono, Giliwa and Sasiyapa (names of some of the Gogodala clan canoes) are given and their speeds calculated. By contrast, the speeds might be given for a given distance and the times calculated. For example, Three C-grade canoes, Kalabali, Alo and Lawi, with speed 2.5 km per hour, 2.0 km per hour and 1.5 km per hour respectively, covered a distance of 3 km in a race. Find the time for each canoe. Or in a B-grade division race, A:ga: is traveling with the speed of 1.75 km per hour, and reached the finishing mark after 2 hours. Find the distance it travelled in that race. 4. For Grades 11 and 12, questions on velocity and acceleration, based on the assumption of travel not being constant, can results in questions on graphs and differentiation. (Galligali, 2001)

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Time In all places, people had a sense of time. Clearly, the following examples were specific to particular villages, but the wide-spread practice of reading time or kinesthetically knowing time is understandably ancient. They could wake in the middle of the night if they had to prepare to travel, for example to sail on a particular wind, prepare food for markets or sharing. Without watches, people in Yombu Selden’s village knew when to get up in the middle of the night to be ready for a trip on a canoe to Tufi for a meeting or plane (Kay Owens, personal experience, 1986). Villagers higher on the mountain than Hobu, Morobe Province, would rise in time to bundle the green leaves they had picked the day before, to walk to the roadhead at Hobu for the Public Motor Vehicle (usually a PMV truck) to drive them to Lae for the opening of the market at 7.30 am (Kay Owens, personal experience, 1981) and in Waima village in the far west of Central Province, people rose at 2 am for the PMV to rendezvous at dawn with other trucks at the next main center for the dangerous trip into Port Moresby (Kay Owens, personal experience, 2016). The morning was often marked by the call of the willie wagtail or the screech of certain insects. People had different ways of marking off days (see the next section). For example, along the Madang coast and around Port Moresby, they took a coconut frond and left only the leaves for the days for their next meeting, and each day they removed one frond (Malalamai stories, Mennis, 2014). The seasons were also marked by the trade winds and the wet and dry. The cycle of the kapok tree and migratory birds were used. They also watched the moon circuits and the stars for deciding on the time of a year. In addition, the position of the sun rising over the mountains in the east would indicate the time of year for those living in upland valleys (personal communication, University of Goroka students, 2006) or in relation to a coastal point for those along the coast (personal communications, Rod Selden to Kay Owens, 1986; Kay Owens, personal experience, Rai Coast). Time was important for food production and gathering, for trade and exchanges for which accounting was important, and for these reasons, counting systems needed to be developed. Counting Counting was used in trading but was also recorded by linguists on first contact. The diary of Mikloucho-Maclay (1975) written in 1871–1883 in the Madang area provides evidence of counting by men from Male village that indicates the use of a 5-cycle method: I took several strips of paper and cut them across. I myself did not know how many there were and I handed the whole lot to one of the Papuans from Male, saying that each piece of paper signified two days. The whole crowd gathered round immediately. My Papuan began to count them on his fingers, but he must have been wrong, or at least the other Papuans decided he couldn’t count and the pieces of paper were given to another. The latter sat down and solemnly called another to help him and they began to count. First he spread the pieces of paper on his knees and at each piece he repeated nare, nare (one). The other repeated the word, nare, bending his fingers, first on one hand then on the other. Having counted up to 10, he closed the fingers of both hands, dropped both hands on his knees saying (word omitted) two hands, then a third Papuan bent one finger. With the second 20 the same procedure was followed, then a third Papuan bent the second finger, the same was done for the third 10, leaving not enough papers to complete a fourth 10 and they were put to one side. Everybody, it seemed, was satisfied. … The people still took the papers home to the village to check the number of days. (pp. 73–74)

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Decisions made when counting, measuring, or comparing were often a group affair, sometimes involving just a few people but other times whole extended families or communities. A large number of people would decide on the number of pigs for a marriage or compensation (20, 200, 400 were numbers mentioned by various students). Discussion occurred to negotiate exchange and improve the making of objects. For example, making a circle for a house in Kavevi village, Eastern Highlands; weaving and house building in Yopnung, Jiwaka Province; bilum making patterns and techniques; house building in Malalamai, Madang; and food exchanges in Pia, Madang were carried out with group discussion. Diversity of Counting Systems Data for Lean’s (1992) work on the counting systems of Papua New Guinea and Oceania were obtained from questionnaires completed by students using their vernacular language, field visits, linguistic data, and first contact data. In the 1880s anthropological expeditions were made to learn more about the peoples of New Guinea and Papuan Islands, the Torres Strait and the Pacific islands. In addition, a large number of records were made by Australian government employees in Papuan areas. Furthermore, missionaries tried to learn the local languages and to teach in these languages. Lawes (1895) provided a Motuan grammar and dictionary. The Standens (Bladon, 1982) provided the Bama people in a remote area of Western Province with literacy in their language. The Summer Institute of Linguistics (SIL) and other linguists are continuing to provide local language records including counting and words associated with space, shape, and measurement. Thus, the oral information by those who can speak the languages indicates foundational knowledge of number and some continuing use of the counting systems, even if there have been some modifications. A greater recognition of the mathematical ways of thinking might preserve other aspects of the mathematical registers of the various languages. There are diverse counting systems, which are best described in terms of their structural pattern, referred to by Lean and others as cycles based on frame words from which other numbers are named. For example, a (2,5) cycle system builds other numbers on 2 and 5 such that 3 = 2+1 and 8 = 5+3. In some (2,5) cycles either 3 or 4 do not follow the pattern but are numerals themselves. Some systems develop into base arithmetic with higher powers of a cycle number (Lean, 1992). However, in many languages, there is not a great deal of emphasis on numbers and if there is need for large quantities then various other methods are used including displays of goods. Table 3.1 provides some further idea of the diversity. (For full details about foundational counting see Owens, Lean with Paraide and Muke, 2018). It is evident that 10-cycle systems have been mostly influenced by Austronesian languages but there were also 10-cycle systems that took on the neighboring 2-, 5- and/or 20- cycle systems. There were innovations in both groups such as the development of 4-cycle (including those which developed 8-cycle systems) and 6-cycle systems. The 2-cycle numeral system, the digit tally, and the body-part tally, were in Australia and New Guinea as part of the cultural knowledge of the ancient immigrants whose descendants now speak the languages which exist today. Similarly, as discussed further below, the 10-cycle system entered the New Guinea region with the original AN [Austronesian] immigrants. …The essential features of the counting and tallying situation which we see in New Guinea and Australia today had already been laid down by the time that the city-states of the Middle East were being established and that, therefore, the origins of counting cannot be sought in these (Middle East civilizations). (Owens et al., 2018, p. 208) However, the systems came from older clever groups of people who deliberately travelled into Sahul (the geographical area now comprised of Papua New Guinea, Australia and surround-

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Table 3.1 Showing the Distribution of Counting-System and Tally Types Austronesian

Non-Austronesian East SepikPapuan Torricelli Ramu

Oceanic

Types

West Papuan

Trans N. G.

Minor Phyla

Total

2

(2)

0

0

0

3

39

0

42

18

(2, 5)

0

1

16

5

86

1

109

(2, 3, 5)

0

1

3

5

17

1

27

12

(2, 4, 5)

0

0

5

3

31

1

40

34

(5, 20)

0

1

2

17

52

7

79

4

(4), (4,8)

0

0

0

1

6

2

9

(6)

0

0

0

0

5

0

5

BodyParts

0

0

0

8

58

4?

70?

45

(5, 10)

2

12

0

3

4

0

22

19

(5, 10, 20)

5

0

0

0

4

3

13

73

(10)

1

8

0

1

2

0

13

3 (10, 20) 2 0 0 0 1 0 3 Note. These are numbers from data collected by Lean covering most but not all languages in Papua New Guinea and Oceania. They exclude 11 West Papuan languages in North Halmahera. Trans N. G. is Trans New Guinea Phylum (Source. Owens et al. 2018, p. 196). ing islands), perhaps travelling and navigating on the sea for the first time. Some systems developed for cultural reasons such as exchange relationships and mythologies as a result of understanding the abstract nature of number and systems, and in some cases as resulting from exchange relations with neighboring language groups (Owens et al., 2018). For example, the Austronesian Atzera language of the Markham Valley dropped their 10-cycle system in favor of the 2-cycle system of their neighbors. Some Austronesian 4-cycle systems could have been influenced by surrounding 5-cycle systems where a hand was the group in both cases but the paired relationships 2, 4 were strong in the Austronesian situation. Whether the people counted or had other ways of discussing large numbers such as displays, there are a couple of notable examples for large number recognition. Most of these are among the Austronesian 10-cycle systems as shown in Table 3.2. However, the Iqwaye on the border of the Eastern Highlands and Morobe, a non-Austronesian and non-decimal system, not only demonstrate their (5,20) cycle system by crouching to touch their toes but they then regard this composite group of 20 represented by a crouched man as 1 as they continue to count to 20 groups of 20 and beyond. The numbers and representations have mythological relevance as well as an understanding of infinity. The interlocking of toes and fingers is significant. On the southern part of New Guinea, on Ndom Island in West Papua, the people have a 6-cycle system. Again they represent small numbers (less than 6 × 6), medium sized numbers (between 6 × 6 and 6 × 6 × 6) and large numbers beyond this with different morpheme combinations. These were innovations for counting and may be fairly recently extended to account for larger monetary developments. Interestingly, there are also 6-cycle systems on either side of the PNG Western Province border with West Papua, suggesting ongoing trading relationships. Other borrowings may be just of single words such as the word for 1 000 or 10 000 (and sometimes not for the same amount) in the Melanesian Islands. The borrowing of a 10-cycle, often with the 5-cycle and/or the 20-cycle has occurred sometime in the past by Papuan language groups neighboring Austronesian groups

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Table 3.2 Showing the Distribution of Counting-System Types Among the Four Clusters of the PNG Languages, West Papua, and Island Melanesia Type

Admiralties/ St Matthias

North New Guinea

Papuan Tip

MesoMelanesian

West Papua

Total

(2)

0

2

0

0

0

2

(2, 5)

0

15

2

1

0

18

(2'', 5)

0

7

5

0

0

12

(5, 20)

1

19

11

1

2

34

(5, 10)

1

20

7

17

0

45

(5, 10, 20)

0

13

5

0

1

19

(10, 100)

23

0

9

41

0

73

(10, 20)

0

0

0

3

0

3

(4)

0

2

0

0

2

4

78

39

63

5

210

Totals 25 Source. Owens et al., 2018, p. 209.

but the reverse has occurred with Austronesian Oceanic groups having 5- and 20-cycles as well as 10-cycles. Counting Objects, Shell Money and Classifiers Paraide’s (2018) work on strings of shell money and large numbers recognizes the adaptability and importance of money in foundational societies especially in the island regions. It was used for exchanges, trade, status, and relationships building. Furthermore, group-counting of specific kinds of objects such as coconuts reinforced multiplicative thinking. One further common feature of Austronesian Oceanic language groups and neighboring Papuan groups is the use of classifiers. These are morphemes attached to the counting morphemes to identify a particular group of objects, people, or ideas that are being counted. For example, a Manus language has different classifiers for a banana, a hand of bananas, and a stalk of bananas. Some languages such as the Trobiand Island language of KilaKiva has 120 classifiers. In some cases, they use the numeral classifiers to indicate hundreds, thousands and tens of thousands. Some classifiers describe shapes such as round objects, long thin shapes and so on.

Intergenerational Knowledge Sharing

There is evidence of intergenerational knowledge being shared over many generations from archaeology and from first contact records. Interestingly, the geography of the areas often influenced what knowledges and skills needed to be learnt and maintained by the communities over generations. The cameo by Charly Muke in the previous chapter and Patricia Paraide in this chapter are indicative of intergenerational knowledge sharing occurring today. Although archaeological evidence indicates that people have lived within the environment of Papua New Guinea for at least 40 000 years, many of the practices used in the past remained in some form until first contact with Europeans. Indeed, most are still practiced with some modification today. These skills and knowledges have been learnt by generation after generation. They have been shared across language groups (Were, 2010) or developed spontaneously given similarities in the environments.

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Pottery As mentioned earlier, pottery-trade supported communities that had poor and small parcels of land for growing food such as in the Bel of Madang on the north coast and the Motu of Central Province in the south of the main island of PNG (Mennis, 2014). Furthermore, catastrophic events resulting from the movement of teutonic plates in both areas led to oral histories of tsunamai and changes in islands. These Austronesian Oceanic groups needed land surrounded by older Papuan (non-Austronesian) speakers. For the Madang area: At both Nunguri and Tilu, we excavated substantial masses of pre-colonial artefacts. These include tens of thousands of broken pottery fragments, hundreds of obsidian stone tools, and dozens of shell artefacts. The excavated pottery is almost all redslipped, spherical pots with out-turned rims, much like the pots being produced around Madang today. Five distinct manufacturing methods were identified in the analysis of the pottery, which seem to fulfill similar functions to today’s cooking pots and water storage pots. These pots were decorated with appliqué (raised blobs of slip in relief), incisions, and carved paddle marks. Using a scanning electron microscope and x-ray fluorescence we have geochemically fingerprinted many of the potsherds and discovered that they were all produced using local ingredients. This means that the sources of clay and temper used by modern potters are very similar to those used by the original potters to arrive on the Madang coast over half a millennia ago. (Gaffney & Summerhayes, 2017, p. 18) In the Madang area, Ham speakers inland on the Gogol river also came from the Yomba island from which the Bel potters came, according to oral history, and speak a dialect of the Bel language. However, they were cut off from the coast and lost their skills of canoe making, fishing and many other skills. However, they retained their pot-making with modifications. For example, they used the coil method and changed some aspects of the design. They traded these pots inland and to the coast (Mennis, 2014). It appears such changes occurred over a short period of time, perhaps a generation, and this change of environment meant not only a loss of these practices but also a loss of mathematical activities and skills associated with them. Similarly, Kranket Islanders, without clay, lost their pottery skills whereas the Bilbil, Yabob, and Mindiri people found clay and did not lose these skills. Another example comes from the early 1907 paper by Pöch (cited in Gaffney & Summerhayes, 2017) at Wanigela, Collingwood Bay, Oro Province. He noted that the pottery in the shell middens were found along with other remains such as skeletons, pig bones and other discarded materials. He noted that the pottery was much older and different from that of the people’s current pottery, which was not as strong. This was in agreement with their oral history that they only settled in the swampland near the coast about a generation ago. From the abundance of clay they still produced new pottery, but interestingly there has not been any continuity and influence of the old style in the modern pottery. String Art This ubiquitous activity has been a pastime that is passed down from generation to generation. Mikloucho-Maclay (1975) noted in the 1880s, a small girl was making string figures at Tengum-mana, Madang. At Gumbu, he noted the girls and women “were doing various designs with a string with the ends tied together, using not only their fingers but their toes” (p. 161). These string designs are numerous and widespread, often associated with cultural stories for each of the steps, and often with a surprising ending (Maude & Wedgwood, 1967). An early extensive

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recording for one area was made by Haddon (1930, reprinted in 1979). However, the designs and stories are specifically culture-based. Understandably, there are mathematical patterns of polymorphs and combinations embedded in the designs, thus allowing people to make up new designs (Vandendriessche, 2007, 2015). Vandendriessche studied string art in the Trobriands and later Vanuatu and made further comparisons. In the Highlands, one student teacher reported how the changes in the string design told the journey to the mountain with different steps being a feature of the landscape (Owens, 2015). The actions were very much in their kinaesthetic or spatial imagery because they would return to the beginning if stuck so they could pick up the physical spatial imagery for moving on with the dynamic changes in the string design. Vandendriessche (2015) also discussed how children were learning and playing with the art of string design. The songs and stories related to the actions were passed on from person to person. It is this area of learning through intergenerational knowledge sharing that we now elaborate.

Processes for Sharing Knowledge

Moieties and Clans Certain knowledge as well as certain lands belong to specific clans within the moieties although the division of the language group and the usual inter-tribal or inter-clan/moiety marriages provided a complex development. Within clans were families. Knowledge, goods, designs, and many other things were owned by one of these sections but ownership came with responsibilities of sharing and relationship building. Importantly, people remembered how the communities were supported by exchanges of brides and goods. There were various ways that people remembered the amounts that were exchanged. Some used a rope that was the girth of the pig, others recalled the 10s and 100s by associating the numbers with parts of the body (Muke, 2000). Sizes of land, their shapes and positions, and values for different purposes were also remembered for division between wives and children. Observing, Discussing, and Communal Valuing The main form of learning was by observing the people of the community, usually of the same gender, and gradually participating in the activities. For example, Gaffney and Summerhayes (2017) noted at Yabob-Up-Top how a young child was watching and was close to the old master potter Yegeg who was assisted by a young woman in selecting the clay. Significant in this knowledge sharing are common explanations, and communal valuing of certain practices. In terms of learning, communal noticing is highlighted. While this practice is mentioned in Western contexts (Towers & Martin, 2014), in PNG aspects of activity and place are commonly pointed out and mentioned during activities. Gesturing and facial indicators (e.g. raized eyebrow and head tilt) are common ways of sharing noticed changes around them or in their activity. Leahy (1994), commenting on his first contact expedition in the 1930s, noted that each time they set up camp with their fishing line boundary, locals gathered and spoke aloud but to no one in particular about what they were noticing about the white people in their midst. Mikloucho-Maclay (1975) and many others have noticed the constant talking among people in the villages; people talk often without looking at others, they say what they notice, they provide explanations, they laugh (or weep) or mimic new actions. As children accompany their parents and older siblings there is constant talk with others. There are things for children to notice and learn about the tasks and explanations. When an Elder in Muke’s Jiwaka village was making some weaving samples for us, all the children gathered around to watch. Another Elder came by and looked on. After a while, the Elder who was watching bent down and pointed to the place which was not following the pattern

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and then pointed to the place several rows earlier where the correction needed to be made, explaining how to fix it. As a young child, Sakopa’s father took her to the bush to cut saplings for the house in Hela Province. He told her to sort the different kinds of wood and how many were needed and how to count them so that they would have the number they needed as he cut the trees. Thus began her links between mathematics and cultural activity (P. Sakopa, personal communication to K. Owens, 2014). During the modelling of a house in Muke’s village in 2006, the Elders discussed the position of various posts, noting the ratio of the height above the wall’s height and the distance to the wall. This required a modification of the position of the post so that the ratio was adequate for the slope of the roof rafter. Each aspect of the making of the house was discussed by the men. Similarly when watching the building of a coastal house in 1997 in Kela 2 near Salamaua in Morobe Province, the young men were learning how to add the morata, the strips of roofing made from a limbom palm stick with sago palm leaves doubled and sewn over it. This was apparently introduced by South Sea Island missionaries and spread around the coast of PNG at early contact times, generally by the pastors who required housing in a village. The men needed to get the spacing, parallel lines, and knots correct to do the job well. They needed to start in the right place to ensure adequate overlap and they talked about this as they worked. Dramatic or imitative play has always been an important way of learning activities with mathematical processing and knowledge. Children imitate their parents and older siblings and others in their communities. Heider observed the Dani children in West Papua: fighting battles with grass spears; using spears and sticks to kill armies of berries, rolled realistically back and forth to imitate warriors advancing and retreating; target practice at hanging moss and at ant hives; hunting birds for fun; building imitation huts and imitation gardens with ditches, dragging around a flower attached to a string, as if it were a pig, and calling it (such) …; and gathering at night around a fire, watching a burning stick fall, and pretending that the person to whom the stick points will become one’s future brother-in-law. (from Diamond, 2012, p. 203) In Chapter 2, Muke recalled that children in his village played with their toy bows and arrows, learning the best angle for making arrows go as far as possible. They also learnt to aim above the animal to allow for the parabolic curvature of the arrow’s flight to hit the animal at a distance. Children would accompany their parents to the garden no matter how far they had to walk and there they learnt to build the garden beds, space the plants, tend the plants, and remove the weeds. All these activities required recognizing materials for the beds, size and shapes, and spacing between beds and plants, as well as how best to place the stalks of sweet potato runners or seeds or cuttings and how to recognize different plants and weeds. Children also learnt how to recognize different landmarks and distances for walking the track to the gardens. When children are asked how many plants in a garden with three kaukau stalks in each mound, they will quickly respond. So if it has 10 mounds, they know there are 30 plants. The amount of yield for the family is well established and the number of mounds needed is quickly assessed for each family, and allowances made for contingencies such as upcoming feasts. Children also collect from plants from the bush, especially nuts, fruits, ferns, green leaves, fungi and medicinal plants. They recall where such plants are found, and what they look like, distinguishing, for example, between edible and poisonous mushrooms, soft and tender ferns, and different kinds of leaves, seeds, bark and flowers. Hence, most training was on the job. Whether it was sailing, making gardens, or finding the right sago palm or tree in the bush, the son learnt from the father as he participated in the activity. Matang (personal communication to Kay Owens, 2006) discussed how they in Kâte would take a piece of string that measured the width of the house to the bamboo clump some distance from

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the house. They would then use the circumference of the bamboo as a measuring unit to cut just the right number of bamboo lengths (these are split to form a rather corrugated floor) for their house. Men discussed the particular sago tree size they would need for making the morata for the house that was currently being built, the area of kunai that needed to be cut to make the roof of a house, the possible tree for one or more canoes, and where that tree would be found, and how many men were needed to transport it to the river or sea (discussions in Malalamai with Serongke Sondo and family, 2006). The amount of material was adjusted so that if the house was to be on say 12 posts instead of 9 posts, they knew to get half as much again of the materials (see Figure 2.15) . In 2014, Kay Owens watched some children collecting fungus from trees. They knew the kinds of places to find the required fungus and also to recognize those which were useful and those which were not. Together they chatted as they collected, pointing out features to be noticed in the formation of the jungle and the nature of the fungus and how much to collect. In 1982, she participated in collecting baloki fern fronds for cooking, learning to recognize which ones were young enough to eat. Many plants are gathered from the bush and processed by most PNG people and the spatial thinking required to find and know when to pick and how to process requires mathematical spatial reasoning. Owens and Kaleva (2008a, 2008b) noted the diversity of measurement concepts and the use of estimation, visualization, and memory of past experiences in measurement practices. In some cases units and occasionally composite units were used, albeit informal units in terms of school mathematics. For example, parcels of sago balls or sticks of fish were examples of composite units of volume or mass. Others used special techniques, even fractions of sticks to match fractions of food for distribution. In other places, specific hand lengths and composites of these are used to measure lengths. For example, fractions of a fathom can be identified across the body (Paraide, 2018). As houses were generally built by a group of people, it was easy to discuss the structure, to line up points, and to mark positions before digging holes. String, sticks, long strips of reeds, bark or leaves were used to space the posts and to get the posts in a straight line. String was also used to get straight edges on drains and gardens. However, it was valorized knowledge, that is held in high regard by the community—know, for example, that adding an extra row of posts could provide half as much again floor space but would require proportionally more materials to build. Sharing of Foundational Knowledge The designs that were shared in some places included designs for artefacts such as the kapkap (Figure 2.8), houses, string and pandanus-leaf bags, and many other items (see Owens, 2015 for many examples). Women tended to learn mostly by sitting and discussing especially as they made string, bilums, tapa and baskets. Again, designs were shared. Sharing designs like commodities brought generosity, obligations and reciprocities. While in many activities there were specialists, others often helped and knew something about the practices. Relationship building was determined also by the lores related to moiety and clan obligations and expectations. The haus man (men’s house) is still a significant place for both Highlanders and coastals. The men who invited us (the Owens) to their villages always spent considerable time at night talking with the other men. There were many decisions to be made and their education was valued by the Elders. This sharing of foundational knowledge, knowledge of the history of the place and relationships between people within and beyond the clan, was always important. The most outstanding men’s houses are on the Sepik River and neighboring Ramu River. These large houses or spirit houses, haus tambaran, contained often two levels, one for more intimate discussions and activities such as initiation rites. They were highly decorated especially

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by carvings and paintings. The lower levels were areas of work, carving, exchanging, discussing, and welcoming visitors. Between Pagwi on the Sepik River and Maprik to the west on the highway into the interior of the Sepik Province, the haus man had a low roof, whereas in Malalamai, it was an unassuming house on the ground level close to the copra houses. In Jiwaka, these long houses, often with rounded ends are divided into four sleeping rooms with a central place to sit around the fire. One large house was said to “sleep 25 men” and “each room was 7 foot by 7 foot.” In other words, a place for six or seven men to sleep close together (so the 25 was a round number to which they were referring). These comments came quickly on top of my question about size and were not calculated but just known. They are centres for spiritual well-being and a place of learning. In these places, men learn the stories of the group, they solve problems by discussion, they pass on knowledge to the next generation, and they make objects such as wooden artefacts and woven hats, baskets, masks, and other objects. The haus man was also used for telling the important knowledge of the village, strengthening the young men and initiating them such as by circumcision or scarification depending on the villages and customs. The young men told us they learnt the skills of making hats and baskets, traps and bows and arrows in the men’s house. They shared how they would make objects. They discussed the village economy, copra and coffee, food supplies, and disputes. Learning Foundational Medicinal Knowledge Mathematical procedures also occur within medicinal knowledge. One interesting study by Sibona Kopi (1997) on traditional beliefs and medicine refers to the selection and training of Motuan babalaudia as medicine healers. Quoting the oral testimony of Mase Dabu in 1993, Kopi recorded: When children are young, they are sent to collect things to make medicine. You send them to the sea, and to the bush to get leaves, bark and grasses which are needed. Some of them are efficient and some are hopeless. Some bring the right amount, others bring too much or too little. Some are careful when they collect the materials, others are inconsiderate and almost kill the plant while collecting certain parts of it. Some are good because of their knowledge of plants and can tell the differences between plants that are almost the same. Some are hopeless and unable to distinguish between the donaiau (tropical ulcers) and toto dika (caused as a result of trespassing). You can tell when children will be able to become traditional healers by watching them. (Kopi, 1997, pp. 147–148) Kopi explained that generally there is some familial connection and some choice by the new recruit in terms of pursuing a role as babalau. Generally the training consists of being present, often most mornings when people come for assistance, to watch and listen to explanations, to follow instructions for mixing medicines and to learn to recognize the ailment and its cause, the medicines and methods and words to use for each ailment. Sometimes this was learnt over a few months and at other times if the person was away from the village for work, it would be over some years. Some people try new remedies while others just follow what their grandparents have taught them. Some specialize while others provide for first aid and common ailments. There is a lot to learn about the plants, preparation for use, and their various uses. Kopi listed over 22 treatments that had been shared, most of which required recognition of specific leaves, trees, and then sourcing saps or processing for use in a particular way. Finding the materials may require careful recognition of the place where the plants grow. Being able to return to that place which could be some distance away in the bush or under the sea requires mathematical spatial skills.

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Cameo by Patricia Paraide on Her Father’s Mathematical Activities and Education My father did not speak English but was well-educated. He had in-depth learning about weather forecasts; counting and record keeping; building houses and canoes; learning about animals, birds, fish, plants, and their habitats; planning of current and future village activities (making gardens/houses, journeys to other islands, marriages, births, deaths, feasting, and anticipated natural disasters); land management; environment management and conservation; Indigenous medicine and psychology; mastering gardening, hunting, and fishing skills; and navigational skills for sailing the oceans to other places near and far. He used this knowledge regularly. Most of these skills involved the application of mathematical knowledge. He learned all of this knowledge through working alongside his father, and other people who were experienced and experts in these skills and knowledge. He observed how his father and other skilled men performed these tasks and applied them to new tasks that he tried to do on his own. He passed on what he learned from his Elders to us his children. For example, in gardening, he knew the spacing between the various crops being planted to allow space for them to grow well. He also knew about specific leaves that produced rich mulch and the quantity needed for the various crops. Our gardens produced well so others approached him for advice on gardening. As I was growing up, I also noticed that my parent’s gardens produced well, and I learned a lot about gardening from working alongside them. My father was recognized as a traditional medicine expert. He passed on this knowledge to his children whom he observed to be capable of receiving such knowledge. I was one of them. He informed me that he first learned of medicinal plants and trees, and the type of environment they thrived in, when he was sent to collect them by experts. This knowledge assisted him to know where exactly to look for the varying medicinal plants needed for future use. He also learned that some plants which grew under the shade had weaker medicinal properties than the same species growing in a lot of sunlight. He also learned about the right quantity to collect to administer to sick people through observing the amount he was asked to collect and the amount of water used to produce the varying strength of the medicine prepared. He stressed that he did not complain when asked to carry out tasks given to him by experts. This led to him being given extra responsibility and hence more knowledge. This was how he was trained to be an expert in medicinal plants. He also pointed out that when he showed an interest in certain knowledge and skills, they were passed on to him readily because he was humble and respected his instructors. He often stressed that people learned skills better when they assisted experts in participating in the tasks at hand, and when they listened attentively to instructions on how to perform them well. He stressed that when unskilled people showed an attitude that they knew better than the experts, and did not do tasks as instructed, then the skilled and knowledgeable people did not pass on additional knowledge and skills to them. In fact, some refused to mentor them at all. My father also learned from his peers when they performed tasks together like building canoes and houses, making gardens and fishing. They showed each other how to perform such tasks as they observed and learned from the experts and experienced people with whom they worked. It was expected that the unskilled people, even when older, talked less but observed more and carefully to what experts did, so they in their turn could apply it when working on future similar tasks. The teaching strategy used by my fathers is similar to scaffolding that is encouraged in modern classrooms. When using this teaching strategy the teacher models or demonstrates how a task is done, or how to solve a problem, and then the teacher steps back, and only offers support to students when they need it. The learners apply the knowledge learned to solve new problems and create something new. This is strengthened through interaction with peers in play or working together and also working alongside those who know more or are more skillful (Vygotsky, 1978).

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Tolai Indigenous education prepared its young people with the skills necessary to survive in their adult lives. How well they mastered these skills was measured by how well they performed tasks in their adult lives. No-one was unemployed in traditional societies. Comment on this Tolai Foundational Learning Smith (1975) viewed this Indigenous teaching strategy positively: The transmission of knowledge between generations of traditional society was efficient because it associated learning with the need to use what was learned. The curriculum was appropriate to the age group and related to local circumstances whether this was the ecology of the lagoon or of forest clearings. (p. 4) Particular bodies of knowledge, such as mathematics, in each culture can only be understood fully by the people within these cultures. As de Abreu, (1995), de Abreu and Cline (1998), de Abreu, Bishop, and Presmeg (2002), Ascher (2002) and Beach (2003) have documented, various cultures have mathematical knowledge that is applied in everyday living. This mathematical knowledge is expressed in the various languages. Barnhardt and Kawagley (1999) and Eglash (1999) stated that, over many generations, Indigenous people have constructed their own ways of relating to the world, the universe, and to each other. Their foundational educational processes were carefully crafted around observing natural processes, adapting modes of survival, obtaining sustenance from the plant and animal worlds, and using natural materials to make their tools and implements. Kawagley (1995) and Cajete (2008) argue that all of this was made understandable through demonstration and observation accompanied by thoughtful stories in which the lessons were embedded. Battiste (2002), added: As diverse as Indigenous peoples are … so also are their ways of knowing and learning. Their stories of Creation and their psychological connectedness to their cosmology play a determining role in how Indigenous peoples envision themselves in relation to each other and to everything else. Knowledge is not secular. It is a process derived from creation, and as such, it has a sacred purpose. It is inherent in and connected to all of nature, to its creatures, and to human existence. Learning is viewed as a life-long responsibility that people assume to understand the world around them and to animate their personal abilities. Knowledge teaches people how to be responsible for their own lives, develops their sense of relationship to others and helps them model competent and respectful behaviour. Traditions, ceremonies, and daily observations are all integral parts of the learning process. (p. 14) However, as Kawagley (1995) and Cajete (2008) have discussed, Indigenous views of the world and approaches to education have been brought into jeopardy with the spread of Western values, social structures, and institutionalized forms of cultural transmission. Western knowledge and teaching strategies have dominated formal education in PNG since the colonial era.

Moving Forward

In modern Papua New Guinea, sharing of intergenerational knowledge has an additional avenue through the school. PNG teachers have reported that they are proud of how their ancestors, their Elders, and their parents, thought mathematically and did mathematics. For them, mathematics is embedded in many cultural activities, although they do not always call it mathematics (Owens, 2014). In various conversations with teachers it became apparent that many teachers could relate a cultural activity with one or more areas of the school mathematics

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curriculum. Recognition of these mathematical activities encouraged teachers and students to observe and question older people and to participate in cultural activities pertaining to mathematics. This is not uncommon around the world (de Abreu, Bishop, & Presmeg, 2002; Rosa & Orey, 2020) but it takes effort. Teachers could be inspired by the Hawaiian mathematics educator, Linda Furuto, who did participate in her cultural activity and learnt the skills of wayfarers from her father, putting time, all her body, thoughts, and spirit into being able to navigate across the Pacific in their sailing canoes (Furuto, 2014). In Chapter 2, Charly Muke provided the example of the bottle top game adapted for early counting, arithmetic, measurement and probability. There is evidence of the intertwining of modern materials into the games of children but the development of the games currently played are often embedded within earlier practices. Together with Patricia Paraide’s cameo, these authors noted how their foundational knowledge, cultural activities and play have enhanced their school mathematical knowledge. They are aware of the need for further research. This has been evident to mathematics education researchers especially since the 1960s. For example, the careful recording of string designs by Haddon (1930, reprinted in 1979) was linked by Vandendriessche (2015) to higher mathematics. In other words, PNG’s foundational mathematics has been noted internationally especially since Bishop’s earlier work. We will look at some of the research carried out soon after Independence in later chapters, particularly Chapters 7 and 11. This important aspect of education was a basis for the elementary schools during the reform period and research by Owens, Muke and others in 2014, described in Chapter 8. However, it is important first (in Chapter 4) to assess early colonial approaches to education to understand the circumstances of Papua New Guinea education. References Amarshi, A., Good, K., & Mortimer, R. (1979). Development and dependency: The political economy of Papua New Guinea. Melbourne, Australia: Oxford University Press. Armstrong, W. (1924). Rossel Island money: A unique monetary system. The Economic Journal, 34, 423–429. Ascher, M. (2002). Mathematics elsewhere: An exploration of ideas across cultures Princeton, NJ: Princeton University Press. Barnhardt, R., & Kawagley, A. (1999). Indigenous knowledge systems and Alaska Native ways of knowing. Anthropology and Education Quarterly, 36(10), 18–23. Battiste, M. (2002). Indigenous knowledge and pedagogy in First Nations education: A literature review with recommendations (Working paper). Ottawa, Canada: Apamuwek Institute. Beach, D. (2003). Mathematics goes to market. In D. Beach, T. Gordon, & E. Lahelma (Eds.), Democratic education ethnographic challenges (pp. 99–122). London, UK: Tufnell Press. Bladon, M. (1982). The song of the Bamu. Lawson, NSW, Australia: Mission Publications of Australia. Cajete, G. (2008). Sites of survivance. American Indian geographies of identity and power: At the crossroads of Indígena and Mestizaje. In M. Villegas, S. R. Neugebauer, & K. Venegas (Eds.), Indigenous knowledge and education: Sites of struggle, strength, and survivance Cambridge, MA: Harvard Education Press. Campbell, S. (2002). The art of Kula. Oxford, England: Berg, Oxford International Publishers. Chalmers, J. (1887). Pioneering in New Guinea: London, UK: The Religious Tract Society. Crawford, A. L. (1981). Aida: Life and ceremony of the Gogodala. Bathurst, NSW, Australia: Robert Brown and National Cultural Council of Papua New Guinea. d’Albertis, L. (1880). New Guinea: What I did and what I saw (Vol. 2). London, Engalnd: Author. D'Entrecasteaux, B. (2001). Voyage to Australia and the Pacific 1791–1793. Edited by E. Duyker & M. Duyker, Trans. E. Duyker & M. Duyker. Melbourne, Australia: Melbourne University Press.

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de Abreu, G. (1995). Understanding how children experience the relationship between home and school mathematics. Mind, Culture, and Activity: An International Journal, 2, 199–142. de Abreu, G., & Cline, T. (1998). Studying social representations of mathematics learning in multiethnic primary schools: Work in progress. Papers on Social Representations, 7, 1–20. de Abreu, G., Bishop, A., & Presmeg, N. (Eds.). (2002). Transitions between contexts of mathematical practices. Dordrecht, The Netherlands: Kluwer. Diamond, J. (2012). The world until yesterday. London, UK: Allen Lane. Eglash, R. (1999). African fractals: Modern computing and Indigenous design. New Brunswick, NJ: Rutgers University Press. Finsch, O. (1888). Samoafahrten.Reizen in Kaizer Wilhelmsland und English-Neu-Guinea in den Jahren 1884, 1885, an bord des Deutschen Dampfers “Samoa.” Leipzig, Germany: Ferdinand Hirt & Sohn. Foley, W. (2005). Linguistic prehistory in the Sepik-Ramu basin. In A. Pawley, R. Attenborough, J. Golson, & R. Hide (Eds.), Cultural linguistic and biological histories of Papuan-speaking peoples (pp. 329–361). Canberra, Australia: Pacific Linguistics. Foley, W. (2010). Language contact in the New Guinea region. In R. Hickey (Ed.), The handbook of language contact (pp. 795–813). Hoboken, NJ: Wiley & Sons. Ford, A. (2011). Learning the lithic landscape: Using raw material sources to investigate pleistocene colonisation in the Ivane Valley, Papua New Guinea. Archaeology in Oceania, 46, 42–53. Furuto, L. H. L. (2014). Pacific ethnomathematics: Pedagogy and practices in mathematics education. Teaching Mathematics and its Applications: An International Journal of the IMA, 33(2), 110–121. https://doi.org/10.1093/teamat/hru009 Gaffney, D., & Summerhayes, G. (2017). An archaeology of Madang Papua New Guinea Working Papers in Anthropology, 5. http://hdl.handle.net/10523/7294 Galligali, M. (2001). The canoe racing event of the Gogodala society. Unpublished report. University of Goroka, Papua New Guinea. Golitko, M., Schauer, M., & Terrell, J. E. (2012). Identification of Fergusson Island obsidian on the Sepik coast of northern Papua New Guinea.(Research Report). Archaeology in Oceania, 47(3), 151. Goodman, J. (2005). The Rattlesnake: A voyage of discovery to the Coral Sea. London, UK: Faber and Faber. Haddon, K. (1930, reprinted in 1979). Artists in strings. New York, NY: AMS. Hallinan, P. (1985). Kula and the traditional canoes of the Trobriand Islands. Paradise in-flight with Air Niugini, 50, 13–14. Kawagley, A. (1995). A Yupiaq worldview: A pathway to ecology and spirit. Prospect Heights, IL: Waveland Press. Kopi, S. (1997). Traditional beliefs, illness and health among the Motuan people of Papua New Guinea. PhD thesis, University of Sydney, Australia. Lawes, W. (1885, revised 1895). Grammar and vocabulary of the language spoken by the Motu tribe. Sydney, Australia: C. Potter, Government Printer. Leahy, M. (1994). Explorations into Highland New Guinea 1930–1935. Bathurst, NSW, Australia: Crawford House Press. Lean, G. A. (1992). Counting systems of Papua New Guinea and Oceania. PhD thesis, PNG University of Technology, Lae, Papua New Guinea. Lineham, T. (1983). A triumph for Tami. Paradise in-flight with Air Niugini, 42, 17–23. Malinowski, B. (1920). Kula: The circulating exchange of valuables in the archipelagoes of eastern New Guinea. MAN, 20, 97–105. Malinowski, B. (1922). Argonauts of the Western Pacific. London, UK: Routledge and Kegan Paul.

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Maude, H., & Wedgwood, C. H. (1967). String Figures from Northern New Guinea. Oceania, 37(3), 202–229. https://doi.org/10.2307/40329593 May, P., & Tuckson, M. (2000). The traditional pottery of Papua New Guinea (rev. ed.). Adelaide, Australia: Crawford House Publishing. Mennis, M. (2014). Sailing for survival. University of Otago Working Papers in Anthropology, 2. http://hdl.handle.net/10523/5935 Mikloucho-Maclay, N. (1975). New Guinea diaries 1871–1883 (C. L. Sentinella, Trans.). Madang, Papua New Guinea: Kristen Press. Moresby, J. (1876). Discoveries in New Guinea. London, UK: Government Report. Muke, C. (2000). Ethnomathematics: Mid-Wahgi counting practices in Papua New Guinea. M.Ed thesis, University of Waikato, Hamilton, NZ. Nakata, M. (2002). Indigenous knowledge and cultural interface: Underlying issues at the intersection of knowledge and information systems IFLA Journal, 28(5–6), 281–291. Nowell, B., & Gilmore, J. (2002). Kakasinga voyage: Keeping culture alive. Paradise in-flight with Air Niugini, 139, 52–54. Owens, K. (2014). The impact of a teacher education culture-based project on identity as a mathematics learner. Asia-Pacific Journal of Teacher Education, 42(2), 186–207. https://doi.org /10.1080/1359866X.2014.892568 Owens, K., & Kaleva, W. (2008a). Case studies of mathematical thinking about area in Papua New Guinea. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Annual conference of the International Group for the Psychology of Mathematics Education (PME) and North America chapter of PME, PME32 - PMENAXXX (Vol. 4, pp. 73–80). Morelia, Mexico: PME. Owens, K., & Kaleva, W. (2008b). Indigenous Papua New Guinea knowledges related to volume and mass. Paper presented at the International Congress on Mathematics Education ICME 11, Discussion Group 11 on The Role of Ethnomathematics in Mathematics Education, Monterray, Mexico.https://researchoutput.csu.edu.au/en/publications/ indigenous-papua-new-guinea-knowledges-related-to-volume-and-mass Owens, K., Lean, G. A., with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Owens, K. (2022, in press). Chapter 7. The tapestry of mathematics – Connecting threads: A case study incorporating ecologies, languages and mathematical systems of Papua New Guinea. In R. Pinxten & E. Vandendriessche (Eds.), Indigenous knowledge and ethnomathematics. Cham, Switzerland: Springer. Paraide, P. (2018). Chapter 11: Indigenous and Western knowledge. In K. Owens, G. Lean with P. Paraide, & C. Muke (Ed.), History of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Parkinson, R. (1907). Dreissig jahre in der Südsee. Stuttgart, Germany: Strecker und Schröder. Reynolds, J., & Coxhead, G. (Eds.). (~1940). The story of the world (illustrator E. Mayo). London, UK: Universal Textbooks. Rivett, J. (2020). It's the economy, and you are not stupid. Sydney Morning Herald, p. 8. Rosa, M., & Orey, D. (2020). Dossier: International perspectives on ethnomathematics: from research to practices, Revemop, 2. https://www.periodicos.ufop.br/pp/index.php/revemop/ issue/view/158/showToc Ross, M., Pawley, A., & Osmond, M. (2003). The lexicon of Proto Oceanic: The culture and environment of ancestral Oceanic society. 2: The physical environment. Canberra, Australia: Pacific Linguistics, Research School of Pacific and Asian Studies ANU. Seligmann, C. (1910). The Melanesians of British New Guinea. Cambridge, UK: Cambridge University Press.

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Summer Institute of Linguistics (SIL). (nd). Ethnologue: Languages of the world. http://www. ethnologue.com/ Smith, G. (1975). Education in Papua New Guinea. Melbourne, Australia: Melbourne University Press. Strathern, A. (1977). Mathematics in the moka. Papua New Guinea Journal of Education, Special Edition The Indigenous Mathematics Project, 13(1), 16–20. Strathern, A., & Stewart, P. (2000). Arrow talk: Transaction, transition, and contradiction in New Guinea Highlands history. Kent, Ohio, and London, UK: Kent State University Press. Swadling, P. (2010). The impact of a dynamic environmental past on trade routes and language distributions in the lower-middle Sepik. In J. Bowden, N. Himmelmann, & M. Ross (Eds.), A journey through Austronesian and Papuan linguistic and cultural space: Papers in honour of Andrew Pawley (Vol. 615, pp. 141–159). Canberra, Australia: Pacific Linguistics, Australian National University. Thurnwald, R. (1936). Pigs and shell money in Buin: Changing economic patterns, 1908–1934. Pacific Affairs, 9, 347–357. Towers, J., & Martin, L. (2014). Building understanding through collective property noticing. Canadian Journal of Science, Mathematics, and Technology Education, 14(1), 58–75. Vandendriessche, E. (2007). Les jeux de ficelle: Une activité mathématique dans certainess sociétés traditionnelles (String figures: A mathematical activity in some traditional societies). Revue d'histoire des mathématiques, 13(1), 7–84. Vandendriessche, E. (2015). String figures as mathematics: An anthropological approach to string figure-making in oral traditional societies. Dordrecht, The Netherlands: Springer. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Were, G. (2010). Lines that connect: Rethinking pattern and mind in the Pacific. Honolulu, HA: University of Hawai'i Press. Whittaker, J., Gash, N., Hookey, J., & Lacey, R. (1975). Documents and readings in New Guinea History: Prehistory to 1889. Milton, Australia: Jacaranda Press.

Chapter 4 Mathematics Education from the Early Colonial Period, Before and After Both World Wars, Until the Early 1960s

Abstract: This chapter covers German, British, and Australian influences on education in the Territories of Papua New Guinea in the nineteenth century, and in the twentieth century before 1975 (when Independence was achieved). Before World War I, Germans supervised formal education in the New Guinea Territory, mostly through missions, and schools spread across the New Guinea islands, Morobe, Madang, and some of the Sepik. Missions also provided education across the Papuan Territory, initially British New Guinea for which the Queensland Government was made responsible. Throughout this period, protectionist policies and distances from places of administration adversely affected the quality and forms of school education in the Territories. After World War I, both New Guinea and Papua were administered by Australia with very limited funding being made available. Decisions were made at the end of World War 1 and at the end of World War II which had important effects on the administration, structure, and nature of formal education in the Territories of Papua and New Guinea. Although the government was keen for education to spread, the opportunities for education were provided mainly by Christian Church missions, and by their missionaries. In individual schools it was the missionaries who defined the ways school education would take place—of particular relevance for this book was the fact that children were usually expected to learn mathematics in their home language. Despite the influence of the missions on schooling, secular policies on education instituted by the Australian Government influenced decisions on who was to be educated, what was to be taught, and what the language of instruction would be.

Key Words: German New Guinea schools · Mission education in New Guinea and Papua · Territories and Administration Post World War I and II MASTA …Day after day, sitting in the classroom, Writing and writing, with a pen Bought from the white man’s store. Sitting quietly listening, with my eyes wide open … Then Christmas comes, I go home. Dad and mam call me Masta They respect me. And the village people respect me. Where have I really come from? Down from Heaven? Oh, Man! I used to be called by name And I am a simple student at school. Miztigi, 1981

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Introduction The previous chapters show that before Independence Papua New Guinean societies featured forms of foundational mathematics which are still in use today. Various aspects of culture, such as “warfare, land usage and acquisition, associated with spatial and numerical relationships were required for communication” (Owens, et al., 2019, p. 71). Mathematical systems changed as they met a society’s needs but there were numerical systems in the traditional PNG cultures which were not taught in schools. The 2-cycle, (2, 5) cycle and (2, 5, 20) cycle (mostly digit tally) and body-part numeral systems, with multiple variations, were part of the mathematical knowledges of the various language groups (Owens et al. 2018). Although this cultural knowledge was recognized by early colonists, their commentaries were often indicative of a colonialist attitude which acknowledged certain aspects of culture but did not acknowledge any level of equivalence with European culture. According to McCarthy (1968): Papua New Guineans learned many things essential to their way of life of which the white man was ignorant. Education is thus a matter of degree, a matter of human intelligence, and often it depends on the environment and customs of the different peoples. (p. 121) McCarthy noted that although people could not write, they could count up to a million, and they could send signals by drum using certain combinations of beats. Colonial values were even evident in comments made on early colonial education. Around Port Moresby, for example, “the bringing up in the fear of the Lord” was considered to be “the beginning of all education” (Thomas, 1886, p. 396, cited in Gash & Whittaker, 1975, p. 119). The caption to a photograph, which appeared in the Sydney Mail in 1922, of boys and girls, holding their small blackboards and papers at an LMS school at Port Moresby stated that “the work of the London Missionary Society is beyond praise,” and “to see 300 native1 children drilling and to hear them singing “God Save the King” is an experience to be long remembered” (cited in Gash & Whittaker, 1975, p. 120) These Port Moresby schools, as noted later, were also using the vernacular language of Motu. Efforts are made in this chapter to provide a history of mathematics education in early colonial times through to the period leading up to Independence. Where possible, we use the voices of Papua New Guineans—albeit the hegemony of the views on education of the colonialists is still an issue in terms of what really happened in education.

Early Contact and Colonial Times

Contact with peoples who came from outside what would become Papua New Guinea had occurred for hundreds of years, at least. There were visits from the Malaccas especially into West Papua and from other western Melanesian groups into the western areas of New Guinea and Papua. The Melanesians in the east came from the Island Melanesia (i.e., the Solomon Islands, Vanuatu, Fiji, and New Caledonia), and many made return visits or kept connected through a chain of trade; the Polynesians came across to Island Melanesia before European exploration. Fijians and Polynesians also came into New Guinea and Papua as a result of European explorations and colonization, often as teachers and pastors for missions but also for trade. Famously, Samoan Queen Emma established a trading empire around Rabaul with her German partners, and there was a German attempt to produce a Utopian race with Germans and Samoans. From The documents at the time referred to “native” for Papua and New Guinean people. However, by the 1970s this term was regarded as a derogatory term and we, as authors (PNGian and Australian), would not use it, even now. We have kept this term in this chapter, often adding “(sic)” to indicate that we are summarizing material written at the time. 1

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1875 onward, Germans, and Methodist missionaries, such as George Brown, brought Pacific Islanders into New Guinea and the London Missionary Society in particular brought Polynesian teachers and pastors to Papua. There were regularly displaced people inside PNG, and this kind of movement of people for various reasons still occurs today. Well-known migrations were associated with: • West Papuans who lived near Aitape for many generations but were hit by a tsunami in 1998; • Groups on Manus Island—probably in the 1600s and 1800s; • Displaced groups were accepted into clans in Chuave (personal communication, Susie Daino) and Jiwaka (personal communication, Charly Muke) from other provinces; • Upwards of 20 000 West Papua refugees, who fled from the impact of Indonesian control, mining, and from the Javanese who were taking over their land—from the 1970s to the present day. There may also have been regular visitors to these shores from south-east Asia, and since the 1600s visitors from Europe often traversed the south seas. Some of the writings of the European explorers—including Mikloucho-Maclay, Bougainville, and members of British Expeditions—have provided evidence that the foundational mathematics described in Chapters 2 and 3 was present, and recollections of other explorers and missionaries have informed this chapter. The Dutch first encountered New Guinea in the 1500s but it was not until 1828 that the Dutch took control of the western half of New Guinea. In 1884, the Germans annexed the northeast (termed New Guinea), and the British took the southeast at the same time. Although Queensland failed to “capture” the southeast in 1883, in 1884 the British took it and called it British New Guinea. And then, somewhat ironically, the British more or less left it to the colony of Queensland to administer. Australia was not an independent nation at that stage, but a set of six British colonies—it did not achieve its own independence from Great Britain until 1901. In 1906 the southeast was formally renamed Papua and was transferred to Australian hands to administer. The missions were first involved in education in 1872 (Lett, 1942), with each mission station having its own school. Neither the German nor Australian administrations were officially involved in providing schooling. Education for Indigenous children was informal and undertaken by those who came into contact with them in their home and community environments (Smith (1975, pp. 11–14). Barrington-Thomas (1976, pp. 4-8), and Dickson (1976, pp. 21-22) also document how schools of a Western kind were introduced by the Christian missions in British New Guinea in 1873. According to Hudson (1971), in 1898, 6 000 Papuans were attending school (p. 23). New Guinea Pre-World War I: German Territory Schooling and church services seemed to be the two main starting points for any mission. So, when the Reverend G. Brown first set up Methodist Missions in New Ireland and New Britain in 1875, after having been a missionary in Samoa for some years, it was not long before Church workers were using local languages in church services and also teaching literacy (in local languages and in German). There is also evidence that some mathematics was also taught at that time. Brown noted that house building required applied skills of practical mathematics, but there is no record of whether he observed that these skills were formally taught or were just learnt by observation (Gash & Whittaker, 1975). Nor is there any indication whether the Church workers in their teaching linked the foundational mathematics within the cultures to any mathematics

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taught in school. However, it seems that mathematics was established as a school phenomenon from a very early date (Owens, Clarkson, Owens & Muke, 2019; Ryan, 1972). For many years there was no subsidy to the missions from the government for anything, including none for schooling. Nevertheless, the Roman Catholic Mission began a boarding school for mixed-race children at Vunapope in 1898 and the Methodist Mission opened a similar school in 1907 at Raluana. The German language was taught at both schools as well as manual training. The German reports indicated that education in mission work that “succeeded though under great difficulties,” and that in the Methodist mission’s schools, “the real desire, not only of the boys, but also of the girls, to learn reading, writing and arithmetic is a good sign” (Dickson, 1972, p. 317). The German government had been considering for some time establishing a more advanced education than that provided by the missions, although it was mainly to be for teaching trades. By 1907 the German administration did open a school at Namanula in East New Britain, which included technical training in the curriculum (Threlfall, 2012). Lessons were in German for the trade subjects taught and some other subjects (Husdon, 1971). The Namanula school had 30 boys in their initial graduation in October 1913 after they had completed six years of education. All of the graduates subsequently worked for the government as assistant teachers, assistant clerks, artisans and printers. Arrangements were made for them to receive further lessons in German and technical subjects (Thredfall, 2012, p. 94). A school opened in 1909 for nine European children and by 1912, there were 15 children including Ambonese and Chinese children. The Lutherans had started a school in Simbang (near Finschhaffen) Morobe Province for their children in the 1890s. Fewer schools were set up in New Guinea before 1900 although the missions such as the Lutheran mission had begun in Morobe Province at Simbang in 1889 with 14 youth who were more interested in pay and new ideas as they worked with the missionaries with new crops and tools than they were in literacy and c ounting. Nevertheless the missionaries moved to other areas setting up stations but with no mention of schools although these are assumed (Wagner & Reiner, 1986) but the evangelists used oral stories, usually in Yabim, a coastal Austronesian Oceanic language, or Kâte, a non Austronesian language. The early missionaries recognized the huge diversity of languages in Morobe and Madang Provinces where they were working on the mainland (see Owens, Lean, with Paraide & Muke, 2018). They understood the distinct differences between these two kinds of languages and that these languages would be more easily learnt and understood than English or German. They often used the local lingua franca, Bel, along the Madang coast (personal communications, Malalamai village2, 2006). In Madang, there were four stations with schools for 90 pupils in 1902 (German Colonial Office 1902, cited in P. Smith, 1987, pp. 28-29) while the Catholic mission had five schools. The Wesleyan mission had 100 village schools and a district school at Raluana in the New Guinea Islands using Fijian and Samoan teachers while a Catholic Society had less than 30 schools. In 1912 the German administration announced it would establish its own system of schools but again mainly focusing on the trades: “Initially there was a nucleus of schools around Rabaul; an elementary school, a technical school and a school of domestic economy” (Radi, 1971, p. 103). Pupils were to be brought from other parts of the Territory and return to their areas as teachers for more schools. There were perhaps less than 600 students before World War I but with plans for expansion. By 1913 there were 180 Europeans of whom 154 were German in the German Territory. The missions were educating about 15 000 throughout the German Protectorate, half in the Gazelle. The majority of the missions established under German rule had no teaching in English. There is a monument to Frierl, the early Lutheran missionary, in this village.

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This is not at all surprising since many of the missionaries could not speak English and used Tok Pisin for general communication (Mair, 1948, reprinted 1970). However use of the vernacular was approved by many experts in native education and was used in many schools. Interestingly, in a departure to the norm, English was used in six government schools and in technical education run by Catholic and Methodist missions. The German administration was proposing a scheme for modified compulsory education to be provided by the missions. The German Governor, Hahl, planned to expand administration schools in a plan prepared in late 1913 but in 1914 World War 1 began. According to Rowley (1971), German educational policies were better than those in Papua. The German language was to be promoted as the language of literacy and as a replacement for pidgin (Tok Pisin) as the language of administration. “Thus the Germans were beginning to tackle educational problems which the Australians did very little to clarify until after the 1939-45 war” (Rowley, 1971, p. 70). Papua Pre-World War 1: British, Queensland, Then Australian Territory Although schools started in Papua at about the same time as in the German Territory, educational development in Papua was different. The first school in Papua was opened in Port Moresby in 1874 by Dr Lawes of the London Missionary Society (LMS). He wrote an extensive Motu dictionary and schooling was conducted in that language. Other missionary bodies soon followed with schooling in the vernacular of their area. Interestingly Lawes’ (1885, revised 1895) dictionary contained counting words and other words that could be used for mathematics like groups and identifying different groups of objects. He continued to upgrade his dictionary over 20 years. At the Port Moresby school in 1883 there were about 70 children present and were divided into nine classes. “The course of instruction was a reading and writing of a very primitive kind” (Gash & Whittaker, 1975, p. 112). The teacher squatted on the floor in the center of the class, listened to the alphabet or words of one syllable repeated alternately by the children. In the higher forms short sentences were being read, which were all of a scriptural nature (Gash & Whittaker, 1975). By the 1890s the government was starting to take an interest in education. MacGregor, the Administrator (Governor) of Papua, wanted every child over 5 years of age and not over 13 years of age to attend the school that was nearest to his home or some other school. There were penalties for breaching the Schools Regulation (No III of 1897) that detailed this requirement. Interestingly this regulation only applied to schools at which English was taught and no parent or person shall be punished for the non-attendance of a child at a school where English was not taught (P. Smith, 1987, pp. 40-41). Other gradual changes were occurring, such as the curriculum seems to have expanded. In 1902 Commodore Erskin visited the LMS school in Port Moresby and saw about 120 children seated on the floor of the schoolroom. The children showed all the interest and obedience expected of European children and were examined by Mr Chalmers: (They) answered questions in geography, counted in English from one to a hundred, and gave the English for several phrases in common use spoken to them by Mr Chalmers in their own language. (P. Smith, 1987, p. 36) At this stage, the administration did not insist that all teaching be given in English. The missions were not teaching in English but they began modifying their stance. The LMS led the way in starting to teach in English in some schools. The native (sic) vernacular could be used in the early years. In fact, it was confirmed in Papua that it was easier for a child to learn initially in a foreign (sic) vernacular, for example Wedau in Oro and Milne Bay Provinces, than in English (Mair, 1948, reprinted 1970). The Anglican Mission in the Northern District of British New

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Guinea prepared a book of words including numbers in the Wedau language but the last two pages were in English. It was a summary in English of the native laws passed by the government. “The list had originally been written by Charles Abel for the children of Kwato. The children were expected to learn the list by heart” (P. Smith, 1987, p. 41). However in 1907 the situation changed. The Royal Commissioners of 1907 were strongly in favor of the teaching of English. This was mainly because of the value in extending “Australian influence and white settlement. They did not recommend direct government participation in education. Instead they argued it was the duty of the missions to teach English” (P. Smith, 1987, p. 49). The Report on the Administration of the Territory of Papua, 1907-1908, (1908, p. 30, cited in P. Smith, 1987, p. 52) suggested that “the government should undertake the education of the people for whose destiny they are responsible”. A scheme was outlined that would lead to a sound education. This included: • that a public school be established at the head government station of each division, with a qualified Australian teacher in charge of each; • that 30 pupils, drawn from different villages in the division, be attached to each school; • these pupils to be educated and trained to act as teachers, and on the completion of their education and training, be established in their various villages as public school teachers on salaries; • that as each batch of pupils becomes fitted to act as teachers their places in the training schools be filled by a batch from other villages; • that before any native pupil is established as a teacher he should pass an examination to show his suitability; • the various missions to have access to the schools at stated times for the purpose of religious instruction. But that the pupils when established in schools as teachers on no account be allowed to give religious instruction. (Commonwealth of Australia, 1908, p. 30, cited in P. Smith, 1987, p. 52) The Australian government up to then had rather half-heartedly urged that English should be taught in the mission schools, but what the government really wanted, and what the country, in view of the white man’s coming, really needed, was “an easy means of communication between the magistrates and planters and miners and traders and the natives, and a very moderate vocabulary of Pidgin English would satisfy these requirements” (Chignell, 1925, p. 130, cited in P. Smith, 1987, p. 50). In 1911 the Australian Parliament considered that “the matter of education must be deferred until pacification and knowledge of the country were much further advanced” (Lett, 1942, p. 141) although Murray, the Administrator of the Territory, wanted more rapid progress. He seriously considering the education of Papuans and native development as an obligation upon the government (West, 1970, p. 145) especially after being asked by Daru chiefs (Western Province) for schools to teach English in their areas (Smith, 1987, p. 55). He planned to pacify Papua, then tax the natives, the proceeds being used to provide schools, especially technical schools. He was considering a fairly high standard of technical education but he ruled out higher education. The money was to be used to subsidise the mission schools where English must be taught. These schools would be “subject to report and examination by inspectors appointed by the Lieutenant-Governor” (Lett, 1942, p. 143). In 1916 he again dispatched a request to the Minister for elementary schooling in English and technical or industrial training together with a plan for finance but again like his earlier request for government schools it came to nothing (Smith, 1987, West, 1970).

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However, the idea of assistance to missions had a more immediate result (West, 1970, p. 183). In July 1914 he proposed a scheme, which was sent to Melbourne and accepted by the Australian government for missions to receive funds, teach in English, and be inspected for progress. Expenditure on education was to be concentrated in particular localities. He argued persuasively that natives (sic) at a distance from mission schools might never see the benefits of their taxes which subsidised schools teaching in English and gaining passes in Queensland examinations (West, 1970 p. 256). That set the tone for many years for various schools to be located at key localities rather than in each village. Nevertheless, the distance from Port Moresby to the Australian government was frustratingly evident and the real situation of Papua was virtually unknown in Australia.

Post World War I

At the end of World War I, the League of Nations (forerunner of the United Nations mandated and handed the German colony to Australia as the Trust Territory of New Guinea. The period of Australian Territories of Papua and New Guinea between the Wars saw independent administrations and several issues that required solutions. The issue of language and conflicts in funding were particular problems. The source of funds could no longer just come from taxes, the missions and administrations differed on language of instruction, and missions needed government funding. Significantly in both Territories there were gold mines and associated activity as well as plantations. In New Guinea the rich gold fields around Bulolo and Wau in Morobe with some finds in the Eastern Highlands were encouraging the development of the sea ports of Lae and Salamaua. In Papua, Misima Island was the main field, a long way from Port Moresby. However, the revenue for infrastructure and administration was not sufficient from the territories themselves nor were recruits particularly experienced (at least in the eyes of the ex-Kenyan reporting administrator, Col. John Ainsworth, to the Senate in Canberra, Australia) (Ainsworth, 1924, pp. 29-30, cited in Smith, 1987, p. 101). This lack of development may have counted against Australia in the onslaught of the Japanese in the Second World War but the poor recruitment practices and punishment of labourers on plantations did continue despite being condemned by the Australian government, again with lack of adequate funding to administer (Hudson, 1971). Mandated Territory of New Guinea The Australian military administration in New Guinea was concerned about the number of Germans in the expatriate population, including the missions, so it closed the German administration schools until a civilian administration set up a system of formal education in both government and mission schools (Owens et al., 2019). The boys in the government school at Kokopo came from many provinces to try to break down inter-tribal hostilities. From 1916 the teaching of German was replaced by Tok Pisin3 in schools which increased in number and in number of pupils. The administration was not concerned with education except that it produced sufficient labourers. The Australian government kept the emphasis on economic development while supporting a different policy in New Guinea from that implemented by Murray in Papua (Hudson, 1971). In 1927 it was announced there would be a government elementary school in each of the nine districts of the Territory, each to be staffed by four fully-trained expatriate teachers seconded from Australian states teaching in English. An elementary school was established at Kavieng but Tok Pisin is a Creole spoken widely across PNG and which began with plantation recruitment in the late 1800s. It has some German language structures with German, English and local language word origins. It is also called pidgin or pidgin English. Although it is similar to the Creoles spoken in Solomon Islands and Vanuatu they are slightly different. Australian kriol is significantly different—despite some similarities it has numerous dialects. 3

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in 1927 it was virtually abandoned. Seven young men from Rabaul were sent to Queensland for further education in 1929. They became the first properly trained Papua New Guinean teachers in Government schools (Griffin, Nelson, & Firth, 1979). However, this move was met by resistance from the expatriate population of Rabaul and elsewhere. The settlers believed that returning students would cause trouble similar to what had happened in India. The white settlers’ opposition put an end to the scheme (Smith, 1987). Existing schools were kept open, but plans to expand were shelved (Hudson, 1971). The problem was that the boys had to acquire a new language at the same time as gaining an education in a trade according to Hudson (1971). The use of English robbed the pupils of the advantages of any education previously received from missionaries who taught in the vernacular (Hudson, 1971). Up to 1927 there were no attempts to co-ordinate the work of the missions and the administration. The large missions remained mainly staffed by German speakers. In 1929 B.J. McKenna, the Director of Education for Queensland, reported on education in the Mandated Territory. McKenna thought the government system was following on the right lines but did not go far enough. He therefore suggested proposals for increasing government participation in education (Smith, 1987). Some of his points included: primary school courses must be adapted to local conditions and a super-primary course for selected students was needed to supply teachers and tradesmen. This was followed by the Griffiths proposals in 1933 for three grades: village school, primary or middle school and high school or college. McKenna noted the hostility of non-official white residents to any system of native (sic) education (Smith, 1987). When Groves was the Administrator for New Guinea, his agenda for education was influenced by what he knew of colonial education in Africa, and by reports that educated Africans had begun to rebel against colonial rule (Ralph, 1978, p. 15). Documents which informed Grove’s plan for PNG education included the Phelps Stokes Commission Report (1919–1924), Victor Murray’s classic, The School in the Bush (1938) and the memorandum of the Colonial Advisory Committee on Educational Policy in British Tropical Africa (1920) which he had often cited during his 20 years as Mandate Administrator of Elementary Schools in Kokopo in East New Britain (1922-1942). In 1936 Groves was responsible for education and developed the education plan by himself in 10 days, with casual conferences with the colonial administrators and others. Ralph (1978) noted that Groves was very proud of his plan when it was approved by the Australian administration. He was keen to see recognition of local languages and cultures within the curriculum. He recommended that village schools prepare children to live in their rural communities, and that the vernacular be the language of instruction. Two reasons were put forward for having schools. The first was to establish a sense of being part of the Territory and to develop a sound administrative workforce. During this time, the debate over whether education should be for an elite, or for all, was in the balance. Some people, including Groves, felt that New Guineans needed education suitable for living in their own village environments. Others such as John Black4 suggested ultimately the nation would become independent—and therefore students should learn to be independent, and the nation self-governing. Higher and more focused forms of education would be needed if this were to be achieved (Bray & Smith, 1985; Cleverley & Wescombe, 1979; Gammage, 1998). However, Walter McNichol, formerly a secondary school headmaster in Victoria, Australia, was appointed the new administrator for Australia, and he did not proceed with the negotiated plans for teaching in the lingua franca. He wanted English in the primary school and for the six years of secondary school. In the New Guinea Territory, “the dominance of German among the expatriates, Pidgin (Tok Pisin) among the locals, and lack of training meant that McNichol did

Gammage (1998) provides a history of Black’s patrols in the highlands and West Sepik areas and his roles after the war including his push for preparation for self-government. 4

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not find a workforce able to teach elementary arithmetic, geography or English to his standards” (Owens et al., 2019, p. 77). Prior to World War II the missions carried out most of the education and the Permanent Mandate Commission of the League of Nations at its last meeting was unhappy with progress in education. The educational expert Miss Dannevig remarked that “she knew of no territory under mandate in which education progressed so slowly” (Mair, 1948, reprinted 1970, p. 225). Territory of Papua When the Native Tax Ordinance was created by Murray in the 1920s, there were few if any, important centers of native5 population without some sort of education. There was a regulation that native children living within a mile of a school should attend. No assistance, monetary or otherwise, was ever given to the missions by the government (Murray, 1929). Taxation provided some revenue and Murray offered to pay the missions some financial assistance if they followed the syllabus designed and approved by the administration. It was 10 shillings for each student passing the two education Standards to a maximum of 250 pounds. In 1919, some basic mathematics was involved. The government provided expectations for Standards 1 and 2. For Standard 1: Arithmetic Mental: Easy addition and subtraction of numbers. Tables: Multiplication tables to 6 times 12. Written: Simple addition, subtraction, multiplication, and short division (Regard will be had in two latter to the multiplication tables set.) Notation of numbers up to 500. For Standard 2: Arithmetic Mental: Mental exercise in the four simple rules and easy money calculations. Tables: Multiplication tables to 12 times 12. Money tables. Measures of length, area, weight (avoirdupois only), and of time. Written: The four simple rules including long division, compound addition, subtraction, multiplication and division. Notation and numeration up to 100,000 Practical application of the foot rule in measurement of yards, feet, inches and halves, fourths, and eights of an inch. (Territory of Papua, 1919, cited in Smith, 1987, p. 59) The quotes from the newspaper in 1922 that started this chapter praised the LMS education in Port Moresby. In 1924, the LMS had a school at Hanuabada with day students and boarders totalling 500 pupils (Chatterton, 1974). There were no classrooms. Technical education was carried out at Kwato and by Papuan Industries but most people had no experience with technical education so it serviced newly introduced technologies like machines and engines. The Methodists had a school on Kiriwina. Murray offered a small grant of 100 pounds plus a subsidy of 50 pounds per annum because he thought agricultural training was very important (West, 1970, p. 270). By 1929, industrial education was seen as more beneficial to the Papuans than knowledge of the 3 Rs. It is interesting to note the funding spent on different mission organizations and that technical education was favored as shown in Table 4.1. Also interesting is that the fund had a fairly large unspent surplus when it closed in 1935 for which no explanation can be found. The LMS received an annual subsidy of 1 000 pounds from the Native Taxation Fund besides the per capita grant and was able to appoint a fully qualified teacher and an expert kindergarten mistress. Two large classrooms, a workshop and a teacher’s This is the term commonly used in the Territories at this time. It is not a term that any of the authors would use as it was regarded as derogatory during their life time in PNG. We would use either the name of the cultural group, “Papua New Guinean,” or “local.” 5

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Table 4.1 Subsidies (in Australian pounds) to Missions for Education from Native Taxation Fund, 1920-1941 Mission Anglican

Per capita 3 045

Special Industrial 1 550

General and Technical

London Missionary Society

3 546

8 640

20 0315

Methodist

1 673

7 185

Missionaries of the Sacred Heart

1 471

17 000

Kwato Seventh Day Adventist

16 000 353

Unevangelised Fields Mission Bamu River Mission* 18 Source. Dickson (1972, p. 318). *Some missions may not have functioned for this full time and had lower numbers. residence were erected. The Roman Catholic Mission also received a subsidy of 1 000 pounds per annum, in addition to the per capita grant on condition that special attention was given to industrial training. The boys were trained in carpentry and ironwork, with drawing as the basis of the training. The Kwato Industrial Association also received a subsidy of 1 000 pounds per annum for special industrial training (Smith, 1987). One of the better known technical schools was at Salamo, the headquarters of the Methodist Mission on Normanby Island. The buildings at Salamo were built by the trainees. Trainees became the artisans, the leaders and instructors in districts to which they returned. The lower grades in the schools were all given some technical work and were taught by doing (Lattas, 2020). The technical approach meant that there was minimally some mathematics such as measuring and working with angles (Lattas, 2000). Record keeping and dealing with money were also likely (Owens et al., 2019). However apart from the minimal Standard Syllabus above there are no records of what was actually taught. Indeed since it appears that much of the trade subjects were taught by doing, echoing the foundational manner of teaching in the village, one wonders how much mathematics was identified and taught as mathematics. However, the emphasis on English literacy was more substantial. The Government invited the mission organizations, in return for financial assistance, to extend and improve educational facilities already existing for the natives. The government chose English as the language of instruction as the most suitable language for the many tribes and languages in Papua as far back as 1907. It directed its officers to use English in speaking to natives. The payment of subsidies was to be limited to schools teaching in English. The government also wanted to avoid Tok Pisin being used as a language of instruction. Smith (1987) suggested: The question of which language to use as a medium of instruction was as much political as it was pedagogical. It involved defining the aims and functions of education which in turn depended on plans for the social, economic and political future of Papua New Guineans. (Smith, 1987, p. 191) On most of the missions the custom was to use the vernacular. In 1920 there was an inspection of schools by a Queensland inspector who found that an improvement in performance over the previous year. English had made significant strides after using a reader prepared by Rev W. J. Saville of the LMS. Arithmetic performance was still weak. At this examination 1 366 pupils were examined and 1 147 passed. However, in 1930, only a few mission schools could attain the level set by the government standards and only 19 schools entered candidates for the government

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examinations. These schools were generally the more important mission head-station schools where most of the pupils were boarders. Meanwhile, some mission teachers travelled beyond the frontiers of Australian control bringing Christian education and some literacy and numeracy to the natives (Owens et al., 2019). The prevailing attitudes of the time are reflected in the editorial of the Papuan Villager by the government anthropologist, F. E, Williams and published by the Administration for the Papuans. It indicates the purpose for school mathematics: The white men know far more than you do. They make and do a lot of things that are quite beyond you. You cannot be the same as the white man; and there is no reason why you should. It is true that there are many of the white man’s ways that you can copy; you can learn to work hard and save money; you can learn to read and do arithmetic; you can learn to buy and sell and be ‘business-like’; you can learn to look after your health and guard your villages from sickness; you can learn to use tools in your gardens; and there are all sorts of useful things you can buy in the stores. But you can never be quite the same as the white men; and you will only look silly if you try to be. (cited in Smith, 1975, p. 9)6 The official syllabus for a five-year primary course was issued in 1931 and revised in 1939. It included arithmetic, drawing, and agriculture with examinations concentrated on English, Arithmetic and Geography7. English was to be the language of instruction. In 1937, Percy Chatterton designed a syllabus that used the vernacular for village schools run by the LMS. Table 4.2 is an extract from the syllabus showing the program for one of the Table 4.2 Timetable for Monday Morning and Afternoon Class A Morning Lessons First Lesson:

Class B

Class C

Write sums.

Write numbers and sums. Number lesson with teacher. Afterwards number Afterwards play with shells, lesson with teacher. stones, or small sticks. --------------------------------------------Drill with all classes-------------------------------------------Second Lesson:

New Testament Writing. Dismiss them. with teacher. Afterwards reading Afterwards writing. lessons with teacher. Afternoon Lessons Teach handwork such as basket making if you can on Monday and Tuesday afternoon. Otherwise, First Lesson:

Number lesson with teacher

Write sums

Second Lesson:

Copy writing

Reading with teacher

Thursday afternoon, all classes came and probably did work parade. Source. Based on syllabus provided by LMS Papua District Committee (1937, cited in Smith, 1987, pp. 76-77). Smith notes this was quoted by H. Nelson (1968) in discussing The Papuan Villager, a national newspaper, in Journal of the Papua and New Guinea Society, 2(1). 7 Geography was likely to be that taught in Queensland, and emphasis on explorers to and in Australia. 6

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three weekly school days (Smith, 1985) indicating that mathematics was part of the morning lesson with more difficult exercises in the afternoons on each day of the week. Meanwhile, although missions had provided education for the children of their expatriate Western families, the Government began schools for Europeans in the 1930s but not for Chinese or mixed-race children who relied on missions or schools run by the Chinese Nationalist Party (Smith, 1987). In 1938 the Rationalist newspaper was angry with Hubert Murray because it said schools should not be entrusted to the missions. Education was in the hands of the missions but subsidised by the Papuan government (West, 1970, p. 213). In 1938 there were five progress levels (standards) roughly equivalent to each year of education, and 77 students examined at the highest level with 2 473 students enrolled. One Papuan teacher Bona Aiaia who spent weekends preparing lessons for his students who were often away and required repetition of material, noted: I know my children better now. They are all different. Some are good at sums, some at mental arithmetic, some at spelling. Some are easy to correct and some are very hard; and I try to find out the best way with them. I find it hard to make them understand that it is wrong to copy or cheat, for they think that, as long as they have the right answer, it does not matter. That is because the people in the village are like that. (Bona Aiaia, 1940, p. 63, cited in Smith, 1987, p. 80) Not only did Bona express his frustrations but he also showed his expectations for his students and his quality as a teacher in recognizing and catering for the diversity of his students. He indicated that students do written work (sums) and mental arithmetic. There were expectations that those who went to school would become teachers and trade workers or return to their villages. When World War II began Papua had no government schools while 416 mission schools with 26 000 pupils were costing the government only 3 100 pounds a year. In the New Guinea Trust Territory the Commonwealth reduced the education grant from 18 000 pounds in 1923 to 5 000 pounds in 1937. It made no grants to 2 566 mission schools with 65 578 pupils whose education cost it nothing (Price, 1965). Due to language difficulties the training of native teachers was often not of a high standard. They were known to “let the lessons degenerate into mere parrotlike repetition” (Mair, 1948, reprinted 1970, p. 163). In 1933 and several following years, 12 Papuans were sent to the University of Sydney for a special medical training course. They did some training in Motu and English in 1st and 2nd years (Lett, 1942). There were special courses in physics, chemistry anatomy, physiology and pathology which must have involved some mathematics. The War Years Michael Somare8 (1975) recalled his experiences during WWII in the Murik Lakes area of the East Sepik Province with a Japanese teacher lasting nine months and the Japanese treated them there well (although they soon needed to be supplied with food by the villagers) before the Australians took them away. The teacher had lived in America so could speak English and quickly taught himself Pidgin. The first thing the children learned was how to count. Bani was the Japanese word for number and the students were shown one pawpaw or one stick or knife and the teacher said itshi. Two pawpaws or two sticks were called ni. So he was taught to count to ten. Then he was taught the vowel sounds and consonants which he repeated and then some words.

Michael Somare became the first Chief Minister of the newly Independent nation in 1975 and spent most of his life as Chief Minister with some period in Opposition. His father was a policeman in Rabaul where he lived for six years so the young Somare showed confidence with the Japanese teacher 8

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During WWII European teachers had left and in some Papuan coastal areas, local teachers continued to teach, with children becoming literate in their language and simple arithmetic (Camillla H Wedgwood, 1944, cited in Smith, 1987). Similarly in New Guinea with stories from East New Britain, Bougainville, and the Lutheran areas of the mainland. Cameo from Nagong Gejammec9 Nagong Gejammec (1973) from Lae in Morobe Province (New Guinea area) tells of moving the Lutheran school at Ampo far up the Busu river behind a large fig tree to continue school during Japanese occupation. Families from Butibum and Kamkumang villages of Lae joined them. He had convinced the Japanese he was New Guinean and all the children were New Guinean and although they did make regular patrols and hassle them, he and the school survived, thanks to the big fig tree protecting them from the bombs and thanks to his politeness and saying Sayonara (goodbye). Other Mission Schools During the War The missions in the highland areas were led by Papua New Guineans and some evangelists were well educated and schools were a regular part of the missions which continued throughout the war despite the internment of all German missionaries and occasional retaliation by Australian government officers at the start of the war and finally all Australian and American missionaries leaving. At Kerowagi, John Kuder (1947) reported: All on their own, and with the help of the local population, the older school pupils had gone out, gathered the children together, and held classes. Without textbooks (except for the books held by the teachers themselves), without paper, pencils, ink, blackboard, or chalk, they had made a beginning instructing their people. (from the April Lutheran Standard, cited in Wagner & Reiner, 1986)

Papua and New Guinea Territory Post World War II

While funding had been severely reduced prior to the war, after World War II, some have posited that a main reason for financial support for educational and other developments in 19451963, was that Australia had a war debt to PNG people and desired to provide a humanistic service to a country that they had started to know. Others gave other reasons for the development and influx of funds such as ensuring security for the nation or the many relationships that had built up over the past 50 years including a number of Australians who regarded PNG as their home. Many of these colonialists, however, expressed poor colonial views seeing Papua New Guineans in a lower role than themselves and available to be used for making money. However, the war also taught a number to value their PNG comrades’ culture, trust and ingenuity. Although the New Guinea mandate was transferred to the United Nations, the Australian New Guinea Administration Unit managed both Papua and New Guinea as one Territory following the single administration unit during the war. Camilla Wedgwood reported on education for the Australian Army’s Directorate of Research and Civil Affairs in 1944 describing conditions in schools throughout Papua. “Her report on the schools in the villages near Port Moresby and particularly in the temporary settlements to which people from Port Moresby villages had been evacuated showed an especially dismal picture” (Smith, 1987, p. 144). Wedgwood noted that mission schools fell roughly into four categories: head-station schools, recognized village schools, vernacular village schools and catechist schools (Smith, 1987). “In addition to these four categories of schools, each of the missions has a center for training teachers and pastors, and Based on his own story as published and which we, the Owens, heard soon after arriving in Lae.

9

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one or more centers for giving technical training” (Smith, 1987, p. 75). She praised the efforts of the LMS in the Port Moresby district. Ron Schardt (2009) tells of trekking in the Huon Peninsula, known for some of the hardest walks in PNG, to visit student teachers in villages during the 1970s when these vernacular village schools or Tok Ples schools were well regarded in Lae education circles despite receiving no government assistance since their language of instruction was a vernacular language. The Lutherans continued the training centers into the next century. In 2014, K. Owens met many teachers who were now in a cooperative of pre-elementary schools in this area, supported by SIL for training. One of these very good teachers did not speak English or Tok Pisin as he had been to school and continued to teach in school in Tok Ples. At least 80% of the children of 12 years and more are fully literate in their own tongue and are able to grapple successfully with simple arithmetic. Few leave school without some reading and writing knowledge of English and ability to work simple sums involving weights and measures and money. (Wedgwood, 1944, cited in Smith, 1987, p. 145) In 1944 Wedgewood noted two dangers that should be avoided given experiences in Africa. Firstly she wanted to avoid producing an educated elite and secondly “introducing hurried makeshift schemes which, while perhaps they may appear to be satisfactory at present and in the immediate future, will actually prejudice later developments” (Wedgwood, 1944, cited in Smith, 1987, p. 152). She also provided many suggestions if education was to go forward in PNG including the making of a linguistic survey of the country. This would be used in discussions about the use of vernaculars in native education. Language surveys and dictionaries had been carried out before 1900 and in the next 30 years e.g. by Ray and Australian government officers but were also carried out in the late 1950s by SIL and continue to today. In 1981, Wurm and Hattori (1981) had made a language atlas but whether this or earlier surveys were ever taken into account in decisions on education is not known. Today, it is considered that there are more than the 11 families that Wurm had identified - some say 23 and others say 40. Two reasons why little thought was given to education until the end of WWII are (i) the cost, given there was very little revenue and (ii) the belief that the mass of the people did not need education. The first Director of Education in Papua and New Guinea from 1946-1958 was W. C. Groves. Twenty-four years earlier he had opened the first Australian administration elementary school at Kokopo. He had then completed some anthropological studies and worked elsewhere in the Pacific. Groves (1946, October) was opposed to: the unintelligent Europeanisation of these people … based on (his experiences in) … the western Pacific. … these native peoples have a unique contribution to make to the sum total of world culture – in their close-knit communalistic social pattern; in their music and in their art and handicrafts, …Our education system, in terms of actual school programs and related activities, must pay due regard to these matters. … Education must be organised to meet the natives in their everyday lives and in their natural environment. To set up schools of the conventional type, therefore, even in the villages, and thus to limit the sphere of educational activities and the direction of the influence of education in the community, is a narrow and altogether inadequate interpretation of the function and purpose of education in such communities. (Groves, 1936, p. 65, cited in Smith, 1987, p.132) However, Smith (1975), Barrington-Thomas (1976), Dickson (1976) and Ralph (1978) were of the view that the suppression of the Indigenous peoples’ intelligence and capabilities was Grove’s agenda, before and after he took up his posting. This view is reflected in Grove’s statement:

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education was to be ‘bound up with peoples’ past … and to provide for their future progress in the directions that will not bring the native into competition or rivalry with Europeans. (Groves, 1936, p. 23) This statement could suggest that the Western education planned for the PNG Indigenous peoples was engineered to keep them below the capabilities of the white population. The decision for the provision of a lower quality education for the Indigenous peoples was influenced by the suspicion that, if the natives were taught well in Western practices and knowledge, they may rival the white population in terms of work knowledge and competency. It is possible that this decision stemmed from the fear that the more knowledgeable the Indigenous people became, the more confident and assertive they could become. Therefore, they would then be in a better position to critique colonial education policies, practices and decisions. On the other hand, Groves came into conflict with governors who were not as interested as him in people’s cultures and abilities and using these in school education (Smith, 1987). It resulted eventually in different schools for Papua New Guinean and non PNG students. Ralph (1968) discussed the school system in terms of race, necessities for Australian children returning to Australia, and concern for appropriate schools for village children. He provided details of A (Asian) schools, mostly in Rabaul area, B for mixed Australian and Papua New Guinean children and European schools. A school children tended to speak Chinese, Malay and Pidgin, B schools tended to use English like the European schools. He explained that gradually these distinctions died out. There was a need for the primary schools to prepare Australian children for an Australian curriculum and those children in the B schools started to attend but also many Chinese business men also wanted their children to attend schools in Australia (they may have lived in PNG for 4 generations) and this did become problematical but also occurred. It was also obvious that some of the PNG children, especially those who did well when starting at a young age, needed to go to secondary education in Australia. Thus funding was available to the Australian public servants and this last group for this education. Eventually there were T schools, that is Territory schools (previously village and village top schools, station schools) and A schools with an Australian (NSW mostly) curriculum. The T school curriculum was a modified curriculum intended to relate to the village situation (Ralph, 1968). When Distance Education (correspondence schools) were available to residents of PNG for both primary and secondary school, this option became available to the Australian children, often supported by teachers or other persons on the ground in PNG. By the 1970s there was a fixed system of International Agency Schools with fairly high school fees and a curriculum based on overseas (mostly Queensland) curriculum. After Independence more and more PNG business families, university lecturers and the like paid for their children to attend these schools. Expatriate children were not to attend Community schools and only a handful of children of missionaries in remote areas eventually found their way into Community or their later counterparts primary education. The Australian New Guinea Administrative Unit (ANGAU) decided in 1944 to enter the field of general education with a central teacher training school opened at Sogeri with 112 ordinary students, many of whom had not passed the Standard 5 examination. Only two or three could it be truly said that they had a working knowledge of English sufficient to enable them to express themselves at all fluently or to follow with any ease a lecture given in English. (Wedgewood, 1944, cited in Smith, 1987, p. 150) The Bishop of New Guinea's Education Policy (1945, cited in Smith, 1987, p. 157) said:

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• there must now be very few children in this country between the ages of 15 and 17 who have passed through Standard 5 or who have reached that Standard; … • Very few of the children in Standards 1, 2 and 3 will ever reach Standards 4 and 5. Their progress had been retarded because they will be too old to go right through school. Non-government schools were re-established relatively quickly after the war. At that time the missions operated more schools and taught more pupils than the administration. Many missions operated a separate education systems both before and after the war with some financial help from the administration. The missions’ educational systems involved more than 50 churches and voluntary agencies (Meere, 1968). The Australian Administration of the combined Territories (PNG) sought the cooperation of the missions. The administration controlled one school for natives near Port Moresby in 1946. It was proposed that as soon as possible, the number of students at the school would reach 200. There were three categories of students at the school including: • those who came to be trained in the technical school, • those who were intending to train as school teachers All students would be instructed in many subjects including English and arithmetic. Short of funds, material and staff the missions were willing to co-operate with the government in providing education but were determined to retain their own school systems (Smith, 1987). In the New Guinea Territory, the situation had deteriorated from the pre-war enrolment of about 90 000 in mission schools and 500 in administration schools (Department of Adult Education University of Sydney, 1969). By 1958 the administration’s system had 526 local teachers, most of whom had been trained in the twelve year period from 1946 (Duncan, 1974). Mission schools could use Motu but the administration schools would only allow English. Overzealous Papuan teachers punished children for speaking to one another in Tok Ples (place-talk). Groves and other Australian education administrators favored village schools teaching in the vernacular in the 1950s supported by radio broadcasts in vernacular, Police Motu, Tok Pisin and English (Owens et al., 2019). The Reverend Percy Chatterton said the vernacular should be used in primary school classrooms. Motu and Roro were to be used as languages of the church and school. Reading and writing in English was introduced once basic vernacular literacy had been established. After World War II the Department of Education started all-English schools. Twenty five years of all-English schools resulted in a generation of school leavers who were not effectively literate in any language. If they went to secondary school their ability to read English with understanding was too limited to be of much use to them (Chatterton, 1974, p. 50). The education system seconded facilities such as an ex-army convalescent camp at Sogeri and individual mission facilities often in ex-army temporary buildings throughout the Territory of Papua and New Guinea. Standards in mission school were generally lower than those in Administration schools due to scarce resources and less well-qualified trained teachers (Fisk, 1966). Where a rural station was staffed by an Australian Patrol Officer there was often a school to which children came from surrounding villages often speaking different languages.

The 1950s and 1960s

In 1952 an Education Ordinance was passed with the aim of establishing control over the non-government schools. This Ordinance took effect in 1953 and the Administration obtained the right to exercise control of the education services through the registration of mission schools and the certification of teachers (Smith, 1987, p. 197). There was also a policy of building administration schools in areas not already covered by the missions. It has been claimed that the 1952

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Ordinance, effective in 1953, was “the greatest single landmark in the development of mission education in Papua New Guinea (Meere, 1968). From our oral history data, one Australian expatriate (J. Parker, personal communication, 2019) reflected, “plans were changing rapidly in terms of when Independence would occur and how to prepare for it. It seemed to be brought forward regularly”. Many Australians were taking up positions to help the nation as teachers, administrators, bankers, business men, and as missionaries. The stories such as those in Taim Bilong Masta (Nelson, 1982) reflect well the affinity of most of these people with the country and its people but in many cases the hardships they endured, in the case of missionaries for their God and the people they loved (Bladon, 1982). The attitudes are evident in the following quotes, showing the impact of Australian colonization: Most Australians went to Papua New Guinea with the best of intentions, but poorly equipped intellectually and by experience to establish good race relations. Few could bend their starched white suits. What John Black calls the ‘master-servant, yes sir no sir, taubada business’ overlays the belief that change had to be gradual. All ‘natives’ would be advanced slowly and uniformly; there would be no black elite lifted above their fellow villagers to challenge the mastas’ hold on all positions of wealth and power … ‘The colonial atmosphere’ Ian Hogbin says, ‘was degrading both to those in charge and to those who were governed’. (Nelson, 1982, pp. 165-166) There were very few people who really saw the people of Papua New Guinea as what they are and what they have become. It was a great shock to a lot of people to realize that they were human beings, just as smart as we are and just as capable of giving and taking friendship. (Australian Dr Andrée Millar, who grew up in Bulolo and worked in Lae, cited in Nelson, 1982, p. 175) I think that the Australians thought that they were going to be here forever. We had to be respectful, and by overplaying that we became weak; we became a lot of humbugs. We had no feeling for our country; and it took a while to eradicate that from one’s minds. (Sir John Guise, member of Parliament and Governor General, cited in Nelson, 1982, p. 165) The missions were still doing the bulk of the education but they had less money than the administration. Hence it was difficult to progress beyond simple reading, writing and arithmetic. There was still insufficient funds for education, about 20% of the country’s revenue, and the administration was reluctant to reduce funds for health. In 1943 John Black (one of the early patrol leaders into the western half of the country) declared that natives must be made ready for self-rule. He dreamed of village councils. In 1946 he instructed each district in Ramu to establish a model elected native council and a village court run by native justices. However “those wanting to put the clock back” to pre-WWII were trying to frustrate those who wanted eventual self-rule (Gammage, 1998, p. 227). The years from 1945 to 1955 were described by the Minister for Territories as “years of poor achievement as far as education was concerned” (Hasluck, 1976, p. 99). Hasluck found it difficult to work with Groves. He was not sympathetic to Groves’ notions of preserving traditional cultures and was unhappy with the lack of haste of the Education Department in building schools (Smith, 1987, p. 186). Groves was an idealist who thought that “a competent teacher should be free to choose the content of the curriculum in his own school” (Smith, 1987, p. 189). This resulted in teachers trying to reproduce the curriculum from their home state. There was no adaption to the local situation. In an arithmetical problem written on the blackboard at Sogeri Central School was the following: “An orchardist plants 1,289 trees, 487 are apple trees, 395 are peach trees, and the rest are plum trees. How many plum trees are there?” (Smith, 1987, p. 189).

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Even at a superficial level, there was no adaptation as there are none of these trees growing in Papua New Guinea. However, there was more emphasis on the teaching of English and middle level technical training as recorded by the Education Report of 1953. Duncan (1974), however, reviews Groves’ educational development far more favorably and notes his extension of schools. In 1951 there were about 100 000 pupils in mission schools and only about 3 000 in government schools (Amarshi, Good, & Mortimer, 1979, p. 153). Hasluck rejected the idea of creating an Indigenous elite. Mair (1948, reprinted 1970, p. 227) noted that fewer than one per cent of adults had completed a full primary education and less than a hundred persons had completed secondary education. The first Papua New Guineans to graduate in agricultural science occurred in 1964 (Amarshi et al., 1979, p. 153). Rowley (1971) noted that Western education takes the most intelligent of the children out of village life, “These form a new elite; [once they complete] … secondary schooling and other post primary training they cannot live by their own efforts in the old subsistence economy” (Rowley, 1971, p. 103). In 1950 white residents persuaded Minister Hasluck to abandon plans to establish a multiracial secondary school in the Territory (Griffin et al., 1979). Only a quarter of children of school age attended school and many would not acquire lasting skills in English. “The round figures in 1951 were over 100 000 in mission schools and about 3 000 in Administration schools” (Hasluck, 1976, p. 86). In 1950, there were 230 pupils in mission and government technical schools with 1 070 at post-primary schools. No Papua New Guinean reached the final two years of secondary education in a government school. “From 1955, 10 students were sent to Australia each year, about 20 students were granted scholarships to attend high school in Australia. Two or three students attempted senior high school courses or equivalents in mission institutions (Griffin et al., 1979, p. 126). Sir Percy Spender told the Australian Parliament in 1950 that “in education the government aimed to provide universal literacy, technical skills and to provide opportunities for communities to develop within their own cultures” (Griffin et al., 1979, p. 124). Table 4.3 shows the numbers of schools, teachers and pupils in the two school sectors: Administration and Mission. Table 4.4 shows the growth in schools in the ten years. This increase continued. More rural children were attending school but may be only for a couple of years and their literacy was rarely maintained although they could understand English, they chose to speak in Tok Pisin or Tok Ples, Table 4.3 Comparison of Administration and Mission School Numbers 1957-1958 School Type Schools Teachers Pupils Source: Price (1965).

Administration 276

Mission 3438

972

2764

17 796

149 675

Table 4.4 Comparison of Numbers in Schools at the Start and Finish of the 1950s Decade Schools

1948-1949 2 463

1957-1958 4 205

Pupils

106 038

171 453

Non-PNG teachers

237

685

PNG teachers Source. Price (1965)

3 316

5 782

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many moving to the cities to find work. This continued into the 1960s. The number of schools did not increase as fast as the number of children starting school. Interestingly, and against the prevailing attitude of the Australian government, Groves and other Australian education administrators continued to favor village schools (primary level) teaching in the vernacular, but now supported by radio broadcasts in vernacular, Police Motu, Tok Pisin and English (Owens et al., 2019). Reading and writing in English was to be introduced once basic vernacular literacy had been established. Strangely perhaps, when the Department of Education started to build their own schools they had a policy of teaching only in English. The conflict on language policy persisted. By 1955 three courses for Papua and New Guinea teachers had been introduced: “A” course students, having completed primary school were given 1 year training to then teach the lower grades of primary schools; “C” and “B” course students with higher qualifications were admitted into 2-year courses. (Griffin et al., 1979) Hasluck still wanted “a system which would provide a sound elementary education in English, matching his policy of gradual uniform development” (Smith, 1987, p. 201). Up till the late 1950s it seems that reading, writing and arithmetic were almost the whole of the curriculum in primary education. Not until 1958 was there an actual plan for the achievement of universal primary education. It involved all children in controlled areas to read and write in English. So by 1958 the policy was to achieve universal primary education, which echoed what Spender had said in parliament in 1952. In 1950, Bladon (1982) went to Balimo. She noted how the Standens had provided hospital treatment for the Bamu and then that they held school for many years. As a newly recruited missionary nurse, after the morning medical care, she had her own class to teach reading and writing in their own language which she learnt as they went. Then she moved to another muddy and poor area continuing with her nursing care and schooling. The deprivation, difficulties, adaptations, and beliefs of the people were discussed by Bladon, a remarkable woman serving in her own small way with similar deprivations and adaptations. Young men made good use of their schooling, went away to work and came back. Young leaders arose among them while others continued with teaching when she left. Nevertheless, there was a problem in that undereducated students do not generally return to their villages and stay in towns (Conroy & Curtain, 1973).

Moving Forward

At the end of this period, it was accepted that Papua and New Guinea would be at least self-governing. The plan included a Legislative Council to assist with governance and to bring decision-making closer to the departments and the people of the country. At this point, many expatriates saw PNG as home on their soldier settlement blocks, entrenched in education and departments and on gradually larger farms such as those owned by the Leahy family in Morobe and the Highlands. Many small businesses such as mechanics, shops, and food shops were owned by expatriates including Chinese and Australians. Many church workers were expatriates. These groups had children born and raised in PNG. Nevertheless, it was gradually recognised that the country was to be run by Papua New Guineans in self-government. With the purpose of bringing the diverse communities of Papua New Guinea together as a unified nation and to introduce a Western type of governing system, efforts needed to be made to educate as many people as possible. New systems would need educated people and so changes needed to be made swiftly in the systems to provide for these changes. The next chapter addresses the years just before and after Independence.

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References Aiaia, B. (1940). I am a teacher. The Papuan Villager, 12(8), 62–63. Ainsworth, J. (1924). Report on the Administrative arrangements and matters affecting the interests of natives in the Territory of New Guinea. Canberra, Australia: Australian Government. Amarshi, A., Good, K., & Mortimer, R. (1979). Development and dependency: The political economy of Papua New Guinea. Melbourne, Australia: Oxford University Press. Advisory Committee on Education in Papua New Guinea (1969). Report, 1969. Australian Advisory Committee on Education in Papua and New Guinea; Chairman: W.J. Weeden. https://nla.gov.au/nla.cat-vn2258617 Barrington-Thomas, E. (1976). Problems of educational provision in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 3–16). Melbourne, Australia: Oxford University Press. Bladon, M. (1982). The song of the Bamu. Blue Mountains, NSW, Australia: Mission Publications of Australia. Bray, M., & Smith, P. (Eds.). (1985). Education and social stratification in Papua New Guinea. Melbourne, Australia: Longman Cheshire. Chatterton, P. (1974). Day that I have loved. Sydney, Australia: Pacific Publications. Chignell, A. (1925). An outpost in Papua. London, UK: J. Murray. Childs, P., & Williams, P. (1997). An introduction to post-colonial theory. London, UK: Prentice Hall/Havester Wheatsheaf. Cleverley, J., & Wescombe, C. (1979). Papua New Guinea: Guide to sources in education. Sydney, Australia: Sydney University Press. Conroy, J., & Curtain, R. (1973). Migrants in the urban economy: Rural school leavers in Port Moresby. Oceania, 44(2), 81. https://doi.org/10.1002/j.1834-4461.1973.tb01228.x Department of Adult Education, University of Sydney (1969). Education in Papua-New Guinea. Current Affairs Bulletin, 43(6). Dickson, D. (1972). Education, history and development. In P. Ryan (Ed.), Encyclopedia of Papua and New Guinea (Vol. 1, pp. 315–323). Melbourne, Australia: Melbourne University Press and University of Papua and New Guinea. Dickson, D. (1976). Murray and education: Policy in Papua, 1906–1941. In E. Barrington- Thomas (Ed.), Papua New Guinea education (pp. 21–45). Melbourne, Australia: Oxford University Press. Duncan, M. (1974). W. C. Groves: His work and influence on native education especially in Papua and New Guinea (1946–1958). Educational Perspectives in Papua New Guinea. Canberra, Australia: Australian College of Education. Fisk, E. (Ed.) (1966). New Guinea on the threshold—Aspects of social, political and economic development. Canberra, Australia: Australian National University. Franklin, K. (2014). Language surveys in PNG (Rev 3-10). https://www.researchgate.net/ publication/340066681_Language_Surveys_in_PNG_Rev_3-20. Gammage, B. (1998). The sky travellers: Journeys in New Guinea 1938–1939. Melbourne, Australia: Miegunyah Press, Melbourne University Press. Gash, N., & Whittaker, J. (1975). A pictorial history of New Guinea. Melbourne, Australia: Melbourne University Press. Gejammec, N. (1973). Mi titsa bilong Niugini. Journal of the Morobe District Historical Society, 1(1). Griffin, J., Nelson, H., & Firth, S. (1979). Papua New Guinea: A political history. Melbourne, Australia: Heinemann Educational Australia. Groves, W. (1936). Native education and culture—Contact in New Guinea. Melbourne, Australia: Melbourne University Press.

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Groves, W. (1946,October) Introductory statement. Paper presented at the Government-Missions Conference, Port Moresby. Hasluck, P. (1976). A time for building: Australian administration in Papua and New Guinea 1951–1963. Melbourne, Australia: Melbourne University Press. Hudson, W. J. (Ed.). (1971). Australia and Papua New Guinea. Sydney, Australia: Sydney University Press. Lawes, W. (1885, revised 1895). Grammar and vocabulary of the language spoken by the Motu tribe. Sydney, Australia: C. Potter, Government Printer. Lattas, A. (2020). Re-analysing the Baining: The mytho-poetics of race, gender and art, Oceania, 190(2), 98–150. 0.1002/ocea.5248 Lett, L. (1942). The Papuan achievement. Melbourne, Australia: Melbourne University Press. LMS Papua District Committee. (1937). Syllabus for village schools in Papua. Sydney, Australia: LMS. Mair, L. P. (1948, reprinted 1970). Australia in New Guinea. Melbourne, Australia: Melbourne University Press. McCarthy, J. K. (1968). New Guinea—Our nearest neighbor. Melbourne, Australia: Cheshire. McKinnon, K. (1968). Education in Papua and New Guinea: The twenty post-war years. Australian Journal of Education, 12(1), 8–9. Meere, P. (1968). The development and present state of mission education in Papua and New Guinea. The Australian Journal of Education, 12(1), 46–57. Miztigi, F. A. (~1981). Masta. In Expressive Arts Department (Ed.), Images of Papua New Guinea (p. 22). Aiyura, PNG: Aiyura National High School. Murray, J. H. P. (1929). Native administration in Papua (Appendix B, pp. 46–50). In B. Jinks, P. Biskup, & H. Nelson (Eds.), Readings in New Guinea history (pp. 129–132). Sydney, Australia: Angus and Robertson. Nelson, H. (1982). Taim bilong masta. Sydney, Australia: Australian Broadcasting Commission. Owens, K., Clarkson, P., Owens, C., & Muke, C. (2019). Change and continuity in mathematics education in Papua New Guinea. In J. Mack & B. Vogeli (Eds.), Mathematics and its teaching in the Asian/Pacific region (pp. 69–112). Hackensack, NJ: World Scientific Press. Owens, K., Lean, G. A., with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Paraide, P. (2010). Integrating Indigenous and Western mathematical knowledge in PNG early schooling. Doctoral thesis, Deakin University, Geelong, Australia. Paraide, P. (2018). Chapter 11: Indigenous and western knowledge. In K. Owens & G. A. Lean with P. Paraide & C. Muke (Ed.), History of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Price, A. G. (1965). The challenge of New Guinea: Australian aid to Papuan progress. Sydney, Australia: Angus and Robertson. Radi, H. (1971). New Guinea under Mandate 1921–41. In W. J. Hudson (Ed.), Australia and Papua New Guinea (pp. 74–137). Sydney, Australia: Sydney University Press. Ralph, R. C. (1968). Education in Papua-New Guinea: Integration—Whither? Australian Journal of Education, 12(1), 30–45. Ralph, R. C. (1978). Education in Papua and New Guinea to 1950. Unpublished book. Rowley, C. D. (1971). The occupation of German New Guinea 1914-21. In W. J. Hudson (Ed.), Australia and Papua New Guinea (pp. 57–73). Sydney, Australia: Sydney University Press. Ryan, P. (Ed.) (1972). Encyclopedia of Papua and New Guinea. Melbourne, Australia: Melbourne University Press with University of Papua and New Guinea. Schardt, R., & Schardt, E. (2009). Mission in motion. Iowa, USA: Authors. Smith, G. (1975). Education in Papua New Guinea. Melbourne, Australia: Melbourne University Press.

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Smith, P. (1985). Colonial policy, education and social stratification 1945–1975. In M. Bray & P. Smith (Eds.), Education and social stratification in Papua New Guinea (pp. 49– 66). Melbourne, Australia: Longman Cheshire. Smith, P. (1987). Education and colonial control in Papua New Guinea: A documentary history. Melbourne, Australia: Longman Cheshire. Somare, M. (1975). Sana: An autobiography of Michael Somare. Port Moresby, Papua New Guinea: Niugini Press. Territory of Papua. (1919). Standards. Government Gazette, XIV(12). The Bishop of New Guinea's Education Policy. (1945). Anglican archives, University of Papua New Guinea, Port Moresby. Threlfall, N. (2012). Mangroves, coconuts and frangipani: The story of Rabaul. Printed by Gosford City Council, NSW, Australia: The Rabaul Historical Society and Neville Threlfall. Wagner, H., & Reiner, H. (Eds.). (1986). The Lutheran Church in Papua New Guinea: The first hundred years 1886–1986. Adelaide, Australia: Lutheran Publishing House. Wedgwood, C. H. (1944a). Report of a patrol in the sub-district of Rigo for the purpose of enquiry into native education (Copy in New Guinea Collection, University of Papua New Guinea). Port Moresby, PNG: Author. Wedgwood, C. H. (1944b). Some problems of native education in the Mandated Territory of New Guinea and Papua (Copy in New Guinea Collection, University of Papua New Guinea). Port Moresby, PNG: Author. West, F. (Ed.) (1970). Selected letters of Hubert Murray. Melbourne, Australia: Oxford University Press. Wurm, S., & Hattori, S. (Eds.). (1981). Language atlas of the Pacific area. Part 1: New Guinea area, Oceania, Australia. Canberra, Australia: Australian Academy of the Humanities.

Chapter 5 Before and After Independence: Community Schools, Secondary Schools and Tertiary Education, and Making Curricula Our Way

Abstract: Before Independence it was recognized that Papua New Guinea needed an educated elite and hence funds were gradually made available for the establishment of senior high schools, an Administrative College, Technical Colleges and Universities. At Independence, in 1975, decisions on education and schooling were influenced by the Report on the new nation’s Goals and Principles of Education spearheaded by Sir Alkan Tololo. Community schools were expected to provide education for village children that would prepare them for both life in the village and for employment in the paid workforce. Vocational education was also recognized as important. This chapter recognizes people who influenced the new nation and the role that mathematics education and pre-Independence policies played at the start of the new nation. This era also saw one significant research project headed by Zoltan Dienes being undertaken in an effort to provide mathematics education appropriate for the multilingual cultures of this new nation. Other studies that were carried out included comparisons of achievement by PNG students, compared to normative data on a range of psychological tests including Piaget’s.

Key Words: Tololo Report · education for village children in PNG · preparing leaders in education in PNG · Dienes No teachers, no schools! The training of teachers is an immediate task and one of magnitude ... It will be necessary to train a large staff of native teachers as soon as possible, not only because the expense of maintaining a large staff of Australian teachers is too great, but also because the natives must be able to man their own educational services if selfgovernment is to be achieved. Murray, 1949, p. 31

An Overview

In 1960, the number of children attending administration schools was 22 143 and mission schools was 173 733 (Price, 1965). Quality forms of education, with more teachers, were needed for the now accepted notion that Independence was coming. In 1960, a 6-month preservice primary teaching ‘E’ course was introduced for Europeans with an entry criterion of three years of secondary education (Hasluck, 1976). The pre-service course for secondary teaching required five years of secondary education. From then on, this was the way most Europeans entered the PNG teaching service. One story of a teacher and then lecturer across this period of rapid development in education captures much of what needs to be recorded. The following are quotations

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and summaries from Roy Kirkby (2019) available in a four-part series provided online during 2019 and 2020.

Cameo from Roy Kirkby The 6-month course itself was most interesting. It combined the basic requirements of the NSW Teachers’ Certificate with the practicalities of teaching in rural areas of the country and with an emphasis on teaching English as a second language (TESL). Even more interesting was the mix of lecturers, some with little or no experience of the country and others – the Territorians – who had lived and worked with the people in a range of settings for many years. What I remembered and found useful I got from the Territorians, in particular from Bert Jones the Principal of the College.

Kirkby taught first in Kompian, Enga Province, with a Tolai teacher who had moved away from his coastal people and into the strange payback society of the Highlands. Kirkby was finding his way in regard to his teaching—whether it should be based on basic ideas about education, to wider societal issues of politics including gender equality: I believed the most worthwhile content should be the 3Rs in English and learning about the capitalist way of economic life. I had fuzzy ideas about democracy, selfreliance and personal ambition. I believed we should strive towards valuing the individual as much if not more than the group. I was strong in my belief and about valuing both genders equally, about being not physically aggressive and valuing reason over physical might. When sent to the Jimi Valley the following year, Kirkby commented: Teaching was improved with the introduction of the new Minenda Series for teaching English and girls came more to the fore as prejudice against them was squashed. I managed to get a few more girls into the school with the help of Corporal Poti who suddenly seemed to realize that his two adolescent daughters in my class were quite bright and, with an education, could have a broader range of options for the future. He was then moved on to Keltiga in the Wahgi Valley, neighboring Danny Leahy and having Pena as an influential clansman. He mentioned a new approach to mathematics promoted by Zoltan Dienes (see later): The real breakthrough for Keltiga came when we became a pilot school for a New Mathematics Project. This was an international project to change the format and content focus of teaching mathematics, sold to the Education Department as a quicker means of developing mathematical literacy in primary school and beyond. Working enthusiastically with me was Kolda, the first local graduate teacher. Kolda took on the new class of 48 children, with an expectation that we would lose half. Then as now many students dropped out of primary school before completing Year 6. Kolda demanded perfect behaviour from the children always and they listened and learnt and did whatever was asked. In a relaxed not fearful way, we soon had a situation where visitors from around the country and from overseas would come and marvel at what these children were doing, all 48 of them since we didn’t lose any! Kirkby’s pictures show that he was involved in the hands-on logic approach, often teaching outdoors as Dienes had encouraged. However, Hammersley, the man in charge around Mt Hagen, wanted Kirkby to bring this version of practical new mathematics to a bigger center. So, Kirkby and his new wife ran numerous workshops teaching teachers how to use the new logic of sets and attributes, real measurement, and showing how the decimal number system could be more easily

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understood using Dienes’ multi-base arithmetic blocks (MAB). In one reality check the principal of Mt Hagen wanted to be sure their pupils were doing as well as the others on the standard school curriculum test, and they actually did really well. More is said about Dienes later in this chapter. So in Mt Hagen the Kirkby couple wrote support materials based on Dienes’ ideas for teachers, because not all teachers could work with the concepts involved. Then from 1967 to 1970, Kirkby was sent to lecture in mathematics education at the Goroka Teachers College (later, the University of Goroka (UOG)), which started as a college for primary preservice courses but was then transitioning to teaching only secondary preservice courses (see Chapter 7). His wife took up a post at the North Goroka Demonstration school next door. Like other Territorian teachers, he developed his own practical curricula in the College to meet the needs of his students. However, the Kirkbys’ abilities to be involved in other activities with students proved invaluable, including teaching both male and female students Korfball, a ball game popular in Europe (for more details, see Kirkby’s online posts). After a year’s further study at the University of Queensland in Australia, he returned in 1972 to take up a position of Senior Lecturer at Madang Teachers College and stayed there until 1975. A few of his former school students from the years he taught in primary schools were training at the college, or were teachers in nearby local schools, which was fun for the couple. Independence was now rapidly approaching, and more national lecturers were taking up positions at the college. He continued his self-education through all this time including his political education. He noted, just as his understandings had been grown; “The many expatriate colleagues and students I knew would continue to evolve an education appropriate for an independent nation with such rich cultures as their foundation” (Kirkby, 2019). This attitude grew among the expatriate Australian teachers before and after Independence (see Eastburn’s comments in Chapter 6). Kirkby left PNG towards the end of 1975 to undertake more study in England. He has returned periodically to PNG over the ensuing years.

Growth in Primary School Enrolments

As was noted in Chapter 4, Paul Hasluck wanted a system which would provide a sound elementary education in which people were literate in some language and, for communication purposes, English was to be prominent. He also advocated universal primary education to support his policy of gradual uniform development for all (Hasluck, 1976; Smith, 1987). He suggested that reading, writing and arithmetic should make up almost the whole of the curriculum in primary education. It was not until 1959 that an actual plan for the achievement of universal primary education was established after Groves had retired in 1958 and Roscoe took over the position of Director of Education with Hasluck’s approval (Hasluck, 1976). He developed a plan by which all children in controlled areas were taught to read and write in English, but at the same time the various degrees of contact that Indigenous populations had had with the colonists in the different regions were taken into account. The T schools were for those students whose mother tongue was not English, and ‘A’ schools for those who had English as a home language. Thus Indigenous students technically were not excluded from A schools.1 (Roscoe, 1958) In 1959 Roscoe recommended “the encouragement of mission schools by raising the rates of subsidy and providing capital grants for buildings” (Hasluck, 1976, p. 225). Hasluck was pleased that he was receiving “the first substantial, constructive and practical proposal I had received in eight years in response to my repeated requests for giving high priority to education. In practice, this was not often the case but when International Schools were established, they were attended by more and more PNG children. 1

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It marks the beginning of a new era in providing schools in the Territory” (p. 225). Indeed rapid growth in primary school enrolments in Government schools did occur in the early 1960s: • In 1960 there were 18 000 Papua New Guineans in Government schools • In 1964 there were 50 000 in Government schools, and • In 1966 there were 65 000. Nevertheless, “the missions were still providing more school places than the Government. High population growth still meant the target of universal education was a long way off” (Griffin, Nelson & Firth, 1979, p. 128). The country’s rapid population growth has continued. At Independence there was a Department on Population Growth which included recognition of the value of family planning, but it was disbanded soon after Independence with concerning consequences in population growth and numbers of children for education2. This continued population growth has meant trying to provide education for all children has continued to be a challenge. Members of the mission from the UN Trusteeship Council headed by Foot in June 1961 “extracted from the Australian representatives the admission that the Commonwealth had done little or nothing to create an ‘educated elite’ ” (Price, 1965, p. 115). The mission was not impressed. They also recommended accelerated efforts should be made in education and industry, as well as in the political arena. However, Margaret Mead (November, 1964) noted that “in other primitive countries the idea has been to educate an elite. This leaves the villages further and further behind. That was the general pattern in Africa, but in New Guinea the idea has been to give everyone some education” (cited in Price, 1965, p. 115). Hence the debate between education for all and that of an elite continued, as it would right up to Independence and beyond. A little time later, Emerson (1964), the education representative on the 1963 World Bank team visiting the Territory made some interesting comments. He recommended a school for domestic servants so there could be an upgrading of the level of service in hotels and restaurants, and this might also lead to an upgrade of general living standards. He hoped such skilled work would attract salaries above the minimum wage. His second recommendation suggested that the T School curriculum was too Western oriented. All that was good in the local community was being denigrated by those implementing the curriculum (at this stage a number of Australian teachers as well as PNG teachers). This meant that language and customs would all be lost by that generation. Emerson suggested, the T school curriculum would give students unrealistic views of what they thought was possible but could not be realized for a developing country. Hence as well as the debate between an education for all or an elite, there was pressure to educate students who were work ready to enter the cash economy, thought of in Western terms. Nevertheless, the system needed to preserve the “good” in the Indigenous cultures, whatever good meant in these terms. With the continuing expansion of primary schools and attendances, there was mounting pressure for post-primary education. Hence it was not surprising that the Education Report for the Australian Government for 1961 recommended an expansion of secondary education facilities (Department of Territories, 1961). By the mid-1960s there were 5359 pupils in high schools (Years 7 through 10) in PNG including some students who had been sent to Australia for their secondary education. Of these, 500 were Papua New Guinea-born Australians. Currie (1964) suggested that by 1974 there should be 30 000 pupils in Government secondary schools with a smaller number in mission schools. That would have demanded a huge increase in numbers, and in the end it proved to be unrealistic.

In 2019, the population of the country had doubled since Independence and this occurred in rural areas dependent on the land as well as in cities but with virtually no migration unlike other countries like Australia. 2

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Cameo from Patricia Paraide

Patricia Paraide, one of our authors, provides a narrative for her place in East New Britain. This case study describes some aspects of her people’s journey through the changes in education before and during the colonial era. This personal reflection adds another dimension to the survey of the literature that forms the bulk of this chapter. Like other PNG and Indigenous peoples, my people had their own system of education before the colonizers arrived. As Tololo (1976, p. 211) has discussed, like most young Indigenous people learning life-skills, our young people, too, learned the life-skills through observation, imitation, and participation. The skills that they learned growing up were useful and appropriate for their adult lives. Smith (1975) made similar observations regarding the teaching strategies used by most of PNG’s Indigenous people to educate their young people. Tololo and Smith acknowledge that everyone in the community contributed to the education of the young. Peers working together and working alongside experienced adults on specific tasks were encouraged through Indigenous education. However, this changed with the arrival of the missionaries. English-speaking expatriates dominated all levels of education before Independence, and they had a lot of influence on the choice of the language of instruction in formal education and the content of formal curricula (Barrington-Thomas, 1976; Smith, 1975). An Australian educationist who visited Rabaul in 1929 was surprised at the hostility of the non-official white residents of the Territory of New Guinea to any system of education. The visiting educationist, Weeden (Advisory Committee on Education in Papua New Guinea, 1969), put this down to: The prevailing belief that any training makes the natives more cunning, generates and develops evil qualities, makes him disinclined to work, and renders him a less pliant instrument in the hands of his master. (Weeden Report, pp. 3–4 cited in Smith, 1975, p. 8) Such views and attitudes created hierarchies, as well as race and class divisions. Also, such views and attitudes led to the suppression of the Indigenous teachers’ voices in educational matters during that era. Childs and Williams (1997) had this to say about the established classifications of different people or races: They have the potential for creating different oppressive hierarchies organized around gender, class, rationality, or any number of other categories, at whose normative center remains the figure of the white, Western, middle class, heterosexual male. (p. 100) Past education materials sighted show that curriculum documents and teachers’ guides, and education policies and guidelines were written in complex English and from colonialist perspectives. For example, there are stark differences between the written English reports on education issues prepared by McKinnon and Tololo (who was also from East New Britain). Their viewpoints on the focus of education for Papua New Guinean children were also different. The education reports and opinions written by expatriates could be barely understood by those not competent in the English language, but were understood by the first-language English speakers. Hence, the colonialists and their texts dominated decision-making on education. In this particular case, the English language was used. Pennycook (1998) described English as the language of power and dominance. Childs and Williams (1997) also offered a similar interpretation drawing on Foucault’s discussions about power and dominance: Power/knowledge involves a more intimate linkage: one does not occur without the other; knowledge gives rise to power, but it is also produced by the operation of power. (p. 98)

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The colonial authorities’ emphasis on English was a powerful way of extending their power and superiority over PNG’s Indigenous peoples. It suited their purposes to suppress the people by implementing the English language policy in schools. As Smith (1975) and BarringtonThomas (1976) suggested, they dismissed the value of all the Indigenous languages and systems of knowledge. This attitude had some bearing on the colonial administration’s rejection of the Tololo Report. This report had a five-year plan and it emphasized vernacular instruction at the lower levels of school. When reflecting on formal education, some (including the authors) are convinced that the integration of Indigenous and Western knowledge, and the use of a language that the students know best in the initial stages of formal education would have empowered Indigenous teachers during the colonial era to teach school subjects more effectively in their own language. The students would have learned better in this language because they were competent in it. However, communication in Indigenous languages was forbidden in schools. Given that missionaries became quite fluent in Tinatatuna, the numbers used in activities were recorded. There was some recognition by the missionaries of the complexity of the numbers used in everyday use, but it took some time before this was properly recorded and valued. The work of Glen Lean (who also spoke Tinatatuna which he called Tolai) and Patricia Paraide’s own work is very significant in this respect (see Paraide, 2010, 2018 and Chapter 3 in this book). There were many more cultural aspects that could have been embedded in the mathematics of the classroom, if the missionaries had had a mind to do this. Teachers In 1960 Minister Hasluck noted that although the education system in New Guinea had over 400 European teachers and some 5400 “native” teachers working in 4 100 schools attended by 196 000 pupils, an important limitation on educational development was finding a sufficient number of trained teachers. In 1960 Roscoe, in line with this thinking recommended an increase in teacher recruitment but the Administrator Cleland wanted to reduce the number because of an inability to process them. But Hasluck reprimanded Cleland. Hasluck was determined to improve education. For him, it was the key to the development of the public service within the Territory in preparation for self-government. In 1963 he noted “that of 540 000 children, only 155 000 were receiving an education of the standard required by the administration” (cited in Price, 1965, p. 125). He also stated that it was more important in the Government’s view for the Territory to be training Papuans and New Guineans using a more appropriate curriculum, rather than educating them with an all-round education. It is strange that at that stage a distinction is still being made between Papuans and New Guineans. One would have thought after 15 years of the two Territories being administered as one entity, and with Independence being thought about seriously for the one entity, such a distinction would have disappeared. Furthermore, it both divided and diminished cultures by giving them categories that did not necessarily reflect the diversities within them (Nakata, 2012). Hasluck’s aim for education seems to have shifted, at least partially, from an education for all, to one that was aimed at producing elite graduates who could take their place in the public service. The target enrolment over the next five years was 350 000 children. The Commission into Higher Education chaired by Currie (1964) concluded that even if an effort was made to have 50% of the 7–14 years age-group or 325 000 children at secondary school by 1975, it would mean doubling the existing effort and enrolling 7500 teachers by 1973 (Currie, 1964). This would have required recruiting teachers at about 25% of the pupils leaving the secondary schools (Price, 1965), which was a somewhat optimistic aim, to say the least.

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By the mid-1960s, in response to the ever-increasing need for teachers, the Education Department was recruiting pupils who gained intermediate standard (that is Year 9). They were then given a crowded course of 12 months before being sent out to teach in primary schools. In September 1960, in the Australian House of Representatives, it was suggested that it was time to consider the establishment of a university in PNG within the next five years (Smith, 1987). The Currie Report published in 1964 as a response to this initiative, specifically noted the advantage of higher education for economic—rather than political—reasons, which was a position held by the World Bank in the mid-1960s (Smith, 1987). Hence planning was undertaken and a University, the University of Papua New Guinea – UPNG, was established in 1967. However, it was inadequately funded and was not in accord with the Currie proposal in a number of respects (Spate, 1966). One of those was the matter of teacher education. As will be noted in Chapter 7, a course for preparing secondary teachers was to have been a crucial component of this new university. However, that did not eventuate for some years after its establishment. The Department in the meantime had to turn the Goroka Teachers College from an institution preparing primary teachers to one preparing secondary teachers, which in turn led to difficulties for many years. By May 1967 there were 143 university students in UPNG’s first year of operation.

Cameo from Daniel and Carrie Luke and Others in the Yarning Circle3 in Sydney

This cameo mainly reflects on the experiences of two colleagues concerning their experiences of being students during the early 1960s. Daniel Luke attended Catholic schools in Milne Bay in the 1960s. His teachers each had the one-year teacher training certificate, that had an entry level of Year 9. Dan reached Year 10, and then went on to Goroka Teachers College to gain his Diploma before teaching in a number of secondary schools. He was then recruited as a staff member of the Mathematics Education Centre, PNG University of Technology (see Chapter 1), helping with research projects. His own schooling was initially in his own Dobu language and included its counting system, because his teacher came from his area. In higher grades he had a teacher from the other side of the island with another language and so he was using both Tok Ples languages, as well as gradually learning English. He did his mathematical thinking at that time in both Tok Ples languages. However, by Grade 5, not finding much Tok Ples language and meaning for school mathematics, such as for multiplication, he thought and worked mathematically in English. He remembered little about the learning of shapes etc. Although there are shape names in his home language, these words mostly were for things like trees and other naturally occurring objects, not for the mathematical shapes he encountered in school. He thought that in the 1960s and onwards, mathematics was said to be as important as English. Dan said, “My elder brother sat me down and told me if you do well in maths and English you will progress.” The conversation in the group at one point turned to punishment. All present related to the requirement to speak in English or be punished. Many of this group like Carrie Luke in the 1960s and Carol Abiri in the 1970s were taught in English by nuns. Younger highlanders who were part of this conversation, were taught only by PNG teachers who spoke English well (some from the same area who taught bilingually and some from the coast who only used English). It was also noted that in the towns and increasingly in rural areas throughout this period that the teachers gradually began to use more Tok Pisin than English, and at times Tok Ples in the villages. Partly the group thought that this change in the urban areas was simply because there were so many The ‘Yarning Circle’ was a meeting with well-educated Papua New Guineans of a range of ages and home places held in Sydney, Australia, November, 2019, which also included three of the authors of this book (K. Owens, Muke and Clarkson). The three-hour long conversation was audio recorded. 3

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cross-cultural marriages between people settling in Goroka, Madang, Port Moresby, and so on. Hence the notion of a common Tok Ples in these urban areas was just not the case anymore, hence Tok Pisin was used. Nevertheless, there was a common agreement within the group that their languages (Tok Ples) needed to be spoken either in school or home, for the languages provided the bases for their cultures.

Mid-1960s to Independence

With different salaries for mission and administration teachers, the whole system was in crisis by the late 1960s (Smith, 1975). The Director of Education cooperated with the mission authorities to set up the National Teaching Service (Education Ordinance) in 1970. The administration agreed to pay the same salary to all teachers yet the missions were able to organize their own religious instruction in schools and advise on other policy matters. The leadership of the education system was taken over by Papua New Guineans and a genuine national education system evolved. In 1976 there were 238 300 pupils in the National Education System. In 1968 it was estimated that about 50% of school-age children attended primary school (McKinnon, 1968). Although this percentage increased, it only remained around 55% to 60% in the years prior to 1976 (Central Planning Office, 1976). The overall increase in children in the population because of the population growth and the difficulties of establishing schools in sparsely populated areas were two factors which depressed the percentage. As well there were significant differences between the provinces so far as educational opportunity was concerned. The percentage of the school-age population attending primary school varied from 34% in Enga to nearly 100% in East New Britain (Central Planning Office, 1976). Another inequality was gender based. There was a much higher proportion of boys than girls enrolled in primary schools with the proportion of girls enrolled in grade 1 remaining below 40% throughout the period (National Department of Education Papua New Guinea (NDOE), 1976). There was also the difficulty of maintaining regular attendance by students, and this was especially the case with girls (Ryan, 1972). Hence it was no surprise that the Department of Education remained under considerable pressure (a) to reduce the inequalities in educational opportunities between the provinces, (b) to increase the proportion of girls enrolling at both the primary and secondary levels, and (c) to expand the system at both levels so that more school-age children would attend school. Although the Department was by now allocated the largest share of the national budget, there were many constraints on it implementing plans to reduce the inequalities, especially financial problems, lack of resources, and a lack of trained teachers. By 1967 there were 620 000 children of school age but only about 224 000 attending school with 143 750 at mission schools and 80 142 at Government schools (Wilkes, 1968). To gain entry to a secondary school, students still had to pass an external examination at the end of primary school, and in fact only 50% of children leaving primary school found places in secondary schools. In 1967 there were more than 11 000 secondary students. Until Goroka Teachers College started producing secondary teachers in the early 1970s (see Chapter 7) all secondary teachers came from overseas. Secondary schooling continued for four years ending with external examinations leading to the School Certificate. All students studied mathematics in each year. Technical secondary education had more than 1 200 apprentices in training across 26 trades, including at the Lae Technical College. The need for qualified trades people was obvious with only 339 qualified tradesmen in the workforce. In the mid-1960s, Mr Barnes, the Australian Government Minister responsible for the Territory of PNG, reflected that back in 1963 “Sir R. Menzies (the then Prime Minister) and Sir G. Barwick ( then Minister for Foreign Affairs) accepted that complete independence was the desirable goal” although this would be through the basic policy of self-determination (Wilkes,

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1968, p. 140). Minister Barnes (on 18th April, 1966) said the Government held the view there should be no rush towards independence or deciding the political future until the Territory had developed a lot further economically and until the people were better able, through the spread of education, to understand the issues which were involved. In his opinion the answers to these problems lay in the adjustment of education, apprenticeships and training policies generally. The gradual devolvement of responsibilities by the Australian Government continued gradually with more Papua New Guineans assuming various roles within the public service, although at that stage these were low and middle level positions. Later (in July, 1970) the Australian Prime Minister Gorton said that the ministerial members of the PNG Administrator’s Executive Council would be given full authority in areas such as primary and secondary education and land use.

Changes in the Education System

Even though there had been various curriculum guidelines issued across the years, they had never been stipulated as mandatory. Hence, as was noted earlier, some schools and individual teachers followed or developed their own courses of study: for example, students in the Port Moresby school were using the Queensland curriculum (Nina, personal communication, 2019). A national system was needed for Papua New Guinea with central planning and a controlling authority, although it was recognized that some local autonomy would be needed, given the wide differences which occurred across Papua New Guinea’s territory and cultures. By the mid-1960s, following the common model in Australia, primary T schools had a seven-year program that included a Preparatory year followed by 6 years of schooling. There were no kindergartens in PNG, unlike the situation in Australia where children often attended kindergarten before commencing school. The length of primary schooling was reduced to six years in 1969 by removing the Preparatory (Prep) year mainly for economic reasons. Basic skills in English and mathematics occupied a much larger proportion of the curriculum than in Australia and hence were taught for a much greater proportion of time than in most Australian schools. Most pupils entered school with no knowledge of English, and most teachers assumed they came with no or few mathematical concepts. It was acknowledged that some children could count but using a different base system (Department of Adult Education University of Sydney, 1969). However, Owens, Lean, with Paraide, and Muke (2018) particularly, and a number of others referenced in earlier chapters have shown the situation was more complex than that. As had been the case for many years, few students gained entry to secondary schools. The four-year secondary education was punctuated by assessments and certificates; an Intermediate Certificate at the end of Form 3 and the School Certificate after Form 4. Mathematics and English were studied in all year levels. All schools had boarding facilities. The re-organization of secondary schools from 1969 was far reaching and involved the introduction of a six-year course, divided into two-year units. The aim was still for 50% of primary school leavers to enter secondary school after passing the externally set Grade 6 examinations. Then 50% of those who completed the first two years of secondary school, and passed the appropriate externally set examinations, would proceed to Forms 3 and 4. A small proportion of these, possibly from 3% to 10%, would proceed to Forms 5 and 6, again after completing externally-set examinations. It was assumed that by the time students reached the final two years there would be nearly a 100% success rate. Sogeri Senior High School was established in 1968, the first of four planned Senior High Schools which would teach two final years of secondary schooling. In 1969 Sogeri commenced with over 80 students with teachers using continuous assessment rather than externally-set examinations (PNG Education Gazette, 1969). Nine institutions teaching diploma, and in some cases degree courses, were considered as tertiary, normal entry being after graduating from the new Senior High Schools (see Chapter 7). Some of these institutions established a Preliminary Year which was designed to bring students

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up to Form 6 (Year 12) level. In order to establish these institutions and the secondary schools, money was being taken away from primary education. This fed into the competing political aim of training an elite, a reversal of the previous dominant policy of building an education system that would cater for all. Growth in School Numbers and Gender Issues During the early 1970s, in the years before Independence, there was a steady increase in the number of schools and enrolments at all levels of the system (see Table 5.1). Enrolments were helped by the introduction of National High Schools teaching Years 11 and 12, first at Sogeri (Central Province) in 1969, and then in 1972 at Kerevat (East New Britain), and Aiyura (Eastern Highlands). Later a fourth senior high school, Passam was established in the East Sepik Province. These Senior High Schools serviced the whole country as boarding schools. Table 5.1 also shows the breakdown between schools belonging to the National Education System (NES) and those who were non-NES schools. NES schools included Administration Schools and all non-Administration schools (usually mainstream church schools) agreeing to the Education Agency Schools requirements. This provided them with teachers’ salaries at least. Within the NES there were still nearly two times as many church-run schools as administration schools. Students up till then had begun school in the Preparatory Year. However, as noted above, the Prep year was phased out and through the early 1970s many students started school in Year 1, aged at least 7. There were some quite old students at each of the various year levels. Table 5.1 also shows figures for Skulanka schools. These were begun in 1973 to encourage more adults into school, but these schools only lasted for a few years. In total there were 28 PNG staff with three-quarters of them being male. These schools ran a two-year secondary course designed for children who had completed a primary education but had not been selected into high school. The program concentrated on practical and outdoor education attuned to community needs (Miller, 1974, p. 1082). Table 5.2 shows the gender breakdown across the various types of schools. In the primary schools there were still far more boys than girls enrolled. Overall, two-thirds of school children Table 5.1 Number of Schools, Number of Enrolments Primary Secondary Secondary Primary PNG Other PNG Other Technical Vocational Skulanka Curriculum1 Curriculum2 Curriculum Curriculum 1970 1557, 2064053 63,8853 59,17785 3,928 9,1575 61,3140 Year

1971 1548, 213320

65,11334

62,19571

3,1103

10,1575

65,3130

1972 1592, 215704

66,11995

66,22861

5,1474

7,2237

73,3778

1973 1587, 218167

65,11943

77,23837

5,2422

9,2724

81,4165

July, 1687, 229527 75, 76533 71,27046 5,1674 9,3015 93,3903 16,728 1974 1 PNG Curriculum designates National Education System (NES) schools, etc. 2 Other Curriculum designates non-NES schools, etc. 3 In each cell of this table, the number of schools is given, a comma then the number of enrolments in each cell. So in this cell there are 1557 schools with 206405 total enrolments 4 Table 5.1 was developed from the Department of Adult Education of the University of Sydney’s Current Affairs Bulletins and the Education Gazettes of the period.

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Table 5.2 Gender of Students in National Education System Schools (NES) (Male, Female)

1970

Total Primary School Male, Female 130685, 75720

1971 26143, 15672 14383, 7417

135899, 77421 5204, 2376 1630, 465

Year

Standard 1 Standard 6 Male, Female Male, Female

1974 27712, 17433 19894, 10237

Form 1 Male, Female

Form 4 Male, Female

Total Secondary Male, Female 13755, 5836

6407, 2929 2693, 1181 18445, 8601

Technical

Year

Skulanka Skulanka NHS Form 6 Form 3 Vocational Secretarial Form 1 Form 2 Male, Female Male, Female Male, Female Male, Female Male, Female

1974 228, 85 1301, 11 500 2884, 1019 364, 86 219, 59 Notes. Total in 1970 includes Prep to Standard 6, but by 1971 there were very few students in Prep during the phasing out and by 1974 there was no Prep standard in NES primary schools. The break down by gender was only available in 1974 for some types of schools. Table 5.2 was developed from the data reported in the Department of Adult Education University of Sydney’s Current Affairs Bulletins and the Education Gazettes of this period. Table 5.3 Teaching Services: PNG and Overseas Staff by Male, Female Year 1971 1972

Primary PNG 4595, 1213 5001, 1382

Primary Overseas 342, 588 350, 648

Secondary Secondary PNG Overseas 105, 34 414, 305 142, 37 511, 389

Technical PNG 33, 19 73, 24

Technical Overseas 184, 45 232, 60

1974 5617, 1488 113, 327 241,111 491, 383 122, 38 250, 70 Note. Statistics collected in 1973 were not compatible with other years so are not shown. Table 5.3 was developed from data reported in the Department of Adult Education at the University of Sydney, Current Affairs Bulletins and the Education Gazettes of this period. were male in the secondary, the technical and the vocational schools. In 1972 Kerevat National High School in East New Britain had 195 male and 52 female students, which was typical for all NHSs. Hence, the gender imbalance for students ran throughout the system. There were some staff who were trying to address this issue. For example, in the early 1970s the headmaster at Henganofi refused to enroll any more boys (now numbering 40) until girls were enrolled in his school. Not surprisingly, there was a huge gender imbalance in the teaching service as well, but not always males outnumbering females (see Table 5.3). PNG male teachers far exceeded in their number the PNG female teachers in all types of schools. In the early 1970s the presence of overseas teachers was still prominent. As for the gender balance for overseas teachers, there were more primary female teachers than males, but not in secondary or technical schools. There was a similar gender profile in Australian schools at the time. It was not just within the school system that gender imbalance was obvious. It was prevalent in the administration of the education system as well. It was not until the early 1970s that the first female was appointed to the National Education Board—Anna Natura, a Catholic representative from Yule Island, and a second woman, Sagilem Kadeu, was appointed to the Teacher Education Board.

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The Developing Mathematics Curriculum

Price (1965) wrote that “after making allowances for lack of home assistance and the language difficulties which pupils suffer, their mentality ranks quite high, except, perhaps in their grasp of numbers” (p. 127). One can hear the colonial thinking still present in such a statement, with expectations that home assistance would be given, even though most parents had not attended school, language difficulties, referring to English although most students knew their own languages perfectly well, and the surprise that “mentality” (cognition?) was “quite high.” Spate (1966, p. 126) noted “the very limited development of counting in some groups.” He recorded the problems with elementary mathematics which was, in his view, essential in basic education. The lack of higher numerals for some language groups and the difficulty of teaching mathematics in a foreign language were viewed as reasons for this. Changes in curriculum planning were occurring at a rapid rate in the mid-1960s, especially in English, and that continued right through to Independence. Inadequacy of English comprehension and expression was considered the biggest single factor limiting success in any subject and at any level including for mathematics. Changes were also occurring in the mathematics curricula. Ken McKinnon (1968), then Director of Education in the Territory, in discussing curriculum development noted the importance of seeking modern ideas, techniques and using curriculum materials. This was, in his view, especially important in the area of the mathematics curriculum. This was a different attitude compared to those held by former Directors and it led to changes. In 1967, a new mathematics syllabus replaced the former syllabus in Preparatory and Standard 1 in primary school. This was based on an experimental research project involving the development and assessment of children’s concepts of space, time, quantity and shape, with tests based on those outlined in Piaget’s child’s conception of number and of geometry (PNG Education Gazette, 1967a). It was gradually extended annually going up one level per year until it was being taught to all levels (Department of Adult Education University of Sydney, 1969). There was also an emphasis on Modern Mathematics involving sets and set notation at the upper primary level with the Department in July 1966 producing a document that provided a glossary of terms explaining the extensive list of words related to mathematics that used sets, and a few less common words related to geometry and statistics that may have been new to some teachers of mathematics. At the same time the Administrator despaired at the low mathematical understandings and having heard of Zoltan Dienes’s4 new innovative ways of teaching mathematics, he invited Dienes to assist in the primary schools with mathematics. Multibase arithmetic blocks, and other apparatus that Dienes had developed over a number of years were introduced in an attempt to help students to discover the logic of mathematical relationships. Some comments on Dienes’ work were made earlier in this chapter in the Kirkby Cameo. When Dienes was lecturing at the University of Adelaide in the early 1970s he introduced his ideas for teaching mathematics into schools in Australia. After he went to PNG he soon introduced his materials and group games there. They were disconcertingly different from anything that was previously associated with mathematics in PNG schools. The Director of Education soon enabled Dienes to share his approaches in classrooms across the nation. On his second day in the country, Dienes was dumped by his driver and told that the school which he was supposed to attend was across a crocodile-infested river. Once on the other side there would be a trek of

Dienes had migrated to Adelaide, Australia, from England in 1961 to take up a position first in Education, and then as Professor of Psychology at the University of Adelaide. He continued there between 1961 and 1964 maintaining the innovative approaches to mathematics education that he had begun previously in England. See Chapter 10 for more on Dienes and also Clarkson (1979). 4

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about two kilometres. Dienes put his box of materials on his head and crossed the river, then walked onto the school. Dienes (1999) was, as his last book is titled, a Maverick. From his childhood, he moved with his family to many countries in Europe and so he acquired quite a few languages. He was aware that language could pose an issue for mathematical structures anywhere, but particularly in PNG. He recalled that one PNG language he came across had no way of expressing “either/ or.” For his logical approach, he needed this idea and so he used sign language: if the answer was possible, he hit his arm and if not, he nodded. When two children gave different responses, and both solutions were correct, this was to be indicated by hitting your arm and nodding. The students were surprised but they quickly learnt the idea and played well at the game. He went to another school, introduced the idea of the materials that were needed and indicated how the children were going to learn, and soon had parents and children working with his blocks. Later the administration organized prisoners to make kits of his equipment for use in PNG schools. Dienes’s enthusiastic teaching was infectious5. His ideas for physical embodiments of mathematical conceptual ideas did not just revolve around wooden blocks, but also involved groups of children in playing games. Having groups learn mathematics together might be common now, but in the 1960s the teaching strategy was quite at the cutting edge of how one could teach. For example, an “attribute train” required the students in turn to place the blocks (the blocks varied by the attributes of color, shape, size and thickness) in a line so just one of the attributes stayed the same. This game, like many others he devised, required a patterning structure to be solved which relied on logical reasoning about attributes. He found that by using physical materials with groups of children, quite young children could be quite logical and could learn to think abstractly. He returned to PNG a few times from Australia in an effort to ensure that teachers were able to understand how to use the multibase arithmetic blocks, the attribute blocks, measuring sticks, balances, and other physical embodiments in his kit. He also encouraged the use of natural materials when they could be used; for example large leaves with vertices held by children to show rotations and reflections, and a pole and stones to act as a sun dial. His key teaching strategy was to engage the children in playing mathematics-relevant games. In 1971 the back cover of the PNG Education Gazette 5(9) pictured children fitting into a closed space. This idea of taking up space is important in measurement, and something young children can learn quite easily. With the new syllabus that followed, mathematics was no longer seen as just arithmetic6. Because of the relatively short time of Dienes’s visits, he had to focus on larger centers such as Port Moresby, Mt Hagen, Lae and Goroka. Nevertheless, he managed to form a committee to continue the work with membership from all key constituents: university staff, state education administrators, head teachers, teachers college staff and members of various missions of very different denominations. Despite attempts to involve teachers college lecturers, Education Department officials, and teachers, the positions often changed, and overall there was insufficient inservicing. Smith (1965) noted some of the number structures that became clearer to students when teachers were using Dienes’ multi-base arithmetic approach, and this meant that “the capacity of children to learn is probably far greater than teachers deemed possible” (p. 107). It was expected that rote learning would be replaced by this better teaching approach so that students would develop a confident understanding of the structure and relations of numbers using the learning Clarkson attended a teachers workshop Dienes led in a Melbourne school in 1972. Dienes had the children sorting out logical issues and performing dance routines which mirrored various algebraic groups and rings in no time. On a Saturday morning (a non-school day) the children were so enthusiastic it was hard to believe they were doing mathematics. Clarkson continued then with even more drive to incorporate Dienes’ materials into his secondary school teaching, convincing his head of department to buy more of Dienes’ equipment. 6 See Chapter 10 for further commentary on Dienes’s contributions to PNG mathematics education. 5

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materials, which would also help them to understand unifying concepts linking geometry and algebra. However, all did not go as smoothly as hoped. Some teachers claimed “they do not understand the aims and content of the maths syllabus being taught”, and students were being taught how to pass examinations without gaining any real understanding of the subject (Department of Adult Education University of Sydney, 1969).There was still a continuing problem with some young Australian teachers not adapting the mathematics problems they used in teaching to the PNG context, which had been noted back in the early 1950s (see chapter 5). The other remarkable issue is that there seemed to be no consideration at the time that the innovations in mathematics teaching were all very Western based. Piagetian theories and Dienes’ materials, which drew on Piaget’s ideas, and the “new math” movement, which although in some ways was separate to Dienes’ work, was largely based on the assumption that children thought in a Western template. The notion that other cultures might see the world differently, and yet just as insightfully, was, apparently, not a consideration. The actual abstractness of Dienes’ ideas was not well understood by the teachers, and any remaining Australian teachers had probably not been trained to teach in this way. It was perhaps not realized that any one child might only be familiar with one mathematical system from their language group and so the idea of using the children’s home background was perhaps misplaced if the teacher used many unfamiliar bases which were outside the contexts of their cultural settings. Dienes stressed a discovery approach to learning through the manipulation of apparatus and material, and which sought to develop basic, intellectual skills. The exciting promise of the mathematics programme is the likelihood of general intellectual development relevant to all areas of learning. The mathematics programme has then been considered a core subject to subsequent developments in other areas. The original Dienes-influenced programme was written in 1968. It covered the first three years of school. Subsequent evaluation led to a decision to rewrite the programme before proceeding with the final three years of the programme. The rewrite is known as Mathematics for Primary Schools. The second year of this programme is now in some schools. ((Iso’amo, 1974, pp. 59–60) One evaluation was carried out by Southwell (1974), who noted the lack of teacher education across the country, the lack of cultural awareness of the syllabus (despite the presence of non-base 10 counting systems), and the heavy reliance on overseas thinking. Her perspective on Dienes’ material (incorporated into the Temlab kits which were supplied—see below) was that insufficient professional development was available especially for persons who were minimally trained for teaching let alone with activity-based logic games, and for a workforce that was rapidly changing. Furthermore, the Dienes-inspired program relied on cards in boxes, and these tended to become quickly disorganized with parts being lost. While the key ideas were incorporated into the upcoming syllabus, the associated academic challenge required far more than rote implementation.

Teacher Education

The education system as it existed in the late 1960s was diverse and complex with over 50 missions, mainly Christian, operating in PNG and most having some sort of school system. Clearly it was these missions which still provided the bulk of primary schooling throughout the Territory. The quality of education provided by the mission schools in particular was variable as were the resources available for each mission. This had been the case for many years as noted in Chapter 5, and nothing much was changing. There was only about a quarter the number of teach-

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ers required for the number of children to be taught (if all school age pupils, roughly 550 000, actually attended primary and secondary schools). Within this milieu it was decided to create a National Teaching Service which would help bring order and consistency throughout the whole system. In 1967 there were essentially three preservice primary school teaching certificates that students could obtain. There were two twoyear courses; one required a student to have passed the Form 3 examination (known as the “C course”), the other only required completion of Form 2 (known as the “B Course”). Both required passes in one of Mathematics or English. There was also a third one-year course available with a pass in both English and Arithmetic (known as the “A course”) (PNG Education Gazette, 1967b). The PNG teaching staff, most of whom had completed the one-year A Certificate, were inadequately prepared by the college courses for a teaching career. It was not surprising then that these teachers were often considered under-prepared, especially in English and mathematics. In the late 1960s there were over 2000 such teachers, hence this systemic problem remained for years. Sadly, it meant that thousands of primary-aged children studied under less than adequate teaching, and few of those students were likely to pass the Year 6 examination and hence proceed to secondary schooling. The growing Government concern with the quality of PNG teaching led to the introduction of a two-year teachers college course, post Form 4, in 1974. Overall, expatriate teachers still accounted for 58% of the teaching service with 1070 teachers (37%) in administration schools, mostly post-primary schools. Most of these had Teaching Certificates from Australia or other equivalent overseas qualifications (Department of Adult Education University of Sydney, 1969). Although the number of teachers colleges through the early 1970s remained more or less the same, there was an increase in the number of student teachers—although not at a rate that was needed to meet the growing school student population (see Table 5.4). The teaching staff in the colleges was dominated by overseas personnel and and there were many more males than females (see Chapter 7 and Appendix 2 for more details). By 1973, it was estimated that Dienes materials had been distributed to between 200 and 300 schools. However, there was some concern that many teachers showed quite a weak knowledge of number relationships embedded in the Western mathematics system, and were having difficulty understanding in any depth what the Dienes approach was all about. The teachers colleges responded to this development in various ways. We have already noted one type of response in the “Cameo from Roy Kirkby” section at the head of this chapter. Another response was when the Port Moresby Teachers College provided a seminar on the new approach to teaching mathematics. A third response was the support of college staff for the 1967 Temlab (the boxed kits which contained Dienes materials and accompanying cards to be used by students). Two expatriate Teachers College lecturers, M. Dunkin at Dauli Teachers College and I. Renner at Balob Table 5.4 Number of Teachers Colleges, Students, and Teaching Staff (Male, Female)

1970

Number of colleges, students 12, 1706

1971

12, 1799

46, 7

96, 70

1972

11, 1936

18, 11

104, 80

1973

10, 1932 12, 10

111, 67

Year

1974

PNG staff Male, female

Overseas staff Male, female

Note: Table 5.4 was developed from the Department of Adult Education University of Sydney’s Current Affairs Bulletins and the Education Gazettes of this period.

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Teachers College, became important in supporting writing teams by running workshops (PNG Education Gazette, 1972). At this time the Department sent some teachers, who had graduated with the Teachers College Form 3 Entry Course, to undertake more studies in South Australia, particularly to learn more about the new teaching methods for mathematics. These teachers also took courses in Education Psychology and Comparative Education, although one suspects these extra subjects were all based on Western ideas and took little account of approaches that might have been consistent with the PNG situation. This Western approach was also evident in relation to the new approach to teaching mathematics.

Teachers’ Seminar

In 1972 the PNG Teachers Association organized a seminar attended by a large body of teachers. This seminar resulted in the following 22 recommendations being forwarded to the Government (Papua New Guinea Teachers Association, 1971). 1. Primary Curriculum – do away with the Standard VI exam because of the anxiety and students not learning. Yearly internal examinations in IV, V, and VI were recommended. 2. There should be one syllabus for all T schools and that should be adapted to needs. 3. Schools needed to prepare children for the changing society. 4. Decisions regarding school education should respect the views of parents and the community. 5. Values and beliefs should be carefully considered and taken into account in teacher education programs. 6. Teacher education programs should incorporate relevant practical knowledge. 7. National unity could be encouraged through school assemblies. 8. Teachers colleges needed to promote true international understanding. 9. Teachers and children should live by Christian principles. 10. There should be loyalty to both the local government and the national Government; 11. The school should recognize rules and ways of living in the community and the laws of Papua and New Guinea. 12. Schools should prepare pupils for full and useful lives. 13. A balanced education agenda was needed, incorporating physical, mental and spiritual development. 14. Children should be taught how to work towards a better society. 15. The education system should promote democracy. 16. Schools should reconstruct cultural heritage by getting students to practice traditional dancing, folk songs, drama and craft and so forth. 17. Syllabuses should use language which was expressed as simple as possible. 18. Priority should be given to English and Mathematics, with Science, Agriculture, Social Studies, Christian Education, Health Education, Physical Education, Art and Craft, Music and Singing included as well. The inclusion of Sex Education should be at the Board of Management’s discretion, and culture at the discretion of the Board of Mission and District Education Board. 19. Issues relating to power and authority need to be addressed. 20. There should be a term examination and only Fridays should be used for outside activities, projects and culture. 21. Schools should run from 8 am to 3 pm each weekday. 22. The age to start school should be, as already decided, 7 years without a preliminary year.

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Many of these points had been accepted for years within the education culture in PNG. However, with Independence approaching it seemed there was a need to reiterate them. Some points clearly showed the Christian mission schools had had an influence—it was clear that the new country of PNG saw itself as a Christian country. Nevertheless, there were points that referenced millennia-old cultural practices. Hence in the view of the teachers, the education system should aim to be a blend of the best of what had been bought by the colonizers with the Indigenous cultures already present. The Government, for its part, accepted most of the points made by the teachers. On a practical level the teachers asked for one curriculum throughout the system and one set of times in which schools would operate. In other words, they wanted one system, not various schools and groups of schools operating under their own procedures. They spelt out in broad terms what they thought should be in a curriculum which varied little from Western curricula throughout the world. However, they asked that the curriculum documents should be written in readable English so that all teachers could understand what was required. As will be noted later in this and subsequent chapters the curriculum documents at Independence lacked this critical ingredient of readability by all, and many teachers were at a loss in grasping exactly what and how they needed to teach. None of the 22 points put forward by the teachers addressed the vexed issue of what should be the language of instruction which had been debated for so many years and was still not resolved. Nor did mathematics get any specific mention except that with English it was classed as one of the two priority subjects. To give added insight into the lot of teachers during this period, two cameos follow. The first is a brief cameo of an expatriate teacher who worked for some 12 years in the villages, well away from the urban centers where many expatriates taught. The second cameo is of one of the towering figures of PNG, Paulias Matane, who began his professional life as a primary school teacher.

Cameo of Trevor Freestone

In 1963, the six-month ‘E’ pre-service teacher education course that Kirkby had taken (see Kirkby’s cameo at the beginning of this chapter) was still being offered at the Malaguna Technical College in Rabaul and Trevor Freestone enrolled in it that year. The recruitment of such teachers was to help speed up education in English so that the communities across PNG could be ready for Independence (Freestone, 2011). The recruits in Freestone’s in-take included several nuns. The students worked hard, learnt to pilfer whatever they could for their anticipated remote postings, played sport with the Technical College students, undertook supervized practicums, and were helpfully inspected when they started teaching. Most of these teachers, as they expected, were sent to vrious remote areas throughout PNG. Trevor Freestone indicated that he certainly had more to learn after his six months training course, but he was willing to learn. He came to understand something of the families of the children he taught, and their cultures. Freestone first taught at Ambunti on the Sepik River. After some years there, he was posted to Pagei on the West Papua border. There he found a Center with good new buildings, a house for himself, and a school surrounded by huge logs. Following that posting he went to Bena Bena, and later to Watabung. In a number of these postings he implemented remarkable developments including supervising the erection of new school buildings in quite short periods of time. He fascinated the children (and communities) with his magic tricks to keep them interested (and at times received threats from the local witch doctors), took the inland children to the coastal towns, and two boys to Australia. He saw his various school projects as a way of preparing the students to be productive in their village environments since he fully understood that only 20% at most of the children from his school would be selected to go on to secondary education, no matter how well the children did. He did not think it worthwhile writing an English–Siane dictionary, or

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teaching reading in the Siane language (for the Watabung children) although he did note that it appeared that those students who learnt to read Siane improved in their English. He put this down to phonetic translation between the two languages. His story showed that it was important for teachers to learn from the local people, to be creative, to lead by example, and provide professional development for teachers coming to the school. In his 12 years of teaching in PNG, he was the only Australian teacher in three of his schools. The exception was at Bena Bena where the headmaster was also an Australian. They both left the school at the same time. Freestone’s service was finished by the Australian Government at the end of 1975, the year of Independence, and he did not return until 2008 to visit family (his in-laws) at Watabung and do some further fund-raising there and in Goroka. It seems there was little his extraordinary experience could influence in the changes that were occurring before Independence, but his legacy was that he gave a good education to the many children in the schools where he taught. There were many other Australian teachers whom we have met who taught in the schools during this period, often first educated as teachers before going to PNG. In all the stories we have heard, sadly there was little commentary about the mathematics curriculum. Language stories and for those who taught in remote areas, issues of safety particularly as Independence approached, dominated, which in many ways is not surprising. However, it does suggest that although mathematics (arithmetic) was always expected to be taught, for many teachers this teaching was at best formulaic and not worthy of their specific interest.

Cameo of Paulias Matane

This cameo has been created by summarizing some of the sections in Paulias Matane’s autobiography (Matane, 1972). Matane was born in the early 1930s in East New Britain in a small bush material house with a well-tied thatched roof and saplings, no windows, and a firm fitting door of planks that slid into slots. His mother was given relief in childbirth with the assistance of the village doctor’s magic and her female relatives to help. There were the appropriate ceremonies of fun and appreciation and food exchange celebrating his birth. During his early months he was able to swing in a bilum designed for babies to lie in. His early years were all spent around his family. The next major event in Matane’s life was the death of his mother. The message of her death was sent on the big talking drum garamut with each signal representing each person, object, action or feeling. There were 14 days of crying, eating, and telling stories, and also performing dance group after dance group. Then came the Tabuan dancer dressed in his leaves and mask. His father, with bird feather headdress and legs decorated, came and then the two big rings of shell money were cut and distributed to the people in payment. The payment was part of a well-oiled economy with shared rules. Matane was always asking questions. But on this occasion his father kept him very quiet and told him he would have to join the Tabuan society and learn from them the answers to his questions when he was older. Matane and his mother’s other children went to live with her relatives after her death but often they felt alone and homeless moving around to different places. Eventually, Matane decided to live in the men’s house. There was much magic for such occasions, as there was for a young man getting the girl he wanted. He learnt dances, how to pig hunt, or rather he was not scared of the stories meant to make him cautious. However, it was quick thinking which helped him kill the pig that was heading towards him. On another occasion he and others organized a spear- and a stone-throwing fight. They were rewarded with some fowls and bandicoots. However, he knew they needed to share with those who did not win—that was part of rules of the community. Then came a Christian pastor who spoke about God or a Father who they could not see. The pastor also wanted to teach the boys and girls to speak and write in English and so started a

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school. He told the children to stand in two lines, but they went into groups, then into circles. The Pastor thought them stupid for not knowing the English words he was using. He then wanted them to wash in the creek, but it was too cold so the bigger boy objected and told the pastor to go away; they would not take orders from a stranger who had no rights on the land. The pastor’s method of instruction was for the students to simply repeat the words he spoke in English. On one occasion, he spoke in Matane’s language but that did not change the students’ attitude toward him. By now they did not like the Pastor and would not go back to “school.” However, the adults supported the pastor so they eventually all had to return to school. In learning to repeat the numbers the pastor was saying in English, often the sounds of their own language became intertwined. Hence reciting the first ten numbers sounded like; one, two, three, pour, pipe, tik, teben, eight, nine, ten. They also learned to play football, which was much better than learning to speak in English. Then came the second World War. The Japanese occupied the land and the men were forced to work for them. However, the boys survived as did Matane’s father. At 17, after the war, Matane went to a “proper” school for the first time with many older boys. They built their big double classroom from bush materials with the help of their parents. By then they were all keen to learn. In the morning we learned how to write, draw, read, do sums, and sing. We loved singing very much and almost all lessons were treated in a musical way. We would sing and spell. … When we did our additions, we used to sing them like this: One and one are two Two and one are three Three and one are four, and so on. We learnt a lot by heart but did not really understand what the things meant. … If we were asked, “If a boy has two bananas, how many bananas will four boys have?” we usually guessed. … Our teachers were hard on us. If one of us made a mistake, he would get a stroke on the back. Many of the students ran away back home but a few struggled on through their difficulties. Two years passed, and I still did not know how to speak English although I could write sentences using the spelling words. (p. 87) By this time Matane really wanted to speak English, but when he confronted the teacher, he was told it would take him years, far longer than it had for him to learn his own language. For some reason being asked about learning English actually made the teacher quite angry. Somehow a little later Matane ended up in more trouble at school. He tried to make amends by taking the teacher betel nut, but that did not help. In the end Matane took revenge by throwing the teacher’s books down the latrine. However, that meant the teacher could not teach, as he could only teach from the books. Through it all Matane persevered. At the end of primary school, there was plenty of dancing and feasting and the names of the boys going onto Kerevat Secondary School were read out. The ten included Matane. He stood up and thanked the teacher and the people, but also said “I will try and become a teacher to teach your children. Even today, I do not know how to speak English” (pp. 91–92). Matane’s ambition was to do better than others. He became a very good student, topping classes throughout his primary and secondary education and won his first prize, $30 worth of books for the Forsyth Prize. He wrote for the Department of Education magazine and was published. One of his aims from then on became to write small, simple and cheap books for people. At the end of 1955 he finished his high school education by completing Form 3. Following secondary school, he gained entry to a Teachers College. As he flew for the first time via Lae to Port Moresby he crossed the Owen Stanley Ranges, by far the most rugged moun-

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tains he had ever seen. He wondered how people could get around down there but also pondered, “Will I be of any help to others? (p. 99). He commenced his one-year preservice primary teaching course at Sogeri Teachers College, 30 miles north of Port Moresby. On completion he returned home to be an assistant teacher. In his second year of teaching he was assigned the top class, a Grade 4 class of boys. No girls had been allowed to go to Tavui as yet, for fear of magic poisoning killing them, or they might meet boys which might lead to the parents not getting their bride price. Matane encouraged the girls to attend school. He pretested the students on their earlier work and then after looking at their results decided he had to spend six weeks covering the earlier work. In PNG the teacher’s work goes way beyond just teaching in the classroom. In the 1950s in Matane’s area, many of the people followed a cargo cult which had an anti-Government stance. They were confused so far as the role of the local Government Council was concerned. Their limited understanding of Christianity meant they thought it was like the council that had tried Jesus. There were difficulties in other areas as well. No matter how hard Matane tried to reason with them, like telling them that taxes help pay for things they need, he could not convince the people that what he was saying made sense. However, he managed to shepherd through a number of projects, such as having roads built. This gained him much respect from the people. Respect also came to him because many of his students, boys and, importantly, a number of girls, were getting through to secondary education. Matane was promoted to headmaster at another school but had to return to his old school as a teacher since there were too few children who enrolled in this new school. But he continued his hard work, including in the school garden. Matane argued that the children were not doing manual work but learning how to garden, because they would need to do that especially if they did not go on to secondary school. At one point he was called away from the garden by the District Education Officer, so he gave the following instructions to the children: Mark rows, three feet apart. How many yards is that, …? (Reply: One yard, sir) Good. Now mark along each row and dig holes, three inches deep and two feet apart. … You will then put two peanut seeds into each hole, cover them with some soil and then pour on some water. When I return, we will record the number of rows in this garden, the number of holes in each row and as there will be two seeds in each hole, we should be able to find out, without counting, the number of seeds in the garden. Anyway, get your bush ropes and start measuring. (p. 108) The District Education Officer told Matane he would go a long way and become a District Education Officer. But Matane protested that it was only for Europeans. The Officer pointed out it was for anyone capable of qualifying for the position. The following year, Matane left his district and enrolled at Sogeri High School to complete Form 4. This was something he was glad to do as it had been hard work doing Geography and English by correspondence. On completion of this study in 1962, he, with 15 other teachers, undertook a six-month course to be a District Education Officer. As the District Officer back at his home had predicted some years earlier, Matane qualified. This was followed by a travelling scholarship provided by the South Pacific Commission which enabled him to visit several countries in the South Pacific. He returned to PNG and took up the position of a supervisory teacher in the Southern Highlands, then the Western Highlands, and then in 1967 back to the Islands as the District Officer in West New Britain. Matane managed to gain a Churchill Scholarship to investigate professional administration in education in Africa for 6 months. In 1969, on his return from Africa, Matane went to Headquarters in Port Moresby to become Acting Superintendent of the Teacher Education Division. He was one of four members to be appointed to the Public Service Board, responsible directly to the Australian Minister for External Territories. It was the highest

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job in the Public Servant to which any Papua New Guinean could be promoted at that time. Sometime later he was appointed to be Director of Lands, Surveys and Mines, and then Permanent Secretary of the new Department of Business Development. An incredible journey not only for Matane but for his people in those 40 years. Although his autobiography finishes at that point, Matane went on to write a crucial report for the Department of Education, outlining a new approach to education, in the mid 1980s. The principles which he articulated in that Report, born out of his experiences in the classroom and education administration, leavened with experiences outside of the education system, proved to be extremely influential in the development of PNG education for many years. We will return to this report in Chapter 6. Change Came with Independence, But Not Easily Although in this chapter some mention has been made of political views in Australia towards PNG in general, and education in particular, it is useful at this point to give a summary spanning the post-war years up to Independence. This will set the scene for a discussion of the contrasting views of Ken McKinnon, the last Australian to hold the post of PNG Director of Education, and Alkan Tololo, then an influential member of Teaching Service Commission who would replace McKinnon as the Director of Education (see Chapter 6 for more discussion of his Report post-Independence and Chapter 10 for more on Tololo).

Australian Policy Makers’ Perspectives on Change

The impetus for gradual change in administration came from Paul Hasluck over his 12-year period as Minister responsible for the Territory beginning in 1952. He firmly believed, based on his understanding of history during the time that Britain was divulging itself of its territories during the 1950s, that change needed to be planned with social and economic advancement before political advancement could occur. “Our method has been vindicated by the remarkable transformation of hundreds of thousands of people in the past ten years from primitive savagery and fear,” said Hasluck in 1960 and published in the Territory statements 1963 (cited in Doran, 2007). Such a statement (“primitive savagery and fear”) also underlines his role as a colonialist. There were increasing numbers in the early 1950s of medical and health personnel, teachers and agricultural extension officers. The number of pupils in schools alone increased from 1952 to 1963 by 760% in Administration schools, although there were relatively few Administration schools. The financial grant by the Australian Government for funding the various aspects of government in PNG increased by 400%. At the end of his tenure, Hasluck proposed and gained the constitutional changes necessary to establish the House of Assembly, a university and an administrative college. At this point, at the end of his tenure, 1963, there was no intention of handing the political reins over to the Indigenous community in the foreseeable future (Hasluck, 1976). The next Minister for Territories was Charles Barnes who was self-effacing but wanted to learn for himself, so visited Papua New Guinea to make up his own mind. In the end he too decided to follow the gradual approach promoted by Hasluck, holding on to the “empire” perhaps, although pushing for greater economic self-reliance. He appointed (George) Warwick Smith from the Department of Trade as Secretary of his Department. Smith was active and at times cut across the opinions of residents in PNG. At times, his emphasis on the development of the economy was at the expense of social and political development. In his own words, through which one can see an old-time colonial attitude rather than a forward-looking approach to Independence, they had a choice of handouts “to provide a modicum towards health, education and the preservation of law and order and burdened by their ancient customs or whether to advance as a modern state financed substantially by its own resources” (submission no. 724 to

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Senate, 1965, cited in Doran, 2007). Productive activities were to be advanced, and so the rate of increase in the number of primary schools slowed even though at that time only 35% of eligible children were attending primary school education. Under the next Secretary, David Hay, a former diplomat, social unrest and criticism of economic development was growing as Papua New Guineans looked to local issues. Resentment gradually had built over various issues. One issue was the earlier decision by Hasluck to cut Indigenous public servant salaries (but not expatriates’ salaries). That needlessly caused concern even though the new salaries were likely to be the salaries under an Independent government. Reactions sometimes included violence such as the deaths of plantation owners or Indigenous people serving in regions other than their own. Significantly it was the young and educated who led the protests against the wage cuts. In a short time, many of these protesters would become significant figures in the PNG’s movement to Independence and later development. These included Albert Maori Kiki, Michael Somare and Oala Oala-Rarua. At this time, Rio Tinto, an Australian international mining company, developed the Panguna Mine in Bouganville to exploit the huge gold and copper deposits there. There was local opposition to the mine from the very start and continuing demands for more compensation for the local peoples, even though Barnes agreed that 5% of royalties would go directly to the local community. His main thrust was that mining should be seen as a national asset, and not be caught up in local politics, and should encourage self-reliance for the embryonic nation. Hay noted the strength of secession movements in both Bougainville and in New Britain. Matters such as the Panguna Mine gradually led to the notion that perhaps elements of Papua New Guinea could break away and become independent themselves, given that by now it was evident that Independence was inevitable for Papua New Guinea. However, in the opinion of Hay and the Australian Government, the push for economic development, of which Panguna was a key component, would not succeed in the way the World Bank Report had suggested, without changes to social and political development. The next constitutional change in PNG was spearheaded by Indigenous leaders. This resulted in measures that did not diminish Australian control but provided the Administrator with an Indigenous Select Committee and Cabinet. The seats in the House of Assembly reserved for Europeans were abolished. Significantly a small group of young, radical, politicians in the Pangu Pati (PNG Union party), who gained election to the Assembly, gently requested change. However, although during the period 1970–1972 Australian policy was still pursuing gradual development under Barnes, this was not sitting well in Australia. The Mautangan Association, a Tolai separatist movement that sought to return alienated plantation land to its customary owners, and which had resisted the central administration for some years, at times violently, grew and opposed local council changes in East New Britain. A land tenure protest in the Bougainville port area, separate to the Panguna Mine issue, was brutally quelled. Protests from the public and media erupted in Australia. Hays called for a heavy police presence, or even bringing in the Pacific Island Regiment (PIR) to quell the dissent (against Australian military advice which said the PIR was not equipped to deal with conflict within PNG). Hays maintained that PNG needed to maintain a strong central government if it was going to progress as a nation. Gough Whitlam, leader of the Federal Opposition Labour Party in Australia, seized on the unrest and the Government’s lack of action to listen to the people. He visited PNG and promoted his own view that independence should be imminent and not contingent on political, economic, social or educational preparedness. He despised some of the racist colonialist culture in PNG. When Whitlam came to power late in 1972, he acted. PNG was given self-governing status in 1973 and gained full Independence in 1975.

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Differences between Tololo’s and McKinnon’s Policies

The national policy objectives and strategies for education since Independence have been shaped by the National Goals and Directive Principles of PNG’s Constitution, and the National Education Act. Prior to independence, as has been noted, there were various statements of policy with educational development strategies being developed by successive governments, and these have continued since Independence (McKinnon, 1976; Matane, 1996 (see Chapter 6); Tololo, 1976; NDOE, 1990, 1996a, 1996b, 2004a). However, prior to Independence, the stark differences between the culmination of what Australia had been trying to achieve in terms of education since the early 1950s and what Papua New Guineans thought should be the way forward after Independence can be illustrated by contrasting the views of McKinnon and Tololo. A year before PNG gained independence in 1975, some of the well-educated Papua New Guineans voiced their dissatisfaction about the type of education that had been provided during the colonial era. As Smith (1975), Barrington-Thomas (1976) and Ralph (1978) noted, under the leadership of Sir Alkan Tololo, an education plan was drawn up. This document is commonly known as The Tololo Report (Department of Education Papua New Guinea, 1974). Tololo was the first Papua New Guinean appointed as Director of Education in 1973, just before Independence, succeeding the last Australian to hold this post, McKinnon who had served as Director since 1966. The Report contained a five-year education plan that was to be implemented when PNG gained its Independence. The plan emphasized an interleaving of community education and formal Western education. Importantly, the use of vernacular instruction at the lower levels of education was to be a priority. Tololo (1976) documented the reasons for a refocus in education services in a later article: Western-type education served a need—an introduced need. Whether it continues to serve needs in the countries in which it was introduced… is another matter…The questions to be asked and answered so that we can shape the education system to the needs of Papua New Guinea are: What sort of country is Papua New Guinea? What sort of country do we want it to be? How quickly do we wish to achieve these goals? (p. 212) Tololo’s view of the education provided during the colonial era was negative and his Committee thought it had not served the interests of PNG well. The current education system has caused divisions and frustrations in society by producing a small elite group. The majority of the people are not well catered for in relevant and appropriate education. (p. 221) His views in some ways aligned with a view of what education should be about in PNG that had persisted for decades. Although at times the Australian Government’s strategies had edged towards Tololo’s view, it had never embraced this stance fully as policy. The demands of the cash economy always predominated, controlled in large part by expatriates. Hence it is no surprise that Tololo was concerned about the provision of an appropriate education that would serve all people of PNG for their current and future social, spiritual, educational, economic and political needs. He cautioned that this needed to be undertaken appropriately: The education system must not be simply an agent for importing a foreign civilisation, but should be based on a study of all aspects of traditional societies and the way they changed… the general approach to education and of the national education system should be one which emphasises meeting the educational needs and problems of the Papua New Guinea society it was set up to serve. (p. 222)

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He noted that the Indigenous voice in education development in PNG had been continually suppressed during the colonial era. Tololo wanted the Indigenous voice to be visible during curriculum discussions and development: I would like to emphasise that it is Papua New Guinea we are considering, and it is Papua New Guineans who must make the decisions, both as to the goals and means to achieve this goal… Suggestions, advice and alternatives are welcomed as a means of sharing and assisting. (p. 213) Clearly, then, Tololo was prepared to consider suggestions and assistance from others to improve the plan outlined in his Report. But he recommended an alternative curriculum to that which had been implemented by the outgoing colonial Government. His Committee viewed their proposal as far more suitable for the needs of the people—well, at least the boys in the unfortunate gender-biased language that was used, in PNG: The school curriculum at all levels should be redirected towards community living (group goal) and living in the community (individual goal), and each level of education should be self-contained. Although one level will be the foundation for the next, its curriculum should be decided by the community needs and living, rather than what is necessary for the next level. It should be relevant; that is, useful and valuable, both to the student completing it and through him, to the community in which he lives. (p. 213) Tololo argued for lifelong education, well before it had become a forceful topic in international educational research from the 1980s. He wanted forms of education which would serve populations, in both the rural and urban communities. He was also aware of PNG being part of the global community and emphasized the importance of an education that would serve such a need. He argued: There will not be then one curriculum which is education, but a variety of curricula to serve needs of both modern and traditional sectors of society. Some of these may not be the responsibility of the Department of Education. This does not mean that they are any less a form of education. If this was to happen much educational effort would not be related to the needs of the community… Curricula developed for various types and levels of education should be aimed at encouraging people to think, to use their initiatives, to be responsible members of their community. It should encourage them to think of needs in the community and be self-reliant. (p. 215) Tololo was suggesting an education system that would be supported by the community to reduce costs, a very different way of supporting education services. In this way he emphasized that communities should be encouraged to take responsibility for the education of their children (see Chapter 10 where we argue for this as a basis for change in language use in schools). He cautioned, however, that: It will require a consistent community-wide effort to accept more direct responsibility for the costs of education and be willing to receive fewer handouts from central government… It would mean looking at new types or forms of education and different ways of serving education. (p. 217) Tololo was well aware of the resistance to the planned changes his Report proposed, and the need to overcome the cargo-cult mentality that Matane (among many others) had encountered. But there would also be resistance in the change of direction for education from educated Indigenous people who had benefited from Western education; those who had indeed found the “cargo.” As well, he feared, there would be no support from the colonial authorities. He observed:

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There is a “cargo-cult” mentality. A consequence of failure to adjust to all sorts of pressure, in a rapidly changing environment, which creates insecurity. There is a belief that formal schooling is a guarantee of material goods... Given these circumstances, it is inevitable that any move for a change in education direction can meet with such resistance. (p. 221) He was not wrong. The colonial authority, the Australian Government, did not support the 1974 five-year education plan, mainly because they viewed it as too costly. McKinnon (1976), in responding to the Report of the Committee which Tololo had chaired, argued: Urgent development tasks in Papua New Guinea require resources so much greater than those foreseeably available that there is no escape from forced-choice ordering priorities. Papua New Guinea cannot have everything it wants in full measure so its teachers must either choose one line of development from among several nearly equally desirable or combine a little of each into a compromise choice. They will need to rigorously and logically be consistent in their selection of priorities, a result not often achieved even in developed countries, in order to avoid wasting resources; in particular there will be some exceedingly difficult choices to be made for allocation of resources in education. (p. 189) McKinnon elaborated his economic argument from a Western economic perspective by suggesting that life back in the village communities would not be desirable enough to entice young people to return there from urban areas: Logical consistency is important since many of the decisions which effectively limit what can be done within the education system are circumscribed by prior choices of political and economic priorities made elsewhere. It is, for example, powerless, to influence young people to stay in rural villages when labour policies provide attractive conditions for urban workers, or to achieve anything really significant in agricultural education unless the land tenure system, marketing policies and transport arrangements convince individuals that they will be well off in rural areas. (p. 189) In making this argument McKinnon failed to address the issue of who was seen to own the land. There was a land tenure system which had been in place from time immemorial. It was not the one the colonial administration had imposed. Secondly most school students did not leave the village when completing Grade 6, since the majority of children who actually went to school did not go on to a secondary education. Neither, by the 1970s, did graduating from primary school guarantee entry to the cash economy as a worker, as it had 20 to 25 years earlier. So, although McKinnon was right to caution that education decisions in part needed to be influenced by other social priorities, he misread the complex milieu that was the context for this education system. Such a decision would have an impact on the curriculum in mathematics as well as in other aspects. From two of the authors’ (Paraide’s and Muke’s) current observations of the people’s lives in the villages, they are in fact at least a little better off than the low-wage-earners in urban areas. Village people can still build their houses on their own land, they do not have to pay rent, and they pay less for gas or electricity if they choose to use these. Much of their food can be sourced from gardens, forests, waterways and the sea. Life may be difficult or seen as not acceptable by others, particularly Westerners, but the villagers can still live well. The younger generations in the rural areas who have missed out on formal employment live in the villages and not in the urban areas. They definitely need the life-skills taught by villagers to be able to make a life in their own communities, as Tololo emphasised. These days, urban employment does not always hold as many attractions for workers as perhaps it once did, given the many economic and social challenges faced by those who now live in the cities and towns.

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McKinnon also argued against the type of curriculum that Tololo proposed. Tololo and McKinnon were working from their quite different views of what they envisaged would be a future PNG society; one from a view informed by a Papuan New Guinean’s lived experience in the villages before moving to the center of politics, the other from a Western urban standpoint and a Western-oriented view of curriculum development. Tololo’s view has been noted above. McKinnon (1976) suggested: Several professional suggestions have been made for a difference between urban and rural curricula and for a rural curriculum shorter than the urban curriculum, or at least for some streaming of bright high school-bound children while the others follow manually oriented courses. In discussions of the National Education Board to date, and there is some evidence to suggest that this reflects general Melanesian opinion, there is very little support for such moves. Several strongly expressed opinions are that the rural children deserve additional years of primary education to offset the inherent advantage of town living, that it would be unacceptable to separate children as early as Standard 4 into those likely and unlikely to be included in the limited high school admission lists, and that Papua New Guinea needs a “national curriculum” through which, irrespective of the future prospects of the students, all will have basically the same diet. (p. 195) McKinnon’s argument supported the thinking of colonial authorities who were not prepared for a change of focus in the formal school curriculum. They were not prepared to explore what ironically was viewed as a sound alternative formal curriculum in the Matane report and reform curriculum that aimed to cater for all students (Tololo’s position), which at one time was supported by Hasluck although never implemented by him. Rather, the Government stayed with a curriculum that aimed for a few passing all the examinations along the way, and finally emerging from the school system to enter formal employment. So, it was no surprise that McKinnon looked toward secondary school expansion into the disadvantaged areas. For him, the illiteracy problems should be top priority—students would need to learn to read, write, and speak English. He stressed that this education focus would enhance economic, political, social growth, and progress; in short, the Westernization of Papua New Guinea. So, the Tololo Report was shelved by the colonial Government until discussions for PNG’s educational reform began again, post-Independence, in the 1980s. Another issue that did no service to the Tololo Report was an issue of language, but not the long argument of what should be the language of instruction in schools. On closer examination, the English used in the Tololo Report was not precise and was not as persuasive as that used in the McKinnon Report, even though it presented a sound alternative education plan. A kind reading of the outcome of the competing Reports was that the problems that were highlighted in the Tololo Report were not fully understood by those who supported education for formal employment. Tololo’s position was that Western education alone did not prepare the students with skills for their adult lives, especially when they missed out on formal employment. The vast majority of students would still live in villages. However, if students did leave their village and subsequently fail to secure formal employment, they would not value life back in their communities and would not want to return. This was because they had little Indigenous education that would have prepared them to live comfortably in that environment. Their Western education, especially if they had progressed to secondary school, did not do that (Matane, 1976; Tololo, 1976). That prediction has come to pass for many young Papua New Guineans. Tololo acknowledged the widening gap between those who succeeded in the Western education and those who did not. The formal education system was already creating the “haves” and “have nots,” which even in the early 1970s was being seen in the communities, and creating frustrations for the dropouts (Department of Education Papua New Guinea, 1974). As discussed

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by Tololo, there really were no dropouts in traditional life, as everyone participated in community work or life. This new group of young people did not belong to either the category of employed or that of village life. They were a new group, sometimes called the “unemployed.” The Report was arguing for a curriculum that would stem this widening gap at its source. Foucault (1978, 1987) and Fellingham (1993) documented the categorization of people or setting them apart from others. These students’ Western education, which often they either had not completed or had not done well enough in, set them apart from others in their communities. They became the troublemakers because both the successfully (Western) educated and the successful village people did not accept them as normal. They were categorized as not normal because they did not have the normal qualities and behavior which were acceptable in either of these two established groups in the community. The Tololo Report also argued for the use of Indigenous languages at least in the early years of schooling, a position that had been argued for by some expatriates since the first years of colonization, but always rejected by the authorities. Pennycook (1998) argued that English was often used as the language of power in many parts of the world. Requiring English to be the language of instruction at all levels of education in PNG gave those who supported education for formal employment great power over PNG’s Indigenous languages and knowledge systems. This further devalued the vernacular languages that had been used for millennia as languages for informal education, and in turn devalued the Indigenous village education, the very thing that many adolescents lacked. It led many young people to lives marked by unemployment. Lather (1991) and Davis (1994) have written on the deconstruction of established practices. This is what the Tololo Report attempted to do—deconstruct the established labels of successful and unsuccessful, well-educated and less-educated, and employed and unemployed people who were beginning to emerge. In summary the report aimed to provide successful lives for students within the Indigenous communities after their formal schooling had been completed, at whatever year level that might be, who did not make it into the cash economy. It was inevitable that there were going to be many students in that situation, given the checks and balances through the many assessment hurdles already built into the system. By contrast, the report argued strongly that this aim rested on the crucial notion that Indigenous languages and knowledge needed to be given equal emphasis in formal education. Tololo, and his all Papua New Guinean Committee, foresaw the problem of unemployment and the accompanying alienation of students from their communities as a consequence of the type of education that was being provided before Independence. However, in 1974 when Tololo presented the report, the colonial authorities still had the power to suppress such Indigenous education initiatives. They still had control over funding for Government services and, to no-one’s real surprise, funding could not be secured to implement the education plan proposed by Tololo’s Committee. The arguments provided by McKinnon won the day. This kind of tug-of-war reflected the power plays that existed between some of the educated Papua New Guineans and the colonial authorities who were supported, it must be said, by a number of Papua New Guineans who had already transitioned into and were powerful in the quasi Western cash economy. In this particular case, the colonizers had the power to block an alternative formal curriculum for the system. They were not willing to provide funds to experiment with pilot programs, which would have been another alternative, one that would have meant going a little way to meeting the aims of the Committee. Another plan, entitled the Papua New Guinea Education Plan, 1976–1980 was drawn up to replace the Tololo Plan. It was strongly influenced by the expatriate Professor of Education at the PNG University of Papua New Guinea in Port Moresby. Again, preference was given to the expatriate view. Even seeking an alternative to Tololo’s plan, drawn up by a legitimate Government committee of Papua New Guineans chaired by Tololo, suggested continuing racial prejudice and

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colonial attitudes. In particular English as the language of instruction at all levels of education was maintained in this plan. This plan generated the one standard curriculum throughout Papua New Guinea which was in place immediately after Independence (Department of Education, 1976). Moving Forward Tololo’s plan was devised to correct a negative trend in PNG’s society which emerged as a consequence of the formal Western curriculum used in schools. Arguably, it was not too early for an educational reform in 1975, as the colonizers argued at the time, and which some Papua New Guineans at the time and subsequently have been conditioned to support. Such justification has been repeated over the years in educational documents to justify the rejection of the Tololo Report. Nevertheless, many public awareness campaigns offering reasons for a possible change in the focus of education, one which would assist the communities to support the initiatives proposed in Tololo’s Report, have been initiated—but with little success. Twelve years after PNG gained Independence, the Matane Report was published (NDOE, 1986). Many regarded this as the birth of education reform (NDOE, 1999a, 1999b, 2002). However, Matane’s Report re-emphasized the issues raised in the Tololo Report (see Chapter 6 for more on this). It is arguable that if the Tololo Report had been respected and financially supported by the colonial authorities in 1974, a relevant curriculum and the integration of Indigenous and Western knowledge for Papua New Guineans, would have begun then. An education system that prepared students for both formal employment and life back in the village communities could have started in 1975. References Advisory Committee on Education in Papua New Guinea. (1969). Report, 1969: Australian Advisory Committee on Education in Papua and New Guinea (Chairman: W.J. Weeden). https://nla.gov.au/nla.cat-vn2258617 Barrington-Thomas, E. (1976). Problems of educational provision in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea Education (pp. 3–16). Melbourne, Australia: Oxford University Press. Central Planning Office. (1976). National Development Strategy, Papua New Guinea Government White Paper, 27th October. Childs, P., & Williams, P. (1997). An introduction to post-colonial theory. London, UK: Prentice Hall/Havester Wheatsheaf. Currie, G. (Chair). (1964). Report of Commission on Higher Education in Papua and New Guinea. Canberra, Australia: Australian Government. Davis, B. (1994). Post-structural theory and classroom practice. Geelong, Australia: Deakin University Press. Department of Adult Education, University of Sydney (1969). Education in Papua-New Guinea. Current Affairs Bulletin, 43(6). Department of Education. (1976). Education Plan, 1976–1980. Port Moresby, PNG: Author. Department of Education Papua New Guinea. (1974). Report of the Five-Year Education Plan Committee, September, 1974 (The Tololo Report). Port Moresby, PNG: Author. Department of Territories. (1961). Report of Committee on the Development of Territory Education and Higher Training in the Territory of Papua and New Guinea, Australia (Chair: J. Willoughby; Department Officer: F. McConaghy). Retrieved from Canberra, Australia: Australian Government.

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Dienes, Z. P. (1999). Memories of a maverick mathematician. Leicestershire, UK: Upfront Publishing. Doran, S. (2007). Australia and Papua New Guinea 1883–1970: Australia in the world, The Foreign Affairs and Trade Files, No. 5. Canberra, Australia: Commonwealth of Australia. Emerson, L. (1964). Points of interest: Visitor's comments. Papua and New Guinea Journal of Education, 2(2), 76–77. Fellingham, L. A. (1993). Foucault for beginners. New York, NY: Writers and Readers Publishing Inc. Foucault, M. (1978). The history of sexuality: An introduction (Vol. 1). New York, NY: Vintage Books. Foucault, M. (1987). The use of pleasure: The history of sexuality (Vol. 2). London, UK: Penguin Books. Freestone, T. (2011). Teaching in Papua New Guinea: The true life story of Trevor Freestone teaching in Papua New Guinea 1963 to 1975: Author. Griffin, J., Nelson, H., & Firth, S. (1979). Papua New Guinea: A political history. Richmond, Vic, Australia: Heinemann Educational Australia. Hasluck, P. (1976). A time for building: Australian administration in Papua and New Guinea 1951–1963. Melbourne, Australia: Melbourne University Press. Iso’amo, A. (1974). Curriculum development in primary education—The Papua New Guinea situation. In Australian College of Education (Ed.), Educational perspectives in Papua New Guinea (pp. 59–75). Melbourne, Australia: Australian College of Education. https:// files.eric.ed.gov/fulltext/ED133808.pdf. Kirkby, R. (2019). Tisa: A teacher's experience in Papua New Guinea 1962–1975. Papua New Guinea Australia Association Journal—Photo gallery. https://pngaa.org/site/journal/ photo-gallery/tisa/ Lather, P. (1991). Getting smart: Feminist research and pedagogy with/in the postmodern. New York, NY: Routledge. Matane, P. (1972). My childhood in New Guinea. Melbourne, Australia: Oxford University Press. Matane, P. (1976). Education for what? In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 57–60). Melbourne, Australia: Oxford University Press. Matane, P. (1996). Pawa na pipel! People and power in Papua New Guinea. Sydney, Australia: Aid/Watch. McKinnon, K. (1968). Education in Papua and New Guinea: The twenty post-war years. Australian Journal of Education, 12(1), 8–9. McKinnon, K. (1976). Development in primary education: The Papua New Guinea experience. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 226–251). Melbourne, Australia: Oxford University Press. Murray, J. K. (1949). The provisional administration of the Territory of Papua-New Guinea: Its policy and its problems. Brisbane, Australia: University of Queensland. Nakata, M. (2012). Chapter 6 Better: A Torres Strait Islander's story of the struggle for a better education. In K. Price (Ed.), Aboriginal and Torres Strait Islander education: An introduction to the teaching profession (pp. 81–93). Melbourne, Australia: Cambridge University Press. NDOE, Papua New Guinea. (1976). Education Plan, 1976–1980. Port Moresby, PNG: Department of Education. NDOE, Papua New Guinea. (1986). Ministerial committee report: A philosophy of education for Papua New Guinea (chairperson P. Matane). Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1990). PNG strategic plan. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1996a). Papua New Guinea national education plan (1995–004) Volume a. Port Moresby, PNG: Author.

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NDOE, Papua New Guinea. (1996b). Papua New Guinea national education plan (1995–2004) Volume b Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1999). Language policy in all schools, Education Circular. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2004). A national education plan 2005–2014: Achieving a Better Future. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1999). National education plan (1995–2004). Update 1. Port Moresby, PNG: Author. Owens, K., Lean, G. A.,with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Papua New Guinea Teachers Association. (1971). The teacher. Noser Library, Divine Word University and National Library of Australia, Boroko, PNG. Paraide, P. (2010). Integrating Indigenous and Western mathematical knowledge in PNG early schooling. Doctoral thesis, Deakin University, Geelong, Australia. Paraide, P. (2018). Chapter 11: Indigenous and Western knowledge. In K. Owens, G. Lean with P. Paraide, & C. Muke (Ed.), History of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Pennycook, A. (1998). English and the discourse of colonialism. London, UK: Routledge. PNG Education Gazette (1967a). PNG Education Gazette, 1(3), 33. PNG Education Gazette (1967b). PNG Education Gazette, 1(1), 70–72. PNG Education Gazette (1969). PNG Education Gazette, 3(6), 65. PNG Education Gazette (1972). PNG Education Gazette, 6(4), 96–98. Price, A. (1965). The challenge of New Guinea: Australian aid to Papuan progress. Sydney, Australia: Angus and Robertson. Ralph, R. (1978). Education in Papua and New Guinea to 1950. Unpublished book. Roscoe, G. (1958). The problems of the curriculum in Papua and New Guinea. South Pacific, 95–96. Ryan, P. (Ed.) (1972). Encyclopedia of Papua and New Guinea. Melbourne, Australia: Melbourne University Press with University of Papua and New Guinea. Smith, R. (1965). New mathematics in the primary T school. Papua and New Guinea Journal of Education, 3(2), 107–110. Smith, G. (1975). Education in Papua New Guinea. Melbourne, Australia: Melbourne University Press. Smith, P. (1987). Education and colonial control in Papua New Guinea: A documentary history. Melbourne, Australia: Longman Cheshire. Southwell, B. (1974). A study of mathematics in Papua New Guinea. In Educational perspectives in Papua New Guinea (pp. 77–89). Melbourne, Australia: Australian College of Education. https://files.eric.ed.gov/fulltext/ED133808.pdf Spate, O. (1966). Education and its problems. In E. Fisk (Ed.), New Guinea on the threshold— Aspects of social, political and economic development (pp. 117–134). Canberra, Australia: Australian National University. Tololo, A. (1976). A consideration of some likely future trends in education in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 221–225). Melbourne, Australia: Oxford University Press. Wilkes, J. E. (1968). New Guinea—Future indefinite? Proceedings of 34th Summer School, Australian Institute of Political Science. Sydney, Australia: Angus & Robertson.

Chapter 6 Key Policies and Adjustments in the Decade After Independence

Abstract: During this period, the educational leaders and policy makers made many trips to other countries such as Africa, to assist with their own educational development. Key policies were developed under the guidance of Alkan Tololo and then Paulias Matane. Among these policies were ones which espoused principles of universal education, gender equality, and the role of examinations for entry into high school, senior high schools and tertiary institutions. These policies were further developed in the reforms of the 1990s. However, despite numerous study tours, it seemed that education was not growing or improving. Funding was taken from the tertiary sector, but it did not reach the primary sector as was planned. Attempts at vernacular education continued at the preschool level outside the system in various provinces but these were not endorsed adequately by the National Department of Education. One result was that a large number of prospective teachers went through an English-only education, possibly with some loss of their own cultural languages and knowledges. Nevertheless, the policies for maintaining schooling for the cultural community remained strong. Key Words: Colonial attitudes · Language of instruction · Matane Report · Two-way mathematics education · Universal education Independence … Tell me My friend and comrade Do you remember The night of Independence When celebration drums throbbed And men and women wept with joy … Did someone tell you That on the morning of Independence The dew on the grass Along the village pathways Would turn into gold? Mori, 1981

Education for Papua New Guinea

By Independence, there were virtually no expatriate teachers in primary schools in PNG. The few who remained were found in some church schools, often Catholic religious schools—more often nuns, but also some brothers. The presence of the nuns was a particular advantage because they encouraged the education of girls. However, there were many expatriates in the secondary schools. Vrtually all the staff in the National High Schools were expatriates, and © Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9_6

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expatriates remained in the National Department of Education Papua New Guinea (NDOE) often holding key positions. In many ways this supported the high standard of education as most expatriates after Independence enjoyed living and working in PNG and they brought high quality educational practices from overseas. However, the presence of expatriates was an issue for many Papua New Guineans. Their presence was seen as a result of the influence of a Papua New Guinean elite wanting the best that an English-language education could bring, but with it came the neocolonial attitude of many of the expatriates who had influence (Cleverley, 1976). The Report of the Five-Year Education Plan Committee, commonly known as the Tololo Committee Report (Department of Education Papua New Guinea, 1974) recommended vernacular instruction at the lower levels of schooling in its five-year plan. However, as noted in Chapter 5, this was rejected and replaced by another plan (NDOE, 1976) that favored English as the language of instruction at all levels of formal schooling (Cleverley, 1976). Weeks, who was Director of the Education Research Unit at UPNG (see Chapter 7) reflecting on this some years later, provided an explanation: Following self-government, in 1974, an all-Papua New Guinean committee was entrusted with the task of drafting the first post-independence 5-year education plan (Smith, 1987). They developed the early seeds of a philosophy of education for Papua New Guinea, and recommended eight years of community or primary education followed by a greatly expanded two-year high school. This plan went through two drafts, but was rejected by the National Department of Education. The Department turned to David Stannard, Dean of the Faculty of Education at the University of Papua New Guinea, to write a final plan, which was approved by the National Executive Council in February 1976 and released in September 1976. Smith was either unaware of the authorship of the final plan, or chose to ignore it, though he does call it a less philosophical and "much more work-a-day document" (Smith, 1987, p. 66). Until the release, and wide dissemination of the Matane Report in 1986, Papua New Guinea thus operated without a coherent, written philosophy of education. (Weeks, 1993, p. 262) This 1976 approach to education continued until the release of A Philosophy of Education in Papua New Guinea (NDOE, 1986b), commonly known as the Matane Report, 12 years after Independence. Many regarded this as the birthplace of the education reforms that began in the 1990s (NDOE, 1999a, 1999b, 2002). However, this later report actually re-emphasized issues raised earlier in the Tololo Report. Until these reforms, English, as the sole formal language of instruction, was the accepted practice for another generation. Even 20 years later in the 1990s, this change of policy contributed to many educated Papua New Guineans criticizing the educational reforms of the Matane Report which placed learning in the vernacular as a foundation of early schooling (see Chapter 8). The main argument offered for changing school curricula in the Tololo and Matane Reports was that the school subjects were based on Western knowledge and were intended for paid employment in which only a small proportion of school leavers participated. As discussed in Chapter 5, Tololo’s report followed by Matane’s, argued for alternative but relevant school curricula for PNG, including vernacular instruction at the lower levels of school as an appropriate basis for young people to participate in life in their communities (NDOE, 1986b, pp. 211-225; Tololo, 1976). However, the problems which were highlighted in the Tololo Report were not fully appreciated by those who supported education for formal employment. Tololo’s position was that Western education alone did not prepare the students with skills for their adult lives, especially in the case of those who missed out on formal employment. Often, when students failed to secure formal employment, they did not value life back in their communities because

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their Indigenous values had been replaced by the Western system (Matane, 1976, p.57; Tololo, 1976, p. 213). As preparation for Independence the Australian Government had sent some Papua New Guineans on various overseas fact-finding missions as a means of broadening their experience and understandings of what was possible. Others were sent overseas to undertake further study. There were also a number of committees that were formed that, often for the first time, had Papua New Guineans as members. The overall aim was to try and ensure that the quality of education, such as it was, did not plummet after Independence. In 1976, a quota system was introduced but although that was was, in principle, abandoned at the end of the year it continued unofficially until 1980. Concern was for the under-representation of girls and rural and remote school students attending high school. Most provinces were unable to sustain affirmative action to ensure that females continued on to high schools and children from remote communities were selected for Grade 7 (Weeks, 1987). The regression to merit selection with no quotas of any kind was exacerbated by the Measurement Services Unit in the National Department of Education, which followed worldwide trends and encouraged selection by basic skills: performance in English and mathematics (the argument being these were the only subjects that correlated with performance in high school). In one study this form of selection was found to produce a negative backwash as teachers began to teach towards what they thought was on the primary leaving examination, and began to neglect other subjects including science, agriculture, social studies, that are examined under the rubric "combined subjects" (Gibson & Weeks, 1990). (Weeks, 1993, p. 265) Quotas continued throughout the 1980s so that a number of schools started to assist students who were unable to gain a place in secondary school (Grade 7) but had attained the required standard. One such school was run by the Lutheran Church in Lae. They were supported by Distance Education and a series of mathematics booklets were produced. These were well structured and supported the teacher in the school. (Graham Cole, personal communication). A number of Graham Coles’s students went on to become teachers or technicians.

International Agency Schools

At Independence there was a need for International Agency Schools. These schools replaced the A (Australian curriculum) schools. The curriculum was generally decided by the teachers and guided by the Agency. The existence of these schools meant that any expatriates recruited to assist with the development of the country—such as in tertiary and upper secondary education or government departments or those who ran businesses—could be ensured of an education equivalent to that from their overseas country. School fees were generally a part of the contracts for overseas employees. Cameo on International Schools by Kay Owens In 1973, at the PNG University of Technology campus in Lae, Taraka Primary School was a split A and T school catering for the children of the university’s staff and the many self-help housing settlements in West Taraka and along the road to the Buso River. There was a well catalogued library, thanks to the university staff and volunteers, and small rooms for the expatriate children whose teachers cared for at least two grades. After 1975, the expatriate children (and some national children) went to Lae International Primary School (LIPS). The T schools were replaced by community schools. My community school teaching experience (1975) was at Taraka where the number of children was increasing rapidly and all the small classrooms as well as the large ones were occupied by up to 45 children. Some classes were held under tents. The

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children in the tents were dismissed if it rained too heavily as the numbers were too great for them to be able to stay dry. In 1992 when I carried out some research at the school, an innovative teacher had the children in one of the small rooms sitting on the floor with a plank on round concrete cores from the universities engineering concrete-testing laboratory. The children set their books on the plank to write. That way she could fit 40 children in the rooms left from the A school. There was a chair and a box of old computer paper for her and myself to sit on. When our own children started school in 1980, there was a choice but after being in PNG for five years we knew that the other school had some children who reflected the parental expatriate business men’s attitudes which conflicted with ours. Mission children also attended this school. Attitudes reflected those expressed in Patricia Paraide’s cameo. The children from PNG University of Technology and a few remaining expatriate public servants, Filipino and Pacific Island parents, mixed-race children, a few other businessmen who had a strong respect for PNG cultures and capabilities, and PNG business families, sent their children to LIPS. The sense of equality across the racial groups was highly evident. The positive approach, family atmosphere, parental support, quality international teachers often with cross-cultural experiences, meant that the classes were stable and the level of education high. There were, however, some disadvantages such as the Australian- or New Zealand-oriented curriculum and a lack of some subject material such as a well-structured PNG social science and sexuality education.

Textbooks and Teacher Professional Development

During this decade a crucial convergence occurred between educational research and the development of Indigenous textbooks. The Indigenous Mathematics Project was undertaken by the NDOE with financial support from UNESCO, and led by David Lancy (1981) and Randall Souviney (1983). The project set out to “document the relationship between environmental and cultural features … and cognitive development. A second goal was the documentation of mathematics learning and instruction” (Lancy, 1981, p. 445). A specific result of the project was that “locally developed textbooks with an appropriate language load would enhance mathematics learning” (Souviney, 1983, p. 183). While Lancy and Souviney, both from the USA, mainly concentrated on the first part of the project, Britt Roberts from New Zealand, led a team which concentrated on the textbook writing. There was a good level of cooperation and mutual influence between the two parts of the overall project to the benefit of both. Britt set out to bring as much of a local flavour to the Years 7 and 8 texts captured by the title of the series Mathematics Our Way (NDOE, 1980, 1981). The pilot materials were trialled extensively with local teachers and reflected the thrust of teaching mathematics internationally that had moved away from the rote learning and memorization to students being active learners and the use of learning materials. Dienes’ lingering influence was felt in this emphasis on learning materials, although his sets of materials were not used. The first series of books drew criticism from many expatriate teachers still teaching in the secondary schools who were not wanting to change their established methods of teaching. The different approach to teaching mathematics embedded in these texts perhaps assumed that teachers in general were more flexible and able to adapt quickly to a different approach. Like previous changes, particularly the changes that Dienes had tried to implement a decade earlier at the primary level (Clements & Lean, 1981; Guthrie, 1980), there were also many issues with making available continuing and relatively easy access to professional learning for teachers. Hence many teachers received the new textbooks with no professional development assistance that might have helped them understand the new possibilities that were embodied in this text. It also did not help that some key mathematics education staff at the Goroka Teachers College gave no support to this new direction in teaching mathematics—they could not see how proper secondary mathematics could be linked to village life. The issue of teachers being under-prepared for their teach-

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ing both in terms of subject competence and in teaching skills still lingers but now other issues may have been at play (Clarkson, Hamadi, Kaleva, Owens & Toomey, 2003). The IMP came to a finish in 1983 with the main overseas leaders leaving. However, Britt in particular had built up a good team of National mathematics educators in the Department who were able to continue writing textbooks. Thus, the Department continued with this thrust of developing text resources for teachers. Through the mid-1980s National members of the Curriculum Branch developed Mathematics Our Way, a text material aimed at students in Years 9 and 10 (NDOE, 1984, 1985), but, interestingly also down to Years 5 and 6 (NDOE, 1986a, 1987). One of the authors of this book, Philip Clarkson, was a consultant in the preparation of all these texts, including the Year 5 and 6 books after he had left his full-time position in PNG. Many of the basic approaches to teaching mathematics at secondary level pioneered by Britt were still present in these later books and gradually the criticism, so harsh to begin with, faded. The crucial remaining issue remained, however, that there was little to no continuing professional learning available for teachers. Within a few years the tide had turned. By 1990 the global publishing firm, Oxford University Press, had been approached and negotiated a contract to publish mathematics textbooks for all of PNG’s primary school years (NDOE and Oxford University Press, 1991a, 1991b). By and large these texts closely mirrored those that they produced elsewhere in the world and were written outside of Papua New Guinea, although some attempt at visiting the members of the Department in Papua New Guinea for consultation did occur at the early stages of writing the first texts. They were multi-colored and expensive. Admittedly the first few pages of the first book included objects to be counted but without words so that students could in fact use their own vernacular counting system. From then on in the main there was only superficial reference to the variety of Papua New Guinea cultures. With the Oxford University Press taking over, the support for a Papua New Guinean self-sustaining production of their own mathematical texts, by their own National writers, for their own schools, was lost (Clarkson, 2016). There is a question that has not been considered in these developments: were textbooks the best resource to use in teaching mathematics in PNG? They are ubiquitous in teaching Western mathematics worldwide. However, given the constraints of cost in providing rural schools with a yearly supply of quality texts over many years, this should have given cause to pause and, perhaps, concern. The more fundamental question, however, is whether textbooks constrain teaching to a specific approach, preventing the intertwining of how and what is taught in schools with how and what is taught in the village. That outlines a problem which defies solution.

Cameo from Kay Owens

In 1975, after teaching for some years in Australian schools, I began teaching at Balob Teachers College in Lae. It was run by the two branches of the Lutheran Church in PNG and the Anglican Church. One class I had was for inservice teachers, to whom I taught the subject “Child Development and Education”. Many of these teachers hardly ever spoke English, even in the classroom. I can recall explaining a few times in Tok Pisin ideas such as Piaget’s preoperational stage of development, assimilation, accommodation, décalage and conservation of number, length and area, although I soon found that using pictures was a good way to go when explaining ideas. About this time, all inservice education was taken over by the Port Moresby Education Institute (then College), which meant that the inservice program Balob had been running ceased. I was also required to teach in a community school (similar to primary schools) for a couple of months which I did and hence gained some good experiences of the sort my college students were going to face when they went into schools. I had already visited some remote areas and seen schools in those areas. The mathematics textbook that we used had a different topic for each day of the week. It was not very satisfactory for consolidating a topic, but the questions were good

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and generated considerable class discussion. I taught all the classes in the school mathematics and health education. As well as giving me good experience, my teaching also served as a vehicle for good inservice opportunities for the teachers some of whose mathematics was not strong. College staff observed and advised our preservice teachers in community schools in town on a weekly basis, and in village schools during block teaching. We provided inservice to school staff when we could. I can recall a teacher in Year 1 getting children to rote learn in a chorus how to spell English words whose meaning they did not seem to know. There was no sounding out of letters. However, Balob Teachers College was strong in teaching students to use sounds for teaching reading. Annie Dommerholtz was a driving force behind this and an inspiration for Yaking Marimyas who in turn became a strong NDOE curriculum adviser for English for many years. Yaking understood the theoretical background and language of phonetics, whole language, and functional grammar, although the documents she produced, like is often the case where experts write, may have been less understood by teachers who lacked continuing teacher education or professional development. Both Annie and Yaking knew that reading in one’s first language assisted students if they were taught with sounds for their reading skills which then transferred to English, but not forgetting other ideas such as sight words, reading in context, and conferencing students to help with grammar especially by comparison of two languages. At the time there was a Basic Skills Test in mathematics for the preservice teachers that the College administered. It always concerned me that the college students were being prepared for this test quite independently of the mathematics education course they were completing as part of their Diploma studies. It was known even then that learning to teach mathematics strengthens the understanding of the fundamentals of mathematics which the basic skills tests purported to assess. Most of the teaching to prepare the students for these tests involved doing many arithmetic exercises. However, the mathematics education course itself was child-centered and involved students in manipulating materials and group work. Michael Smith and Auhave Sarufa were two of the lecturers at the time with good experience in primary teaching and teacher education. They were influential in building a worthwhile preservice program. Later many primary/community school lecturers in mathematics education had trained as secondary school mathematics teachers and had little experience of the primary school teaching methods. Sadly, this gradually led to a college program that was heavily influenced by the traditional way of teaching mathematics; the ways secondary teachers assumed would be the best ways to teach in primary classrooms. When I first went to Balob, I was given a list of topics which made up the health education course. I remember that it did not include nutrition. Fortunately, I was a trained health education teacher and was studying my Master of Education degree in Health Education at the time, and so I was able to write a comprehensive course and set of materials. I was like many expatriate teachers and lecturers who completed various upgrades to their qualifications by distance while in PNG, not only to keep them busy in remote areas, but also to bring them up to date with education research and theory. Being used to group work in these classes meant I easily transferred group work to mathematics, something that was fairly new at the time. By the early 1980s, there were various Advisory Committees to prepare the different levels of the national school syllabuses and the NDOE also set up an Advisory Committee to prepare common national Teachers College syllabuses. The Curriculum and Assessment Unit was tasked with developing the various external examinations. At that time, around the world, there was considerable discussion about writing syllabuses and lesson plans using behavioural objectives. Certainly including behavioural objectives gave much needed detail to a syllabus that in the past had often consisted of a few general statements of content. However, for mathematics, using the behavioural objectives approach meant that the syllabus writers broke down the mathematics into far too many small bits of information. This resulted in the syllabus documents and teacher’s guide being quite substantial, and hence intimidating for teachers to read as a guide to preparing lessons. The many specific objectives also tended to mean the teachers taught the bits, but often

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failed to teach for the connections between the bits, and hence the overall understanding of what a topic was really was often lost. This led to “recall of knowledge” tests similar to the Basic Skills tests. These objective-based syllabi proved to be of little use to student teachers who s tended to copy out the objectives from the teachers’ guide for a specific topic and thought that that was all that was needed. Having copied out the objectives diligently, they failed to elaborate on them, or link them, or provide extra detail for their lesson plans in terms of how they might best be taught. Hence, they had little ownership of what they were doing and subsequently in what they were teaching. When I began at the College there were some excellent science cards for teachers to use. Then there was a new science curriculum and teachers were advised to throw away the “old stuff” and implement the new. What a waste of good resources. Similar things had happened in mathematics. In Chapter 5 it was noted the enthusiastic and important approaches that Dienes had introduced in terms of physical involvement, group activity, recognition of different base arithmetics, attributes of shapes, and logic. Sets of wooden materials and cards were developed and made available to teachers who had undertaken some training in their use either at College or at Inservice sessions. These ideas were still penetrating teacher education many years later. However, in the schools because students found multi-based arithmetic calculations using leaves and sticks and stones1 fairly difficult, the multiple arithmetics topic was dropped from the primary school curriculum. Again, it was drill on the different bases rather than providing an appreciation of the composite groupings whether it were 3, 4, 5 or 10. Perhaps too much was lost; the Temlab cards and materials were discarded—if they had not already been lost. Thankfully, the notion of using concrete materials in general was still encouraged, especially bundles of 10 sticks for the composite unit of 10. From the late 1970s new textbooks were developed by the Curriculum Unit. In the high schools, it was Mathematics Our Way (see earlier in this chapter). They were PNG focussed. I also noted books used in the early 1990s which, although prepared by the Curriculum Unit, proved to be difficult for teachers in primary schools to use, largely because their proper use was not understood by many teachers. and they did not hold together for very long in the rough conditions of rural classrooms. Many teachers found that in general they set too high a standard of mathematics. An unintended outcome was that they led to very textbook-oriented teaching for both secondary and then primary classrooms. Eventually they too were replaced by textbooks associated with the Curriculum Reform Project and the outcomes-based education approaches (see Chapter 8). With class sizes increasing and a new emphasis on behavioural objectives and tests, individual learning and performance began to replace group work in mathematics. The teacher orator was taking over, and in the best classrooms, with good demonstration of mathematical concepts using concrete materials. Then emerged an over-emphasis on textbooks, especially the Oxford green community school books. Although they were potentially good, many of them were were soon lost. A teacher often only had access to one student book, and no teachers’ guide. Another might have a guide but only eight textbooks to be shared around 45 students. Worksheets were no longer a possibility, particularly in rural schools: because the jelly stencils were gone, and so too were the colored sheets and the roneo machines. The Gestetners were breaking down. On top of these equipment difficulties, in rural schools there were often multigrade classrooms, not because that was seen as good teaching practice, which it could be, but simply because of necessity through a shortage of teachers. Multigrade classes meant more differentiated resources were Using sticks and stones ,etc., to represent ones, tens and hundreds was clearly at variance with what Dienes had used. His mult-base materials were all made of wood of the same color, and the differences were shown by different volumes. Using different natural materials (stones, leaves, sticks, etc. for the different place values) could cause much confusion. BUT many, students nevertheless found the multi-base arithmetic blocks very hard even when using the original Dienes materials. 1

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needed if they were to provide a positive experience for students. Many years later when computers started driving printers for worksheets, paper and printer cartridges were too expensive. There were few alternatives to textbooks. Primary schools were called community schools at Independence. The change in name implied that all subjects were expected to be taught in such a manner that students could remain in their communities at the end of primary years of school and still feel a part of those communities. Sarufa was an excellent college lecturer promoting this approach. In many schools there were frequent mathematics lessons geared to counting and solving problems in the gardens. Recognizing the counting words and other mathematical words of the language patterns of the village was another important aspect of making schooling community-oriented. Health education in the school certainly had a focus on health in a village community, featuring knowledge of tropical diseases and how they spread, three food group nutrition with local foods, breastfeeding and not bottle-feeding (fortunately, legislation banned bottles without a prescription), first aid, family planning, and sexually transmitted diseases. The policy of deciding on whether village practices were harmful, helpful, or neutral was a key approach in health education (Owens, 1977). Between contracts with Balob Teachers College, I taught mathematics on contracts at the PNG University of Technology2 (Unitech). More details are given in Chapters 7 and 11.

Concerns About Standards of Education

Weeks (1987) called it “an obsession” (p. 266). From 1977, there were committees to consider the standards of education, desiring the school results on tests to remain as high as they had been and equivalent to overseas education despite the changing demographic of school students and leavers, and the change by which all Government community school teachers were now nationals with two-year diplomas. It led to overseas study tours for comparisons of education and to curriculum changes made by syllabus committees, often with an overseas advisor. There were several projects to improve numeracy and literacy including school magazines, books for disadvantaged schools and then school libraries, and eventually the World Bank Advisory group. However, expenditure on education dropped by nearly 40% between 1978 and 1988. One tragedy of the 1980s in Papua New Guinea was that though resources available to higher education declined3, they were never redistributed to primary education— instead, they were intended to pump-prime agriculture, but diverted to law and order enforcement. Between 1983 and 1991 the per capita expenditure on education in Kina thus declined from K31.68 to K26.32 (at 1983 prices) and the expenditure per pupil from K269.68 to K207.68 (National Department of Education, 1991a, p. 13). A change of Government in mid-1989, with new advisers with different priorities and perspectives facilitated a slow reversal of the restrictive policies of the 1980s. (Weeks, 1993, p. 268) Secondary education also required a lift in funding and in development if standards were to be maintained but these were held constant in the hope that primary education would increase (Education Sector Committee, 1984). Weeks (1987), citing Dore’s (1976) theory, considered that the late start to development in PNG (see Chapters 4 and 5) ultimately led to a strong desire for obtaining qualifications, which See Chapter 7 for details of teaching mathematics at the PNG University of Technology, and the Mathematics Education Centre 3 Universities struggled to pay salaries and into the 1990s the Registrar at PNG University of Technology often had to ring Treasury and beg for funds for salaries. 2

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in turn often led to money and recognition (noted in a number of cameos in this book where much is made of wanting to succeed in schooling so parents and their wider community would not be let down, and some personal standing would also be gained). Weeks noted in particular the drive to secure educational certificates, since these were more and more needed to enter particular occupations. Hence school qualifications became valued in terms of the positions that could be accessed within the wider society. Hence, the greater emphasis on examinations was accepted as part of this system, and that has often been at the expense of other purposes for education. This situation continues to this day. In the 2020s people now want to gain the status of a doctorate, since those with doctorates or higher degrees are called upon by their families to stand for parliament, and where the opinion of people with education is powerful in the community, for better or worse. Of course, at a more local level people with recognized education qualifications now have standing that perhaps was not afforded in the past. For example, Saxe (2012) told the story of the opinion of a teacher being accepted without question in an Oksapmin community—although, as related in the cameo for Matane in Chapter 5, this is not always the case. Ross (1984) conjectured that the standard of teaching was probably the main reason for low mathematics achievement since Independence. He showed that 1.2% of post Grade 10 students were reaching even the minimal expected standard for Grade 8 students using his Mathematics D2 test and their scores formed a normal distribution compared to the skewed assessment in Kenya and Tanzania on the same test, with 60% and 52% reaching the minimal standard, respectively. That same low standard in PNG had occurred since Independence. He recorded from Government reports that the minimum entry requirements for students entering teachers colleges were not met by 25% for secondary and 87% for primary teacher education. In 1981, a Ministerial Committee of Enquiry into standards chaired by Simon Kenehe investigated the public’s concern over the alleged problem of “falling standards of academic achievement” in schools. Details of the Kenehe Report covered a wide range of issues associated with quality and standards of education, particularly, the apparent lack of performance and accountability, and weaknesses in the school system (Kenehe, 1981). The study revealed a variety of perceptions of standards, and large gaps in knowledge and skills in the various components of the system of education. The World Bank when approached by the Government for financial help on this matter responded to the Government’s concerns on low standards in schools by placing strong emphasis on qualitative improvements to educational planning and production of curriculum materials (NDOE, 1984; cited by Weeks, 1993). However, despite the Government’s efforts, McNamara (1983) asserted that “political ambitions appear to be outstripping administrative capacity” (p. 20). He continued that weaknesses existed in administration, planning and incompetent officers. Sometimes, he asserted, teachers’ good efforts to address central directives associated with curriculum content and balance, methods of teaching, assessments and evaluation and school procedures, had been frustrated by irresponsible and incompetent administrators in the provinces (McNamara, 1983). Anderson (1984) pointed out that reducing levels of confusion and incompetency amongst the provincial officers was essential for planning and maintaining any level of efficiency. But he recognized that the high turn-over of public servants could have contributed to a lack of professional knowledge within the provincial education offices. Furthermore, Tapo (2004) asserted that a “general lack of support from the Government made it difficult for the provincial education offices to sustain any level of efficiency in the school system” (pp. 46–47). Equity and Education By 1986, leaders in education were concerned about the state of education in PNG. Idealistic views were set by political leaders and policy makers without due regard for the cultural and linguistic diversity of the nation (Rogers, 1986). With these difficulties accumulating and the system apparently not able to cope, Paulias Matane was asked to chair a Committee to explore the state of play in education and suggest ways forward (see Chapter 5 for a cameo on Matane’s early life and career). The Matane committee (NDOE, 1986b) reported that, in general, teachers’

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skills and knowledge, including both subject and pedagogical content knowledge, were lacking. Also, irrelevant curricula, and lack of community support, had had a profound effect. However, the Matane report did more than detail deficiencies of the system. It set out what it was about in the title used; A Philosophy of Education for Papua New Guinea. The report proposed a radical philosophy of education based on a notion of “integral human development.” This philosophy was for every person to be dynamically involved in the process of freeing himself or herself from every form of domination and oppression so that each individual would have the opportunity to develop as an integrated person in relationship to others. This implied that education should aim for integrating and maximizing: socialization, participation, liberation, equality (NDOE, 1986b, p. 6). The report was particularly concerned at the loss of relevance of education for the majority of students, and the lack of early childhood education opportunities. It recommended that the language of instruction be in the language spoken by the child—that is, one of the more than 850 languages which existed in Papua New Guinea. This last point was a daunting challenge, but it was underpinned by a strong maintenance approach to Papua New Guinea cultures and ways of doing things. The report was not fully appreciated at the time of its release, but it formed an important prologue for much of the policy documentation and literature on education in Papua New Guinea until 2014. In 2004, Michael Tapo, then Director of Teacher Education and later Secretary of the Department, commented, in his doctoral thesis on the Matane Report, by first noting its emphasis on the need for a strong connection between the school and the local community, perhaps echoing what had been written in the Tololo Report 10 years earlier. He then commented on some of the implications for teachers in that Report. It is significant that Tapo was basing his thesis on the Matane Report given his role within the education system. We therefore quote extensively from it to indicate the importance of the Matane report, even 20 years later. The Matane Report (1986) pointed out that the school curriculum has to emphasise home and local community as being the child’s first agency of socialisation. Children are first to learn about their culture and traditions and to learn about respect, cooperation and justice. … Teachers are to integrate traditional values and skills in modern Papua New Guinea and therefore should use the local knowledge from the community. The following elements in Matane’s report relate to teachers. These are: 1. Be committed to a policy of community involvement, both in the school and in out-of-school situations; 2. Have knowledge of and respect for traditional aspects of Papua New Guinea cultures; 3. Have knowledge and skills to stimulate active involvement of children in the community and of the community in the school, and 4. Have knowledge about the global community and be able to relate this to the concerns it shares with Papua New Guinea. … To achieve these aims, a teacher should be: 1. Competent in the knowledge areas associated with the above aims; 2. Aware of the characteristics, needs and abilities of children at different stages of development and have the skills to be able to attend to these in the classroom and wider situations; 3. Aware of and familiar with the body of modern educational theory and able to select from it alternative teaching strategies suitable for his or her children's needs; and 4. Skilled in the application of such basic techniques of teaching as the use of advanced questioning, motivating children, relating new learning to previous knowledge, teaching in relation to context, and varying learning through materials, methods and settings (Matane 1986, pp. 22–27). A teacher with these qualities will

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assist children to achieve the aims of oracy and literacy, numeracy and graphicacy, social development, resource development, and spiritual development. (Tapo, 2004, p. 61) However, lack of expenditure on education continued to thwart the achievement of goals for education. There was no percentage increase in students gaining a primary or secondary education. It remained around 73% and 16% between 1973 to 1993 (Weeks, 1993). The shortfall in teachers exacerbated this. There were only 800 primary teacher graduates in 1991, and 520 in 1993 with a shortfall in teacher numbers being over 2200 (NDOE, 1991a).

Gender Equity

When the Australian administration were providing schools, they encouraged girls to go to school. Occasionally the principals would not take any more boys until the number of girls increased. Many aid projects from Australia, because of the Tololo and Matane plans incorporated reference to gender equality, either in seeing achievements, female PNG partners, or strategies to improve gender equality. It has been a recognized issue in education from the beginning but during this period, only one significant conference on the theme was held (Wormald & Crossley, 1988). Naomi Wilkins’ (1995) unpublished extensive research provides an overview of policies on education especially related to gender equity. One study, conducted at the PNG University of Technology in Lae, analyzed data gained from a questionnaire and interviews with successful graduates and current students. University student records were also accessed to ascertain attrition rates. Wilkins also qualitatively analysed two graduates’ opinions on tertiary education. She incorporated ideas for improving students’ introductions to university life through activities on getting to know each other and also personnel who might facilitate their road through university. The graduates emphasized the importance of keeping physically fit, completing homework/lab reports, regular study particularly in the morning after a good sleep, making a study timetable and managing time well, keeping notes and revising them, studying for tests and examinations, and self-assessment of progress as well as reading for relaxation. She also discussed student motivation such as developing a competitive climate, goal setting, using other people as role models or themselves as a role model, living according to Christian beliefs, maintaining family responsibilities, accepting encouragement from lecturers and friends, aiming for prizes and awards, attending relevant seminars and completing projects. Cost of education has often been highlighted as a major issue for many parents and families. Warner Smith (cited in Sukthankar, 1995) reported that some female secondary trainees at Goroka Teachers College selected the college rather than a university under pressure from families to get a job quickly. Families were more likely to give an education opportunity to a son which was indeed the motivation for Komhiol’s study mentioned below. Women were generally given a secondary status, although they were regarded as important for bride price and in matrilineal societies for land inheritance and power (often associated with sorcery). The examinations that reduce numbers of students at each level of education result in only 10% of Grade 10 graduates moving onto senior high school, and of these only 10% are female. It was suggested that girls needed to be provided with special places at the start of school and at each stage where examinations may have been reducing the proportion further. The universities also needed to retain their female students. The lower grades in first year compared to second year at university might relate to the lack of surety moving from a small National High School campus to a large university campus. There were other adjustment issues in establishing a place for themselves without the cultural expectations or restrictions. Importantly Wilkins noted the difficulties for female students especially in terms of their lack of safety (Sukthankar, 1995). Interestingly, some campus places were a concern for more students later on than initially such as the library at night, or the coffee shop and the playing field during the day,

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but their concern about failure, males and other fears decreased dramatically. A matter of concern was the fact that the fear of attack, rape and drunks had increased threefold from initial to later impressions. Needless to say, this was due to knowledge that such things occurred frequently. There have been studies on gender equity especially through the University of New England (Dinah Ope, Marita Kapa Semosa, and in 2020 Teng Waninga Komhiol).

Decentralization, and Provincial Equity

In some ways, education in PNG has always been decentralized with missions often educating along their own guidelines. At Independence, allocation of teachers in schools was a provincial matter. Some schools such as the Seventh Day Adventist (SDA) schools and the Lutheran Tok Ples schools were registered but taught in their own ways. However, in the 1980s there was a formal intention to have planning at a provincial level and for new high schools, for example, to be supported if the province indicated that there was a lack of financial viability to do so, yet the need was there. However, only a couple of provinces could actually establish a new high school from their own funds. Then there was an effort to have Provincial Planning Administrators but the course that was provided at the University of Papua New Guinea (UPNG) was not taken up by more than 4 of the 17 administrators. There were four regional expatriate advisers to assist, but this did not seem to have much effect. Plans were not comprehensive or forthcoming. Plans were not implemented even with further directives. The Provincial Education Boards in all but one province had not met by 1988, even though they had been set up in 1983 (Weeks, 1989). Weeks’ (1989) case study of the situation in East New Britain revealed that there was a very high percentage of students attending school and a very high proportion going on to secondary schooling when compared to other provinces—except for Manus which also achieved well in this regard. The overall standard in East New Britain continued to rise from 1982 to 1988, despite the fact that the province has four areas that can be classified as disadvantaged with lower school enrolments (two interior areas—Pomio and Bainings—and two small islands). The national Government was wanting increases in secondary schooling especially from these kinds of areas. However, the planning indicated that there would be considerable cost to provide this kind of schooling for so many. The province, therefore, dropped the preschools and vocational centers in order that they might manage their budget. It was noted that the new Grade 7s in the primary schools would break the provincial budget for teachers’ salaries alone, and there were not enough secondary school teachers. Basically, the idea was not workable without further funding and so the plans had to be abandoned. The province was considered advantaged and received little national funding to expand. With salary increases for teachers, resources available for important aspects of school improvement declined. This case study indicated that there were considerable issues for the Government to overcome in passing on messages, funds and planning for workable change.

Moving Forward

In many ways the Matane Report was a continuation of what the Tololo Committee had envisaged in the year before Independence (see Chapter 5). That earlier Report was not implemented, and although the Matane Report was officially adopted by the Government, it has, at times, seemed at the time to suffer the same fate. However, the Tololo and Matane way of thinking kept reasserting itself and in the 1990s was finally given a chance to bring a new direction to PNG school education. However, appropriate funding was not made available. Those developments will be reviewed in Chapter 8 when what came to be known as the “Reform years” will be discussed. Clearly there were many issues in relation to the education system which needed attention throughout the decade following Independence. However, the number of students in the school system continued to grow. Remembering that education was not compulsory for children, the gross school enrolment rate was 70% in 1987 across all levels of schooling. In 1987/88 there

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were 2470 Community Schools with 373 989 pupils enrolled; 123 Provincial High Schools with 52 000 pupils; and four National High Schools with 1800 pupils. Only 32% of children continued their education beyond Grade 6, and of those who entered Grade 7, only 66% completed Grade 10 (Crossley, 1994). Although these numbers are low, they are a marked improvement on the numbers in the 1960s before Independence. At this stage, despite efforts to have two-way education or glocalization that met local needs and global expectations, there was still a concern that the mathematics education received by many, probably most, children was unrelated to their cultural backgrounds. Too many teachers were teaching by rote methods and students were only able to recall facts rather than apply them in practical situations, or in their school tests, or in university studies. References Anderson, B. (1984). Efficiency in provincial education administration. NDOE, Papua New Guinea. Clarkson, P. (2016). The intertwining of politics and mathematics teaching in Papua New Guinea. In A. Halai, & P. Clarkson, (Eds.), Teaching and learning mathematics in ultilingual classrooms: Issues for policy, practice and teacher education (pp. 43–56). Rotterdam, The Netherlands: Sense Publications. Clarkson, P., Hamadi, T., Kaleva, W., Owens, K., & Toomey, R. (2003). Final report. PNGAustralia Development Cooperation Program. Baseline survey: Interpretative evaluation report of the Primary and Secondary Teacher Education Project. Melbourne, Australia: Australian Catholic University. Clements, M. A., & Lean, G. A. (1981). Influences on mathematical learning in Papua New Guinea: Some cross-cultural perspectives (Report No. 13). Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Cleverley, J. (1976). Planning educational change in Papua New Guinea: A comparative study of the 1973 and 1974 five-year plans for education. Comparative Education, 12(1), 55–65. https://doi.org/10.1080/0305006760120107 Crossley, M. (1994). The organisation and management of curriculum development in Papua New Guinea. International Review of Education, 40(1), 37–57. https://doi.org/10.1007/ BF01103003 Department of Education Papua New Guinea. (1974). Report of the Five-Year Education Plan Committee, September, 1974 (The Tololo Report). Port Moresby, PNG: Author. Dore, R. (1976). The Diploma disease: Education, qualification and development. University of California Press. Gibson, M., & Weeks, S. (1990). Improving education in Western Province. Division of Educational Research, National Research Institute, University of Papua New Guinea. Guthrie, G. (1980). Stages of educational development? Beeby revisited. International Review of Education, 26, 411–449. Komhiol, W. (2020). Analysis of congruency occurring in policy development and implementation practices to promote gender equity in pre-service teacher education in Papua New Guinea. PhD thesis. University of New England. Lancy, D. (1981). Indigenous Mathematics Project—An overview. Educational Studies in Mathematics, 12, 445–453. McNamara, V. (1983). Learning to operationalise policies: Factors in the administrator development lag. Administration for Development, 20, 19–41. Mori, J. (~1981). Independence. In Expressive Arts Department (Ed.), Images of Papua New Guinea (p. 12). Aiyura. Eastern Highlands Province: Aiyura National High School. NDOE, Papua New Guinea. (1976). Education Plan, 1976–1980. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1984a). Interim report of the Education Sector Committee: Medium term development strategy (chair , S. Roaekina). Port Moresby, PNG: Author.

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NDOE, Papua New Guinea. (1984b). Mathematics our way: 9. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1985). Mathematics our way. 10. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1986a). Community school mathematics Grade 6. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1986b). Ministerial committee report: A philosophy of education for Papua New Guinea (chairperson P. Matane). Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1987). Community school mathematics Grade 5. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1991a). Education sector review: Volume 2, Deliberations and findings. Port Moresby, PNG: Author NDOE, Papua New Guinea. (1991b). Education sector study. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2002). National curriculum statement. Port Moresby, PNG: Author. NDOE and Oxford University Press. (1991a). Community school mathematics, 1: Melbourne, Australia: Author. NDOE and Oxford University Press. (1991b). Community school mathematics, 1—Teachers' resource book. Melbourne, Australia: Author. Owens, K. (1977). Curriculum development in health education with special reference to Papua New Guinea. Master of Education Long Essay. University of Sydney, Sydney, Australia Owens, K., Lean, G. A., with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Roberts, A. M. (1977). Mathematics in Provincial High Schools 1976. Papua and New Guinea Journal of Education, 13(1), 7–20. Rogers, C. (1986). Philosophy of education in Papua New Guinea. The setting of ideals and goals. Papua New Guinea. Journal of Education, 22(1), 3–14. Ross, L. (1984). The quality of Grade 10 school leavers’ pool. In P. Clarkson (Ed.), Proceedings of the Fourth Mathematics Education Conference (pp. 189–198). Lae, PNG: Mathematics Education Centre, PNG University of Technology. Saxe, G. (2012). Cultural development of mathematical ideas: Papua New Guinea studies. Cambridge University Press. Smith, G. (1978). Counting and classification on Kiwai Island. Papua New Guinea Journal of Education, Special Edition: The Indigenous Mathematics Project, 14, 53–68. Smith, P. (1987). Education and colonial control in Papua New Guinea: A documentary history. Longman Cheshire. Souviney, R. (1983). Mathematics achievement, language and cognitive development: Classroom practices in Papua New Guinea. Educational Studies in Mathematics, 14(2), 183–212. https://doi.org/10.1007/BF00303685 Sukthankar, N. (1995). Gender and mathematics education in Papua New Guinea. In B. Grevholm & G. Hanna (Eds.), Gender and mathematics education, an ICMI Study. Lund University Press. Tapo, M. (2004). National standards/local implementation: Case studies of differing perceptions of national education standards in Papua New Guinea. PhD Thesis, Queensland University of Technology, Brisbane, Australia. Tololo, A. (1976). A consideration of some likely future trends in education in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 221–225). Melbourne, Australia: Oxford University Press. Weeks, S. (1987, 12 March). Oro's high school selection unfair. Times of Papua New Guinea, p. 27. Weeks, S. (1989). Problems and constraints in educational planning in Papua New Guinea: A case study from East New Britain Province. PNG Journal of Education, 25(1), 57–79. Weeks, S. (1993). Education in Papua New Guinea 1973–1993: The late-development effect? Comparative Education, 29(3), 261–273. Wormald, E., & Crossley, A. (Eds.). (1988). Women and education in Papua New Guinea and the South Pacific. UPNG Press.

Chapter 7 Higher Education for Mathematicsa and Mathematics Education: Research and Teachingb

Abstract: Universities are institutions at which tertiary mathematics is usually taught. Some also provide programs for preservice secondary mathematics teachers, and courses for teachers of younger children as well as other forms of adult education. Most of the preparation of primary teachers in Papua New Guinea, however, has been placed in the hands of teachers colleges, some of which are now associated with, or part of, a university. The various relevant departments in these institutions see their predominant role as one of teaching. In the past there were some research institutions which dealt with mathematics education exclusively (e.g., the Mathematics Education Centre at the PNG University of Technology) or in part (the Education Research Centre at the University of Papua New Guinea), but these ceased operation some years ago. At present, some research is conducted in both mathematics and in mathematics education in Papua New Guinea through the National Research Institute, which has also conducted a number of national and international projects. That is also true of the Glen Lean Ethnomathematics Centre at the University of Goroka. This chapter reviews the various universities and teachers colleges in Papua New Guinea, and their contribution to mathematics and mathematics education teaching and research.

Key Words: Glen Lean Ethnomathematics Centre · Higher education providers · Mathematics Education Centre · Mathematics education research · Mathematics education teaching · Mathematics research · Mathematics teaching · Teachers colleges · Universities

a We thank Chris Wilkins with Benson Mirou, and Deane Arganbright, for much of the content of this chapter. The section on the history of teacher education in PNG is based on the work of Dr Pam Quartermaine. Pam assisted the PNG Branch of the Australian College of Educators and encouraged many staff, including Kay Owens, over many years. She arrived in PNG in 1955 at the age of 21 and devoted her next 38 years to community schools and to teacher education as a teacher, a Teachers College lecturer, an Inspector, and an administrator in the Waigani Central Office. She touched the lives of many students, colleagues and members of the community in Rabaul, Dregerhafen, Goroka, and Port Moresby in such a way that these relationships often turned into lifelong friendships. She had an abiding interest in the needs of females in education and was Dean of Women Students at Port Moresby Teachers College. She worked for the National Government maintaining very cordial working relations with churches which operated colleges and schools. Pam studied education in Western Australia, the UK, the USA and Tasmania and travelled widely, broadening her knowledge and perspective. She left PNG in 1993 to live in Perth, and in 2001 she completed her PhD thesis on Teacher Education in PNG. In 2006 she returned to East New Britain to teach at an international school, but left prematurely because of serious ill health. Pam had a deep and enduring affection for PNG and its people. (Eulogy by Neville Robinson, cited in PNGAA, Sept, 2007). b

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In mathematics as elsewhere, the expansion of knowledge is a dialectical process where neither naive creativity nor critical consolidation can function alone, each of them establishing that the other may become operative. Høyrup, 2002, p. 4041 Introduction Using a historical approach, this chapter reviews mathematics and mathematics education teaching and research in Papua New Guinea up to the present time. In the 1960s Australia recognized that an independent nation would require tertiary qualified graduates to be part of that nation’s higher education structures. It provided a number of scholarships for students such as Bernard Narakobi (a lawyer and politician) and Ron Elias (a businessman and electricity commissioner) to study at universities in Australia. It also provided some scholarships for secondary school education at such schools as Woodbine and Kinross. In addition, it began the University of Papua New Guinea and some national high schools (the first of which was Sogeri National High School) and the Administrative College—which became a separate institution. In the early 1970s, the universities restricted the number of overseas students to 10% of their enrolments. Most of the overseas students were from the Solomon Islands and Vanuatu, with a very few from Fiji, Samoa or Tonga. The Law degree at the University of Papua New Guinea was recognized by Australia but not by Fiji. Violence or perceived violence discouraged islanders going to PNG unless a scholarship required it. During 1986–1987, there were 300 overseas students, half of whom were from the Solomon Islands and a sixth were at the Pacific Adventist University. The universities needed to produce adequate numbers of graduates for the professions, so they set up extension studies with open and distance education courses being provided for adults. Mathematics courses provided a significant part of the post-primary education for many students in the universities. As will be discussed in Chapter 11 of this book, with reference to Musawe Sinebare’s (1999) doctoral thesis, it became apparent in the 1980s that with the number of institutions providing computing courses there needed to be regulation with respect to higher education. At that time, tertiary institutions besides the two universities were under various departments. The government set up a Commission for Higher Education (CHE) which set standards and parameters for the universities and other institutions. In 1987, there were 67 post-secondary institutions (Murphy, 1987). The CHE needed to establish equity, excellence and efficiency, and so a number of institutions needed to amalgamate. CHE was charged with the task of overseeing deletion of courses to avoid unnecessary duplication when numbers were low. All institutions needed to consult with the CHE as well as relevant government departments. There were institutions for fisheries, forestry, nursing, business studies, medicine, law, teaching, and many other professions. Most eventually amalgamated. or aligned with a university, or became a university. For example, in the 21st century many Catholic teachers colleges aligned with Divine Word University; Forestry colleges became part of Papua New Guinea University of Technology (Unitech), and the University of Goroka separated from UPNG. A number of other national departmental institutions amalgamated or became universities. For example, Vudal University was created in 1999 in East New Britain with the charge to concentrate on agriculture. The universities will first be discussed and then the teachers colleges as a group. When free education began, and the elementary schools were struggling to have teachers trained and registered, several provinces or education providers set up their own colleges. However, the government reminded them that all colleges and all courses had to be registered. We provide information on some, but not all, of the colleges, especially in relation to the courses they offered in mathematics and mathematics education. Thanks to Brian Greer for sharing this quotation.

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The University of Papua New Guinea (UPNG)

General Introduction The University of Papua New Guinea (UPNG) commenced teaching in 1966 and today it has approximately 15 000 students. Since it commenced during the period of Australian administration it is no surprise that the structure and the ethos of this university largely mirrored universities in Australia. The various undergraduate degrees were normally three years of study, and in time masters and doctorate degrees became available. Within each undergraduate degree students normally took a major. A mathematics major was normally part of a Science degree, but could also be part of an Arts degree. The Faculty of Education was an exception to the rule. Only starting in 1970, it offered programs for preservice secondary teachers and taught an undergraduate B.Ed. degree. It also offered a graduate diploma for students who already possessed an undergraduate degree, and in time offered higher degrees. There are two aspects of the University that are of interest to this volume—the teaching of mathematics within the Science Faculty and the teaching of mathematics education within the Education Faculty. Mathematics Teaching From the beginning, mathematics has always been a key component of the structure of the Faculty of Science at UPNG. Across the years it has had various names as it broadened to include statistics and then computing. At the time of writing, mathematics sits within the School of Natural and Physical Sciences, and students can pursue majors in mathematics, statistics and computer science in their undergraduate degrees. The units on offer seem to be what one would expect to see in most university undergraduate offerings in mathematics, statistics and computer science across the world. From the start the traditional format of teaching in Western universities of lectures and tutorials has been followed. Throughout much of the 1960s and 1970s most staff appointments were expatriates. One USA expatriate, Deane Arganbright, taught first at UPNG in the mid1970s, returned in 1989 as a Fulbright Scholar, and held the position of Head of Department from 1995 to 2000. He finally taught at the Divine Word University (DWU) in the 2000s (see below for his time at DWU). The following cameo2 has Arganbright reminiscing on his experiences of teaching mathematics at UPNG across those years and provides insight into what and how mathematics was, and still is, taught. Of course, this is only one view from an expatriate. Others, both nationals and expatriates, no doubt, would have various other recollections which would emphasize other aspects of teaching mathematics at UPNG. Cameo from Deane Arganbright Lecturer in the Mathematics Department 1975-1976. After a rather frantic time preparing for my relocation to Port Moresby from the USA, and after a lengthy air journey with my wife and two small sons, my family and I arrived for a two-year stay. On the day after arriving, I began my teaching in what was the second or third week of the second semester. I taught two courses (units) that semester. The first was the mathematics course for preliminary year students. This was taught by one lecture a week followed by a tutorial session for each class. The second course was first-year mathematics. This involved two or three one-hour lectures each week, and then two one-hour tutorial sessions with each class each week. Kay Owens and Philip Clarkson have a full version of Arganbright’s reflections.

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In those early days of the University there were very few senior high schools in the country, and so the University ran a one-year Preliminary Year program for the most able students who had completed Grade 10 in schools. The Preliminary Year was designed to get students ready in one year for the standard University program. As I recall, in the mathematics course students were provided with locally produced and printed materials briefly discussing topics and techniques in secondary-level maths with some worked examples. Each day I would give some explanations and then walk around the classroom helping students as they worked exercises. This seemed to me to work reasonably well—and was not dissimilar to the way that I had been taught. I greatly enjoyed both my Preliminary Year class and my first-year mathematics class. The latter also used locally produced materials and notes. The format was not very exciting, but it seemed to serve well for the time. I was able to maintain contact with several of those students, and from my observations they generally fared well in their further studies and future employment. There are two amusing incidents that I recall from my lectures with the Preliminary Class. Both involved topics dealing with trigonometric and geometric measurements. One day I set an assignment by saying, “Every day in walking from my office in Arts II to the Science Lecture Theatre, I pass that large gum tree outside. Your assignment is to determine the height of that tree.” The following day I saw students using meter sticks to measure the lengths of shadows of the tree and of the sticks, while others used protractors to estimate angles of elevation. But then on the next day, I was greatly surprised to find a number of the male students climbing barefooted as high as possible on the tree, holding a long stick with a rope hanging down, reaching to the top of the tree, and then measuring the rope! My first thought was that if this had occurred in the USA I would probably have been disciplined for having students doing something possibly dangerous. However, in PNG it seemed that everyone was adept at climbing. My second thought was that in the future I needed to be more specific in giving my directions. In any case, those using that approach were still awarded full marks. A week later I presented another task which required indirect measurement. Since the class met in the Science Lecture Theatre, there was a large lab table at the front of the room. So I said, “Suppose that this table is an island, that the floor is a body of water, and I need to know how far it is to the mainland at the door.” So I made some measurements on the table and approximations of angles from ends of the table to the door, followed by calculations on the board in relation to the distance to the door. With about 10 minutes left in the class, I said “Let’s see if this works.” So I jumped on top of the table (quite an unexpected thing for their lecturer to do) and then said, “Now let us see if I can swim to the door in the number of steps that we have calculated.” So I then jumped down and paced off the trip to the door, and. sure enough, it worked out (I had not checked in advance). I then said “Well, I made it to land, and I hope that you too will be successful.” Then I left and did not come back. For a few seconds there was dead silence until they realized that I was not coming back. Then there was uproarious laughter as they exited. While I am generally more serious in my teaching approach, with this group some of these unusual things worked quite well, and I subsequently became close friends with several from the class. One of these was Moses Woruba from East Sepik. I found out years later that he was only six years younger than me, the reason being that in his time for any education he had to go away to a boarding school in PNG, and to do that he had to be old enough to grow his own food. During the last three semesters, I taught Linear Algebra, Calculus, and Operations Research. Most of these units used standard textbooks. Typically, overseas editions of the texts were available at a lower cost. My style of teaching was essentially the same as in the USA and was normal for lecturers teaching mathematics at UPNG—lectures to explain concepts and then tutorials for problem solving, developing the mathematics, and providing applications. My favorite class during this time period was the one in Operations Research for final-year students. Professor McKay (then Head of the Department) had asked me to teach either FORTRAN programing or Operations Research, and I chose the latter. We used a standard respected text, and I had truly superior students. I presented the same level material that I had recently learned at Iowa State University.

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Those two years were in many ways the highlight of my career. The experience of teaching at UPNG opened lots of new doors, and I made lifelong friendships with both students and academics. I felt that I, along with the strong international staff at UPNG, was able to contribute to the educational development of a newly independent nation. Unfortunately, I cannot recall all of the mathematics staff from those years. Professor Max McKay was the Head of Department, and younger Australian mathematic academics at UPNG at that time included Peter Scobie, Lance Bode, Wilson Sy and Neil Roberts. These were excellent teachers, fine colleagues, and continued teaching elsewhere after leaving UPNG. Michael O’Reilly from Northern Ireland was a bit older and had significant experience in Africa prior to coming to UPNG. He later taught in South Africa before moving to Minnesota in the USA, Bernard Duzchek was from Europe and later taught at Bond University in Australia. During my final semester at that time, Raka Taviri, a Papuan, joined the Department after completing a Master’s Degree in the UK. He was the first national citizen to achieve an advanced degree in mathematics and at times served as Acting Head of Department. It was a great time with dedicated colleagues and many high quality students. One last extremely interesting academic in the department was John Renaud. He had come to PNG from Australia as a high school teacher. By 1974 he was a senior tutor at UPNG and then remained there for decades, becoming skillful in computing and increasing his insights into some areas of pure mathematics, ultimately earning a PhD and having papers included in research journals. He was truly a unique and exceptional individual who contributed greatly to the university. One of the things that an American, like me, had to adapt to was the method of awarding of grades at the end of a semester. In the USA, with the exception of multi-section classes taught by several lecturers, lecturers are entirely responsible for determining grades by themselves. However, following Australian traditions, in PNG grades are suggested by their lecturers, and then go through several levels of oversight and possible adjustment by departments, faculties, and the academic board. This took getting used to. Another tradition in most USA universities is that academics usually set their own examinations, and did not have others review or modify them. At UPNG each of us setting examinations had to have another inspect and approve them. I found this to be a very good idea, and I followed it in my own classes for the rest of my career, and strongly encouraged it for others when I served as a department head. In the initial years of UPNG those teaching primarily third- and fourth-year classes were required to write examination questions well in advance so that they could be sent to Australian universities for review. The idea was to maintain an acceptable level for the examinations. Over the years this practice has been dropped, but at the time it served to ensure the quality of the teaching at the young UPNG. Fulbright Professor, UPNG, 1989. At the instigation of the Department I applied and was appointed as a Fulbright Professor at UPNG for 1989. The head of the Mathematics Department at the time was Associate Professor Om Ahuja. In the years since my first time at UPNG, there were significant changes in the academic staff. The Australian and Western academic staff would largely be supplanted by academics from the Subcontinent, India and surrounding countries. It appeared to me to be the case that the primary interest that the Department had for me was to help to integrate computing into the Department’s curriculum. Consequently, one of my first tasks was to write a document outlining what might be done in this area in the future. I also ran short courses for graduate students in computing, designed and co-taught a short course in programming aimed at the surrounding business community, and short courses in the relatively new Microsoft Office suite of programs, especially EXCEL It was important that the courses

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showed how what was taught could be used in mathematics teaching. As well I undertook revisions of a local text that was to be used for first-year mathematics. Among the academic staff that stayed at UPNG for a long time and were there in 1989 were Dr. John Renaud (Australia), Assoc. Prof. Sekkappan (India), G.A.S. Gunasekera (Sri Lanka), and Dr. Don York (United States). Also by this time several high-quality Papua New Guineans had joined the departmental staff. These included: Dr. Cecilia Nembou, Mr. Vincent Malaibe, and Lakoa Fitina. All of these also achieved high executive positions in universities in subsequent years. Professor and Mathematics HOD, UPNG, 1995–2000. During 1994, again at the instigation of the Department of Mathematics, I applied for a three-year renewable position as Professor and Head of the Mathematics Department. I won the position and subsequently served a second three-year period. During my interview, the somewhat more traditional overseas Professor of Chemistry insisted that the way that mathematics should be taught was to start writing on the blackboard from the left side of a room, continuing to fill all of the boards, and then start over. I assured him that while I could teach that way, my style using computers when possible had proven to be more effective in educating students, while additionally providing them with computing skills that were highly valued in the educational, governmental, and business communities. I spent the bulk of this time at UPNG reorganizing and building up the computing aspect of the Department. Apart from teaching in the area we also networked the system and gradually obtained more up-to-date machines. After six years we had quite a robust computing section within the Department. The Department’s computer classes were standard UPNG units, so they contained assignments, tests, and a final examination. From my earliest days of teaching at UPNG, finding appropriate and affordable textbooks was a recurring problem. During the first two decades of the University’s existence it seems that the Mathematics Department was able to rely on a nice range of less expensive international editions of books published in the Western world and sold in the University’s Bookshop. However, over the years that source dried up, and cheaper books published in Asia, some of which were unauthorized versions of other books, came into use. In addition, members of the Department itself were writing and producing local texts. My knowledge of these books is largely restricted to my years at the University, but here are ones for which I still have copies. These were used as the texts for the main campus, and often for remote campuses as well. Often the initial versions of books were later updated and modified by other and newer members of the Department. What I still have are: John Renaud (1990). Elements of Computing. John Renaud & G.A.S. Gunasekera (1996). Foundation Mathematics 1. Deane Arganbright (1995). The UPNG Introduction to Computing. [revised edition in 1998] Deane Arganbright (1998). Spreadsheets for Statistical Methods. Deane Arganbright (1999). The UPNG Introduction to Computing, Computer Applications Software Supplement. Deane Arganbright (1999). Applied Finite Mathematics for Papua New Guinea. Deane Arganbright (2000). Fundamentals of Calculus. Mac Dandava & Billy Kaleva (2003). Foundation Mathematics 1 (revised edition). Nori Fadri (2003). Applied Finite Mathematics, Course Outline and Guide. During this time the Mathematics Department had many well-qualified national and overseas academics who were also excellent teachers. These included Dr. Cecilia Nembou, who had a PhD in Operations Research and Large Data Sets, and who could teach a wide range of units throughout the curriculum, including in Operations Research. Later she became Deputy Vice

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Chancellor and then Acting Vice-Chancellor. She held similar positions at Divine Word University where, before she retired in 2020, she ensured that mathematics research occurred at DWU. Our Department’s statistics offerings were especially strong. Mr. Vincent Malaibe who had earned strong Master’s statistics degrees from two highly regarded overseas universities, in the United States and New Zealand, taught both statistics and mathematics. He later became the University’s Registrar and later again Acting Vice-Chancellor. Dr. Sekkappan had a PhD in Statistics and was a well-organized and first-rate teacher of statistics. Both were well-respected by staff and students. Moreover, graduates who had them as teachers were in high demand upon graduation. Mr. Malaibe and Mr. G.A.S. Gunasekera designed and taught the introductory mathematics classes. They also served on major national committees for National High Schools and university selection. Ms. Nori Fadri also was proficient in teaching first-year mathematics, and later took over the leadership of that unit. Mr. Billy Kaleva, another holder of a Master’s degree, could teach a wide range of topics and became Head of Department after I left. Dr. Donald York, from the United States, was a very dedicated and effective teacher, and able to teach a wide and diverse range of themes in mathematics. He was particularly skilled in computing. At UPNG he and I worked together in publishing a journal paper. Years later we served together on a panel at an annual meeting of the Mathematical Association of America, discussing the topic of living and teaching overseas. On the panel we were able to share our experiences in Papua New Guinea. We remain in contact to this day. John Renaud and Michael Morgan (Canada) were also teaching during this period and Lakoa Fitina (PNG) completed a PhD in Computer Security at the University of Wollongong and later became the Registrar of UPNG. We had little change in staff, providing a sense of stability and continuity. However, many of these individuals also had strong interpersonal and administrative abilities, and the University’s administration later co-opted a number of them to work in the administration, where they advanced rapidly, as I have noted. This was good for them and for the central administration of the University, but it caused the Department to continue to rely on overseas appointments for some years. The present day. The inclusion of statistics and computer studies into the teaching purview of the Department during the 1980s and 1990s established a reputation for the Department which has remained to the present day. It would appear that all staffing positions are now filled by nationals. Fitina has returned as Professor after a number of positions in the other universities and at least one staff member is enrolled in a PhD program. With Fitina as Professor, the number of research projects is likely to increase. Faculty of Education From 1970 the Faculty of Education enrolled students who wished to become secondary teachers. Primary school preservice programs were run by various teachers colleges (see below). Students enrolled in either an undergraduate degree in Education, or a Graduate Diploma program after they had completed an undergraduate degree. Clearly notions associated with liberal arts approaches informed these programs: First, students studied at some depth their specialist subject areas in their early years of their B.Ed. or in their undergraduate degree if doing a diploma. Then the later education units were built on that understanding, and through some general education courses with specialist curriculum methods courses the students would be prepared to teach in schools. Sadly, many students (both in PNG and Australia, for that matter) saw the Education component as an add-on to the real learning that took place in their major area of subject specialization. Hence most students at the conclusion of their studies went to schools having given little thought as to what had been taught in the Education units, and wishing to start teaching their

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subjects as they had been taught at University, or how they had been taught in their own schools during pre-University days. From the inception of UPNG the Government urged the University to begin teaching Education, and later to increase the number of Education graduates given the number of secondary schools that were opening throughout the country. The lack of secondary teachers was a crucial issue, with virtually all teaching being undertaken by expatriates. However UPNG was unable to meet either demand mooted by the Government. Hence the Department of Education determined that the Goroka Teachers College (later the University of Goroka, see below), which had started as a primary teachers college in 1961, would change to teach a program for secondary teachers. Hence, for some years after 1967, Goroka was the only provider of secondary teachers. This story will be taken up below when discussing the University of Goroka (for more details see Guthrie, 2001). During the majority of time that UPNG taught Education the Mathematics Methods lecturers were expatriates, mainly from Australia. Most had already taught in this area elsewhere and hence much of their units mirrored what was taught in Australia. Given that most of these lecturers only remained in PNG for short periods of time, little crafting of their teaching reflected PNG culture and society. At this time, the National High School teachers continued to be taught in a Diploma of Education program at UPNG when Goroka Teachers College and the secondary school teachers formed part of the UPNG Faculty of Education (see University of Goroka for that development). The Education Research Unit (ERU) was established at UPNG in 1974 with Professor Sheldon Weeks as the foundation Director. Weeks came to the ERU from a university position in East Africa after completing graduate studies at Harvard University. Within the structure of the University the ERU was a research center aligned to the Faculty of Education. It held an annual conference with an accompanying set of refereed conference papers. The conference became known at an international level. Within the conference proceedings, however, only a few papers dealt with mathematics education—there was a much greater focus on science education papers, many of them looking at Piagetian stages. More recently, Weeks published papers on PNG education from a sociological perspective (see, e.g., Weeks, 1989, 1990, 1993).

Papua New Guinea University of Technology (Unitech)

General Introduction In the early 1960s, the Australian Federal Government was becoming seriously concerned about post-secondary education in PNG. A committee chaired by George Currie recommended that an Institute of Higher Technical Education be established which would be self-governing with Council responsible for operations to the limits of its finances. Unlike UPNG, which was a liberal arts university from its inception, the Institute began life in 1965 by an Act of the House of Assembly. It was originally named the Papua New Guinea Institute of Higher Technical Education and was sited in Port Moresby. This went some way to meet what the Government and industry saw as a need for practical technological education in the country. In earlier chapters this has been noted as a recurring theme from early colonial times. It was intended that the Institute would be independent but close to UPNG, a liberal arts university, and somehow expected to share certain facilities like residences and library. The Institute’s first Director, Duncanson, suggested that the planned form of co-operation was unworkable. He also realized that the idea of having the highest qualification as a diploma and not a degree was to not learn from other countries’ mistakes. It would not ensure a flow of professionals and could lead to the necessity to depend on the employment of professionals from overseas. The assumption had been that diplomas were all that the PNG population were capable

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of achieving and would meet the technical needs of the country. It would rely on the employment of expatriate professionals, but did not give due consideration to the nearness of independence or to the availability of expatriate professionals at that time. This colonial thinking was short-sighted but fortunately Duncanson with the support of the Chancellor, Alkan Tololo, and the Chairperson of Council, J. A. Louis Matheson, was able to secure legislation by which first an Institute of Higher Technical Education and then a University of Technology would be created. The first mathematics courses were developed in a rush for the start of 1968 with very new staff from overseas. It was hard to recruit a suitably qualified Head of Department. The initial intake of 31 students in 1967 included 11 in Civil Engineering, 14 in the first year of Surveying, and 6 into the second year of Surveying (the Surveying course moved with its Head from the Department of Lands). Another issue was the housing of the Administration (The Director, Administrative Officer and a Secretary shared one small office) at a Technical school which had two heads. The Australian Public Service system allowed an incumbent to appeal an appointment which meant that for a while there were two heads but this other person would not negotiate making things difficult. Fortunately, after the appeal, Duncanson held the position. The Institute moved to Lae in 1968 solving many problems, but no buildings were ready for the start of the academic year. Students were accommodated at the showground and staff in Government houses in town. However, some staff still had to commute from Port Moresby— although the early morning pickup sometimes failed. Choosing to develop the Institute on the far side of the Busu River from the town caused problems. When the causeway across the Busu River disappeared in a flood, students were at first unwilling to help, but with staff pitching in they also moved stones to repair it. At the time they were not aware it would again be washed away a number of times (of course, long-term residents of Lae knew that this would happen) before finally the Australian Army built a concrete causeway as the Army Barracks were on that same side of the river as the Institute although further out of town. Monies from Canberra were generally not enough for the development of the new campus. Staff were called on to do far more than their job descriptions but the camaraderie that built up between staff and students to make it all work was strong. For example, Neville Quarry, Head of Architecture helped in various aspects of design of buildings and general layout of the campus. A long-term resident horticulturalist, Andrée Millar, managed to provide continuity by planting the gardens. The planned dual carriageway entrance never occurred but a Utility building was quickly designed and built. Matheson was hoping that the student accommodation built around a quadrangle would encourage communication between students from the various provinces throughout their years of study. Fortunately, a couple of buildings in the early years were able to reflect PNG culture, the Haus Kopi, the doors of the Architecture and Business building, and the columns on the large lecture theatre. Later murals on the Haus Kopi and library also assisted in reflecting PNG culture. Engineering was to be the key component of the University’s academic agenda. Planning was difficult with many matters having to be referred to the Australian Government in Canberra, and/or the House of Assembly, if changes to the Act were required. Nevertheless, Duncanson was able to set up separate degrees for three different engineering areas on the grounds that the level of incoming students’ backgrounds and the differing amounts of knowledge required for civil, electrical and mechanical engineering meant they should not be combined in one course. He overcame issues associated with a lack of facilities, clearly underestimated by the Currie Committee, and the unsuitability of establishing the institution next to UPNG. He needed high attaining graduates from schools, but the higher achieving students continued to prefer to go to UPNG, and over the years continued to do so despite the quality of the degrees at Unitech. Very soon (1970) what had started as an Institute of Technology was upgraded to award degrees in Technology and Commerce and called the Papua New Guinea Institute of Technology and in 1973 a further amendment of the Act changed the name of the new institution to its current title.

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One advantage of starting as it did was that the courses were practical from the beginning. Furthermore, the interest of staff and students to assist in the development of the rural areas of PNG, and the creation of testing laboratories such as that for civil engineering and later the chemical and biological analysis laboratories, meant that a number of villages and businesses in PNG were quickly served by the University. From 31 in 1967 to 950 in 1975 to 1500 in 1977, it now has about 3000 students enrolled. In 1972, the 19 graduating students included two expatriate students and a female, Cora Mati. Its first female degree graduate, Taita, in 1974 was pictured on the 1975 calendar (Figure 7.1). One of its early graduates (1973), Sam Andrew, worked for a private firm of consulting engineers Willing & Partners; he would subsequently complete a Master of Science in civil engineering at the University of Dundee, Scotland, and would serve as an academic for a short time at Unitech. Unitech’s second Vice-Chancellor was Dr Sandover but during his time, despite excellent staff like Professor Jack Woodward, expansion and increasing numbers, new buildings, more staff, new degrees, an impressive Artefact Collection, improved curricula, and the establishment of the Mathematics Education Centre, there was unrest by the students and staff (even to the extent of having The Retorter—the official weekly news sheet was The Reporter). This was probably unavoidable, given the challenges of self-government then independence, lack of qualified nationals for the staffing needs, quite different approaches to administration of the first two ViceChancellors, a new faculty, and technician level courses, contract-only staff, and the change from Institute to University. The government then appointed national Mat Tigilai as Vice-Chancellor who brought the university students and staff together but at the end of his contract, he was

Figure 7.1 (Left) First female degree graduate from the PNG University of Technology; (Right) Sam Andrew an early Civil Engineering graduate and academic.

(Source. PNG University of Technology) (Source. Lae Nius, 4(47), p. 1)

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expected to return to the public service, which he did. The fourth Vice-Chancellor was Alan Mead, another expatriate, but one who achieved much. First, Mead gained respect from staff and students, negotiated with the national and nonnational unions, developed a single pay scale for all staff, but with additions for expatriates, developed and raised funds for extensive upgrading of qualifications for national staff through internally run courses (especially for administration staff) and overseas degrees. In particular, he appointed three national Deputy Vice-Chancellors, one of whom—Moseley Moromoro—was to succeed him after completing a masters degree overseas. In 1980, 15 national academic staff took up postgraduate studies overseas, with 25 the next year, then 33, and then about 43 the following year. This resulted in a significant upgrading of qualifications. He restructured the Registrar’s department and reduced committees, and expanded the academic offerings of the University by bringing the Faculty of Agriculture to Lae from UPNG. There was now Forestry and Food Technology in Natural Resources, and the Appropriate Technology Development Institute was placed on a firm footing. Mead considered the Government’s National Planning needs, working well with the Office of Higher Education. Finally, he increased the cultural elements of the graduation ceremonies with a local dance group leading the academic procession, encouragement of national dress as well as academic dress, and a party, celebratory atmosphere for graduands, staff and families. Importantly, he increased the professorial and research levels of the academic schools with appointments such as Rex Coates in Civil Engineering and Ken Lyons in Surveying which was moving into the digital age. Other staff rose to the occasion such as Andrew Taylor in Languages, Eric Cousins in Civil Engineering, Walter Wong in Mechanical Engineering, and Sid Patchett in the library (Mead, 1988). Moromoro’s leadership was exceptional in the following years until he returned to industry. Unfortunately, the finances of the new country and the need for monies to go into universal education rather than universities saw the closure of the Mathematics Education Centre. Misty Baloiloi was another graduate of the University who returned to lead the University when monies were a major problem. During his time, he had study leave to complete his doctorate. The first national Registrar was Tess Chan, another Unitech graduate. By 1990, the University was being run by Papua New Guineans. They were still recruiting academics from overseas, sometimes with considerable debt to the Government. For example, staff from Mid-Western Universities, USA, were appointed through a loan from the World Bank that the Government had to repay. Repaying this loan, along with the loss of monies supposedly being diverted to community school education, made it difficult for the University to pay staff. There were some further issues with staffing, and another expatriate ViceChancellor who worked well with staff and students. The University has continued to develop and provide a high standard for the nation across all the technological professions. Mathematics at the PNG University of Technology Mathematics topics were selected for the particular technological profession once a sound review of school mathematics was undertaken in the early years of the university in the preliminary year and with links to technology in the first year. Hence quite advanced mathematics was covered especially for the engineers. Staff. Mathematics at Unitech began as part of the Department of General Studies and that continued until the early 1970s. Its first Head of Department was Michael Deakin, seconded from Monash University, and he was followed by R. Morris Leigh who brought a technical background, then Associate Professor David Tombs followed by Associate Professor Peter Jones who had previously been a member of the Mathematics Learning Project and came from Swinburne University where he returned to become Professor of Mathematical Science in Statistics.

Professor Kathleen Collard followed Jones. She came from the UK and was highly respected during her time from late 1983 to 1987. She encouraged relevant problem solving and also research. She had been decorated for her service to students with disabilities and had also

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encouraged females in mathematics and had worked in Africa. She was followed by the longserving Associate Professor Glen Lean (1989–1990), who was writing his doctorate on PNG counting systems, and then Professor Ted Phythian. Phythian ensured the Computer Science degree was well started. Dr Chris Wilkins has been the longest serving academic in the Department and possibly the University as an expatriate, and he gave substantial input into the statistics section and later the computer sciences. Two long serving secretaries were Agnus Solo and Betty Agum, with Ginom Naya and Sogaedi Idagaela providing the printing materials, the last three serving for more than 30 years. Henry Korim and Janos Suagotsu, Wilson ToVorika (who became Pro Vice-Chancellor Administration), and Atawe Koigiri (who also had a stint in National politics) were early national lecturers. Kathleen Collard, John Lynch and Terry Fairclough prepared Samuel Kopamu, John Gesa and Geori Kravia to complete final year mathematics at UPNG and then to obtain masters degrees in mathematics in overseas institutions. John returned and lectured at Unitech for many years. Sam gained a doctorate and taught at UPNG and Unitech, became Head of School at University of Goroka, and briefly Acting Pro Vice Chancellor then Research Director at Pacific Adventist University. Benson Mirou has been a long-term national lecturer and Head of Department. In the first part of its growth expatriates came from many different countries (Poland, USA) providing a wonderful mix of experiences and expertise. Some had come from developing countries (e.g., India and Philippines, e.g., Lilian Semille-Dube) or worked in other developing countries. The many Australians, New Zealanders and British staff were interested in people and cultures as well as mathematics and maintained long-term friendships among the staff and students. During the 1990s, the Department offered degrees in Computer Science and this became the focus of the Department although it continued to service the mathematics subjects, with computer usage for the technologies as a focus. Chapter 11 gives further details about computer technology in PNG. Mathematics Teaching The Mathematics Department at Unitech was one of the foundation departments of the institution. From the start, staff in the Department provided service mathematics teaching to all degree programs that were housed within the other Departments. However, within five years of the formation of Unitech it became evident that students were not coping with the traditional ways of teaching mathematics in tertiary institutions and they often struggled with content knowledge. In an attempt to rectify this, the Mathematics Learning Project (MLP) was brought into being. The Project was set up in 1972 with the aid of the Nuffield Foundation, the R. E. Ross Trust and other philanthropic bodies. Its core aim was to appreciate and manage the diversity of students’ mathematical ways of learning. They then developed and published in-house text material for the teaching of mathematics, based on the ideas of mastery learning and modified programmed instruction while keeping the topics holistic. There were several equivalent tests to complete for each unit so students could revise if need be. Glen Lean, a long-term member of the Mathematics Department, was the key person who lobbied for the project but it had the whole-hearted backing of departmental staff. Early evaluations at the University showed marked improvements in results (Booth, 1975; Jones, 1975). Being part of a first study into how Papua New Guinean students learn mathematics, the self-paced approach was also used at several provincial high schools but the students there made less improvement (Progress Report, 1975 for the Mathematics Learning Project). At the end of Project funding, the Mathematics Education Centre (MEC) was developed and funded mainly by the University with some initial funding coming from the British Council (see section below). Problem solving was beginning to be a focus of mathematics learning and

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Dube (1981) undertook to study how students at the University took to learning through problem solving. She found that the group problem solving success rate was near-perfect and the end-of- session examination had a high success rate. Many short units of work for each mathematics subject taught in the preliminary year (entry students from year 10, as in UPNG) or year 1 were devised by the project staff and departmental staff such as Robert McKibbon, Pak Yung, Glen Lean, Peter Jones, Lesley Booth, Stuart Hosking, Duncan Rasmussen (seconded from Coburg Teachers College—which later became part of RMIT), and Peter Jones seconded from Swinburne Technical College (see above as he became Head of the Department, 1978–1982). For the publication of the units a small print room with two offset machines was set up and printed the units to order. The early versions of the units were short with students working on them individually; they went from one to the next within a subject and students were deemed to show mastery if they gained at least 90% on a short test for the unit (they could revise and sit for a second or third equivalent test, if that was necessary in

Figure 7.2 Graduation PNG style at PNG University of Technology, 2003.

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this self-paced system). However, as time went by, revisions of the original units moved away from the initial approach. One module of work was devised for the whole of the term, answers were provided at strategic points for the various exercises students were to complete, and the group tended to move through the unit as one group. In effect the units became more textbooklike. Nevertheless, since the material was under a rotating two- to three-year revision cycle, as the competence of the beginning students rose, the units were modified accordingly, and the writers were always able to adapt the content to new and relevant material within the immediate knowledge of the students. Importantly, each mathematics course was geared to the technology it was intended to serve. This meant that different units needed to be prepared for surveying, accounting, architecture, natural resources, communications, electrical engineering and other engineering students. In late 1983, the new Head of Department and Professor arrived. Kathleen Collard had had a distinguished career as a mathematician. After studying with Max Born, she took up a post at Oxford, then as Head of the Mathematics Department at the University of Ibadan, Nigeria. Later, she returned to Nottingham University, before retiring and moving to PNG. She began by taking the Department away from the use of small booklets (noted above) and introduced the notion of real problem solving into the work that students undertook. She also lobbied for and in 1986 began the teaching of the Post Graduate Diploma in Engineering Mathematics, which became the first stand-alone program that the Department offered. The tradition of staff-developed textbook material is still noted in the most current website for the Department3, although the principles of mastery learning and self-guided units of work are no longer used to frame the teaching approach. Now, the normal university format of lectures and tutorials predominates. As Bray (1987), Murphy (1987) and Wilkins (2010), have pointed out, the universities were set up without much consideration being given to future costs or what they would be expected to achieve. Thus, as Wilkins points out, the self-paced learning modules were somewhat reliant on small staff-student ratios. According to Murphy (1987), the Australian government was keen to provide funding if that would generate a large number of graduates. However, the need for professionals in the country also meant that lower levels of recruitment and slow progress were tolerated for too long. Some of the modules were modified to contain more relevant problems for a particular technology or for engineering. Appendix 4 provides examples, 1970s–1980s—Unit 131 Formulas 1970s–1980s—Unit 245 Errors 1985—MA141 Algebra Revision—Chris Wilkins 1986—Mathematical Modelling for Architects combined a number of smaller modules on statistics (descriptive statistics, probability, t-tests, multivariate analysis, chi-square analysis) and added new areas such as queuing theory, statistics and design – Kay Owens 1986—Geometry and Trigonometry for Architects. This module covered symmetry, conic sections, 3-dimensional drawings, mensuration 1 and 2, polar coordinates, 3-dimensional geometry applications to sun and wall directions, vectors, center of gravity, first moment of force; the centroid: first moment of area; theorems of Pappus: surface area and volume; stability; shear and bending moments; second moment of area; perpendicular and parallel axis theorem, applications of second moment of area: stress under compression, bending stress, deflection – Kay Owens 1988—Mathematical logic and flow charts for Quantitative Methods 1 Module 1 – Chris Wilkins

Website of the Unitech Dept. of Mathematics and Computers: accessed on 6 May, 2020 https://www.unitech.ac.pg/ departments/mathematics-computer 3

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1990—Mensuration and proportion via problem solving for MA180 Mathematics for Natural Resources—Rod Selden 1991—Algebra, Calculus, Word Problems for MA111 Quantitative Methods 1 During the remainder of the 1980s and throughout the 1990s the offerings of the Department increased, but in computer science rather than in mathematics. Given mathematics graduates were coming from UPNG it was deemed unnecessary to have two such programs; hence the move into computer science where there was no undergraduate programs in the country. During this period the other critical issue was the burning down of the Mathematics Department building and all its resources in 1999, and the building would not be replaced for more than 10 years. During this time, the staff had a small office, a borrowed computer laboratory and classrooms and offices scattered elsewhere. It was a sorry sight when Owens visited Unitech in 1999 and Clarkson visited in 2001. Many other details of the development and history of the Department, noting the many successful graduates and national staff that have taught there, are given in the history written by Dr. Chris Wilkins4, himself a member of the Department for some 40 years. It should also be noted that although Chris does not refer to his own immense contributions to the teaching of higher mathematics in PNG across this long period, it should be acknowledged that he has kept the place going along with Ben Mirou during this period of time. Today the Department continues to teach a great range of units required by other university departments as well as offering its own four-year undergraduate degree in computer science begun in 1993, as well as studies at masters and doctoral level in both computer science and mathematics. There was difficulty getting scholarships for nationals to complete higher degrees overseas. However, after the fire and 10 years of temporary accommodation in which a number of staff left for better situations, they had a new building which barely replaced the out-of-date building and was too small—with no gains for the staff. It was the Chinese who finally built the replacement. Staffing fluctuates, with some appointments staying for longer than others. Staff have managed to achieve funding for masters degrees, but not necessarily for doctoral studies. From frustration at the lack of support for scholarships overseas, especially in Australia, Benson has now started his doctorate working with lecturers in other Unitech departments. Research continues with research papers published each year. Heavy teaching loads limit the amount of time staff has for course development and research. However, besides applied mathematics interests, theoretical mathematical interests of staff include topological rings, groups and lattices. Some members of the Department are working in cryptography. Supplementary to this, the Department has reactivated weekly departmental seminars in which new results from research are discussed. (This was a feature of the Department through the 1980s up until the fire.) Senior staff members are giving lectures for young members of the Department, and postgraduate students are engaged in a wide range of research projects. Mathematics Education Centre The Mathematics Education Centre (MEC) was the successor to the Mathematics Learning Project (MLP) (see above), but had a more diverse set of aims. The establishment of the MEC was made possible with aid from the British Council, over the period 1978–1979, which funded the salary of the initial Director, Alan Edwards. Throughout its life the Centre worked closely with the Department of Mathematics. However the Director always reported directly to the Vice Chancellor. Website giving the history of the Department of Mathematics and Computing, Unitech accessed on 6 May, https:// www.unitech.ac.pg/?q=departments/mathematics-computer/about/history 4

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Initially, as well as the Director, the Centre staff were a secretary (initially Carrie Luke), and a resource manager (Dan Luke an ex-secondary school teacher)—both Dan and Carrie have contributed cameos to this volume. There were two printers in a print shop which was moved from the Department to the Centre. As had the MLP, the MEC continued to work with the Department on the production and printing of many unit booklets used in the preliminary and first-year mathematics units. However, the MEC broadened its scope by working with the local schools, the technical colleges and the education colleges. It embedded itself within a number of the national Government departments by providing professional development opportunities for their staff. In 1980 Philip Clarkson came from Monash University to succeed Edwards as Director of the MEC. From then on the MEC was a fully stand-alone research center of the University, within and funded by Unitech. Its brief was not only to continue its various roles up to that point, but also to carry out mathematics education research at all levels of the education system. This meant that the Centre was to provide advice and assistance to tertiary-level institutions (including the two universities, teachers colleges and other tertiary training colleges). Of particular note was the assistance it provided to the Schools of Nursing, the four National High Schools and, as it had done in 1978 and 1979, to schools in the Morobe Province as well as technical colleges, the local Balob Teachers College, and the governmental department education units (Clarkson, 1985). Other University departments, such as Chemistry and Physics, also supported the National Department of Education (NDOE) and the National High Schools. During this time, as well as the staff appointed earlier, a joint appointment at lecturer level with the Department was made—namely Peter Sullivan, who has since published widely in mathematics education, and was appointed Professor of Mathematics Education at Monash University. Two Research Officers were also appointed—Wilfred Kaleva and Naomi Wilkins. Wilfred later obtained his doctorate from Monash University, lectured at UPNG and UOG, became the First Director of the Glen Lean Ethnomathematics Centre at UOG, then Head of the Mathematics and Statistics Department at the University of Goroka, before taking up a Research Officer position with the National AIDS Council Secretariat. He undertook many consultancies in the Department of Health and the Department of Planning with his statistical and research background. Naomi Wilkins upgraded her Diploma of Teaching to a BEd through UPNG and wrote an MPhil thesis on educational policy. She later rose to be Deputy Registrar of Unitech. A number of staff within the Department also worked with the Centre on various research issues, providing professional development sessions for teachers and participating in the examination process for the National High Schools. These staff included Kay Owens, Rod Seldon, Chris Wilkins, Peter Jones and David Shield. Among the various international visitors to the Centre were Claude Gaulin from Canada; Kath Hart, Susan Pirie, Alan Bell and Brian Wilson from England; Beth Southwell (on three occasions), Ken Clements, Gilah Leder and Peter Clarkson from Australia; and David Lancy from USA. The Centre published two series of reports; the research reports series and the technical reports series. Annual research conferences at Unitech which focused on mathematics education were also hosted and included published conference proceedings. Other conferences were also held in conjunction with other entities; for example, in July 1983 a Problem Solving Through the Curriculum conference was jointly conducted with the National Institutions Division, NDOE. Professional development workshops for technical colleges and other colleges housed in various Government departments, initially begun by Edwards, were continued. Rather than simply taking school mathematics and encouraging the teachers to teach that, we encouraged them to analyze just what mathematics was needed for their area of work and teach that with specific work-related examples (Clarkson & Sullivan, 1982). With impetus from Pirie on nursing mathematics, Sullivan went further with these notions and working with the Schools of Nursing (Sullivan, 1981, 1984; Sullivan & Clarkson, 1982), he carried out research into the mathematics

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of nursing and wrote his MPhil thesis based on that work. This was a thorough look at current practices (nurses fulfilled many more roles in PNG than in Australia), areas of difficulty, how best to present the mathematics to nurses so it was understandable, motivating, effective, and relevant. Kay Owens later extended this into an inservice series that was trialed in six hospitals and was reprinted with UNESCO funding for the Health Department over many years (Owens, 1986–1987, 1987). Other details of the Centre’s operation, particularly the research carried out and the close cooperation forged with the NDOE, have already been given in Chapter 1 under Clarkson’s contributions to mathematics education in PNG. Although the position of Director was advertised internationally after Clarkson moved to Australia in 1985, no substantive appointment was made. The Centre finally ceased operations in mid-1986, due to financial constraints within Unitech. Mathematics and Mathematics Education Research at Unitech Mathematicians during the 1980s and 1990s continued with their fields of research. John Lynch, Kathleen Collard, Terry Fairclough, Phil Sheridan, Don Lewis, Bob Lockhart, Steve Bedding, and Graham Snedden produced research papers. Others like Basil Landau, David Shield, and Rod Selden also shared the findings of their investigations in seminars. As Peter Jones said, “I was able to recruit staff with doctorates who also had teaching degrees and were excellent teachers.” Following the introduction of the PNG Journal of Mathematics, Computing and Education, many mathematical papers were prepared by numerous staff, especially staff from overseas. Research on visuospatial reasoning (including spatial abilities and visualisation), and calculators and computers in mathematics learning is summarized in Chapter 11. Mathematics education research was particularly strong with at least 33 research reports and 11 technical reports being published, as were the first three conference proceedings on Research in Mathematics Education in Papua New Guinea. Other books were produced by the Mathematics Education Centre (see Appendix 4 for some titles). In 1979 the Mathematics Learning Project supported a special issue of the PNG Journal of Education which included a wide range of papers contributed by many authors. Publication was supported by UPNG’s Education Research Unit and by the Department of Education. Lesley Booth carried out research, particularly on algebra. The mastery learning approach taken to remediate post-secondary students’ basic arithmetic was particularly important for Unitech (see e.g., Sullivan, 1983). One of the interesting reports was Clements and Jones’ (1983) description and analysis of the education of Atawe Koigiri who was a lecturer at Unitech, and later a Member of Parliament. The paper provided details on Atawe’s upbringing in a remote part of the Eastern Highlands and how he achieved in mathematics despite the apparently limited counting system of his cultural group. Clarkson wrote several reports on what kind of errors students made, how these differed from Australian students, to what students attributed their success or difficulty with mathematics, and whether this related to their errors (e.g., Clarkson, 1983b, Clarkson & Leder, 1984). Another area of particular interest to researchers was that of language, especially learning mathematics in a second language. This area is covered extensively in Chapter 10 of this book (see, e.g., Jones, 1981, and Clarkson’s work listed in this later chapter and elsewhere in this book). Gender Studies There was also some research by Naomi Wilkins and Penny Murphy, and others, on female students and policies. It is worth noting some findings from these studies. Murphy (1987) noted that at the time official widespread and centrally controlled formal education in PNG was only three decades old. Murphy continued:

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Despite this, approximately 64 percent of the relevant age group are now enrolled in primary schools, 17 percent in secondary schools and 5 percent in higher education. … 15 000 in higher education … about one third … were female. Ninety-one percent of all students were supported at public expense … either through scholarships (80%) or public sector employers’ training allowances (11%). (pp. 83–84) Equity of provincial representation5 was largely overcome at the University but this was not the case for females who were more likely not to commence or to remain in school due to the force of traditional roles and attitudes. These were exacerbated after Independence since the teaching positions were nationalized at primary and secondary level. The role model of male teachers was also demeaning of women to the extent of being violent to women (Cox, 1982). At the tertiary level, women may not have been allowed to speak or their ideas were undermined in various ways. The chances of women’s rights to education and within education roles were not well appreciated except by tertiary-educated women. Keeping women at University through education was also difficult. An attempt at Unitech through discussions on the importance of relationships, the role of and consent for sex, and family planning ran for several years. The writing and issuing of Lifestyles for Tertiary Students (Owens, chairperson, 1987) was reprinted yearly until 19966. Other Departments at the PNG University of Technology The Department of Mathematics and Computing continues to provide mathematics courses for different degrees, they themselves producing graduates who need to be strong in applied mathematics. They cover an extensive range of Engineering (Mechanical, Electrical, and Mining) as well as Surveying and Valuation, Architecture and Building Science. In addition, the Appropriate Technology Centre for Development and Innovation (ATCDI) continues to provide technology including mathematical understanding, to many in PNG. Its first Director was Lukas Romaso, a Unitech graduate. It publishes and provides books appropriate for PNG such as LikLik Buk on health and Bridges of Papua New Guinea. They often work with the other departments including the Mathematics, Computing and Applied Physics Departments where solving real problems becomes a focus for students. Hydroelectricity was provided to Fasue and after floods destroyed this system, it was restored in 2015. More recent projects have included the Mongi River and two computer science students (females) working on a hydro scheme. Another arm of the University is the South Pacific Institute for Sustainable Agriculture and Rural Development (SPISARD). In time, Vudal College developed into a university (see below). Forestry also concentrates on adding value to resources through manufacturing and sustainability.

It should be recalled that some remote and Highland areas were not receiving education as early as Port Moresby or Rabaul or other coastal cities. 6 The Unitech Women’s Group (national staff and students supported by expatriates) initiated these discussions led by health educator and mathematics lecturer Kay Owens and the National University doctor and counsellors. It was supported by many staff and allowed for small group discussions involving many students. This worked well until sadly some male PNG staff diminished the effect by their sexual violence against women. The book was distributed for about a decade, when data on sexually transmitted diseases was by then out of date—the book provided the first information about HIV AIDS, but records were not being kept or action taken for about 10 years in PNG so without those data, the book clearly needed updating. There were also no facilities for national staff or students to have care of their babies near the lecture area, so they were dependent on relatives who often lived in their small one-room accommodations with them, often at a distance. 5

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University of Goroka

General Introduction As noted above (in the section on UPNG) the University of Goroka started life under the control of the Department of Education (DOE). There were requests from DOE to UPNG to begin teaching in education from its foundation, and when they finally did begin, the quest to ensure that the number of secondary teacher students completing their courses met the country’s demand for secondary teachers, created tension. Hence, in 1967 Goroka Teachers College changed from enrolling preservice primary teachers to enrolling preservice secondary teachers who took a two-year diploma program aimed at qualifying them to teach years 7 through 10 in secondary high schools (Guthrie, 2001). Goroka, in 1972, wanting to increase its profile, began talks with Unitech on a possible amalgamation. Startled by this occurrence UPNG also entered negotiations with Goroka, which resulted in Goroka joining with UPNG and becoming a quasi sub-Faculty of the relatively new UPNG Faculty of Education. However, given the difficulty of travel and lack of funds, there was never a true amalgamation. During the early 1980s when Clarkson regularly visited both UPNG and the Goroka Teachers College, it seemed that Goroka was operating quite independently of UPNG except for staff records and salaries and that neither the University nor the College staff wanted it any differently7. Cameo from David Shield I had been teaching in Applied Mathematics at the Australian National University before I went with my wife, Jenny, and family, to teach at Goroka Teachers College in 1976, not long after Independence. We were there until 1979. Students studied in three main subject areas for Grades 7 to 10, Mathematics, Science and English: hence, they graduated to teach in one or more of these areas. They could also do Expressive Arts, Home Economics, and Social Studies. They also studied a subject called Professional Studies (Education). Many students came from Grade 10 and completed a preliminary year before doing the two-year Certificate course for secondary teaching. Ross (see Ross, 1982, 1984) was giving his test that he used in Africa to students on entry—I’m not sure how that made them feel but it seemed to me to be mainly testing English comprehension. He administered this test for many years. I loved teaching both the mathematics and mathematics education subjects. We had new books prepared by the Department under the guidance of Britt Roberts – Maths Our Way. It was so well connected to PNG with practical introductions to the topics. This meant we could use concrete examples and materials for teaching, basically demonstrating how to teach well, and use the textbook well. Area started with comparing the extent of two regions and reasoning whether they differed in size—both were covered with thumb prints and a decision was made on which was “the bigger.” This emphasized that the thumb print was being used as the unit of area as a measure of area and how that was related to multiplying the lengths of the sides of the rectangle. We compared this with village approaches which were by eye, using the diagonal or adding the length and breadth of a rectangle. The students did not need to do higher levels of mathematics like calculus because they would not be teaching that—calculus was a topic in the National High Schools curriculum. In 1980 I went from Goroka to the Mathematics Department at Unitech. We continued to use concrete materials for teaching but the level of mathematics was higher. There were many materials in the Mathematics Resource Centre that were developed for the Mathematics Learning Project and the Mathematics Education Centre. I liked to show with the simple rectangular off-

Staff salaries continued to be paid out through a clumsy system at UPNG Port Moresby for decades.

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cuts of wood how you could build an arch keeping the center of gravity over the previous block (Personal communication, May 2021). The Next Period of Development at the University of Goroka The pressure from the Government for more secondary teachers did not let up and finally in the late 1980s to the early 1990s UPNG had to act. It did so in 1992, by amalgamating the Faculty and College, and planned to move the Faculty from the main campus in Port Moresby to Goroka, against staff opposition on both sites. With amalgamation, a degree course was added in 1995 to the suite of programs offered at the College, with higher degrees following in later years. However that was not the end. There had been growing political pressure for the Highland Region of PNG to have a university established there. This finally led to the re-badging of Goroka to become the University of Goroka (UOG) in 1997. Today UOG teaches some 4000 students in various programs8. Besides teaching undergraduate and postgraduate programs in education, UOG also teaches undergraduate programs in Arts, Business, Science, Health, Agriculture, and Hospitality among others. There is a steady flow of research projects in all areas, with ethnomathematics (Kaleva, Matang, and Muke and visiting scholars), mathematics curriculum (Kaleva and Matang) and lattices (Kopamu) emanating from the Division of Mathematics, Computing and Ethnomathematics. Mathematics and Mathematics Education Teaching In the early days of the College, teaching followed the style used in the primary colleges and schools. Students were allocated a group and that reasonably small group moved through the timetable in school-like tutorial type teaching. However, as the number of students grew, the style of teaching reverted more and more to the lecture and tutorial mode used in most tertiary institutions. From a mathematical point of view this meant that students had to rely more on their own study abilities, which were not always good for learning deep knowledge. Funding issues also eventually led to less practicum experience, which had been the hallmark of the programs at UOG and had been acknowledged as a real strength of their programs by the schools. By 2016, it consisted of an early teaching experience assessed by a university staff member who attended one of the student’s mini-lessons at a nearby secondary school, followed by a practicum in their final year somewhere in the country that may or may not be assessed by a staff member. Since 2015, mathematics and mathematics education units have been taught by the Division of Mathematics and Computing. Ethnomathematics was added to the title of that Division, although the Centre had been established in 2000 within the School of Science and Technology. Students can complete a major in mathematics within a four-year B.Sc. undergraduate degree. The various units in the program are like those in similar degree programs in other institutions9. A two-year B.Sc. program in Mathematics Education is also offered as an In-service degree. This is pitched at secondary teachers who already have a Diploma of Teaching and wish to return to upgrade their qualification to degree standard, or at those who have no teaching qualification other than a degree with some mathematics. Although most of the units in mathematics are similar to those elsewhere, there is also an elective research unit in the second year in ethnomathematics10 and there is an opportunity for postgraduate students to complete a research study in ethnomathematics. An examination of the various diplomas and degrees offered in education UOG website accessed on 14 May 2020 http://www.uog.ac.pg/home/history BSc program at UOG accessed on 14 May 2020. https://sites.google.com/a/uog.ac.pg/official/schools/school-of- science-technology-academic-programs/bachelor-of-science-major-in-mathematics 10 BSc In-service 2-year degree accessed on 14 May 2020. https://sites.google.com/a/uog.ac.pg/official/schools/school- of-science-technology-academic-programs/bachelor-of-mathematics-education-in-service-secondary 8 9

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shows that there are surprisingly few units in mathematics education. One is offered in the Division of Mathematics, Computing, and Ethnomathematics and the other by the Division of Education. Over the years more and more national staff have filled the academic roles at UOG. Now there are probably no expatriates on an academic staff that numbers about 90, although the NGO CARE continued to supply nursing lecturers for many years. Many staff have upgraded their qualifications. UOG certainly has taken advantage of such programs as the Australian Aid Virtual Colombo Plan (VCP) Project ran in the mid 2000s with Charles Sturt University and TAFEGlobal, which offered Master programs through UOG for teachers college lecturers and NDOE personnel. A course was developed by Kay Owens and then led by Steve Pickford who had been the team leader for the Primary and Secondary Teacher Education Project. The CSU and UOG staff were highly competent and understanding of students and 39 out of 40 students graduated (1 was ill) partly due to the tutoring by UOG staff at the colleges. Issues included poor internet at UOG to use the CSU databases and possibly none at the colleges. UOG still does not have access to library journal databases for research and study. The PNGIE computer laboratory was used for the final workshop when students were finalizing their research studies on the topic they chose, usually in the area they were teaching. Clarkson was part of an evaluation team but the World Bank VCP program was seen at a higher level to have issues especially around intellectual property and the funding of courses such as this one were not run again11. A similar course was later held at DWU and DWU attracted a number of senior educators into their Leadership course, but some completed a master’s degree at UOG. The turnover in teachers college staff suggests a similar course needs to be re-run as many of them do not have a master’s qualification, and very few a doctorate. A steady flow of doctorates have eventuated from UOG staff mainly through Australian awards, but UOG has to agree to support the students and that is often a sticking point. Sometimes resignations occur (although reinstatement is also likely). There still remains uncertainty of employment for some staff, despite their seniority. The Glen Lean Ethomathematics Centre Glen Lean had been a foundation member of the Department of Mathematics in what eventually became Unitech. Over a 20-year period he had collected via student surveys at Unitech, and his own travels throughout PNG, a vast data bank relating to number systems used throughout the country. He finally collated this material and used it as the basis for his doctorate, which included some five additional appendices, each a small volume in itself, in which various number systems were detailed (Lean, 1992). He had earlier circulated to all secondary and tertiary institutions in PNG, his databases by Provinces and a bibliography (18 booklets) which were updated and formed his first four appendices—the fifth was on Oceania (West Papua and the Pacific). After Lean passed away soon after a special graduation held in Melbourne in 1995, his academic executor and supervisor, Professor Alan Bishop, made the decision to have the electronic copies transferred from Apple format to PC format and had many of Lean’s collated papers sent to his two PNG students, one at UPNG and one at UOG. As circumstances had it, one lot—most of the PNG materials—were damaged at sea but the others were received by Dr Wilfred Kaleva. These comprised materials which encouraged UOG’s then Vice-Chancellor Mark Solon, Head of Mathematics Musawe Sinabare, and lecturer Wilfred Kaleva to start the Glen Lean Ethnomathematics Centre at UOG. Kay and Chris Owens were visiting UOG on study leave and long service leave and catalogued all the papers and organized them into filing cabinets. When the Centre was set up, these were transferred. The National Science Foundation (NSF) in the United States, through Nancy Lane at Pacific Resources in Education and Learning VCP funds were then used to support Australian students to study in the developing countries.

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(PREL) supported Kay and the Director Rex Matang with the assistance of Japanese volunteers Kiyu, Mara and other volunteers to employ research assistants such as Martin Imong, and three staff members’ children who had completed secondary schooling (Grade 10 or 12). They set up electronic versions of Glen Lean’s counting systems data into a Filemaker Pro database which would be available and searchable on the web with password and server protection as well as the majority of the papers collected by Glen Lean, his thesis, many projects on PNG mathematics by students, and papers written by a team of ethnomathematicians (Kaleva, Matang, Owens and Muke). Being old photocopies, scanning and cleaning carried out by Kay Owens, Anna Johaness, her daughter, and others, was tedious work. The USA university to house this digitisation for PREL also had difficulties in doing this so the GLEC site was a worldwide source for about five years. Unfortunately, lightning destroyed the database server so Kay Owens transferred the files to Word documents by Provinces. She had already developed an EXCEL summary spreadsheet from the database. The University has not been able to install these onto the website since 2009 but they were available on disk in the Centre and library and to students on their intranet for a number of years. At the time of writing the website has not been re-established on the universities latest website although it is listed as a Research Centre of the University. The Centre attracted a number of overseas researchers, although some plans fell through due to competitive funding difficulties. Rex Matang (the Director) organized for Geoffrey Saxe to open the Centre in 2001 (see Figure 7.3), and he made one further visit to PNG in 2015 with the Centre’s support, negotiated by Kay Owens and UOG staff. Owens supported Rex with his doctoral studies on young children learning mathematics in the vernacular compared to learning in Tok Pisin and/or English. He studied and modified materials from Dr Bob Wright, his supervisor, through Southern Cross University (SCU) who supported him well with his fieldwork.

Pictured: Rex Matang, Geoffrey Saxe, and Wilfred Kaleva

Figure 7.3 Opening of the Glen Lean Ethnomathematics Centre, University of Goroka.

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Unfortunately, Rex Matang died as he was about to return to SCU to finalize his thesis. Kay, with Rex’s son and brothers’ permissions, was able to use his draft data and chapters to prepare a journal article in his name, adding to his preliminary work. It supported the value of learning basic arithmetic concepts in the vernacular (Matang & Owens, 2014). Anthony Pickles carried out a fascinating study of gambling in Goroka and the ways of thinking and cultural expectations on gambling approaches and understanding of chance. He was also able to analyse the ideas of counting based on historical data and attitudes. Eric Vandendriessche knew of the Centre through communication and studied string figures in the Trobriands and intended to visit the Centre but was prevented because of travel issues. Wilfred Kaleva and Kay Owens began a two-year measurement study and prepared documentation for the website. Over 250 students and staff completed the measurement questionnaire to supplement the students’ research reports on ethnomathematics gathered over 10 years. Several other researchers supported this project, especially Rex Matang, Charly Muke, Zuzai Zhizoke, Serongke Sondo and other staff at Madang Teachers College. A number of UOG students and staff provided long interviews on their cultural mathematics and Summer Institute of Linguistics (SIL) personnel also shared knowledge. The provincial files did not make it onto the GLEC website but the material is available in several papers (Owens, 2012b; Owens & Kaleva, 2008a, 2008b) and formed the basis of many papers on visuospatial reasoning and identity from culture to mathematics (Owens, 2012a, 2013, 2014, 2015). A summary of research from the Centre has been provided by Owens (2016). A research project developed by UOG’s Mathematics, Computing and Ethnomathematics staff and Kay Owens working with another Australian teacher and linguist, Cris Edmonds-Wathen, was on developing a model, through design research, on elementary school mathematics education that combined work on early childhood mathematical thinking, culture and language in ethnomathematical thinking, early childhood teaching, and mathematical thinking. This project won an Australian Development Research Award to run from 2014 to 2016. Vagi Bino, Charly Muke, Serongke Sondo, Susie Daino, Maclean Pikacha, Smith Wendell, Priscilla Sakopa, Geori Kravia, Mindeng Goreka, Jo Gimisimo, Kila Tau, Eastern Highlands teachers, Jiwaka teachers and Inspectors, Rai coast teachers, Cooperative Elementary School Principals and teachers, Madang Province, Tubuserea, Waima school teachers, Council chairs, and nearby teachers have been involved together with Kay and Cris. The manual for teachers was used for professional development for teachers with support from inspectors and teachers college and curriculum unit staff. A series of papers were written as the design developed (e.g., Bino, Owens, Tau, Avosa & Kull, 2013; Kravia & Owens, 2014) and several on language by Edmonds-Wathen, Owens, Bino and Muke (see Chapter 10; Owens, Edmonds-Wathen, & Bino, 2015) as well as reports for Australian DFAT who funded the research. A further aspect was the use of a laptop computer powered by a solar panel and lithium battery with the manual as a self-instruction unit, numerous videos to support the modules, on the assessment used in the program, and on cultural mathematics, the early readers on key mathematical concepts using cultural experiences, suggestions for activities and games, and examples of learning plans. The material on USB keys was left with the PNG Institute of Education, the NDOE Curriculum Unit, UOG, and Madang Teachers College and Australian Aid DFAT. All the teachers and inspectors involved had drafts while some inspectors had the final product. It has also been shared in presentations made in the Solomon Islands and Tonga.

Divine Word University

General Introduction Divine Word University (DWU), a church-related Catholic university, commenced operation in 1996. The main campus is in Madang with four other campuses in different parts of the

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country. Madang and two other campuses teach education: the Wewak Campus is a continuation of St Benedicts Teachers College, which amalgamated with the university in 2003; and the Rabaul Campus is a continuation of the OLSH Teachers College, which amalgamated with the university in 2015. There are also campuses in Port Moresby (business) and Tubabil. As well as various undergraduate programs the University offers postgraduate programs at masters and doctoral levels. DWU teaches about 3 000 students. It is noticeable from the University’s website that the vast majority of staff are nationals, with a sprinkling of expatriates in some senior roles12. Mathematics Teaching The main teaching of mathematics occurs within the Faculty of Business and Informatics, in the Department of Mathematics and Computing Science13, although some mathematics is taught by other Departments, specific to their programs. For many years the Department of Mathematics and Computing Science has been headed by Dr Peter Anderson. In the section above dealing with UPNG, reminiscences of USA expatriate Deane Arganbright were related. In November 2008 he returned to PNG at the invitation of DWU. Cameo from Deane Arganbright The Department at that time (i.e., in 2008) was led by Dr Fitina who had earned a PhD in Computing Security at Wollongong University in Australia, and was a most able mathematician as well as being an outstanding computer scientist. He had been a member of my Department at UPNG from 1996 to 2000. His goal was to create the first university program in PNG leading to a degree in computer science (as opposed to computing applications), while also including a strong mathematics emphasis. DWU already had a very well designed and functioning Department of Information Sciences under the outstanding leadership of Dr Peter Anderson, who also had a PhD in Mathematics. In the first year of my stay I taught the first-year units to a group of approximately 30 students. Courses included discrete mathematics for computing, programming in C+, calculus, mathematical modelling, statistics, and some other units taught in other departments. The modelling class was taught using the text, The Active Modeller: Mathematical Modelling with Microsoft Excel, which I had recently written with Erich Neuwirth of the University of Vienna. Our publisher provided us with 30 copies of that book and a similar number of copies of their top-rated calculus book. During the second year I taught classes in algorithms and linear algebra in addition to the standard classes for the new first-year students. The unit in algorithms was not a success. In my view such a class would also include programming of the various algorithms that we would study. However, I quickly found that most students were still having difficulties writing even elementary programs on their own. As a result, I decided that I needed to reteach programming skills. With a few exceptions, this proved to be less effective than I thought it would be. Most students still could not write even fairly elementary code, instead copying code from online sources, sharing work, etc. Consequently only a few received a pass in the class and Dr. Fitina re-taught the class as a true algorithms unit in the next semester. During both years my mathematics and statistics classes seemed to go well, with a good number of students showing a talent for these subjects. The use of EXCEL for mathematics, my own special area, seemed to allow students to feel at ease and be creative. The website of DWU. Accessed on 10 May 2020. https://www.dwu.ac.pg/en/ Department of Mathematics and Computing Science at DWU, PNG. Accessed on 10 May 2020. https://www.dwu.ac.pg/en/index.php/study-dwu/faculty-of-business-infomatics/dep-ofmathematics-and-computing-science 12 13

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The long term question of how to find future Papua New Guinean academic staff in mathematics, computing, and information science was an issue that concerned us. Fortunately, Dr. Peter Anderson had already taken on that challenge successfully in the area of Information Science. He would select his best students who had completed their degree to become tutors in his Department, allowing them to gradually take on more responsibilities. He would then send the best of these to get master’s degrees from abroad. The year after I left (2010), Dr Fitina also left. The Department hired Dr. Cupp from the United States to take on much of the teaching, while Dr (and, by then, Professor) Anderson became Chair of the Department as well as of Information Science. He continued the process of hiring the top graduates in both areas as tutors and then sending them overseas for advanced degrees. At least two of these have now received advanced degrees from Poland and are teaching at DWU. Sadly Professor Anderson died towards the end of 2021. Before leaving Divine Word University in 2010, I was greatly honored to be named as Professor Emeritus of DWU for both my work at that University and for my other years of contributing to the nation. Mathematics Education Teaching The other Department at DWU that is of interest here is the Department of Science in the Faculty of Education. The Department offers mathematics and mathematics education units to undergraduates in the preservice program for primary school teachers. The undergraduate education program is a three-year Diploma in Teaching. After some years of teaching, students may return to complete a fourth year to obtain their education degree; B Ed. This had been a typical format for primary preservice programs in Australia up to the 1990s. Embedded within this program are typical mathematics education units covering the various aspects of mathematics teaching from Pre-Elementary to Grade 8. Most of the staff have at least a masters degree in education. From staff profiles and an inspection of DWU’s research journal, it appears to be the case that very few articles deal with mathematics or mathematics education.

Pacific Adventist University

The Pacific Adventist University (PAU) began in 1997, an upgrading from an earlier tertiary college, with the main campus at Port Moresby and other campuses in Rabaul and the Solomon Islands. It has about 1000 students. In both the B.Sc. and the B.A. undergraduate programs a major and minor in mathematics are offered14. As well, PAU teaches a f our-year undergraduate preservice program for secondary teachers at Port Moresby, and diplomas and degree programs for early childhood and primary teachers in Rabaul. These programs have the normal mathematics/mathematics education units which are common in such programs. Most staff hold masters degrees and some doctorates. The number of research publications may increase in mathematics and mathematics education with Dr Kopamu recently taking up a research position at the University.

Description of programs and units that PAU teaches: https://www.pau.ac.pg/uploaded_assets/504291-Academic_ Bulletin_2020.pdf?thumbnail=original&1582498606 14

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PNG University of Natural Resources and Environment

Vudal University began as a Faculty within Unitech, but after some years it was designated as a stand-alone university eventually changing its name in 200415. Its first Vice-Chancellor Professor Wong had previously taught Engineering at PNG University of Technology, and he provided the new University with a good managerial start. It is now ranked with the other top universities as PNG University of Natural Resources and the Environment (PNGUNRE) (UniRank, 2021). It included a campus at Popondetta (it too had university status in its own right for a while) in Oro Province, but there is now some concern regarding its current and future status. PNGUNRE staff are all nationals, except perhaps one, and the Pro Vice Chancellor has a doctorate. It has had high profile chancellors including Sir Alkan Tololo, former prime ministers Julius Chan and Rabbie Namaliu, and their first female Vice Chancellor Margaret Elias and currently Prof. Kenneth Simbuk. Like other universities, their fourth year includes a research project. There is some overlap with the PNG University of Technology in Agriculture and Forestry and with Divine Word University in Tourism and Business. However, Fisheries is only covered at this University. Needless to say, all of its courses require mathematics.

Western Pacific University

Under Prime Minister O’Neill, a new university was set up at Ialibu in the Southern Highlands with construction between 2016 and 2018 and governance set up for first enrolments in 2021. The focus of this University is providing for the knowledge economy of the 21st century with emphasis on Information and Communication Technologies. The students build on Foundation Year Studies. For example, there are courses in Visual Communication, International Business Management, and Computer Networking. The Executive Chairman of the Council is Father Yan Czuba (former President of DWU), The President is Dr Janet Ronga supported by experienced educators such as Dr Ludy Salonda, Caroline Kupi, Dr James Yoko (formerly ProVice Chancellor at UOG), and Dr Musawe Sinebare (former Vice Chancellor of UOG). Its goal is to encourage students to bring high quality standards into these necessary fields for PNG and an export knowledge industry and to have future overseas students particularly from the Pacific.

Primary Teachers Colleges

General Introduction Teacher education was begun by expatriate teachers, often South Pacific Islanders or persons from the countries sending early missionaries to the territories. They taught Grade 6 graduates then Grade 8 and finally Grade 10 national teachers how to teach. These Government courses were often of an internship-type. Interestingly, one such course held in Rabaul was different and recruited from Australia, training prospective teachers in short six-month courses. The missions gradually expanded their primary teacher training program into residential colleges while the Government established colleges in Madang, Goroka (although that reverted to secondary teacher courses, see above) and Port Moresby. By self-government, the College system had succeeded to a high degree with virtually all Community school teachers being nationals with only expatriate teachers in high schools and the National High Schools. Some expatriate high school staff, like Colin Meek and Rod Selden, later became lecturers at the PNG University of Technology, with Colin also teaching at the University of Goroka. For many years the primary teachers colleges, as mission agencies, continued to have expatriate staff. When these were replaced, the primary teachers colleges often recruited nationhttp://www.unre.ac.pg/ is their new website with the history, courses, and staff listed.

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als from the University of Goroka or other universities for teaching subject areas. So in mathematics, staff were often recruited with no primary-school training. Incidentally, many secondary teachers who had some mathematics in their degree did not have teacher education before they started teaching. When they could, they often went to the University of Goroka to complete a postgraduate Bachelor of Education course. From the 1970s until 2000, the government community and then primary teachers colleges were Madang Teacher College in Madang, and the Port Moresby Teachers College (which became the PNG Inservice College and then the Institute of Education). The church-sponsored colleges were Gaulim Teachers College for the United Church, which was located outside of Rabaul in East New Britain, Our Lady of the Sacred Heart College in Rabaul, Dauli Teachers College for evangelical churches in Tari, Balob Teachers College for Lutherans and Anglicans in Lae, and two further Catholic colleges, Kaindi Teachers College in Wewak and Holy Trinity Teachers College in Mt Hagen. The Seventh Day Adventist (SDA) church also had a college, not initially under the auspices of the Department of Education. Sometime later in this period, Enga Teachers College in Wabag began as a government college. After 2000, when it became clear that the system was failing, various institutions in different provinces began specifically to train elementary school teachers. Some of these became registered. Each of the original colleges has between 300 and 600 students and offers the Government-stipulated three-year Diploma of Teaching, now generally conducted over two years. Within this program there is room for each religious affiliated college, if they so desire, to add their own specific religious/religious education units. No matter which college students graduate from, all are eligible to teach in any primary school in the country. In the attempt to upgrade the colleges that began in the early 2000s and continued over a number of years, Madang Teachers College became associated with the University of Goroka which already ran a B.Ed in primary education. It appears however that little of substance has come from that collaboration especially in terms of finance, research and upgrading of staff. The various Catholic colleges have become aligned with Divine Word University, first Wewak and then Holy Trinity. Staff at the colleges were encouraged to complete their Master of Education from DWU and that appears to have happened. Some email communication in particular is now possible across the campuses which hopefully will bring closer integration in the future. Balob Teachers College, Martin Luther Seminary and Lae School of Nursing are all located in Lae, and were hoping to join together as a university but this has not yet eventuated, apparently due to financial issues. In 2018, there were 18 approved teachers colleges recognized by the Department, there being at least one in most of the provinces (Kombra, 2018). Those recognized are: 1. Dauli Teachers’ College, Hela Province 2. Holy Trinity Teachers’ College, Western Highlands Province 3. Enga Teachers’ College, Enga Province 4. Melanesia Nazarene Teachers’ College, Jiwaka Province 5. Simbu Teachers’ College, Simbu Province 6. Madang Teachers’ College, Madang Province 7. Balob Teachers’ College, Morobe Province 8. St. Benedict’s Campus of DWU (Kaindi), East Sepik Province 9. Gaulim Teachers’ College, East New Britain Province 10. Rabaul Campus of DWU (Kabaleo) OLSH, East New Britain Province 11. Sacred Heart Teachers’ College—Bomana, NCD 12. Sonoma Adventist College (SDA)—Campus of PAU, East New Britain Province 13. Papua New Guinea Education Institute (PNGEI)—For Elementary Teacher Training, NCD 14. AOG Jubilee Institute of Higher Education, NCD 15. Rev Maru Teachers’ College, East Sepik Province

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1 6. St Peter Chanel College of Secondary Teacher Education, East New Britain Province 17. Asia Pacific Institute of Applied Social Economic and Technology School (APIASETS), NCD 18. Callan Studies National Institute, East Sepik Province. Callan has always focused on methods for teaching students with special needs. History of Teacher Education William Lawes, the LMS missionary, started a college for teaching locals in 1883 how to teach literacy and arithmetic. He insightfully bought a printing press for literacy readers as well as the Bible passages he was translating. The missions continued to provide teacher education at a number of small colleges. There have been up to 34 mission or government institutions, as shown in Appendix 2 which gives the affiliations, names, inceptions and reductions or amalgamations. In 1954, Lewis, an Englishman, recommended to the Australian government that there be funded annual meetings between principals of colleges and head office, working closely with college lecturers to design teaching materials and to help mission colleges maintain standards (Quartermaine, 2001). The annual meetings and cooperation were a hallmark of teacher education in PNG until the 1980s and were aptly managed by Quartermaine and Liriope in their roles as facilitator, mentor, and administrator. The Weeden report (Advisory Committee on Education in Papua New Guinea, 1969) was particularly instrumental in setting up the Teacher Education Committee to report to the National Education Board. They were preparing for Independence. In 1990, the Higher Education Plan, referred to a partnership between church and state for teacher education. The Committee for Higher Education’s first agenda covered mainly finances, but the provision of support and the need to meet church requirements were not carefully considered. Integral human development was at the forefront of the philosophies driving teacher education. Australian educators whose own system was influenced by the British, brought their theories to PNG. There were also people on major advisory committees who had British colonial experience, i.e., Beeby, Currie, Weeden. One of the consultants in 1994 (and 1996) was Dr J Turner, of the University of Manchester. His 1960s publication about Christian education in Nigeria was distributed by the PNG head office (27/9178) for college staf readings. (Quartermaine, 2001, p. 44) Teachers colleges were the poor relatives of the universities, so education at UPNG for secondary school teachers was better funded until funds began to dry up in the 1980s. By this time primary teachers were expected to have Year 12 and towards the end of the 1980s it was anticipated that there would be a three-year diploma instead of the two-year certificate. After all, it was recognized that there were high demands on teachers. The Association of Teacher Education was a leading body calling for change. The kind of teacher which the new program hopes to produce is one concerned about his or her personal, professional and intellectual development. Such a teacher recognizes individual differences among children and is “prepared and skilful enough to adjust the learning environment to meeting individual needs” (Tetaga, 1989). Equally, such teachers need to be concerned for the development of moral and spiritual values within the school environment and outside of it. The type of teacher needed to perform the above functions is seen as one who is a “self-reliant, independent professional, genuinely interested in the community in which he or she serves, and

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committed to education for resource development” (Tetaga 1989). This teacher is also one with a critical thinking approach to the curriculum and to the practice of teaching (NEB/ATE 90, p. 1). However, staff saw the advisory document as being imposed on them from above, as they previously had had input into the curriculum through the advisory committees. External aid from Australia was by the 1990s tied to programs to ensure that aid money achieved its goals. However, this meant that Australian universities were competing for funding to run these projects and the projects were disjointed and did not meet central requirements of education. Australian educators’ influence was high, as the Primary and Secondary Teacher Education Project and the Curriculum Reform and Implementation Project did not initially talk to each other. This was seen as another form of colonization. In 1995, there were 1714 students in teachers' colleges with only 40 more men than women. Of these about 60 percent were aged between 20 and 24 years, and 25 percent between 15 and 19 years. A few older teachers continued to start courses. In summary, the evolving teacher education policies … were the 1946 creation of a Department of Education for both Papua and New Guinea and overseas … children but with on-going racist attitudes; the long-term influence of the positive 1950s policies of the Minister for Territories, Mr Paul Hasluck the late-1960s consolidation of church and government teachers' colleges, and the “team” philosophy or partnership of the principals with the Teacher Education Division; the Weeden Report document resulting in the combined Teaching Service and the National Education Board of the 1970s; the standards (of English) debate, or college-entrance levels, leading to teacher education research in the 1980s and the ad hoc NEB creation of the Association of Teacher Education. Professional and academic aspects were then led by the Professor of Education between 1990 and 1994 towards a NITE [National Institute of Teacher Education] until confounded by the [Commission for Higher Education] CHE and ultimately overtaken by foreign aid and loans policies of the late 1990s. (Quartermaine, 2001, pp. 46–47) The Teaching of Mathematics and Mathematics Education The mathematics offered by the Colleges is taught with the view that students need to learn mathematical content. However, what mathematics should be taught, and how it is taught can generate debate. In the 1970s and 1980s, there was a Basic Skills Mathematics subject. It was not integrated with how to teach mathematics. It was often intended to help students who did not remember their rote-learnt primary or high school mathematics to have an opportunity to revise their basic mathematical knowledge. For some, they were learning primary and early high school topics for the first time. Most students were Grade 10 graduates at this time. With the reform of the college courses in the 1980s, this subject was dropped as Grade 12 was now the entry requirement. However, the content is still presented in most—but not all—colleges as a formal set of ideas that are there to be memorized, rather than a set of ideas on which students need to work. This is partly due to most staff coming into the colleges with secondary school teacher education and teaching experience, or just with a university mathematics background. However, staff like Mea Dobunadobu at Balob Teachers College have been exceptional in providing concrete experiences, group work, and cultural mathematics as part of the learning experience. For most college staff, however, mathematics too often remained a set of concepts to be learnt, that might be useful in the future but not particularly in everyday life. Teacher talk and exercises were common practice.

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Sometime ago Vagi and Green (2004) noted that in Papua New Guinea: Generally, curricula developed for training purposes have been grounded in worldviews, beliefs and norms of those who design, develop and teach the curricula. … [college staff] will generally have been influenced by and implemented a mono-cultural Western mathematics curriculum (i.e. a very traditional approach to the training and learning of mathematics) to primary students. (p. 315) A reflection on the PASTEP Evaluation Project some 15 years ago, with which some of the authors were involved (Clarkson, Hamadi, Kaleva, Owens & Toomey, 2004), attests to the above view, and we believe it still holds true at the time of writing. We take one example from that project which exemplifies some of the difficulties that mathematics teachers in the Colleges face. The PASTEP program sought to energize and support teacher education in Papua New Guinea as staff grappled with the curriculum changes of the 1990s. One of its core aims was the critical notion of bridging. Bridging was the term coined for the gradual movement from teaching using the language that students used at home and in the elementary school grades (PreElementary, Elementary 1 and 2) to using English as the language of instruction in the primary school. This movement in language was to occur mainly in the first year of primary school (Grade 3) when it was envisaged both languages would be used but with more and more emphasis being given to English, and if needs be on into Grade 4 (Muke, 2012; Paraide, 2010). Clearly, if the notions of bridging were to be taken seriously in schools, then it was a change that had to occur across all of the curriculum, not just during the teaching of language, nor should bridging just be seen of interest when societal issues such as the multiple languages in PNG were being thought about in relation to the classroom. If the whole of the curriculum was to be impacted by bridging, this issue would also need to have an impact on what went on in mathematics classrooms. If that was the case, and it clearly was an expectation of PASTEP, then one would have expected that at least this notion would have been explored fully during the years beginning teachers spent at teachers college. It was somewhat surprising, then, in discussing this issue of bridging with various college staff during the visits of the PASTEP Evaluation team, and backed up by results of surveys conducted, that there was little to no appreciation of the need to address this issue across the college curriculum (Clarkson et al., 2004). The staff felt that their curricula were already too full. This issue of overloaded curricula was exacerbated when the three-year diploma course was taught over two years as a financial saving for the country, at the instigation of a past expatriate college principal16 who had moved to head office in Port Moresby. However, most staff and students reported that the increased pressure was too much for both teachers and learners grappling with so many new concepts and ideas. Individual discussions on this issue of bridging were held on three separate occasions with three different mathematics lecturers in three different colleges. None of them saw any need to include bridging in their units. Even if this was important, they argued, there was no room in their curriculum to deal with bridging. Indeed they saw no reason at all why language (and cultural) issues should be dealt with at all in mathematics. They seemed to be unaware of reports and conference publications from 1978–1984 from the Mathematics Education Centre at Unitech (see above), more recent ones from the Glen Lean Ethnomathematics Centre at UOG (see above), previous research work of Clarkson (1983a, 1992), Lean (1992) and Saxe (1985), and the ongoing work at that time of Kaleva (1995) and Matang (1998)—although they knew the last two lecturers. The PASTEP ethnomathematics subject provided an example from Australia of a digittally system (5, 20 cycles) and a body-part tally system where students were asked to provide similar examples from their own language group, but the staff were not able to provide possible examples from PNG, and other areas of mathematics including geometry were not addressed. His college tended to have the best Grade 12 students and he had tried this in his college.

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The best work from the Curriculum Unit in terms of examples of patterns from different cultural groups, was covered in the pre-elementary support document on Patterns. However, this was unknown to college staff or teachers to whom we talked. The idea of sharing knowledge with teachers that the recently revitalized Mathematics Association planned to do would seem to be exceptionally important but teachers argued that there needed to be materials and professional development especially to address such issues. A focus like this might also encourage a deeper understanding of school mathematics topics and serious thought needed to be given to having culture and language as a compulsory unit for mathematics teachers. It should be noted that the VSO supporting English in teachers colleges and schools considered the mathematics program (revised after PASTEP) as rather difficult for the lecturers and students. This might reflect the difficulties of gaining quality students into the colleges or an emphasis on the abstract nature of the mathematics being presented rather than practical everyday, cultural mathematics. Preparing for their practical teaching experience, students were good at copying the objective for the lesson from the syllabus but unable to extend and write the steps of a lesson plan that involved activity, students talking, and representing the mathematics in diagrams and words, nor any problem solving. Occasionally, one example of the questions to be done was written down e.g., 12 + 13 =; or Vagi had 12 shells and she was given 13 more, how many did she have then? These single-answer questions were not the kind of problem solving envisaged for learning. There was little to no sense in differentiating for different students even within a multigrade classroom. It was as if the worst of the teaching from the 1960s by expatriates was still what was remembered two generations later. Having said that, early counting (in English) did show similarities to programs in Australia like Count Me In Too (a NSW program). This position of all the college lecturers in general, and those teaching mathematics in particular, was actually surprising to us. Curriculum Guidelines had been published for the teachers colleges (Baing, 1998) and were to be implemented in 2000, giving much needed time for the college staff to be well prepared for the curriculum change, unlike earlier curriculum changes for colleges (Norman, 2003). These guidelines had a gestation period of four years, starting in 1994, and were written when the new syllabi for schools were also being prepared. There is clear reference for the notion of bridging in the mathematics school syllabi both for the elementary and primary schools (NDOE, 1995, 2003, 2004). But in the college curriculum bridging rarely got a mention. One place where it was mentioned was in the mathematics-science strand, but there it was relegated to one of nine possible topics that lecturers could choose from in the one mathematics methodology unit (Baing, 1998, p. 84). It would appear from Clarkson’s discussions with the college mathematics lecturers in 1999, 2001 and 2003 that none of them saw any need to deal with the topic of bridging. The above has dealt with the issue of bridging and by extension that of language and culture in the mathematical preparation of beginning primary school teachers. In more general terms, the college curriculum guidelines noted above were, for the most part, following those that had come before. As far as mathematics was concerned the guidelines were virtually all to do with mathematics per se, and there was very little to do with cross-curriculum issues that PASTEP had highlighted, such as gender issues and bridging. The PASTEP Evaluation Project also found a worrisome sign even when it came to mathematics per se. The test results in general across the five colleges at which we worked suggested that most graduating students had a poor command of the mathematics they had studied in college. We have been unable to discover any other research results that suggest that this has changed. In 2003, Kay Owens provided the PNG Institute of Education with the material on counting systems from the GLEC. She also led inservice programs at several colleges and repeated these in 2007 (on measurement) and in 2014 (on elementary mathematics taking account of culture and early childhood mathematical thinking and learning). It was well received and fascinated the staff who knew nothing of the counting systems across the country. But, the overall message was that

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what Owens had to say did not make a lasting impression on what they would do when teaching the college mathematics courses. Staff turnover seemed to occur and the teachers struggled to look beyond the fact (or what they perceived to be “the fact”) that “the syllabus” did not really allow them the freedom to incorporate this kind of mathematics into their teaching. Probably the college staff were not concerned by the changes that occurred with the 2013 government announcement that English would be the teaching language for all subjects and all years (Clarkson, 2016). The notion of bridging for the mathematics lecturers was now of no concern to them. The reference to the children’s languages as they started school, or their experiences encoded in those languages that appeared in earlier syllabi, is mentioned but not elaborated in the content of the mathematics syllabus, with the clear implication that all teaching must be in English (NDOE, 2015). It is Western mathematics, and only that matters. A selection of curriculum documents can be found in the Appendices 2 and 3.

Technical Colleges and Institutes of Technology, and Vocational Centers

The push by industry and business for technical schools to be funded by the government to allow a skilled workforce to grow in the country has been noted in earlier chapters of this book. From the early 1900s various technical schools were begun by the missions, and even in the primary schools not only was the curriculum skewed to skills that would be valued by the cash society, there was often quite specific teaching time given to manual work skills. As time progressed, technical education became an established component of the education system ranging from technical colleges preparing students to work in trade-related occupations, vocational centers and community centers mainly based in villages and concentrating on agriculture and other village-based enterprises, and business colleges preparing students to work in offices and administrative positions. There are more single-purpose colleges such as the various schools of nursing but many of these have amalgamated with a university. A scan of the internet at the time of writing indicates that most major cities in the country have a technical college, with many also having a business college, and some a school of nursing. With the majority of people still involved in subsistence farming and/or in local village enterprises, some vocational and community centers operate and play important roles in informal education throughout the country. It seems churches often keep them running. It appears that in most of these education institutions the mathematics taught is now related to the work that students are likely to move to, or in which they are already engaged, rather than just replicating the school curriculum.

Distance Education and Flexible Learning Institutes

Finally, it should be noted that for many decades the University of Papua New Guinea (UPNG) has established and coordinated centers in different towns for distance education preparation. It has had a strong staff of mathematics lecturers such as Dr. Francis Kari (who holds a doctorate in ethnomathematics) and Joel Silas (with a master degree and further studies and who has also taught mathematics at UOG, DWU and now at UPNG). There is also a Flexible Learning Centre at the University of Goroka. These have been important centers for recruiting and supporting incoming students. Kari (1998) considered what the students thought of mathematics and how it might link to mathematics in culture. Male students were more likely than female students to agree that their home languages and their cultures had mathematics and that it was possible to discuss school mathematics using words of their language.

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Concluding Comments

For some years after Independence, students were paid a living allowance to attend a university or a teachers college, but this soon disappeared, and students were then required to pay fees to attend university. By the 1990s, PNG was gaining income from mining but was still relying on World Bank and Asian Development Bank loans, Australian Aid and aid from other countries such as Japan. More monies were being taken away from the tertiary sector to fund the early years of schooling, although it often seemed the funds did not quite make it there. Infrastructure could not be maintained or used effectively. The need for more money was driven by the ever-increasing population of school children (health care improved lives but family planning was poor) which eventually flowed on to tertiary institutions with lectures being conducted to more than 300 students at the University of Goroka, and to whole cohorts elsewhere in the tertiary sector. There were now large tutorial classes, with usually 20 to 40 students. Institutions built originally for much smaller numbers of students and rooms for much smaller class sizes could not cope. The residential necessity for students was now spilling over into the nearby villages where rooms, often without power or any services, were provided to students. Churches still supply much of the infrastructure for their teacher education colleges. They continue to have representation on relevant high-level committees but the national government controls the curricula and salaries. Even so, there remains much dissatisfaction, which has plagued the sector for years; “Some bureaucrats themselves behave remarkably like politicians. They call themselves public servants but behave like public masters; they set their own policies, and are barely accountable to anyone” (Bray, 1987, pp. 140–141). Paper work is extensive but fails to keep account of practice, and there is political instability which, rightly or wrongly, affects manpower and action. It is clear that the tertiary institutions have nationalized over the last 50 years. In the 1960s virtually all academic staff were expatriates, and even in the 1980s most academic staff were expatriates with some nationals. Now, in all institutions, virtually all academic staff positions are held by nationals with few or no expatriates in the normal staffing profiles. This is also true for the departments and faculties that teach mathematics and/or mathematics education. It is also noticeable that most academics now hold at least master degrees with a significant minority having completed their doctorates. Overseas institutions have awarded most of these higher degrees, although there are now some staff who hold a higher degree from one of the PNG institutions, a trend that is being maintained. One occurrence has been the duplication of subject areas and perhaps too little cooperation between the universities and teachers colleges, something which may reflect Australian universities as they compete for students and funding. In the colleges now without the regular Advisory Committees representing and providing an avenue of meaningful cross-college discussions between staff, there is less sharing or sense of ownership. Since travel is so difficult and costly, it is hoped that electronic communication might strengthen collaboration and sharing of research and teaching practices. The principle of using electronic communication without costing the universities and colleges enormous amounts of money for personal downloading will be the key to this venture. Nevertheless, the Mathematics Association under Sam Kopamu has striven to bring about cooperation and sharing of mathematical research papers since 2014 and hopefully that is beginning to bear fruit. Since the beginning of tertiary education in the mid-1960s, the primary goal of the various institutions has been clearly that of teaching as effectively as possible. Most of the teaching has followed a traditional style, with the exception in the early years of staff developing their own text materials. Nevertheless, it has met the country’s needs. Although research has been an aspiration for all the universities, in mathematics there has been relatively little of it. One exception

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is the research efforts over the last 30 years of Sam Kopamu17 who has continued with his research on lattices having made a start under Kathleen Collard, John Lynch and Terry Fairclough, as well as the group of statisticians at the PNG University of Technology in the 1980s. Apart from the lack of resources able to be devoted to research in the institutions, many academics who might have become effective researchers have soon exited university life and taken up far more profitable positions in industry. Cecilia Nembou18, who is a high profile mathematician, recently retired. Most staff receiving doctorates have not easily been able to continue with research which they often claim is due to having heavy administrative and teaching loads, lack of funds and a lack of access to a supportive research group. Currently at Divine Word University there are a couple of staff completing doctorates in mathematics overseas. The follow up to PNG Journal of Mathematics Computing and Education, the South Pacific Journal of Pure and Applied Mathematics and the revived Mathematics Association with its two previous, and planned, third conference helps to keep research alive. The other two local journals, Contemporary Studies in Papua New Guinea (DWU) and the new Journal of Melanesian Perspectives (UOG), could cover mathematics education papers but so far only the former has covered such topics. The role of research centers would seem to be critical for extending research in mathematics and mathematics education (Clarkson, 1985). Overall there appears to be little recent research carried out in mathematics education. As with mathematics the real emphasis has been on teaching, not research. There was a real thrust of activity starting in the early 1970s, reaching a peak in the 1980s but falling away through the 1990s to the early 2000s. The authors of this book have continued with some research over the years, but there seems to be very little ongoing research now. Sadly, with a lack of ongoing Indigenous research there is little to contest the continued push to have schools teach Western mathematics in a manner suited to Western classrooms—in a style not necessarily well suited to PNG classrooms. References Advisory Committee on Education in Papua New Guinea. (1969). Report, 1969: Australian Advisory Committee on Education in Papua and New Guinea; Chairman: W. J. Weeden. https://nla.gov.au/nla.cat-vn2258617. Baing, S. (Ed.) (1998). National curriculum guidelines for the Diploma in Teaching (Primary). Port Moresby: Staff Development and Training Division, NDOE, Papua New Guinea. Bino, V., Owens, K., Tau, K., Avosa, M., & Kull, M. (2013). Chapter Eight: Improving the teaching of mathematics in elementary schools in Papua New Guinea: A first phase of implementing a design. In J. Pumwa (Ed.), Mathematics digest: Contemporary discussions in various fields with some mathematics—ICPAM-Lae (pp. 84–95). Lae, PNG: PNG University of Technology. Also published in South Pacific Journal of Pure and Applied Mathematics, 2014. Booth, L. (1975). Investigation of teaching methods and materials I: Investigation of a '”learning for mastery” approach in elementary coordinate geometry Progress Report 1973–1975. Lae, Papua New Guinea: PNG University of Technology. Bray, M. (1987). Afterword: Ideals, realities and relevance. PNG Journal of Education, 23(1), 137–149. Clarkson, P. (1983a). Types of errors made by Papua New Guinea students. Educational Studies in Mathematics, 14(4), 355–368. Although now at Pacific Adventist College, at the time of writing the following profile was available: http://www.uog. ac.pg/schools-1/science-technology/mathematics-computing-ethno-mathematics/dr-samuel-kopamu 18 https://en.wikipedia.org/wiki/Cecilia_Nembou 17

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Clarkson, P. (1983b). Error analysis and attribution theory. Mathematics Education Research Report 31. Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Clarkson, P. (1985). The role of a research centre in a developing country. Research in Mathematics Education in Australia, 3, 1–8. Clarkson, P. (1992). Language and mathematics: A comparison of bi and monolingual students of mathematics. Educational Studies in Mathematics, 23, 417–429. Clarkson, P. (2016). The intertwining of politics and mathematics teaching in Papua New Guinea. In A. Halai, & P. Clarkson, (Eds.), Teaching and learning mathematics in multilingual classrooms: Issues for policy, practice and teacher education (pp. 43–56). Rotterdam, The Netherlands: Sense Publications. Clarkson, P., Hamadi, T., Kaleva, W., Owens, K., & Toomey, R. (2004). Findings and future directions: Executive summary, final report and recommendations (The Baseline Survey of the Primary and Secondary Teacher Education Project). Melbourne, Australia: Australian Catholic University. Clarkson, P., & Jones, P. (1981). Learning mathematics in a second language; and Mathematical language and the learning of mathematics in Papua New Guinea. Reports 19. Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Clarkson, P., & Leder, G. (1984). Causal attributions for success and failure in mathematics: A cross cultural perspective. Educational Studies in Mathematics, 15(4), 413–422. Clarkson, P., & Sullivan, P. (1982). Vocational mathematics: A handbook for course development. Lae, PNG: Mathematics Education Centre, Papua New Guinea University of Technology. Clements, M. A., & Jones, P. (1983). The education of Atawe. In I. Palmer (Ed.), Melbourne studies in education 1983 (pp. 112–144). Melbourne, Australia: Melbourne University Press. Cox, E. (1982). Policy, practice and the place of women. PNG Journal of Education, 23(1), 107–118. Dube, L. (1981). Research on the teaching of problem solving at the Papua New Guinea University of Technology: Retrospect and prospect. In P. Clarkson (Ed.), Research in mathematics education in Papua New Guinea (pp. 38–42). Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Guthrie, G. (2001). Then and now: Secondary teacher education 20 years later. PNG Journal of Teacher Education, 5(4) & 6(1), 1–8. Høyrup, J. (2002). Lengths, widths, surfaces: A portrait of Old Babylonian algebra and its kin. New York, NY: Springer. Jones, P. (1975). A self-paced learning program for first-year mathematics at the Papua New Guinea University of Technology (pp. 59–69). Lae, Papua New Guinea: PNG University of Technology. Jones, P. (1981). Learning mathematics in a second language; and Mathematical language and the learning of mathematics in Papua New Guinea. Mathematics Education Centre Report 16. Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Kaleva, W. (1995). Ethnomathematics, and its implications for mathematics education in Papua New Guinea. In O. Ahuja, J. Renaud & R. Sekkappan (Eds.), Quality mathematics education in developing countries (Proceedings of the South Pacific Conference on Mathematics and Mathematics Education). Delhi, India: UBS Publications. Kombra, U. (2018). Education pipeline. Quarterly Newsletter of the Department of Education PNG, 5(1). Kravia, G., & Owens, K. (2014). Design research for professional learning for Cultural Mathematics. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.), MERGA37: Curriculum in focus: Research guided practice. Adelaide, Australia: Mathematics Education

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Research Group of Australasia. https://merga.net.au/Public/Public/Publications/ Annual_Conference_Proceedings/2014_MERGA_CP.aspx Lean. G. A. (1992). Counting systems of Papua New Guinea and Oceania. PhD thesis, Papua New Guinea University of Technology. Matang, R. (1998). The role of ethnomathematics and reflective learning in mathematics education in Papua New Guinea. Directions: Journal of Educational Studies, 20(2), 22–29. Matang, R., & Owens, K. (2014). The role of Indigenous traditional counting systems in children’s development of numerical cognition: Results from a study in Papua New Guinea. Mathematics Education Research Journal, 26(3), 531–553. Mead, A. (1988). 1979-1982 “A transition period.” In Tupela Ten Yia: Three personal stories by Dr Duncanson, Dr Sandover, Dr Mead (pp. 45–66). Lae, Papua New Guinea: PNG University of Technology. Muke, C. (2012). Teaching mathematics in the language bridging year in PNG schools. PhD thesis, Australian Catholic University. Murphy, P. (1987). Equity, excellence and efficiency in higher education: Implications for policy, planning and management. PNG Journal of Education, 23(1), 83–97. NDOE (1995). National education plan 1995–2004. Waigani, PNG: Author. NDOE (2003). Cultural mathematics: Elementary syllabus (and teachers guide). PNG: Author. NDOE (2004). Mathematics: Lower primary syllabus (and teacher guide). Waigani, PNG: Author. NDOE (2015). Mathematics: Elementary Syllabus. Port Moresby, PNG: Author. Norman, P. (2003). Curriculum reform under program 2000 for primary teachers colleges in Papua New Guinea. In A. Maha & T. Flaherty (Eds.), Education for the 21st century in Papua New Guinea and the South Pacific (pp. 93–98). Goroka, PNG: University of Goroka. Owens K. (chairperson). (1987, reprinted yearly until 1996). Lifestyle for the tertiary student. Lae, Papua New Guinea: PNG University of Technology. Owens, K. (1986–1987). Drug calculations, Ordering drugs, Solutions, Maternal health mathematics: Inservice series for nurses and health workers with tutors' manuals. Port Moresby, PNG: Department of Health. Owens, K. (1987). Mathematics for health workers in Papua New Guinea. PNG Journal of Education, 23(2), 229–246. Owens, K. (2012a). Identity and ethnomathematics projects in Papua New Guinea. In D. Jaguthsing, L. P. Cheng, & S. F. Ng (Eds.), Mathematics education: Expanding horizons, Proceedings of 35th annual conference of Mathematics Education Research Group of Australasia. Singapore: MERGA. Owens, K. (2012b). Papua New Guinea Indigenous knowledges about mathematical concepts. Journal of Mathematics and Culture (on-line), 6(1), 15–50. Owens, K. (2013). Diversifying our perspectives on mathematics about space and geometry: An ecocultural approach. International Journal for Science and Mathematics Education. https://doi.org/10.1007/s10763-013-9441-9 Owens, K. (2014). The impact of a teacher education culture-based project on identity as a mathematics learner. Asia-Pacific Journal of Teacher Education, 42(2), 186–207. https://doi.org /10.1080/1359866X.2014.892568 Owens, K. (2015). Visuospatial reasoning: An ecocultural perspective for space, geometry and measurement education. New York, NY: Springer. Owens, K. (2016). Culture at the forefront of mathematics research at the University of Goroka: The Glen Lean Ethnomathematics Centre. South Pacific Journal of Pure and Applied Mathematics, 2(1). Owens, K., Edmonds-Wathen, C., & Bino, V. (2015). Bringing ethnomathematics to elementary teachers in Papua New Guinea: A design-based research project. Revista Latinoamericana

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de Etnomatematica, 8(2), 32–52. http://www.revista.etnomatematica.org/index.php/RLE/ article/view/204 Owens, K., & Kaleva, W. (2008a). Case studies of mathematical thinking about area in Papua New Guinea. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Annual conference of the International Group for the Psychology of Mathematics Education (PME) and North America chapter of PME, PME32 - PMENAXXX (Vol. 4, pp. 73–80). Morelia, Mexico: PME. Owens, K., & Kaleva, W. (2008b). Indigenous Papua New Guinea knowledges related to volume and mass. Paper presented at the International Congress on Mathematics Education ICME 11, Discussion Group 11 on The Role of Ethnomathematics in Mathematics Education, Monterrey, Mexico. https://researchoutput.csu.edu.au/en/publications/ indigenous-papua-new-guinea-knowledges-related-to-volume-and-mass Paraide, P. (2010). Integrating Indigenous and Western mathematical knowledge in PNG early schooling. Doctoral thesis, Deakin University, Geelong, Australia. Quartermaine, P. (2001). Teacher education in Papua New Guinea: Policy and practice. PhD thesis, University of Tasmania. Hobart, Australia. Ross, L. (1982). The use of algorithms in Year 7. In P. Clarkson (Ed.), Research in mathematics education in Papua New Guinea (pp. 186–207). Lae, PNG: Mathematics Education Centre, PNG University of Technology. Ross, L. (1984). The quality of Grade 10 school leavers pool. In P. Clarkson (Ed.), Proceedings of the Fourth Mathematics Education Conference (pp. 189–198). Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Saxe, G.B. (1985). Effects of schooling on arithmetical understanding: Studies with Oksapmin children in Papua New Guinea. Journal of Educational Psychology, 77(5), 503–513. Sinebare, M. (1999). Private computer training in Papua New Guinea: From chaos to order. PhD thesis, University of Wollongong, Wollongong, Australia. https://ro.uow.edu.au/ theses/1794 Sullivan, P. (1981). Investigation into the mathematics of nurses-in-training in PNG. In P. Clarkson (Ed.), Mathematics education research in Papua New Guinea 1981 (pp. 121–128). Mathematics Education Centre, PNG University of Technology, Lae, PNG. Sullivan, P. (1983). Remediation of basic arithmetic weaknesses in post-secondary students. Mathematics Education Centre Research Report 28. Mathematics Education Centre, PNG University of Technology: Lae, Papua New Guinea. Sullivan, P. (1984). An investigation of the mathematics education of nurses in Papua New Guinea. Mathematics Education Centre Research Report 32. Mathematics Education Centre, PNG University of Technology: Lae, Papua New Guinea. Sullivan, P., & Clarkson, P. (1982). Practical aspects of drug calculations. Mathematics Education Centre Research Report 24. Mathematics Education Centre, PNG University of Technology: Lae, Papua New Guinea. Tetaga, J. (1989). From the Secretary. Papua New Guinea Education Gazette, 22(9/10), 3–4. UniRank. (2021). Top universities in Papua New Guinea. https://www.4icu.org/pg/ Vagi, O., & Green, R. (2004). The challenges in developing a mathematics curriculum for training elementary teachers in Papua New Guinea. Early Child Development and Care, 174(4), 313–319. Weeks, S. (1989). Problems and constraints in educational planning in Papua New Guinea: A case study from east New Britain Province. PNG Journal of Education, 25(1), 57–79. Weeks, S. (1993). Education in Papua New Guinea 1973–1993: The late-development effect? Comparative Education, 29(3), 261–273. Wilkins, C. (2010). History of the Department of Mathematics and Computer Science. https:// www.unitech.ac.pg/?q=departments/mathematics-computer/about/history

Chapter 8 The Reform Period: Major Changes and Issues in Practice

Abstract: Structural changes to PNG’s education system were made following the Matane Report, and advice from the World Bank. The modifications to school structures and curriculum were complex, each having an impact on mathematics education. The desire for universal education encouraged the development of elementary schools which were intended to be operated in vernacular languages and to provide remote area schools for children where no previous education had been close at hand. However, insufficient funding and training were available for this reform to be effective, with teacher education becoming a major issue. At the same time, administration was a provincial matter with a number of projects requiring aid funding. Primary schools were intended to go from Grade 3 to Grade 8, and there was to be further growth in secondary schools which would be able to teach to Grade 12, with provincial high schools supplementing National High Schools. The syllabuses mimicked overseas trends but were limited in size. Although support documents for teachers were prepared, after a few years many of these could not be found in the schools. The syllabus committees for teacher education, and school administrators, also modified some of the curricula, including those for science, mathematics and English. Sometimes that introduced further issues. Mathematics textbooks and teachers’ guides came and went, and class sizes continued to grow. The growth, especially in secondary education, resulted in an increase in the student/lecturer ratio in teacher education. Meanwhile there were still issues with infrastructure, especially in relation to finance and selection of personnel. Technology and facilities for secondary teacher education were greatly improved. This was often a result of overseas aid, but also of funds provided by the PNG Government.

Key Words: Aid agencies · Elementary schools · Language of instruction · Outcomes-based education · Reform of education in PNG · Vernacular education Even if policy called for a culturally-based curriculum, the official curriculum was not culturally based. Even if the official curriculum was not culturally based, … there would have been no guarantee that the cultural dimension would be implemented, because it is the teachers’ beliefs and values that really matter since these are transmitted to students. ... The curriculum needed to change as well as incorporation of ethnomathematical ideas into teacher-education programs. … Although the trial IMP project showed successful results (Souviney, 1983), the lessons or ideas and the wealth of information gathered from that project were never fully incorporated into the primary mathematics curriculum. … There is now overwhelming evidence that suggests that mathematics is not culture-free. ... Despite constraints to practice, in spite of contextual factors and the content being school mathematics, the teacher with cultural mathematics orientation views about the nature of mathematics was able to portray the methods of solution in mathematics as negotiable. Kaleva, 1998, pp. 270-271 © Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9_8

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The Birth of Education Reform

Education reform in Papua New Guinea (PNG) began in the 1990s. The general goal of the reform was to provide a suitable school curriculum that would prepare students for both formal employment and for life in the communities should they fail to secure formal employment. The former school curriculum’s major focus had been to prepare students for future employment, and it did not emphasize the need to prepare students for life in their communities. Despite every effort being made to assist community schools to focus on the community, the desire for employment and the associated focus of schools tended to alienate students from life in their communities. Consequently, when students failed to secure formal employment, they often found it difficult to survive in their own communities. They had difficulties readjusting to community life because there were no well-established links between the students’ community ways of knowing and the formal school subjects which they had studied. This was a serious education lack because these students had wasted their time in formal schooling which had focused solely on preparing students for formal employment. The majority of students could have also been learning how to live off their land, seas and rivers, and the informal economy. Their inability to secure formal employment after formal schooling was often viewed with scorn by members of their communities who recognized that the students had not mastered life-skills which would have enabled them to progress in their communities at the end of their formal schooling (Matane, 1976; Tololo, 1976). Before the education reform, formal education for the majority of students ended at the end of Grade 6. Furthermore, by the 1990s most of the students who progressed to secondary school failed to secure formal employment or to further studies at the end of Grade 10. This was demeaning for the young people especially when they were viewed as “useless” after 14 years of formal schooling. The reform curriculum aimed to correct the trend by which all students basically were given the same curriculum diet (McKinnon, 1976). The Tololo Report (Department of Education Papua New Guinea, 1974), as it is commonly referred to, discussed the type of education that would be suitable for PNG after it gained independence. However, the recommendations were not implemented at Independence due to neocolonial attitudes. Twelve years after Independence was achieved, a committee chaired by Sir Paulias Matane ─ who is regarded as the father of PNG’s educational reforms ─ produced a report, A Philosophy of Education in Papua New Guinea. This report was used as the basis for major curriculum reforms (National Department of Education [NDOE], 1999b). Matane’s Ministerial Committee Report (NDOE, 1986) emphasized the notion of integral human development. It stated: This philosophy is for every person to be dynamically involved in the process of freeing himself or herself from every form of domination and oppression so that every individual will have the opportunity to develop as an integrated person in relation to others. This means that education must aim for integrating and maximizing socialization participation, liberation, and equality. (p. 6) The previous form of Western education caused dissatisfaction in many people, and there was some demand for an education system which suited the people of PNG by utilizing available resources and opportunities to enable citizens to become productive members of their societies (Matane, 1986). This was summarized by one of the four National Objectives assigned by the National Executive Council to the NDOE: To develop a schooling system to meet the needs of Papua New Guinea and its people which provides appropriately for the return of children to the village community, for formal employment, or for continuation to further education and training.1 NDOE (2002) summary is provided in a Staff Reporter fact sheet https://www.pngfacts.com/education-reform-in-png/ papua-new-guinea-education-policy-framework towards the end of the reform of syllabuses and school structures 1

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Research Studies The Indigenous Mathematics Project (IMP) and its associated studies across the country involved researchers, linguists, and teachers like Smith (1978). Not only were many ideas brought together in terms of cultural mathematics, but people like Bob Roberts and Randall Souviney worked alongside NDOE officers preparing materials for use in schools that were likely to ensure that school mathematics would reflect cultural activities. There were two important research studies carried out during the mid-1990s when Professor Alan Bishop moved to Monash University in Melbourne, Australia, and Australian awards were provided to two lecturers associated with UPNG. Wilfred Kaleva was not only interested in the beliefs of teachers in regard to ethnomathematics but he also looked at some of the students’ views. Frances Kari was interested in his distance education students’ views on cultural mathematics and the mathematics they were undertaking at UPNG. Both used questionnaires and analyzed data generated by student responses, and both interviewed some of the locals who participated in the studies. Table 8.1 illustrates the diversity of views of preservice teachers which Kaleva suggested was mainly due to lack of alternative perspectives being discussed at university level, if not before. Given that Bishop (1988) had suggested there were many everyday activities involving mathematics such as counting, designing, explaining, measuring, playing and locating, how did teachers perceive everyday activities that were more associated with local cultures. In all the activities, the teachers considered there was mainly some mathematics with a surprising number of students seeing activities as containing no mathematics. Perhaps due to the

Table 8.1 Responses to Traditional and Non-Traditional Activities Activity Non-Traditional Activities pilot flying an aeroplane carpenter building a house estimating the height of a tree measuring the height of a student selling (betel nut) buai the teacher counting the number of students in the classroom. Traditional Activities children playing a traditional game making patterns on bamboo walls woman weaving a mat painting a haus tambaran villagers building a traditional house villager using the stars to navigate by canoe from one island to another building a canoe the warrior counting his arrows using own counting system Source. Kaleva, 1998, p. 211

No Maths Some Maths Lot of Maths % % % 1 1 3 4 8 10

2 29 70 81 82 78

97 70 27 15 10 12

42 34 27 25 17 17

58 56 64 60 64 55

0 10 9 15 19 28

16 13

70 74

14 13

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collective nature of the game category, it was the highest number of “no mathematics” or because it was not sufficiently sophisticated, which is similar to often-expressed views that ethnomathematics is only related to simple mathematics. This type of thinking was often evident in responses to interview questions. There was also a reasonable number of teachers who asserted creative activities were not to be associated with mathematics, while low-level mathematics such as counting was only seen as having a little mathematics. Kaleva’s (1998) data also indicated that many student teachers did not really understand what mathematics was. Their beliefs about mathematics and about ethnomathematics needed to be challenged. For that reason, Kaleva introduced Mathematics, Language and Culture, a new subject into the mathematics courses at the University of Goroka. D’Ambrosio’s approach to ethnomathematics provided the greatest influence on the content of the new subject—the primary aim was to convince students of the need to integrate local cultural practices and school mathematics, and not to leave them as disparate and unrelated pieces of knowledge. Although it was an elective research subject, it allowed over 250 students to engage in some aspect of their culture, to see mathematics within the activity, and to relate it to the secondary curriculum (some primary school teachers related it to that level). Although some of these connections were to be associated with the end product, there were some that really engaged with the “making” process and hence developed mathematical thinking “in practice.” Nevertheless, Kaleva’s (1998) research suggested a greater emphasis on problem-solving processes and knowing other ways of thinking about mathematics could be an advantage. In some cases, students were encouraged to see that many areas of the curriculum could be related to cultural mathematics. Above all, the students were proud of the fact that their ancestors and elders had used mathematics even “if they did not call it mathematics”. Owens (2014) has shown that students who have a sense of belonging to their cultural groups are more likely to recognize that cultural mathematics can be an important way of establishing one’s own identity as a mathematical thinker. It also encourages greater self-regulation in learning, with all the processes that are used by villagers in creating houses, canoes, weaving, designs and the other processes mentioned above or in Chapters 2 and 3. Frances Kari (1998, 2005) studied the views of distance education students and made some comparisons between male and female students. Typical of responses in the interviews about the usefulness of their mathematics course, one student said: It seems useful [pause] I say “some” because some maths is also abstract ... some is useful for my science courses. I mean it is useless to my people in the village [pause] yes some … well back in the village they use their own maths (Kari, 2005, p. 29) The males were more likely than their female counterparts to claim that their mother tongues have words for mathematical ideas. This result seems to suggest that the dominant males were more inclined to articulate positive attitudes towards activities which promoted their dominance in society (Sukthankar, 1995)—which does not necessarily mean that they have those qualities that they claim to possess. Kari (2005) recommended: Utilizing the home-culture mathematical ideas to supplement and complement Institute of Distance and Continuing Education (IDCE) mathematics, and vice versa. This proposition suggests that the mathematics curriculum should be valued and implemented from two perspectives. It is recommended that the first approach is termed the “ethnomathematical approach,” and it should emphasize the dominance of home-culture learning situations, with the formal mathematical learning situations introduced to reinforce and link the concepts as tool and object of learning. Moreover, some examples of linguistic conventions for mathematical knowledge and skills are taken from the author's home culture, with the purpose of justifying the viability of the proposition. The second approach is labelled the “integrated approach,” which involves formal mathematics learning situations as the focus of attention, with the

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home-culture learning context being used to demonstrate the link between both contexts. (p. 34) Kari provided several examples of ethnomathematics from his own culture, and showed how integrating with the mathematics provided for students in IDCE classes was likely to provide greater learning and insight into both cultural and the IDCE mathematics. These studies generated PNG evidence in line with studies around the world that had looked at alternative, often Indigenous, mathematics and had seriously considered the benefits of integrating the mathematics. It was this knowledge together with their own experiences as people of PNG that encouraged the committees chaired by Tololo and Matane to pursue the reforms in education and to encourage children to experience and articulate their cultural mathematics initially, and then to transition by connecting these together as they went through school. Early primary school, in particular, was seen as a critical time for melding both home and school mathematics. Kari not only valued ethnomathematical approaches but also integrating approaches based on a two-way valuing of both culture and school mathematics (see the quotation at the start of the next chapter). Numbers of Qualified Mathematics Teachers Previously, secondary teacher education students prepared to teach two and in many cases three subjects but now they would prepare for just two. Students prepared for mathematics, and English, science or social science. However, many people teaching mathematics had completed a degree with some mathematics but had not completed any teacher education. This led the University of Goroka to offer postgraduate courses for these teachers. A number of mathematics graduates also taught in the primary teachers colleges but again with no teacher education qualifications or appropriate school teaching experience. Some completed postgraduate studies on primary education. Rozoki Karkar at Balob Teachers College was one who transitioned well to understand primary teacher education. He was well mentored by Mea Dobunadobu, an experienced teacher and then a lecturer, who taught from a practical ethnomathematics perspective. He also appreciated the importance of statistics for secondary schools. Kaleva (1991) also noted the advantage of knowing new mathematics like statistics and applying it when evaluating in schools, evaluating news, and to assist with trials in farming. National and International Committees The following are some of the committees that led to the reform that began in the 1990s but continued to evolve into the early 2000s: • The report of the Five-Year Education Plan Committee, commonly known as the Tololo Committee Report (Department of Education Papua New Guinea, 1974); • A Philosophy of Education in Papua New Guinea (NDOE, 1986), commonly known as the Matane Report; • Jomtien Declaration of Education for All (UNESCO, 1990), a signed declaration between approximately 150 countries to improve human resources, reduce poverty, and improve literacy rates—PNG sent a delegation and learnt from others as well as making a presentation on their own situation (Weeks, 1993); • Education Sector Study (NDOE, 1991b); • Education Sector Resource Study (NDOE, 1995); • Papua New Guinea National Education Plans (1996 and 2004a, 2004b); • Language Policy in All Schools—Education Circular (NDOE, 1999a); • Primary Handbook (NDOE1999c); • National Language Policy (NDOE, 2000b); • National Curriculum Statement (NDOE, 2002); and • National Assessment and Reporting Policy (NDOE, 2003c).

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Clearly, Papua New Guineans wanted to improve their education system using a range of possible strategies. In particular, a goal that considered the worldview of peoples of PNG, namely the goal of integral human development (IHD) that targeted the following areas: • Access to nine years of basic education for all children; • All children to begin their formal school learning at the age of six in a language they use and understand; • Strengthening of all areas of the curriculum with improvements in standards and relevance and increased emphasis on relevant practical skills for life; • Expanded access to secondary and vocational education. Based on this, the education reform was carried out in two main areas: structural change and curriculum change. When the education reform began in the early 1990s (NDOE, 1996), support agencies began to realize that the educated Indigenous people had to contribute to education decisions as their improved Western education gave them the power to critique information presented to them. It began with Milne Bay and West New Britain being pilot provinces for this reform. For this reason, Milne Bay had vernacular preschools, and by the time Paraide (2002) investigated literacy and numeracy the impact of the reform could be well assessed at all levels. The Matane Report was readily accepted and used to develop relevant and appropriate curricula to suit the economic and social needs of PNG (NDOE, 1986). Structural changes were implemented before developing the reform curricula (NDOE, 1990, 1996, 1999b). Elementary school was added to the new education school structure, but in some places that decision was not immediately implemented. The Education Plan aimed to provide all six-year-old children with an opportunity to enrol in Elementary Prep by 2012 and to complete three years of Elementary education, and in Primary School there would be space available for all children to complete an education from Grade 3 through to Grade 8 (NDOE, 2004a). Basic education, as the current priority in education, was largely influenced by the World Bank, and has been a priority since before Independence (Smith, 1975). During Paraide’s (2010) study, and when this book was being written, no documentation was found that refers to when basic education will cease to be a priority and when a refocus on other sectors of education, using available limited resources, will begin. Currently, basic education is still the main focus.

Structural Changes for the Reform of Education

Strongly advised by the World Bank and by information gathered in various comparative trips that education officers made to other developing countries, the education system began changing. The structure of the new system can be compared with Figure 6.1 and is presented in Figure 8.1. The main difference is the introduction of elementary schools for Pre-Elementary, Elementary 1, and Elementary 2. The elementary schools were intended to meet three of the main educational issues: 1. Children would first learn concepts in their home language and learning numeracy and literacy in this culture; 2. The link between school and community was encouraged so that its relevance to village and/or cultural life was visible; and 3. More children would be able to attend school because one was available in their own community (language group usually).

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Figure 8.1 Structure of school after the Reform. However, in many places, the primary schools remained as a mix of elementary school and primary school (with top-up classes for Grades 7 and 8). The structural change also allowed secondary schools to provide Grades 11 and 12, so that more students were not only moving into secondary education but also continuing into the two highest levels of school education. This provided a stronger basis for the recruitment and entry level to the tertiary courses as well as for employment.

Impact of the Reform

In the former education system, less than 40% of Grade 6 students were able to continue to Grade 7 because of limited places available in the provincial high schools (NDOE, 2002), which included boarding and day provincial high schools. The other students who did not gain places in high school were labelled by their own people as failures at 11 and 12 years of age. The focus was on formal employment, which was viewed as important by some educated Papua New Guineans—a view strongly influenced by the colonial authorities (Department of Education Papua New Guinea, 1974). The Tololo Report (Department of Education Papua New Guinea, 1974) focused on the need to prepare students with appropriate life-skills and formal instruction in vernacular languages at the lower levels of education. By 2005, the transition rate from Grade 6 to Grade 7 had increased to 74%, largely as a result of the education restructure (NDOE, 2004a, 2005). Schools were under the direction of school boards comprising community members. These boards were significant in keeping the schools going but at times disagreements arose between teachers and boards. For the elementary schools, boards had to provide the buildings for the school and for the teacher housing. Elementary schools only ran in the morning. The teachers had also to manage their gardens, which were often their cash crop gardens needed for survival.

Implementing Change

By 1993, PNG educators recognized the enormity of the task of implementing change. The Goroka Campus Seminars was held and its proceedings formed part of the first issue of the PNG Journal of Teacher Education, published in 1994. One of the focal issues was teaching Grades 7 and 8 in a primary school context. It was strongly recommended that good teachers be provided with professional development and good textbooks and curriculum be available to implement the

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course, rather than expecting the load to be carried by newly graduate teachers who had covered the teaching of Grades 7 and 8 at the teachers colleges (Simpson, 1994). There was also the need for teachers colleges to change their courses to prepare teachers for Grades 3 to 8, but of course that took time to implement. This came just after the decision to make the primary teachers college courses three years. When Kay Owens visited a number of teachers colleges in 1992, she noted there was uncertainty about what was to happen at teachers colleges. One thing to be decided were the subjects to be taught, as there were up to 14 possible different subjects. Eventually, teachers specialized in one of two groups of subjects, namely social sciences and literacy or sciences and numeracy, with a number of the subjects combined into basic technologies. For some years Grades 7 and 8 were taught in both primary and secondary schools with parents more likely to consider the latter a better preparation for Grades 9 and 10. However, it was not long before the three years diploma was crammed into six sessions over two years as a cost-saving measure. That decision increased teacher and student fatigue, and contributed to content superficiality. Concern was expressed about students having practice teaching experiences in Grades 7 and 8, or seeing good practice, so the demonstration schools, historically Grade 1 to 6 schools, needed to consider how the demonstration teachers could be skilled for quality teaching at Grades 7 and 8. Meanwhile, the notion of a National Institute of Teacher Education that provided accreditation and directed curriculum for the teachers colleges was expected but it did not eventuate due to conflict between various education groups. At the same time, there was a loss of the advisory committees representing the colleges which was a regrettable loss of communication between the college staff and their input into curriculum for the colleges and schools (Quartermaine, 2001). Meanwhile the elementary schools were supported by various NGOs such as SIL until the curriculum was prepared but in general there was a slow start in the organisation and teacher education for this sector, a major concern for many. Participants at the Goroka Campus Seminars conference formed groups to discuss the implementation of each subject area. The mathematics group noted that the existing Grades 7 and 8 textbooks prepared students for life as well as for further studies, and that they were adequate for the primary schools with the same number of hours. However, there was concern that many of the earlier-trained primary school teachers would need professional development to use the materials well. In addition, although participants felt streaming was possible in primary schools to emphasize practical mathematics for one stream and more academic work for the other, many felt that in a streaming regime many parents would be dissatisfied with the placement of their child. Hence it was felt that parents needed to be educated about the purpose of the practical nature of the mathematics, and that the upgraded schools were likely to produce better quality students. They wanted to discourage the use of the term “top-up” and suggested “extended school” for the primary schools with Grades 7 and 8. However, they also emphasized that the community should make the decision about whether a primary school would go to Grade 8. Interestingly, a decade later, there were numerous primary schools without Grades 7 and 8. Richard Zepp, from the United States, was then at the University of Goroka as a Visiting Scholar and his enthusiasm was somewhat contagious. He also involved UOG students in finding out about mathematics in a range of occupations including subsistence farming. He published students’ projects on Mathematics, Language and Culture and this inspired future students who prepared reports. He assisted Matang and Sapau (Zepp, Matang, & Sapau, 1994a) to use the test given to Goroka students for many years by Ross to compare the performances of Grade 8 students in community schools with those in provincial high schools. The latter did significantly better, with a mean score roughly double the other but this was not surprising as they had been selected already for the provincial high school. Interestingly, the top-up teachers in Madang Province who had been teaching Grade 7 for a year did surprisingly better on the test during their Lahara (professional development session over summer) than those in Port Moresby without this experience. However, the mean score was still less than 50% correct, suggesting further professional development was needed. They also found that students from national high schools outperformed students from provincial high schools but this is not surprising given that the former had already achieved a posi-

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tion in a national high school by scoring higher in the entry assessment test. Furthermore, after 1992 the UOG students were doing better on the same test. This was a particular issue for teachers of subjects other than mathematics, and yet there were some mathematics preservice teachers who had low scores. One thing teachers really appreciated in the Lahara sessions was sharing with each other and hearing new ideas from them (Zepp, Onagi, & Raju, 1994b). However, lack of expenditure on education has continued to hold back progress in PNG education. There has been no percentage increase in students completing a primary or secondary education—it remained around 73% and 16% respectively from 1973 to 1993 (Weeks, 1993). A shortfall in teachers could be held responsible for that. There were only 800 primary teacher graduates in 1991, and 520 in 1993 with a shortfall in teacher numbers of over 2 200 (NDOE, 1991). This led to the building of a large lecture theatre at UOG but this generally meant large lecture classes (350 students) and less opportunity for examples of good teaching practice. Attendance at lectures was not always high. Nevertheless, students were expected to complete the assignments, tests and examinations and these were generally of a good standard. Conscientious staff were overworked coping with these numbers too. Mathematics was covered by students doing examples with full answers being provided later. Quality mathematics education in terms of good questioning and whole-class discussion was provided in tutorials and computer laboratories but it is not clear how much of this was translated into school classrooms. One key issue was the reduction in the teaching practicum. Besides an early practicum experience, with a difficult-to-use assessment tool, to small groups at Goroka Secondary School and others nearby, there was only one long practicum in the final year (observation, 2015). Again, funding as well as security for students and supervisors were issues, but these opportunities were available across the country with good organization behind them and many dedicated lecturers.

Financial Issues for Teachers and the Impact of Fees or No Fees

During this period, teachers often travelled to provincial offices to sort out pay problems, communicate with the Department about maintenance, get materials, and pay for power (if available). This was a major issue for the Department as they were then missing from the classroom. Although computers and systems were somewhat in place, and in many cases pays did reach teachers’ bank accounts, it was then necessary for teachers to stand in long queues at the bank to get the necessary cash for items from the village trade store, for transport, or for medicines. However, many schools did not have power or telephones or access to these although some received solar panels as part of an aid project. Matters improved by 2010 when more ATMs became available to get cash and to purchase mobile units and power units. No Fees In 2002, in order to encourage more people to move into and through school, a Free Education policy for all levels of education came into effect. However there were still inbuilt barriers with examinations for moving through school especially beyond Grade 8 and for selection to specific schools. Abady (2015) summed up the issues relevant to fees in terms of its impact on the diversity of PNG schools: When schools are directed to implement government policies, the implementation is expected to create a change in the organisation and the running of the school, … and school policies and practices.… Implementing the Free Education Policy (FEP) means that the government pays subsidy for each student. … the value of that amount does not take into consideration the location, availability of materials and resources, availability of class teachers or the general cost of providing education (NRI [National Research Institute], 2004; Swan & Walton, 2014). … Unlike a school in the city, a school in a remote area of Papua New Guinea does not have to pay for the use of

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power. Unlike a boarding school, school leaders in a day school do not have to be concerned about buying food for the students. …FEP has produced a significant increase in enrolment and as a result many schools find they have inadequate resources (Guy, 2009). Bray (2007) points out that education is never free as families still have other costs to deal with, such as school uniforms and resource materials. Again in PNG parents are asked to assist their school by buying resource materials for their children. In many of the practical lessons, in a boarding school, students are required to bring their own working tools … Clearly the increase in class size may influence the practices and policies of the school if it is accompanied by a lack of learning and teaching materials and tools and equipment. … The role of Papua New Guinea secondary school leaders is to carry out directives stipulated by the National Education Department and the Provincial Education Board. Despite this, school leaders have created their own school policies where they see them as pertinent for the smooth running of their institutions (Guy, 2009). Carrying out state directives is not an easy undertaking for school leaders …[and] 85 percent of the total population lives in rural areas, making it difficult to implement policies effectively. (Abady, 2015, pp. 22–23, 26) Interestingly, free education once again became a political platform on which O’Neil campaigned in 2012—see Chapter 9 of this book and Paraide (2015) on fee-free education (FFEP). However, despite these policies, education is still not universal across PNG, but more opportunities do exist for rural and remote students and for those from families with no formal or regular income. One key issue for resources is the lack of books, syllabuses and teachers’ guides in a school. This can seriously affect not only the teaching strategies for mathematics but the number and diversity of problems that the student might carry out if traditional exercises for mathematics are considered the normal practice, given that few school teachers know about inquiry and problem solving2. It has been well documented in PNG as elsewhere that those who can afford additional education costs have access to education (Ahai & Faraclas, 1993; Guy, Paraide, & Kippel, 2001; Kippel, Paraide, Kukari, Agigo, & Irima, 2009; Paraide, 2015), and those who have educated parents are more likely to do well in examinations and be selected for further studies or formal employment. Consequently, even with the implementation of FEP or later FFEP, the gap widens.

History of Curriculum Changes

The educational reform clearly identified the value of basic numeracy, literacy, and general skills, and other specific life-skills which students can build on in their future lives (NDOE, 1996, 1999c, 2000c, 2003a, 2004a, 2004c, 2004d). It also attempted to deconstruct the divides between education for employment and informal economic activities. The notion of informal employment had never been addressed during the colonial era, as the focus of education then was on formal employment (NDOE, 1976). Initially during the reform period, the elementary school curriculum used the old Grades 1 and 2 syllabuses but soon new materials introduced literacy for beginning reading through sounds or phonics in their own language. “Shell” books were created telling a village picture story to which the vernacular language could be added on the page following the guides (which were in English). Problem solving was a major focus worldwide and in the mathematics at Unitech in the 1980s as mentioned, Chapter 7. 2

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SIL assisted many village schools with language materials and their own literacy programs were adapted. However, mathematics looked like the old Grade 1 mathematics except that children were able to count and use a few other words in their own language (often Tok Pisin, Motu or another local lingua franca) depending on the background of the teacher even though they were expected to be from the language group. The elementary school syllabus was later called “Culture and Mathematics.” The elementary school curriculum was intended to be integrated and often themes were used to bring the curriculum material together around a cultural activity. The development of the reform curricula began in 1992. The Elementary curriculum was developed first, followed by lower primary, upper primary, and secondary levels, respectively. A great deal of external influence dictated when, how, and the length of the development of the reform curriculum. Donor agencies, using their monetary power, replaced the colonial powers, and dictated to the recipients when and how to implement educational initiatives. The donors tended to select companies that knew how to write project documents competently in the English language. The companies decided on particular foci, how the proposed curriculum was to be presented, and the training strategies to be used to prepare teachers to implement the reform curriculum. Consequently, the school curricula at these levels still had a strong Western influence, even though Papua New Guineans were involved in their development. The elementary curriculum was first implemented in 1993, in Milne Bay Province, because that Province was the first to volunteer to take on the new initiative. In 1997, the Elementary Teacher Education Support Project (ETESP)—an AusAID-funded project—continued training for elementary school teachers. The donor had the financial power to dictate what should and should not be done in the project. The power of domination over the less powerful, as discussed by Smith (1975), Barrington-Thomas (1976), Fellingham (1993) and Said (1978, 1993), was evident here. The donor selected the company to manage the project and decided when the project should commence. The recipients had minimal influence on such decisions. The local management or steering committees that were established acted merely as “rubber stamps” for approval of curriculum documents before development commenced, and development and completion of curriculum documents were advanced and completed. They generally had only limited influence on the actual development of the curriculum documents. Even though much of the power remained with the donor agencies regarding the implementation of educational projects, the Indigenous people were able to make some contributions in terms of local knowledge during the development of school curricula. Evans, Guy, Honan, Kippel, Muspratte, Paraide, and Tawailoye (2006) found that some communities, especially in the rural areas, had curriculum committees which supported the development of teaching and learning support materials at the Elementary level. The effectiveness of school curriculum committees varied across the communities. Their level of success depended on sound community support and strong school leadership. Evans et al. (2006) found that some of the teachers in the study sites had forgotten some of their Indigenous mathematical knowledge, and had to relearn it during their elementary school teacher training. The lower primary curriculum was first implemented in 1997 in Milne Bay Province, followed by the upper primary curriculum in 2000 (NDOE, 1996, 1999c). The initial reform school curriculum prescribed teaching to set objectives (NDOE, 1998a, 1998b), which had been the common teaching strategy in PNG for a while. In 2002, the revision of the elementary, lower primary, and upper primary curricula began (NDOE, 2002, 2003a, 2003b, 2003c), and was led by the Curriculum Reform Implementation Project (CRIP) team. CRIP was also funded by AusAID and was managed by an Australian company. This company brought in its own Western experts in curriculum, so that they had the power to influence the revised curriculum, which was different from its predecessor (NDOE, 1998a, 1998b, compared to 2003a, 2003b). All Australian advisers were expected to live in PNG and relinquish their Australian curriculum positions. The PNG officers were swept along during this process. Approval was given by the NDOE, primarily because it had the funding to support the

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development of the revised elementary and primary curricula. Initially the teachers colleges were not informed about these new approaches, and indeed the NDOE Curriculum and Assessment section was not informed for some time, as the teams did not communicate for one year, despite buildings being in the same location—the CRIP team was engrossed in their planning of targets and measures for accountability. Each PNG curriculum writer was supported by an expatriate (Australian) adviser with experience in curriculum writing and with experience in PNG or with Aboriginal communities. However, this was the era of the outcomes-based education (OBE) worldwide and this new approach to curriculum writing moved away from having many small objectives toward more expansive ideas of the subjects being introduced. During the revision period by CRIP this change in the outcome-based education approach occurred. The PNG elementary and primary school teachers had to modify their current teaching practices because the experts changed the former curriculum from using objectives to prepare lessons to a modern, updated version of outcomebased education (NDOE, 2003a, 2003b, 2003c). Unfortunately, the desire to keep syllabuses short and lack of funding to support teachers’ guides also meant that many teachers failed to know how to teach to reach the outcomes. This was often not only due to the lack of teachers’ guides and professional development but also to a slow beginning for preservice teachers. Nevertheless, a guide for teaching Cultural Mathematics, and another one that targeted the topic of Patterns and Mathematics which had a strong link to cultural objects, were prepared by Barbara Sparrow for elementary school teachers. However, teaching approaches in the reform encouraged the integration of Indigenous and Western knowledge, and the use of vernacular languages, as the languages of instruction in elementary and lower primary grades Although the initial materials prepared in the 1990s and those prepared by CRIP in the 2000s had teaching strategies with strengths and weaknesses, those who opposed the introduction of the outcome-base education (called OBE in PNG) tended to focus more on its negative aspects. Education for Rural Living In discussing the value of vocational and technical subjects in formal schooling, Guy, Bai, and Kombra (2000) highlighted the fact that the majority of school leavers return to rural environments, and many participate in the informal sector of the economy. Their discussions on vocational and technical subjects complement vernacular and bilingual instruction and relevant curricula in PNG’s formal schooling (Guy, 1999). However, by 2005, equipment for these schooling avenues were seriously depleted with an emphasis on literacy and school mathematics. Inclusive Education As part of the emphasis on universal education, and in line with world-wide trends, in 1994 a policy for Inclusive Education was approved (NDOE, 1992, 1993). Pokana (2013) used questionnaires and interviews with teachers, teacher educators and Provincial and National Education Officers to indicate that there was little support in PNG for implementing the policy. There was minimal inclusive education at the teachers colleges, although for a number of years there was an expatriate (sponsored by CBM, an international organisation for the blind) and national lecturers at the University of Goroka. Highlights in the country were the Special Education Resource Centres that provided education for students, and support for teachers (in Goroka, Lae and Callan supported by the Catholic Church in Wewak). These had huge impacts and provided rich experiences for the teacher education students. Callan was a particularly valuable centre as student teachers gained exceptional experience and supported the community for many decades in this area although 2021 reports from DWU lecturers at the Wewak campus suggest its benefit may not have been supported recently by the new expatriate leadership of this campus. Pokana (2013) pointed out that there was no support, however, in the curriculum documents, including the mathematics curriculum documents, for inclusive education.

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Teacher Education

The ETESP project had a great deal of influence over the PNG NDOE. Decisions on a suitable mode of teacher training for elementary school teachers during its two-and-half year lifespan were made by the project management, with some consultation with the NDOE. Elementary school trainers were trained at the Papua New Guinea Education Institute. The trainers’ major role was to support elementary teachers during the duration of their training in schools. The primary objective of this particular project was to train sufficient elementary school teachers in all the provinces to provide for the elementary school education expansion in PNG within two and a half years. A mixed mode for teacher training was created to train a large number of teachers in a short span of time. Although this teacher training strategy was viewed as cost effective, with hindsight it can be argued that it did not prove to be an effective strategy for PNG. The ETESP project ended in 2001 but funding for elementary school teacher training continued through another AusAID project. Consequently, the power still remained with the donor agencies that were operating in PNG. This included eligible people who had been selected by their communities to be trained to be elementary school teachers. There were four selection criteria: 1. They needed to be living in the community; 2. They could read, understand, write, and speak English well; 3. They could speak, read, and write well in the selected language of instruction for the elementary school; and 4. They were grade 10 graduates, at least. The fourth criterion may initially have been subsumed under the third, but as time went by and the certification for qualification went on, this became a major hurdle for those wishing to become elementary school teachers. For this reason many elementary school teachers were never paid. It may have been because of their unorthodox schooling (e.g., they may have always been involved in Tok Ples schools3), or the school needed to be inspected (and some of these schools were not easily reached by inspectors, or they were unwilling or unable to go there), or the training was unavailable. At first the teacher was not paid but trained under the principal. It was intended that there would be a gradual increase in pay as the teacher learnt more under supervision and passed units of training. Most principals, at least initially, completed a year at the PNG Education Institute (PNGEI), but most of the elementary school training was taken by the lecturers from PNGEI in remote centers over two-week blocks followed by submitted assessments. This was another hurdle for getting full pay and it generated a particularly difficult situation if there was no training. Later there were provincial trainers who used the PNGEI materials (self-instruction units, (SIUs)), however, these were not always printed and available. The mixed mode of teacher training was used to train the elementary school teachers over a period of three years (NDOE, 1996, 1997, 2004a). The funding for provincial training needed to come from the provinces, and once again this was sometimes difficult to achieve. There were, however, inspectors who supported the remote and rural teachers and encouraged training and helped with the assessment of the children. They knew the schools, often having to walk into schools where cars (if roads were passable) or dinghy did not go. Class sizes could occasionally reach 75 per teacher, and 45 was not uncommon. Evans et al. (2006) found that elementary school trainers’ support for the elementary qualified and trainee teachers was inadequate. This situation could be viewed as a sad legacy from some powerful donor agency Owens had heard that the Tok Ples schools in the Huon Peninsula (Morobe and Madang Provinces) were continuing and in 2014, she led an inservice session in Sialum Madang at which one of the highly regarded teachers could not speak English or Tok Pisin but was good at teaching literacy and other subjects in his Tok Ples, having only attended school himself, studied to teach, and taught, in Tok Ples. 3

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strategies. On the other hand, studies have also found that committed trainers often supported their teachers well, when money was available (Owens, personal observations; Paraide, 1998, 2002; Paraide et al., 2002; Evans et al., 2006). Concerns about Teacher Education An important education conference—The 12th Extraordinary Meeting of the Faculty of Education, a biennial event of the University of Papua New Guinea—was held in 1991. At that conference, Avalos and Neuendorf compiled papers not published elsewhere that became the basis for a book that expressed the hopes and concerns of many leading educators in PNG. Those in attendance raised particular goals and hopes necessary for the new education era. Two important aspects which were identified were: 1. Formal education was most likely to be effective if it developed from the child’s own home and cultural background, was facilitated by their home language, and prepared them for change in their place wherever that might be in PNG (the village, the town, the profession); 2. Education, especially mathematics education, should prepare students for the quickly changing world of technology and the professions, through inquiry that was tailored to PNG. One paper in the collection, though, on history and social sciences, particularly noted that little regard was given to the available knowledge of teachers (for example, there was little chance that the teachers would know, as the syllabus materials required, who was an obscure British Protectorate administrator who only stayed in PNG a few months) and that expatriates could attempt to prepare for PNG but miss the mark in knowing what is valued by Papua New Guineans such as ordered timelines being more valued than relevance and relationships provided in oral histories (Macpherson, 1991). Narakobi (1991) emphasized that teachers needed to instill Melanesian and Christian values as well as to teach the “content” of their subjects. Matane (1991) also emphasized the importance of teaching values espoused in home and local communities—as places to learn about respect, cooperation and justice—in a child’s first education. He emphasized that the community must be involved in school and that school administrators, and teachers, not only need to know about and respect traditional cultures in which the schools are placed, but also to stimulate students to be active participants in their communities’ development. He also noted that they needed to be aware of their role in the global community and its impact on PNG. However, numerous papers raised concerns about teacher education and the inservice education of teachers. This was a particular concern when it came to being able to use computers. Sinebare (1991) provided a strong outline of needs for school administrators and teachers to become computer literate in their preservice courses, especially when schools had purchased computers. He pointed out that all preservice teachers at the University of Goroka were to be offered the opportunity to enrol in a Computer Studies course—and not just those doing mathematics. There was a gap between computer users (most teachers) and computer scientists (some mathematics teachers might be in this category). Kaleva (1991) pointed out the need to revise mathematics curricula so that they would be up-to-date and to include, for example, statistics in order to prepare students for future professions and for carrying out—at all levels—village-level research on crops, health and other activities for future development. He was in no doubt that it was important for there to be computers used in mathematics classes in schools. He also emphasized that teachers were to provide problem solving and inquiry and be facilitators of learning rather than just mediums through which students would to be asked to memorize rote procedures. They should provide instruction on practical processes and organize procedural resources and ways forward which would help meet challenges and changes over time. Thus, they would need not only a significant level of mathematics but also mathematics education, on how to provide this kind of mathematics in schools. This, and the use of computers, would best be experienced during practical sessions.

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More than 10 years later the essence of these concerns for values were still being echoed. Tapo (2004), then Secretary of Teacher Education, maintained that a teacher should “assist children to achieve the aims of oracy and literacy, numeracy and graphicacy, social development, resource development, and spiritual development” (p. 61). He went on to discuss standards of education and the role of teachers in providing purposeful education, especially for rural students: Curriculum strategies which attempted to encourage more “doing” (skills and practice) on the part of students and teachers were discarded in favour of more theoretical and abstract knowledge. Despite this shift of emphasis, there is still a mismatch between the interpretations of national standards set in the Education Department and those in schools. (Tololo, 1997, quoted in Tapo, 2004, p. 66) Primary and Secondary Teacher Education Project (PASTEP) At the end of the decade the Australian Aid project, Primary and Secondary Teacher Education Project, was funded to the tune of $45 million but driven by the PNG NDOE. This was a five-year project so the developments within the teachers colleges were fairly significant. Each expatriate adviser worked alongside a Papua New Guinean adviser or developer of curriculum and they mentored lecturers in each discipline field. Quality teaching approaches such as group work and new topics and ways of teaching both at school level and university were being incorporated into the changes. It was under the direction of Dr Steve Pickford (formerly at Madang Teachers College but then at Charles Sturt University, Australia), with the involvement of numerous staff from Charles Sturt University and other universities in Australia. If the reform was going to be embedded in the teaching cultures of the country, then clearly the aims and way of teaching envisaged in the reform needed to be imparted to beginning teachers as they began their professional journey in college, and in their first few years of teaching in school. The mathematics specialist had previous experience with Australian Aboriginal communities. There was a week on cultural mathematics in one of the mathematics courses (subjects) for the primary school teachers colleges but there was also an elective course which was later not offered with reduced time for teacher education. In this elective Australian Aboriginal counting systems (a (2, 5) cycle system and a body-part tally system) were used to encourage the student teachers to consider their own systems. However, knowing more about the counting systems and other mathematics of PNG would have been more appropriate. In fact, every teachers college, university, senior high school, library, and education departments had received copies of Glen Lean’s Provincial Summaries and there were papers in the PNG Journal of Education and Mathematics Education Research Journal by Owens (2000, 2001) that could have been used as references. With the inadequacy of the professional learning program for teachers in the schools, the impact on beginning teachers in the colleges became crucial. Shortly after the commencement of this project, an independent evaluation of the PASTEP Project was implemented and led by Philip Clarkson with Kay Owens as another member of the five-member project team which included two Papua New Guinean teacher educators—Wilfred Kaleva and Theresa Hamadi— and another Australian teacher education leader, Ron Toomey. Some of the results of this evaluation project are pertinent in understanding why the reform as it applied to mathematics education did not fulfil the promise it held. In particular, the key ideas of cultural mathematics and the language of mathematics that the child brings to school were not apparent in the mathematics education college teaching and in the perceptions of graduating college students (Clarkson, 2016; Clarkson, Hamadi, Kaleva, Owens, & Toomey, 2003). Towards the end of the period, Professor Toni Downes from Western Sydney University led an Australian Aid feasibility delegation which recommended the setting up of a large secure, computer laboratory in every teachers college. This too was carried out and some strong, aircon-

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ditioned buildings were built to accommodate the necessary facilities. Thus, students as well as staff now had access to electronic resources. The future of these facilities is discussed further in Chapter 12 of this book.

Elementary and Lower Primary Mathematics Curricula

The elementary and lower primary mathematics curriculum documents state that space, measurement, number, pattern, and chance are to be taught at those levels of schooling. These mathematical areas are present in the Tolai culture (Paraide’s culture), Yu Wooi (Muke’s culture) and from our other research (see Chapters 2 and 3; Owens, 2014, 2015, 2016b, 2022 in press; Owens, Lean with Paraide & Muke, 2018), and most probably in most if not all Papua New Guinean cultures. Lipka, Wahlberg, George, and Ezran (2004) provided evidence that the practical application of Indigenous mathematical knowledge is likely to strengthen students’ understanding of mathematical concepts, particularly if they are integrated into classroom mathematics. The PNG reform curriculum caters for such integration (NDOE, 2003a, 2004d). However, Paraide (2010) found that teachers were not yet adequately prepared to integrate Indigenous mathematical knowledge with formal mathematics. She also found that teachers were not yet fully aware of the fact that mathematical knowledge that was applied in students’ Indigenous ceremonies and everyday activities can be used as stepping stones for teaching formal mathematics. The recognition and use of Indigenous mathematical knowledge within schools strengthens understanding of concepts and processes of mathematical thinking. Indigenous knowledge is valued and the student’s identity strengthened along with taking the initiative to explore further both school and cultural mathematics. A stronger bonding between the teaching of Indigenous and Western mathematical concepts, skills, and knowledge could enhance students’ in-depth understandings of similar and new concepts, skills, and knowledge in both worlds, and encourage further exploration of the mathematical knowledge that they have gained. This could elevate Indigenous mathematical knowledge in PNG, as Battiste (2002) and Nakata (2002) reported was the case in their Indigenous knowledge situations in Canada and Australia, respectively. One must stress caution when noting how Indigenous mathematical concepts appear to be similar to Western mathematical concepts, as Ascher (2002) and Battiste (2002) did in their discussions of Indigenous knowledge in South America and Canada. For example, caution needs to be used when noting how the Tolai counting system of coconut relates to Western multiplication, division, addition, and subtraction operations. Although there are similarities there are also distinct differences to the Western base number systems. When counting coconuts in Tinatatuna, coconuts are always tied in twos, then grouped in fours, then grouped in sixes, then the sixes are again grouped in twelves, and finally the sets of twelves are grouped in sets of ten for oral recording purposes. This complex counting system cannot be easily translated into the Western concept of base counting systems and taught using Western teaching methods, or taught in isolation from the gathering of coconuts because only coconuts are counted using this method (Paraide, 2018). Crucially, as Ascher (2002) stated, Indigenous mathematics cannot be separated from the people’s cultures and the ways particular mathematical concepts are used. The elementary and lower primary mathematics syllabi and the teachers’ guides do not detail how cultural or Indigenous mathematical skills and knowledge can be linked to the strands listed in the documents. There are no examples from any language groups detailing how Indigenous mathematical concepts and knowledge can be integrated into the mathematical strand in the syllabus. Yet, as Kaleva (1998) confirmed, teachers often do not know their own or their students’ Indigenous mathematics, and have difficulties in linking cultural mathematics to formal mathematics. Practical examples may encourage Papua New Guinean teachers to research their

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own mathematical knowledge, and they could use as a stepping-stone when teaching formal mathematics, especially at elementary and lower primary levels of education. No studies have been undertaken of the trainers’ level of understanding of PNG’s Indigenous mathematical concepts and knowledge, and of how these might best be linked to formal mathematical teaching and learning environments. Furthermore, no study has occurred to date of the trainers’ competency to facilitate the integration of Indigenous and Western knowledge for elementary school teachers. These issues could affect how well Indigenous and Western mathematics can be integrated at the elementary level, and could affect how well the dynamic two-way movement from one body of knowledge to another is conducted. Research suggests that sufficient examples of teaching concepts and practical demonstrations showing how to apply a particular teaching strategy in classroom situations is a preferred training strategy by teachers (Guy, Paraide, Kippel, & Reta, 2001). Papua New Guinean teachers learn better through this type of presentation, because they can visualize the concepts and understand them better, rather than through print information alone. Curriculum documents provide some examples to show the links between knowledge areas and subjects that are equally important to the students’ understanding of the integration of knowledge and how they complement each other. However, the integration between Western and Indigenous mathematics is minimal. The teaching strategies used for the integration of Indigenous mathematics into formal mathematics should be the main focus in the Papua New Guinean mathematics syllabi, given that cultural bonding in all subjects is encouraged at various levels of education. One of the few attempts to integrate cultural knowledge and language in elementary schools was developed through a design research project (Kravia & Owens, 2014; Bino, Owens, Tau, Avosa, & Kull, 2013; Owens, Edmonds-Wathen, & Bino, 2015).4. Several professional development sessions were held in Madang and Central Provinces by Owens, Bino and EdmondsWathen. Sondo, Muke, Daino, Kravia and Sakopa also participated in bringing this ethnomathematical approach to rural and remote schools in Madang, Jiwaka, Simbu, Eastern Highlands and Hela respectively. In particular, teachers were supplied with a manual in hard copy or on a small computer connected to a lithium battery and solar panel which also contained videos of ethnomathematical practices, assessment tasks and ideas for teaching. The manual brought together early mathematical learning with culture and language and ways of teaching through inquiry learning. It provided a number of game activities, and assessment tasks on counting and early arithmetic, space and measurement. However, the new government was beginning to stipulate changes in education and further funding was not available to extend this potentially valuable program. The final edition of the teacher’s manual5 was set out as a self-instruction unit (SIU) and a computer resource (set out like a website) (Owens, 2016a) and was made available to PNGIE and the curriculum unit in 20166. The final version incorporated examples of lessons in the format of the new revised draft elementary school teachers’ guide (with scripted lessons). There were at the time no real SIUs covering mathematics, so this was a significant opportunity for PNGIE. Importantly, in this professional development which provided one of the few if not the only opportunity available to teachers, the researchers particularly considered the language of mathematics. This generally required considerable discussion since the teachers had not really considered the implications of culture and language on their thinking about school mathematics which

This project was funded by an Australian Development Research Award (2014–2016). This manual has been well received by teachers in Solomon Islands and Tonga as well as in PNG. 6 Members of the Curriculum Unit participated in the first workshop and provided input on a regular basis to the project, especially Kila Tau and Philippa Darius. 4 5

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they had appeared to learn more by rote than by meaning (Edmonds-Wathen, Owens, Bino, & Muke, 2018; Edmonds-Wathen, Owens, Bino, 2019). There has been a struggle concerning the implementation of the education reform in PNG— especially the implementation of vernacular and bilingual instruction. One group of well-educated Papua New Guineans hold the view that English instruction at all levels of education would prepare students better for life in a globalized economy in which English is the global language, and thereby improve the standard of education in PNG. Another group is of the view that vernacular and bilingual instruction at the lower levels of education can assist the students to understand and learn basic concepts and knowledge better, and thereby enable mastery not only of vernacular literacy and numeracy skills, but also of English literacy, formal numeracy and other skills needed to function well in the global community. These opposing views have proved to be major hindrances in the implementation of vernacular instruction at the elementary level, and bilingual instruction at the lower primary level. Many educated Papua New Guineans do not realize—and may be not willing to acknowledge— that a wealth of Indigenous knowledge is embedded in the vernacular languages. Much of these knowledges are similar to the formal subjects. For example, the knowledge which is embedded in the Tolai language is similar to that which Warren (1991) and Warren, von Liebenstein, and Silkkerveer (1993) highlighted. Ascher (2002) also identified similar mathematical knowledge used by the Indigenous peoples in South America. The debates about the best language of instruction at the elementary and lower primary levels occur among influential educated Papua New Guineans. The majority of people simply want a relevant education that will prepare their children for possible formal employment, or a worthwhile life back in their communities. In relation to the reform, Paraide’s (2010) thesis, located vernacular languages and Indigenous knowledge at the same level as English and Western knowledge in the elementary and lower primary levels of education which were stimulated by the literature concerning the arguments about the values of Indigenous education and vernacular instruction in formal learning institutions. Language of Instruction in Formal Schooling The learning of reading and writing skills in the English language was supposed to be gradually introduced in Grade 3, with an increasing use of English as the students advance to Grades 4 and 5. The teaching of English literacy skills began in Grade 3, and continued in the upper primary and secondary schools. The language of instruction at the upper levels of education was—and still is—English. It was anticipated that when the English literacy areas were taught and supported well in primary and secondary schools, the students would be able to master sufficient English reading and writing skills in order to participate well in the Grades 8, 10 and 12 National Examinations, which are in English. The critics of the former education reform have not supported vernacular and bilingual instruction at the lower levels of formal schooling because they are of the view that this teaching strategy will not lead as well to mastery of English language literacy skills. Ethnomathematics Rauff (2003) described ethnomathematics as “a new field of study which lies at an intersection of anthropology, education, and mathematics” (p. 2). Ethnomathematicians view all signs of counting, measuring, designing, patterning, modelling, sorting, or reasoning as evidence of mathematical ideas. Rauff (2003, p. 2) added that these ideas, whether implicit or explicit, past or present, and irrespective of cultural setting, are grist for ethnomathematicians’ mills. Ascher (2002) stated that ethnomathematics is a powerful tool for understanding other cultures:

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Local mathematics can be detected, for instance, in the work of artisans and craftsmen, as well as in the lives of farmers, fishermen, healers, storytellers, and street merchants. It manifests itself in beadwork, games, hairstyles, maps, painted designs, songs, and woven goods. (p. 2) According to Ascher (2002), the Iqwaye people of PNG, use fingers, toes, and spaces between toes as tools for counting to numbers much larger than 10, 20, or 28 (in the Hindu- Arabic notation). These form the basis of a sophisticated numbering system that can count to numbers of infinitely large sizes. Paraide added to this discussion by describing number and measurement knowledge that is used by the Tolai people from East New Britain. The Tolais also count numbers of indefinitely large sizes and use measurements and other mathematical knowledge in their lives (Paraide, 2008, 2010, 2018). Ethnomathematics has gained increasing recognition during the past few decades (Ascher, 1994, 2002; de Abreu, Bishop, & Presmeg, 2002; Rauff, 2003; Zaslavsky, 1973). Ascher (2002) described ethnomathematics as a “research program in the historical and epistemological foundations of mathematics with pedagogical implication” (p. 1). In part, that involves mapping the vast diversity among groups of people in the field of mathematics, the ways that numbers and other mathematical ideas are understood and conceived, the methods of reasoning, and the systems that people adopt to model and find patterns in their own social and natural environments. Understanding cultural mathematical concepts can assist learners and teachers to comprehend abstract mathematical concepts (Prince, 1969). Ascher supported this view in this statement: When one views cultural practices from a mathematical perspective, understanding is deepened, vague descriptions are clarified, and the sophisticated conceptual underpinnings of those practices are revealed. (p. 2) Mathematical knowledge has been used by many people around the world for millennia and is reflected in the work of artisans, craftsmen, farmers, fishermen, healers, storytellers, and traders. It has manifested itself in beadwork, games, hairstyles, maps, painted designs and woven goods (Ascher, 1991, 2002). Patricia Paraide emphasizes that from an Indigenous perspective, she knows that mathematical knowledge is one of the dominant features in Indigenous people’s lives because it is applied in all practical activities. For example, number and measurement are only two areas of mathematical knowledge that are used extensively by her people, the Tolais.

Formal Integration of Indigenous and Western Knowledge

Battiste (2002) discussed the integration of Indigenous and Western knowledge in her own country, Canada, and viewed this as a belated act of elevation of Indigenous knowledge. However, Tololo (1976), Smith (2012) and Young (2001) have argued that many Indigenous people have now generally accepted that Western knowledge and languages are more valuable and important than their own Indigenous knowledge. They assert that such acceptance is primarily a consequence of the dominant focus on Western knowledge and language in formal education settings, such as schools, during and after colonial rule, and the desire for an education that is on par with the global community, an outcome of globalization. As Ascher (2002), de Abreu et al. (2002), Beach (2003), and Bishop and Seah (2003) found students in their studies, just like Papua New Guinean students, come to school with some Indigenous knowledge, including mathematical knowledge and skills, which they use in their home environments. Indigenous Papua New Guineans’ mathematical knowledge is embedded in their languages (Kaleva, 2001; Paraide, 2008). Some Indigenous mathematical concepts are similar to Western concepts (de Abreu, 1995; de Abreu & Cline, 1998; de Abreu, Bishop, & Presmeg, 2002; Ascher, 2002; Beach, 2003; Bishop, 2004). However, Indigenous mathematical knowledge, as a body of knowledge, has been suppressed and ignored for many years, first by the

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colonial authorities, and subsequently by p ost-Independence authorities by accepting curricula inherited primarily from past colonial influences, within which Western mathematical knowledge has been dominant. Bishop (1995), de Abreu (1995) and Ascher (2002), stressed that, as a consequence, Indigenous mathematical knowledge remained ignored and disconnected from formal Western school mathematics, even by Indigenous teachers. The history of formal mathematical education has generally been dominated by those who are wedded to the idea of decontextualized knowledge and skills. Western mathematical concepts have been taught to Indigenous students with no links to the mathematics knowledge that they already have (de Abreu & Cline, 1998; dÀmbrosio, 2001; de Abreu et al., 2002; Bishop & Seah, 2003; Bishop, 2004). However, attempts are now being made to address this inadequacy in some reform curricula around the world (Battiste, 2002). In the 2020s, PNG’s educational documents reflect this change from what had been developed by the beginning of this century (Department of Education, 1998b, 2000c, 2003a, 2004d). Mathematical concepts which were taught in isolation, in a foreign language, and understood (or not understood) and taught from a Western perspective, have dominated past, and much of current, formal mathematical teaching (Beach, 2003; Bishop, 1995, 2004; DÀmbrosio, 2001; de Abreu et al., 2002). We, the authors, believe that Indigenous languages, knowledge, and teaching strategies began to lose their value in PNG during the colonial era because they were excluded from the formal curricula when English and Western mathematical knowledge and teaching strategies were made compulsory. The latter’s dominant position in almost all formal education elevated its value and status, which made it more visible than cultural mathematics Indigenous languages, and cultural knowledge and teaching and learning strategies. Consequently, students began to view their Indigenous knowledge, languages, and teaching and learning strategies as second class. A further consequence was that English became the only accepted language of instruction. Furthermore, Western knowledge and teaching strategies became established practices in formal education. From a post-colonial perspective, English and Western mathematical knowledge were imposed on formal education because these were powerful agents for changing the Indigenous people’s views of the world. They were also used as a form of control and domination. Education became institutionalized (Fellingham, 1993), and Western experts were brought in to teach the curricula. English became the language of power (Pennycook, 1998), and Western knowledge and teaching strategies were held to be more superior (Smith, 1975). Indigenous languages, cultural knowledge, and teaching strategies were forbidden and classified as belonging to primitive and uncivilized people (Barrington-Thomas, 1976; Smith, 1975). Over time, the negative views associated with Indigenous languages, systems of knowledge, and teaching strategies became embedded in educational practices, and currently still stand as stumbling blocks in the implementation of vernacular and bilingual education, and the integration of Indigenous and Western knowledge in formal education. Ethnomathematical teaching strategies on the other hand consider the parallels between traditional Indigenous perspectives and Western mathematical knowledge and concepts, as well as the teaching of mathematics, as part of particular cultures (Ascher, 2002; Rauff, 2003). Indigenous mathematics cannot be separated from its social settings because it functions as a part of total cultural pictures. Ethnomathematical projects encourage the marriage between Indigenous and Western mathematical knowledge in formal education. As Ascher (2002) stated: Numerous similar projects teaching Western mathematics in traditional settings are underway from Papua New Guinea to the inner cities of the United States. All recognize that the understanding of local mathematical knowledge can both validate a child’s native culture and provide a bridge to modern Western mathematics (p. 2). The emphasis on the total cultural perspective has led mathematical educators working in non-Western settings to recognize the importance of ethnomathematics in their own work (de

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Abreu et al., 2002; Ascher, 1991, 2002; Beach, 2003; Bishop & Seah, 2003; Rauff, 2003). LaraAlecio (2002) added that with the recognition of Indigenous mathematics, mathematics educators can integrate culture into the formal curriculum and develop students’ competence and confidence using ethnomathematics. Moll and Diaz (2002) also argued that ethnomathematics calls for the reconstruction of mathematics curricula everywhere, to achieve cultural compatibilities. The NDOE, through the curriculum reform, emphasized the teaching of cultural mathematics at elementary and lower primary levels of education in PNG (NDOE, 2000c, 2003a, 2003b, 2004a). The curriculum documents use “cultural mathematics” instead of enthnomathematics to emphasize that mathematical knowledge is embedded in PNG’s languages, cultures and daily activities. These documents were intended to assist teachers to encourage students to link and appreciate the mathematical knowledge that they already have, with those in formal education. Currently, minimal literature has been cited relating to the interaction of Indigenous and home mathematical knowledge and formal mathematical teaching and learning in PNG. De Abreu et al. (2002) have discussed some of the difficulties that Indigenous students face in formal learning situations including: • Different levels of mathematical understanding; • Different cultural and economic backgrounds; • Different methods of mathematics application, that various students already have when they enrol in formal learning environments; and, • Some of the difficulties that students have in relating to concepts in formal mathematical learning situations. For example, one of de Abreu et al’s studies (2002) showed that migrant students who are involved in informal trading at home have difficulties linking addition, subtraction, multiplication, and division knowledge to their formal mathematics learning. Paraide has noted that Tolai students who use a complex counting system at home have similar difficulties with the transition similar to de Abreu et al’s migrant students. Some Tolai children have difficulties with addition, subtraction, multiplication, and division operations in formal school mathematics although they confidently use these concepts competently in all of their trading activities and counting after school. De Abreu et al. (2002) also claimed that the language of instruction used in formal mathematical learning is one of the contributing factors to an inadequate understanding of mathematical concepts. Students may not have sufficient knowledge of vocabulary in the language of instruction to be in a position to understand the knowledge that is communicated by the mathematics teachers and other students in class. de Abreu et al. (2002) identified previous methods of teaching mathematics, the level of mathematics taught, and the work demand on students in previous mathematics learning experiences, as having an impact on how students participate in, build on, and explore their existing mathematical knowledge. de Abreu and Cline (1998) acknowledge the differences in mathematical practices in different cultural groups, and caution that mathematics learning can be problematic in classroom situations. Their study revealed problems associated with transferring and generalizing knowledge from one cultural context to another. They highlight that inquiry-oriented mathematics is more conceptually organized than traditional school mathematics, which emphasizes students’ verbal communication to the exclusion of other forms of communication. Furthermore, inquiry-oriented mathematics which prizes argumentation as a means of communication will differ significantly within cultures where children have to observe and only occasionally speak if they want to learn. In such learning contexts, students do not challenge their Elders or teachers. Furthermore, schooling often features high stakes-testing environments which are very different from students having the flexibility to choose when to engage in learning, how to engage in learning, and when to display their knowledge in an out-of-school context.

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de Abreu et al.’s (2002) findings are similar to the mathematical learning problems identified in PNG classrooms by Paraide (2002, 2008). PNG has many cultural groups and each has its own mathematical knowledge and practical application of such knowledge. In many cases, what they already know is not related to what they are learning in formal education. The teaching strategies in the formal and informal environments are also different. Cole, Gay, Glick, and Sharp (1971) suggested: We want to emphasize our major conclusion that cultural differences in cognition reside more in the situations to which particular cognitive processes are applied than in the existence of a process in one cultural group and its absence in another. Assuming that our goal is to provide an effective education for everyone … our task must be to determine the conditions under which various processes are manifested and to develop techniques for seeing that these conditions occur in the appropriate educational setting. (Cole et al., 1971, p. 233) The practical applications of mathematics can enhance understanding of mathematical concepts if they are linked to formal mathematical learning. Lipka et al. (2004) noted that during their study of Eskimo students’ working with patterns Elders would be working with geometrical relationships such as folding a square to check the sides are equal and the right-angles permit symmetry. They could do similar things for a rhombus and a rectangle thus “showing” proof of the shape required. That example shows how traditional activities can be related to Western mathematical concepts in geometry. The observation process of learning mathematical concepts is also present in the Tolai culture in applying number and measurement knowledge (Paraide, 2010). Chapters 2 and 3 in this book provide many more examples from PNG cultures. However, without further research funding and valuing of cultural mathematics by all in PNG, there will not be progress in this way.

Moving Forward

Did the reform come too late to PNG? And did the use of minimal teaching information in the syllabus contribute to the reform failing in the eyes of many, and in terms of literacy testing? There is no doubt that research both outside and in PNG has shown that learning mathematical concepts in your home languages in early education is valuable but without the necessary teacher education of elementary school teachers and pay that recognized the added value, this level of education was not likely to succeed. By the time the reform occurred, there was another 10 years of education with Western mathematics and English or Tok Pisin only. With lack of preservice and inservice education on bilingual or multilingual education, there was also likely to be issues. Issues associated with replacing English with Tok Pisin rather than quality English as well as quality local vernacular languages still remained. There was not the expertise, time or perhaps interest in ensuring that the community’s mathematics was discussed and incorporated into the school curriculum across the country. A greater level of ownership and vision was needed for this to occur. Pockets of quality integration of Indigenous and Western mathematics occurred. However, this needed to be a focus at the beginning, and not something on the periphery 12 to 15 years later at the end of the reform period. References Abady, R. (2015). The impact of the Papua New Guinea Free Education Policy on the school executive's decision making in the management of class size. Master of Educational Leadership thesis, University of Waikato, Hamilton, New Zealand. https://hdl.handle.net/10289/9822

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Ahai, N., & Faraclas, N. (1993). Rights and expectation in an age of debt crisis: Literacy and integral human development in Papua New Guinea. In P. Freebody & A. Welch (Eds.), Knowledge culture and power: International perspectives on literacy as policy and practice (pp. 19–29). London, England: Falmer Press. Ascher, M. (1991). Ethnomathematics: A multicultural view of mathematical ideas. Boca Raton, FL: Chapman & Hall/CRC of Taylor and Francis. Ascher, M. (2002). Mathematics elsewhere: An exploration of ideas across cultures. Princeton, NJ: Princeton University Press. Avalos, B., & Neuendorf, L. (Eds.) (1991). Teaching in Papua New Guinea: A perspective for the nineties. Port Moresby, PNG: University of PNG Press. Barnhardt, R., & Kawagley, A. O. (1999). Indigenous knowledge systems and Alaska Native ways of knowing. Anthropology and Education Quarterly, 36(10), 18–23. Barrington-Thomas, E. (1976). Problems of educational provision in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 3–16). Melbourne, Australia: Oxford University Press. Battiste, M. (2002). Indigenous knowledge and pedagogy in First Nations education: A literature review with recommendations (Working paper). Ottawa, Canada: Apamuwek Institute. Beach, D. (2003). Mathematics goes to market. In D. Beach, T. Gordon, & E. Lahelma (Eds.), Democratic education ethnographic challenges (pp. 99–122). London, England: Tufnell Press. Bino, V., Owens, K., Tau, K., Avosa, M., & Kull, M. (2013). Improving the teaching of mathematics in elementary schools in Papua New Guinea: A first phase of implementing a design. Mathematics digest: Contemporary discussions in various fields with some mathematics— ICPAM-Lae (pp. 84–95). Lae, PNG: Papua New Guinea University of Technology. Bishop, A. (2004). Mathematics education in its cultural context. In T. Carpenter, J. Dossey, & K. Koehler (Eds.), Classics in mathematics education research (pp. 200–207). Reston, VA: National Council of Teachers of Mathematics. Bishop, A., & Seah, W. T. (2003). Values in mathematics: Teaching the hidden persuaders? In A. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Second international handbook of mathematics education (pp. 717–765). Dordrecht, The Netherlands: Kluwer. Bray, M. (2007). Governance and free education: Directions, mechanisms and policy tensions. Prospects, 37(1), 23–35. Clarkson, P. (2016). The intertwining of politics and mathematics teaching in Papua New Guinea. In A. Halai & P. Clarkson (Eds.), Teaching and learning mathematics in multilingual classrooms (pp. 43–55). Rotterdam, The Netherlands: Sense Publishers. Clarkson, P., Hamadi, T., Kaleva, W., Owens, K., & Toomey, R. (2003). Final report. PNG- Australia Development Cooperation Program. Baseline survey: Interpretative evaluation report of the Primary and Secondary Teacher Education Project. Melbourne, Australia: Australian Catholic University. Cole, M., Gay, J., Glick, J., & Sharp, D. (1971). The cultural context of learning and thinking. London, England: Methuen. de Abreu, G. (1995). Understanding how children experience the relationship between home and school mathematics. Mind, Culture, and Activity: An International Journal, 2, 199–142. de Abreu, G. (2002). Towards a cultural psychology perspective on transitions between contexts of mathematical practices. In G. de Abreu, A. Bishop, & N. Presmeg (Eds.), Transitions between contexts of mathematical practices (pp. 170–189). Dordrecht, The Netherlands: Kluwer. de Abreu, G., & Cline, T. (1998). Studying social representations of mathematics learning in multiethnic primary schools: Work in progress. Papers on Social Representations, 7, 1–20.

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de Abreu, G., Bishop, A., & Presmeg, N. (Eds.). (2002). Transitions between contexts of mathematical practices. Dordrecht, The Netherlands: Kluwer. DÀmbrosio, U. (2001). What is ethno-mathematics, and how can it help children in school? Teaching Children Mathematics, 7(6), 308–310. Department of Education Papua New Guinea. (1974). Report of the Five-Year Education Plan Committee, September, 1974 (The Tololo Report). Port Moresby, PNG: Author. Edmonds-Wathen, C., Owens, K. D., & Bino, V. (2019). Identifying vernacular language to use in mathematics teaching. Language and Education, 33(1), 1–17. https://doi.org/10.1080/0 9500782.2018.1488863 Edmonds-Wathen, C., Owens, K. D., Bino, V., & Muke, C. (2018). Who is doing the talking? An inquiry-based approach to elementary mathematics in Papua New Guinea. In R. Hunter, M. Civil, B. Herbel-Eisenmann, N. Planas, & D. Wagner (Eds.), Mathematical discourse that breaks barriers and creates space for marginalized learners (pp. 257–275). Rotterdam, The Netherlands: Sense Publishers. Evans, T., Guy, R., Honan, E., Kippel, L. M., Muspratt, S., Paraide, P., & Tawaiyole, P. (2006a). PNG Curriculum Reform Implementation Project—Impact Study 6. http://hdl.handle. net/10536/DRO/DU:30010511 Fellingham, L. A. (1993). Foucault for beginners. New York, NY: Writers and Readers Publishing Inc. Guy, R. (1999). Reforming education: Attending to relevance and quality. Port Moresby, PNG: National Research Institute. Guy, R. (2009). Formulating and implementing education policy. In R. May (Ed.), Policy making and implementation: Studies from Papua New Guinea (Vol. 5, pp. 131–154). Canberra, Australia: ANU E Press. Guy, R., Bai, W., & Kombra, U. (2000). Enhancing their future. Port Moresby, PNG: NDOE, Papua New Guinea. Guy, R., Paraide, P., & Kippel, L. (2001a). Mi lusim school: A retention study in primary and secondary schools. Port Moresby, PNG: National Research Institute. Guy, R., Paraide, P., Kippel, L., & Reta, M. (2001b). Bridging the gaps. Port Moresby, PNG: National Research Institute. Kaleva, W. (1991). Teachers and school mathematics in the 1990. In B. Alavos & L. Neuendorf (Eds.), Teaching in Papua New Guinea: A perspective for the nineties. Port Moresby, PNG: University of Papua New Guinea. Kaleva, W. (1998). The cultural dimension of mathematics curriculum in Papua New Guinea: Teacher beliefs and practices. Doctoral thesis, Monash University, Melbourne, Australia. Kippel, L., Paraide, P., Kukari, A., Agigo, J., & Irima, K. (2009). Follow-up retention study: Students’ absenteeism and withdrawal-contributing factors. Port Moresby, PNG: National Research Institute, Port Moresby. Kravia, G., & Owens, K. D. (2014). Design research for professional learning for Cultural Mathematics. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.), MERGA37: Curriculum in focus: Research guided practice. Adelaide, Australia: Mathematics Education Research Group of Australasia. https://merga.net.au/Public/Public/Publications/ Annual_Conference_Proceedings/2014_MERGA_CP.aspx Laycock, D. (1976). Pidgin’s progress. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 177–186). Melbourne, Australia: Oxford University Press. Lipka, J., & Adams, B. L. (2004). Culturally-based math education as a way to improve Alaska native students' math performance. Working Paper 20. Appalachian Collaborative Center for Learning, USA (ED484849) (ERIC Ed.).

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Lipka, J., Wahlberg, S., George, N., & Ezran, D. (2004). Transforming the culture of schools: Yupìk Eskimo examples. Canadian Journal of Science, Mathematics & Technology Education, 2, 287–304. Macpherson, J. (1991). How our history may guide our future. In B. Avalos & L. Neuendorf (Eds.), Teaching in Papua New Guinea: A perspective for the nineties (pp. 213–224). Port Moresby, PNG: University of PNG Press. Matane, P. (1976). Education for What? In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 57–60). Melbourne, Australia: Oxford University Press. Matane, P. (1991). Teachers for the future. In B. Avalos & L. Neuendorf (Eds.), Teaching in Papua New Guinea: A perspective for the nineties (pp. 139–145). Port Moresby, PNG: University of PNG Press. McKinnon, K. (1976). Development in primary education: The Papua New Guinea experience. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 226–251). Melbourne, Australia: Oxford University Press. Nakata, M. (2002). Indigenous knowledge and cultural interface: Underlying issues at the intersection of knowledge and information systems IFLA Journal, 28(5–6), 281–291. Narakobi, B. (1991). Issues in education and development. In B. Avalos & L. Neuendorf (Eds.), Teaching in Papua New Guinea: A perspective for the nineties. Port Moresby, PNG: University of PNG Press. National Research Institute (2004). Papua New Guinea: Public expenditure and service delivery. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1986). A philosophy of education for Papua New Guinea (chairperson, P. Matane). Port Moresby, PNG: Government Printer. NDOE Papua New Guinea. (1990). PNG strategic plan. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1991). Education sector review: Volume 2, Deliberations and findings. Port Moresby, PNG: Author NDOE Papua New Guinea. (1991). Education sector study. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1992). Relevant education for all. Education sector review. Port Moresby, PNG: Facilitating and Monitoring Unit, Papua New Guinea. NDOE Papua New Guinea. (1993). National special education plan and policy guidelines (pp. 1–31). Port Moresby, PNG: Author. NDOE Papua New Guinea. (1995). Education resource study. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1996). Papua New Guinea national education plan (1995–2004) Volume a. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1997). Elementary handbook. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1998). Lower primary language syllabus, grades 3–5. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1999a). Language policy in all schools, Education Circular. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1999b). National education plan (1995–2004). Update 1. Port Moresby, PNG: Author. NDOE Papua New Guinea. (1999c). Primary handbook. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2000a). The state of education. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2000b). National literacy policy. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2000c). Lower primary mathematics: Bridging to English resource book. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2002). National curriculum statement. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2003a). Cultural mathematics: Elementary syllabus . Port Moresby, PNG: Author.

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NDOE Papua New Guinea. (2003b). Elementary school teachers’ guide. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2003c). National assessment and reporting policy. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2004a). A national education plan 2005–2014: Achieving a better future. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2004b). The 2004 state of education. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2004c). Lower primary language syllabus. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2004d). Lower primary mathematics syllabus. Port Moresby, PNG: Author. NDOE Papua New Guinea. (2005). The 2005 education report. Port Moresby, PNG: Author. Owens, K. (2014). The impact of a teacher education culture-based project on identity as a mathematics learner. Asia-Pacific Journal of Teacher Education, 42(2), 186–207. https://doi.org/ 10.1080/1359866X.2014.892568 Owens, K. (2015). Visuospatial reasoning: An ecocultural perspective for space, geometry and measurement education. New York, NY: Springer. Owens, K. (2016a). Mathematics for elementary schools. Manual and tutor’s manual (4th ed.). Owens, K. (2016b). The line and the number are not naked in Papua New Guinea. International Journal for Research in Mathematics Education. Special issue: Ethnomathematics: Walking the mystical path with practical feet, 6(1), 244–260. http://sbem.iuri0094.hospedagemdesites.ws/ojs3_old/index.php/ripem/issue/view/99 Owens, K. (2022, in press). Chapter 7. The tapestry of mathematics – Connecting threads: A case study incorporating ecologies, languages and mathematical systems of Papua New Guinea. In R. Pinxten & E. Vandendriessche (Eds.), Indigenous knowledge and ethnomathematics. Cham, Switzerland: Springer. Owens, K., Edmonds-Wathen, C., & Bino, V. (2015). Bringing ethnomathematics to elementary teachers in Papua New Guinea: A design-based research project. Revista Latinoamericana de Etnomatematica, 8(2), 32–52. http://www.revista.etnomatematica.org/index.php/RLE/ article/view/204 Owens, K., Lean, G. A., with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Paraide, P. (1998). Elementary education: The foundation of education reform in Papua New Guinea. Papua New Guinea Journal of Education, 34(1), 1–18. Paraide, P. (2002). Rediscovering our heritage: An early assessment of lower primary learning in the reform curriculum (DER Report number 76). Boroko, Papua New Guinea: National Research Institute. Paraide, P. (2008). Number in Tolai culture. Contemporary PNG Studies: DWU Research Journal, 9, 69–77. Paraide, P. (2010). Integrating Indigenous and Western mathematical knowledge in PNG early schooling. Doctoral thesis, Geelong, Australia: Deakin University. Paraide, P. (2015). Challenges with the tuition fee free education policy implementation. Papua New Guinea. Contemporary PNG Studies, 23, 47–62. Paraide, P. (2018). Chapter 11: Indigenous and western knowledge. In K. D. Owens & G. A. Lean, with P. Paraide & C. Muke (Ed.), History of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Pennycook, A. (1998). English and the discourse of colonialism. London, England: Routledge. Pokana, J. S. V. (2013). The barriers and facilitators of institutionalisation of inclusive education policy and practices in Papua’s education system. Doctoral thesis, University of New England. Australia.

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Prince, J. (1969). Teaching mathematics in PNG secondary schools. PNG Journal of Education, 12(2), 19–26. Quartermaine, P. (2001). Teacher education in Papua New Guinea: Policy and practice. PhD thesis, University of Tasmania, Hobart, Australia. Rauff, J. (2003). The varieties of mathematical experience. Natural History, 112(7), 2–9. Rogers, C. (1986). Philosophy of education in Papua New Guinea. The setting of ideals and goals. Papua New Guinea Journal of Education, 22(1), 3–14. Said, E. (1978). Orientialism. New York, NY: Pantheon Books. Said, E. (1993). Culture and imperialism. London, England: Chatto & Windus. Saxe, G. (2012). Cultural development of mathematical ideas: Papua New Guinea studies. New York: Cambridge University Press. Simpson, A. (1994). The grades seven and eight transition to primary schools in Papua New Guinea. Papua and New Guinea Journal of Education, 1, 9–13. Sinebare, M. (1991). From totems to tools: Computer literate teachers for the nineties. In B. Avalos & L. Neuendorf (Eds.), Teaching in Papua New Guinea: Perspective for the nineties (pp. 233–249). Port Moresby, PNG: University of Papua New Guinea Press. Smith, Geoffrey. (1975). Education in Papua New Guinea. Melbourne, Australia: Melbourne University Press. Smith, G. (1978). Counting and classification on Kiwai Island. Papua New Guinea Journal of Education, Special Edition, The Indigenous Mathematics Project, 14, 53–68. Smith, P. (1987). Education and colonial control in Papua New Guinea: A documentary history. Melbourne, Australia: Longman Cheshire. Smith, L. T. T. R. (2012). Decolonizing methodologies: Research and indigenous peoples. New York: Zed Books. Souviney, R. (1983). Mathematics achievement, language and cognitive development: Classroom practices in Papua New Guinea. Educational Studies in Mathematics, 14(2), 183–212. https://doi.org/10.1007/BF00303685 Sukthankar, N. (1995). Gender and mathematics education in Papua New Guinea. In P. Rogers & G. Kaiser (Eds.), Equity in mathematics education: Influences of feminism and culture (pp. 135–142). Washington, DC: The Falmer Press. Swan, A., & Walton, G. (2014). Tuition fee-free education: Insights from PNG school surveys. http://devpolicy.org/presentations/Presentation%20by%20Tony%20Swan.pdf Tapo, M. (2004). National standards/local implementation: Case studies of differing perceptions of national education standards in Papua New Guinea. PhD thesis. Queensland University of Technology, Brisbane, Australia. Tololo, A. (1976). A consideration of some likely future trends in education in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 221–225). Melbourne: Oxford University Press. Tololo, A. (1997). Chair of Ministerial Committee's Report and recommendations on the operation of the National Training Council, Ministry of Industrial Relations, Department of Labour and Employment. Port Moresby, PNG: National Training Council Secretariat. UNESCO. (1990). World declaration on education for all. Jomtien, Thailand. https://bice.org/ app/uploads/2014/10/unesco_world_declaration_on_education_for_all_jomtien_thailand. pdf UNESCO. (2011). World data on education: Papua New Guinea. http://www.ibe.unesco.org/ fileadmin/user_upload/Publications/WDE/2010/pdf-versions/Papua_New_Guinea.pdf Warren, M. (1991). Using Indigenous knowledge for agriculture development. https://documents. worldbank.org/en/publication/documents-reports/documentdetail/408731468740976906/ using-indigenous-knowledge-in-agricultural-development

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Warren, M., von Liebenstein, G., & Silkkerveer, L. (1993). Networking for Indigenous knowledge. Indigenous Knowledge and Development Monitor (Online Journal), 1(1), 1–4 http:// www.ciesin.org/docs/004-205/004-205.html Weeks, S. (1987, 12 March). Oro's high school selection unfair. Times of Papua New Guinea, p. 27. Weeks, S. (1993). Education in Papua New Guinea 1973–1993: The late-development effect? Comparative Education, 29(3), 261–273. Zaslavsky, C. (1973). Africa counts: Number and pattern in African culture. Boston, MA: Prindle, Weber & Schmidt. Zaslavsky, C. (1998). Ethnomathematics and multicultural mathematics education. Teaching Children Mathematics, 4(9), 253–262. Zepp, R., Matang, R. A., & Sapau, M. (1994a). Mathematics content knowledge of community schools. Papua and New Guinea Journal of Teacher Education, 1, 25–34. Zepp, R., Onagi, G., & Raju R. (1994b). Teachers' perceptions of high school in-service workshops in Papua New Guinea. Papua and New Guinea Journal of Teacher Education, 1, 35–39.

Chapter 9 Revising the Reform: Standards Based Education

Abstract: Although the modifications to school structures and to curricula under the reform were complex they were under-funded. Major issues arose in terms of literacy, and hence mathematics. Teachers seemed powerless to improve the situation because of fundamental flaws in the implementation process and lack of teachers’ guides to supplement the small syllabuses. As time went on, new teachers’ college lecturers were not sufficiently educated to assist preservice teachers on quality teaching for the outcomes in the syllabuses. Neocolonial attitudes were arising in the Parliament and among the people. Expatriate volunteer advisers came in promoting overseas trends from their own country. Globally, at this time, education institutions were emphasizing Standards. PNG selected Japanese aid and advisers for mathematics education. The draft syllabuses were heavily loaded with performance standards. Drafts for Pre-Elementary, Grades 1 and 2 were available with so-called scripted lessons from 2015. By 2021, syllabuses for a Standards Based Education (SBE) for Grades 3 and above were developed. By the end of 2021 new syllabuses and teachers’ support material were produced and online, making their availability more widespread. A new structure of 1, 6, 6 years in each level of schooling was finally set, but the communities and schools now had to adjust available school sites to suit the revised structure although the Government stated that it would set aside funding for new schools. Elementary teachers, of whom only a third had a secure position, would need to train to be offered positions in the primary schools for Grades 1 and 2. Neocolonialism was driving this revised system.

Key Words: Curriculum change · Implementation of policy · Language policy · Tuition free-fee education · Structural change. The volume of linguistic conventions for mathematical ideas in pre-technological societies is incredibly extensive. Results of research studies nationally and internationally show the need to link home-culture learning situations with those of formal mathematics learning situations; and vice versa. The ethnomathematical approach has been described and … the other approach that is designed to utilize mathematical knowledge and skills in both contexts, … the integration approach. … For example, … the concept of “line” and “straight line” in a geometry lesson, the concept of line in home-culture contexts should also be taught and compared with what is being taught, in order for learners to “visualize” the integration and translatability of one system to the other. This integration of mathematical ideas from the two learning situations, if implemented with proper preparation, is most likely to facilitate meaningful translation of mathematical knowledge in both contexts. Kari, 2005, pp. 36–37

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Introduction Although teacher education for elementary school teachers was a major concern for PNG educators, lack of funding continued to limit this endeavour. Despite some programs through SIL for literacy, and the provision of materials for mathematics provided to the PNG Institute of Education from the Elementary Mathematics Teachers’ Project (Owens, 2016), there has been little success in providing the necessary training for elementary school teachers. The introduction of English as a further language was also not understood by teachers and did not seem to be achieved. Indeed, the level of English literacy across the country was falling (Czuba, Homingu, Malpo, & Tetaga, 2013). In 2011, UNESCO reported a lack of qualified teachers including mathematics teachers, in PNG, and in 2012 the Prime Minister announced, in his election campaign, the abolition of school fees to enable education for all in elementary and primary schools. The Prime Minister promised that places in schools would be made available to all who passed examinations for secondary education (Grades 9 and 10) or senior secondary education (Grades 11 and 12) but were not in school due to school fees. He also announced that English was to be the language of instruction from the beginning, in Pre-Elementary. Furthermore, outcomes-based education (OBE) was blamed for holding back advances and would be eliminated with the former objectives being re-introduced. Some experienced PNG educators feared, however, that with no school fees, the number of students in schools would be out of control and would put extra pressures on the University of Goroka (UOG) and the teachers colleges. It could be argued that the failure of the reform was largely because of a lack of support documents for the syllabuses and a lack of funding for teacher education, especially at the elementary level. That said, there could be little doubt that the education world was moving globally toward standards-based education, and this was introduced in PNG, with most provinces taking on the new approach. Some thought that the language in the classroom must be standard too— English only—and this was promoted. It was decided that elementary Cultural Mathematics would become just Mathematics. The concept of cultural mathematics was disparaged. In terms of curriculum for mathematics, Japan provided the most input into a curriculum which initially had a heavy emphasis on detail within the assessment section of each draft elementary syllabus, and provided details on teaching that the OBE syllabuses had not provided. The UK VSOs had attempted to script lessons especially for elementary school teachers for literacy and that idea was also taken up by the mathematics curriculum writers. In 2015, a draft syllabus and teachers’ guide for elementary teachers was prepared. In 2017, lower primary school, Grades 3 to 5 syllabuses standards based along with teachers’ guides and then Grades 6 to 8 were released. Other online materials from 2003 and 2006 were still available. There have also been structural changes with Grades 1 and 2 returning to primary schools and finishing at Grade 6. The expectation is for 11 years of schooling for all children. In 2021 it was not yet compulsory and it was recognized that this would not be possible until more teachers were trained and schools built.

Disquiet and Politics Create a Change to Curriculum and Language of Instruction

In 2013 the policy on the use of vernaculars as the languages of instruction in the elementary and lower primary levels of education was changed—the policy reverted to English as the language of instruction at all levels of education. The former educational structure and the education reform structure were shown in Figures 6.1 and 8.1 respectively. By 2016, early childhood was still not part of the school system and the intention was for elementary school to run for one year followed by six years in primary school (Grades 1 to 6) and six years in high school (Grades 7 to 12). The so-called 1, 6, 6 system. There were still the National High Schools, Grades 11 and 12. This structure is further discussed later in the chapter.

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During the implementation of the OBE curriculum for the education reform there was opposition by some of the educated elite to the use of vernacular instruction in elementary schools, and bilingual education in lower primary levels of education. They were supported on this by many primary school teachers, some educated parents, and other stakeholders. Paraide (2010) in her review of the literature pertaining to this issue has pointed out that, before Independence, English had been imposed by the colonizers as the language of instruction at all levels of education. At the time when Independence was achieved, Tololo (1976) described the resistance to formal vernacular instruction in the following way: There is a semi-magical reason why many people think that speaking and writing in English like Europeans will result in the possession of many goods and much power. (p. 220) The view that speaking and writing English will lead to formal employment, a better life and the inheritance of power is supported by some parents and students. The introduction of vernacular and bilingual instruction and the integration of Indigenous and Western knowledge in formal learning were conscious attempts by the National Department of Education to prepare students to participate in their communities if they do not continue on to post-secondary education or secure formal employment. Paraide (2010) argued that because English and Western knowledge are now accepted as normal, better, and more valuable by many Indigenous Papua New Guineans, there is opposition to the implementation of vernacular and bilingual instruction at the lower levels of education. After the education reform began these conservative views were often given space in the two national newspapers, the Post-Courier and The National. For example, during the period of the reform before 2013, a Morobe Province political party spokesman stated that the use of vernacular languages in lower levels of school contributed to a decline in educational standards in PNG. He further stated that students were unable to speak English well as a consequence of vernacular instruction at the lower levels of education (Korugl, 2008, p. 4). The Post-Courier also reported that the Morobe Provincial Education Adviser had invited University of Goroka’s academics to present papers on the strengths of vernacular and bilingual instruction to counter the critics (Rai, 2008, p. 4). Some school teachers have always been unwilling to accept the rationale of the reform curriculum that encouraged the incorporation and integration of Indigenous and Western knowledge in schooling because of their negative views on the use of vernacular languages for formal instruction. These views were supported by other strong voices from people with limited understanding of the rationale for using first languages for formal instructions at the lower levels of education. For example, a school principal, also from the Morobe Province, maintained that vernacular and bilingual instruction were the causes of weak mastery of the English language (Nebas, 2008, p. 4). Such statements were consistent with the view that some teachers were reluctant to implement vernacular and bilingual instruction in elementary and lower primary levels of education. The number of influential people who held that viewpoint outweighed those who recognise the value of vernacular and bilingual education. Politically, a decision was made to revert the policy on language of instruction to be English at all levels of education. However, our research suggests that this required appropriate teacher preparation and language consultation work for each language as well as consistent implementation. Where these factors occurred, progress was good but this was rarely achieved. Cameo from Kay Owens During Rex Matang’s visit to a Markham Valley school for his project in which he aimed to compare early mathematics students’ mathematical understandings if taught in their vernacular, or in Tok Pisin or in English, I was able to carry out some interviews with him for Wilfred

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Kaleva and my project on measurement in PNG. We called a meeting to discuss my project and Rex asked if the community had any other questions about education. Our conversations were in Tok Pisin with the community who gathered, who were mainly men but there were a few women. One man asked about the children learning in Tok Ples because that had happened when he had gone to school—and then to high school where he had learnt in English. We asked if he had expatriate teachers at high school, and he said he had. Rex then explained that there was a lot of research suggesting that children learn to read and understand mathematical concepts if they learn in their home language, but this was the purpose of his research to find out if this was true for PNG languages. He mentioned that in his initial research with his own language of Kâte, the children who learnt in their Tok Ples did in fact do better than those who learnt in Tok Pisin or English and Tok Pisin (Matang, 2006, 2008). As I interviewed some men about their mathematical practices, I noted a woman was loudly beating tulip tree inner bark to make tapa, clearly letting me know that women as well should be interviewed about their mathematical activities. When all the school data was taken into account, Rex found that those who learnt with Tok Ples did almost as well as those who learnt in English on a Western mathematics set of counting and early arithmetic tasks. However, those who learnt in English were usually from well-educated families who also spoke at least some English at home or they were from a class of students with disabilities requiring them to learn with additional effort and a good teacher in smaller classes whose home languages were quite diverse compared with those used in most village elementary schools (Matang, 2005; Matang & Owens, 2014). The Continuing Debate on Vernacular Languages for Instruction and Cultural Content In 2007, Unage, one of PNG’s academics, referred to an article published in The National newspaper which discussed students’ lack of mastery of the English language in primary school (2007). This article suggested that the lack of mastery was the consequence of vernacular instruction at the Elementary level. Unage also referred to a Member of Parliament’s call on the floor of Parliament to review vernacular instruction in elementary schools. His justification for such a stand was that “students in the rural areas were going backwards with the exclusive use of vernacular in elementary schools” (Unage, 2007, p. 25). On the other hand, Unage supported bilingual instruction, but suggested that bilingual instruction should begin at the elementary level. He believed that bilingual instruction beginning at the elementary level would result in better mastery of the English language. The PNG population has divided opinions on vernacular and bilingual instruction. Some are of the view that these teaching strategies will not achieve the mastery of English literacy skills. Others are of the view that bilingual instruction should begin much earlier than Grade 3. Teachers also have divided opinions on the effectiveness of the vernacular and bilingual language teaching strategies. As noted in newspaper reports, it is generally viewed that only the English language can do this well. Those teachers who have negative views about vernacular and bilingual instruction continue to use English as the dominant language of instruction. Guy and others (2001, 2003) found that some lower primary teachers in some schools did not support teaching in vernacular languages and, therefore, continued to use English as the dominant language of instruction. Matsuura (2008) argued that Indigenous languages support progress towards sustainable development and a harmonious relationship between the global and local context. If they are literate in both their first and second language then this will enable them to return to their rural communities to progress and improve their standards of living. Formal curricula, which were in place in the past, had not adequately prepared many students, especially in rural and remote villages, for community life (Matane, 1976; Tololo, 1976; NDOE, 1998, 1999, 2004a, 2004b).

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Those who opposed the educational reform initiatives were unconvinced that the previous curricula had such shortfalls. Undoubtedly, well-publicized expressions of negative views on the effects of vernacular and bilingual instruction hindered the implementation of the language policy and the integration of Indigenous and Western knowledge in formal learning situations. Cameo from Patricia Paraide At the present time some educators and other stakeholders hold the view that learning in a child’s first language is not progressive in this modern era. However, from my experience, learning in Tinatatuna and also in English as a second language during my early years of formal schooling did not seem to have any negative effects on my cognitive development. I always did well in my formal subjects in primary, secondary and tertiary education. From my personal observations, I also mastered reading and writing skills in the English language better than some of my relatives and friends who went through an education system where English was the exclusive language of instruction. I did very well in the Grade 6 National Examinations. Furthermore, my learning was not hindered by my participation in current global community developments. My mastery of the English language skills is largely the result of the use and availability of a wide variety of reading and writing resources in the English language, the continuous encouragement to read English literature and write meaningful texts in English, and the English audiodrama presentations (with mainly first language speakers of English actors) that were used to teach the English language in primary school. My interaction with first-language speakers of the English language in primary and secondary school, and my continuous exposure to and interaction with the English language in my later years of formal education improved my English literacy skills further. This experience led me to identify with the Matane philosophy behind education reforms whereby vernacular and (then) bilingual teaching are advocated. My experience of the resistance to this approach by educated PNG citizens, especially teachers, contributes to my commitment to advocate for the use of languages that children know best in formal instruction when necessary. The matter of the efficacy of the educational reforms, both in general and also specifically in relation to mathematics education, were the focus of my research. The Binary Divide The binary divide between English and Indigenous languages, and Western and Indigenous knowledge was visible in the developmental documents of the formal education system in PNG, before and after Independence. Smith (1975), Barrington-Thomas (1976) and Ralph (1978) discussed the established strategies and attitudes in education that set apart Indigenous peoples from the colonial population. For example, English was the preferred language of formal instruction. They were of the view that the philosophy that “white people” should remain superior to the “black natives” provided the background to education decisions made by Groves, who was the first Director of Education in PNG to plan the development of the school system in the post-war years. Other perspectives and factors such as political goals, economic indicators, computer technologies, standards set by the global community, and donor agencies, have had great influence in shaping or even dictating the policies that have driven PNG’s curriculum, pedagogy, programs, and educational research. Papua New Guinean views, Indigenous knowledge, and teaching and learning strategies have almost been invisible. The colonial legacy, which gave birth to the view that Western types of curriculum and teaching strategies are better than those of the Indigenous people, has needed to be deconstructed. The view that Western types of curriculum and teaching strategies are superior to those of the Indigenous peoples is currently proving to be a difficult one in PNG’s educational circles, primarily because Western teaching strategies are now engrained in most formal learning institu-

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tions as well as in the minds and practices of those teachers who have been trained in many of our education institutions. The binary between Indigenous and Western knowledge, and English and Indigenous languages of instruction, can be deconstructed. Ascher (2002) in her discussions of Indigenous mathematics in North America and PNG highlighted the fact that Indigenous and Western societies share common bodies of knowledge such as number and measurement. These are used in peoples’ lives, which makes both bodies of knowledge valuable. For example, the use of number and measurement are used by different people, like the Tolai people, for their own particular needs and purposes in their particular environments (Paraide, 2008). Many of these are on-going and like all practices modified to meet new circumstances. With the issue on languages used in formal instruction, the early missionaries effectively used Indigenous languages to teach Christian principles and values and to provide basic education which brought changes to the people’s lives. Therefore, it can be argued that Indigenous languages and English which are used as languages of formal instruction can both be effective if students are able to understand well the knowledge and concepts that are presented in them, and are able to apply them in different and new situations. Paraide (2010) argued for the deconstruction of the binary divide between Western and Indigenous education and to clarify the values of Indigenous knowledge and vernacular instruction in PNG’s formal education. The integration of Western and Indigenous knowledge in school curricula is a notion that was considered in colonial and early post-independence education. In 1955, the Australian Minister for Territories, declared the blending of cultures as one of PNG’s educational objectives. Barrington-Thomas, (1976), Giraure 1976), Matane (1976), and Tololo (1976), in their discussions on PNG’s education before and after Independence, argued for a formal curriculum that would serve the majority of the PNG student population. However, the blending of cultures and relevant school curricula was not adequately addressed until the period of education reform beginning in the 1990s (NDOE, 1999, 2002a, 2002b). Although the implementation of the educational reform varied in each province it was noted by Guy et al. (2001, 2003) and Evans et al. (2006) that there were constraints on the implementation of the reform. These constraints included: • inadequate awareness among all stakeholders of the reform curriculum; • inadequate teacher preparation to teach the reform curriculum; • insufficient or unavailability of elementary and lower primary curriculum reform materials in the schools to support teaching; • inadequate Tok Ples print materials at school level to support Tok Ples literacy teaching; • unwillingness of teachers to implement the policy on language of instruction at elementary and lower primary levels; • lack of administrative support for the implementation of the language policy and curriculum reform at the school level; and • insufficient linkages between formal learning and students’ everyday home activities.

Task Force Report for the Review of Outcomes-Based Education (OBE)

Although some of the above points were noted in the task force report for the review of OBE (Czuba, Homingu, Malpo, & Tetaga, 2013), there are others worthy of comment. It should be remembered that OBE was not the central purpose of the reform. That happened to be the approach taken in many countries at the time including developing countries and new syllabuses were needed since the overall structure was changing and the need for vernacular languages in elementary schools needed to be addressed. It should also be noted that with the 2013 review and

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the move away from OBE and vernacular language of instruction, the new curriculum took on the next worldwide type of curriculum, namely standards based education (SBE). However, there was minimal reference to the whole of the reform agenda, and an emphasis was placed on OBE and language of instruction. Inquiry was mentioned only briefly and this was important for mathematics. The Task Force concluded, almost as an aside, that the arguments for maintaining vernacular languages and using them in schools were strong. They suggested further research be carried out for PNG (to our knowledge little was done about this by the Government but there were some studies in Morobe Province on the use of vernacular phonemic awareness, and SIL has a proven track record of successfully teaching literacy in vernacular languages). However, they saw these two ways of teaching occurring within standards-based education. It is interesting that the connection between outcomes-based education and standards-based education was glossed over. That is, standards set the level of achievement of the criteria for outcomes. This is evident in the new teachers’ guides with an attempt to show marking for different levels of achievement of an outcome. Small tables now replace pages of the draft syllabus which has benefits but may now be less clearly presented (See Appendix 4). The task force review covered the following: • A comparison with South Africa was made in terms of teachers’ difficulties in understanding new curricula and resourcing books and materials but it failed to mention the emphasis in South Africa of the use of vernacular languages in multilingual classrooms (Adler, 2002; Setati Phakeng & Moschkovich, 2013). However, this interesting comparison seemed a continuation of colonial and postcolonial comparisons since before self-government. • An expert panel across the PNG sectors was needed for framing the standards- based curriculum, perhaps reminiscent of the value previously placed on PNG teacher educators’ participation (Quartermaine, 2001). • A visionary leader as Secretary of Education who was not also Chairman of the National Education Board (NEB) was needed. Thus the political and administrative key leadership positions would be separated. • Early childhood learning should be identified and emphasised as a core for education. It should be repositioned within Department of Education. This does not appear to have happened and early childhood education, in the private sector, has grown with less than adequate standards in some cases although other providers are well resourced and teachers well educated. • Teacher training should be extended from two years to three years and it “must be relocated in the Office of Higher Education as directed by the National Executive Council decision of 54/1995” (p. 3). Interestingly, with the reform, a three-year diploma was established but because of funds it was taught over two years—see Chapter 7 in this book for more details.

Standards Based Education

The syllabuses for mathematics were prepared with Japanese advisors and funding. At first the draft documents contained some new ideas, with the emphasis being on assessment tasks. Appendix 3 provides some examples of draft elementary level syllabuses and the published primary school syllabuses and teachers’ guides. The reform syllabuses followed overseas trends but were limited in size due to a lack of adequate funding and to the ideal of promoting the key ideas of Western mathematics clearly. However, they wanted to have as few outcomes as possible for teachers to assess. The syllabuses lacked the necessary guidance and teachers’ guides were soon unavailable. There also emerged

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disquiet among teachers who felt that there was too much emphasis on assessment. Textbooks came and went and class sizes continued to grow. The VSO advisers (Voluntary Service Overseas) from the United Kingdom identified many of the issues, especially problems associated with teachers being involved in the implementation of curricula. For literacy, they encouraged scripted lessons for the teachers’ guides indicating that teachers could just read the pages for the class to learn. Another issue was that teachers colleges had too many staff without Masters degree being given responsibilities in the implementation of the revised (post-PASTEP) curricula. One issue was how to ensure that teachers across the country could understand the syllabus and implement it to the required standard. The elementary curriculum writer for mathematics chose to prepare a teachers’ guide based on a learning progression supplied by the Summer Institute of Linguistics (SIL) some years before, when no materials were available. It had little to do with the new syllabus that was prepared basically by the Japanese advisor. Japan also used scripted lessons for their lesson study program in which teachers support each other by planning together and observing each other and improving on the jointly prepared plans which all the teachers in the school or schools could use. This was quite different to supplying teachers a written set of scripted lessons that could be read out as so-called teaching to meet the standards. The lessons that were prepared were too short and needed extension for inquiry as would be appropriate for the additional time in school for mathematics. However, the scripted lessons which were provided were first drafts. Modifications to the structures associated with early childhood then developed. The 2015 Syllabuses for Elementary Mathematics Standards were quite different to the final online syllabuses, and from 2017 the Lower Primary Syllabus Standards based Grades 3-5, and the Upper Primary Syllabus Standards based Grades 6-8. Earlier syllabuses were still available and for higher years these earlier syllabuses (from 2003–2006) remain the only ones available. These earlier syllabuses that were made available during the reform period made reference to cultural context. However, a look through the new teachers’ guides and syllabuses does not reveal more than the use of PNG names (most of the time) and money as relevant to cultural context. It was hoped that the syllabus writers and teachers’ guide writers for Preparatory and Grades 1 and 2 and for future revisions of the other syllabuses, including the higher grade syllabuses, would already be familiar with many PNG cultural activities and inform themselves of the mathematical activities provided in Chapters 2 and 3 and other writings by Kay Owens, and the student reports of the Glen Lean Ethnomathematics Centre at the University of Goroka (see recognition of these in Owens, 2014, 2016—copies of many of these are held by Owens). In addition, the materials prepared for teacher education of elementary school teachers included a Self-Instruction Unit on Teaching Mathematics, early reading books for mathematical concepts such as The Frog and the Kangaroo Jump that is about the group of 10 and estimating size of numbers or Seli Measures Area, and games and videos of cultural activities for stimulation, videos of giving assessment tasks and of good lessons, all of which were available at UOG, Madang Teachers College, PNGIE and the Curriculum Unit, and with many teachers and the team (including Susie Daino, Charly Muke, Vagi Bino, Serongke Sondo, Kay Owens, and Cris Edmonds-Wathen). Critics of the OBE syllabus had argued that it was not detailed enough, and that teachers’ guides were not available to teachers who needed help, or that the syllabus was too hard to read, but there was little improvement in these respects with the SBE syllabus. For example, there were words like “multiplier and multiplicand” and “quotient and partative (sic.) division” (note the spelling error and inconsistency of adjectival form). More meaningful words could be used that come from an everyday situation such as “sharing equally”—e.g., giving each person an equal share of nuts— and “equally marking off”—e.g., taking a length of pandanus and marking off equal lengths until the roll is finished, or taking the bag of nuts and marking them off in piles of 20 until the bag is used up.

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In the standard-based syllabus material, there was an emphasis on procedures despite attempts to encourage learning through problem solving or inquiry and discussion (“extend” was used in the objective or standard). However, teachers need considerable professional development to understand that extend could mean inquiry or problem solving. The elementary manual, to which Owens and colleagues referred, provided the inquiry approach which can also be adapted from Murdoch (2019). Extend or go further, and then act and reflect, are important for each learning experience. The sequencing charts provided a list of content phrases. There were some difficulties in interpreting the materials, which should have been expected given the lack of relevant background knowledge of many of the teachers. The idea of a tape diagram or a conceptual model (see Xin, 2012) was new but this was not illustrated by an example. However, in later grades it was mentioned without details being given; occasionally an area model would have been more appropriate, but examples would also have been needed if they had been used. Spelling was occasionally inconsistent, especially for words like “metre” and “centimetre” (correct PNG spellings) but often “centimeter” and “meter” were written. Occasional “errors” also led to confusion such as an assessment task or blackboard example where there were three additions of three-digit numbers which showed incorrect answers, perhaps some instruction for the item was missing. Another example was the equating of 100 cm3 with 1l although L was recommended for litre and 1000 cm3 would be the correct amount anyway. Another issue was evident: “there are units in this chart that are seldom used in our country [Japan],” referring to the incomplete “deci-” (NDOE, 2019, p. 19). Interestingly, weight was the topic tackled in measurement. Occasionally the material was adapted to PNG, but fairly poorly. For example, “Weight of 10 toea coin............1 □” (NDOE, 2019, p. 155). Although this was perhaps a first edition in which corrections could be made for further printing or online, nevertheless such poor presentation could be confusing to teachers. Some of the objective statements in the syllabus were so simplified that they were difficult to garner how they might be taught. There were some confusing simplifications like “division by 1 and 0”, whereas the latter is not really feasible. Although, it seemed that the cost of producing the material might have hindered the use of sufficient space on paper to do a better job, nevertheless color was used throughout. However, there were many occasions in which cultural experiences could have been used, but were not used, when virtually unknown experiences were mentioned. For example, To experience and understand that symmetric figures are used in their daily life by finding them in their surroundings. Symmetric figures like the ones in tiled figures and tiled patterns can be found in the beauty of balanced figures in everyday life. (NDOE, 2019b, p. 29) Decorative tiles are not commonly seen in PNG except in hotels in the cities. Many of the reflection symmetry examples (NDOE, 2019b, p. 39) are not common. There are ample examples in PNG foundational mathematics in which point-symmetry could be illustrated. Even in a Western textbook one can find diagrams such as those shown in Figure 9.1a, a curve that is common in sand drawings and Kula art, in Milne Bay Province. Each of the two panels of the tapa cloth from Oro Province in PNG in Figure 9.1b, the carved wooden plate from Tami Island, Morobe Province in PNG, and the cover of the syllabuses, all have point symmetry. Bilums have a variety of symmetries in their designs, mostly translation symmetry and reflection symmetry, but point symmetry is also seen. The geometric shapes mentioned in the syllabuses are often straight-edged and derive from Euclid’s Elements, from ancient Greece, but also seen in ancient Asian cultures. No mention is made of classifications of shapes from PNG cultures (for example, in Kula art – Campbell, 2002; Owens, 2022 in press or the Wahgi shields discussed in Chapter 2).

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Figure 9.1 Point symmetry in a cultural context.

However, the details of the guided lessons was appropriate for encouraging teacher preparation but also assisting the busy teacher with a lesson already prepared. Background information given to teachers was also detailed although more images might have assisted in some cases. Well explained mathematical processes for mathematical reasoning were given and in many cases these were incorporated, although not evidentially, in the guided lessons. The teachers’ guide had apparently received sufficient funding for Grade 6 material to have 98 lessons. This was unavailable for funding for the OBE materials, it seemed.

Equity and Social Justice

The issue of gender equality was also mentioned in the corporate plan (NDOE, 2019a) in terms of the departmental wing under which gender equity would sit, but also in terms of recognizing gender-based violence and the need for all teachers to have knowledge and education around gender equity, HIV aids and gender violence, especially in schools. However, there were no details other than possible funding sources. The issue could almost have been missed. For the reform curriculum, one issue was overcoming perceived conflict between traditional views of learning and learner-centered approaches which were seen to need further resources (Neofa, 2010). Bondo (2009), however, felt that the reform curriculum would interest students, reduce drop-outs, and benefit mobility of different cultural groups. The curriculum for International Education Agency schools (1997) (see Appendix 4 for an example) has been well detailed in accord with not only PNG students’ backgrounds but international approaches and are

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more readily available online and in hard copy. It was easier to read than some overseas documents on which it was based. In Appendix 4, a sample from the Elementary Schools is given. It seemed that resourcing and the spread of purpose of OBE did not occur for administrative reasons. Since the 1970s learner-centered teaching and going from the known to the unknown were phrases well used by teachers. They were, however, not always put into practice. While criteria-based assessments were regarded in a poor light, it was not recognised that this was indeed on the standards-based agenda, where there was a section on “standards expected on criteria.” Assessment was also seen as an area that was not properly developed under OBE and this has been a major change in the new curriculum. In fact, for a topic in the draft Grades 1 and 2 documents, more pages (previously not allowed for the OBE curriculum) were devoted to assessment than to unpacking the trajectories or inquiries for learning content. This did not happen in the rather reduced lessons of the published teachers’ guide. It is also worthy of note that trajectories or learning progressions were the main way in which, around the world, curriculum with outcomes have been further developed but only with the proviso that children may learn mathematics in a different order or way than that set out in progressions. One aspect that was not emphasised for learning progressions was that each aspect was worthy of note for teaching, although not necessarily in the order given, rather than just the final outcome. Nevertheless, the task force document did say “A blended approach to teaching and learning, combining the best of the outcomes- and objectives-based models will allow pupils to continue to enjoy high quality classroom experiences” especially in terms of books and resources (Czuba et al., 2013, p. 3). The new syllabus documents also suggest assessment for learning encourages teachers to adjust to students’ learning. Appendix 4 provides sections of the revised curricula that indicate some of these approaches.

Teacher Quality and Ways of Overcoming Difficulties

There was widespread dissatisfaction with elementary school teaching which is not surprising given the lack of training. Many teachers did not have opportunities for training. No two year course was available until UOG set up an Early Childhood Diploma. Early Grade Reading Assessment (EGRA) was particularly damaging in terms of poor reading levels as assessed by the EGRA tool. One report on a well-funded project called READ showed that in two provinces with around 4% of children as good readers in primary school, the introduction of reading materials and teacher training for elementary and primary schools increased the level to 25% (Patrionis, 2013). Low reading levels particularly occur when students are learning to read English when this is not their home language and the teacher is not trained to teach bilingually. Students were also behind in literacy and numeracy assessment for the Pacific Islands (PILNA1) but that could be expected given the greater number of languages in PNG. Again it seems that there was slow production of support materials for teachers, student books, assessment development, and integration of existing syllabuses and materials. The Capacity Needs Analysis (CNA)2 conducted for the task force on the OBE and future education planning did not find issues with OBE per se. However, before its introduction, the criticism was the irrelevance of content rather than the objectives. Curriculum Reform and Implementation Project (CRIP) training was good but limited to too few (Nongkas, 2007). Teachers colleges and universities were finding widespread dissatisfaction with students’ entry levels and there was poor alignment between high school and university curricula. Since examinations were aligned with criteria for OBE, a new task force on examinations was set up to revitalize the measurement and assessment unit of NDOE. PILNA – see https://eqap.spc.int/PILNA; https://eqap.spc.int/sites/default/files/EQAP/Reports/PILNA%202015%20 Regional%20Report.pdf 2 CAN – see http://www.educationpng.gov.pg/QL_CNA/ 1

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Perhaps a key issue was the need for a properly trained teaching force and since it was felt that this was still lacking, it was recommended that materials (scripted lesson plans and assessment tasks etc) for literacy and numeracy be given to teachers so they did not need to plan their lessons, assessment tasks, and resources from scratch (Agigo, 2010). Interestingly, the idea from the UK VSOs brought from England the idea of scripted lessons for English and mathematics at the lower levels and this was added without any query. In other words, assessing the deliberately limited number of outcomes was seen as requiring a large number of indicators to measure a child’s progress. This was seen as too much for teachers to do. It seems teachers are more comfortable with assessing individual progression levels albeit if they are in fact equivalent to indicators to determine a child’s progress. New syllabuses would provide assessment tasks. To overcome a number of issues, a recommendation was made to introduce a literacy assessment for teachers, including phonemic awareness. Cameo from Kay Owens From 1981 to 1983, and in 1997, 2001, and 2006, and then from 2014 to 2016, I had opportunities to lead inservice sessions or to be a lecturer at a PNG teachers college or at the University of Goroka. One aspect that always took some time but was quite achievable through group planning was to write a good learning plan for a class. I always found teachers could take a syllabus, copy out the topic and subtopic, and copy out an objective, outcome and indicator, or a learning expectation from the syllabus. Then they would start their plan with an introduction with “what do you know about …” and give the topic of the lesson. Then they might write down a phrase about a learning activity as the body of the lesson. Then they would give the conclusion such as ask students how they felt about the lesson today. However, the critical part of writing the learning plan was missing. In our elementary teacher program, teachers participated in a learning experience and then as a class wrote down what the plan would be like. (In shorter inservices, we would deconstruct my exemplar learning plan). This was followed by each small group writing a plan with much prompting about filling in details such as questions, extending the problem solving or inquiry lesson, and listing resources. The elementary school project team encouraged introductions that helped with tuning into the topic enthusiastically such as visiting a part of the village where they could see mathematics in practice (teachers have given ideas such as the garden, collecting kaukau and sharing, visiting the village market, looking at the plants in the village nursery). Key ideas for learning plans would include building a model or other representation, and group work on the problem to be solved. It did not take long for teachers to make some good suggestions from their reading and previous experiences of quality teaching. However, a superficial experience and a recognition that teachers need time to prepare lessons (the reason why, in fact teachers were supposed to stay at school for an hour after classes) may have led to introducing scripted lessons. As a teacher, I would find it difficult to be given another person’s script for a lesson, too many words for me to remember. I would rather work from key points and key questions. One VSO commented the teachers were expected to read from the plan!

Revised Structure of Education

After a few years of uncertainty, the structure of education was fixed at 1, 6, 6 (see Figure 9.2). Early childhood education had begun but it seemed to remain outside of the Department of Education. There is now one year of Preparatory presumably at village schools. Primary schools incorporate Grades 1 and 2 (previously called elementary) up to Grade 8. There is not an easy alignment with the syllabuses from one section to the next. High school has 2 + 2

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Figure 9.2 Change of structure of school education. years, that is junior high school being Grades 7 and 8 followed by Grades 9 and 10 with two years for high school and senior high school for a further 2 years. The plan also provided details on numbers of schools and students. It provided numbers of teachers by gender with ratio of the number of students per teacher. Some of the information is given in Tables 9.1 and 9.2. The Government spent more this time on education for the tuition fee free roll out compared to the earlier attempts. At all levels, the number of male teachers was greater than the number of female numbers and the number of students and teachers at the elementary level was high, despite the lack of teacher education. The numbers of teachers at elementary level do not include the considerable number of teachers who had not been inspected, or had not gained a teacher education certificate, or passed Grade 10, and were therefore not included in Tables 9.1 and 9.2. In some cases, schools were not registered but the community were supporting their schools and teachers. Other information in the Report fails to indicate the number of teachers who did not have teachers’ guides, syllabuses, or teaching materials. However, it was known that this was a problem, and that a new roll out of teachers’ guides and syllabuses was well overdue—whether or not these were modified or changed significantly as occurred with the SBE syllabus. With the population of PNG doubling since Independence, there was a huge need for improved education provisions. In particular, stronger rural and remote education was urgently needed. However, providing quality education at this level was fraught with difficulties and expense. The reduction of elementary level to one year perhaps reflected this difficulty. Nevertheless, the plan did set out aspirations, and also a time frame for implementation of this ambitious review.

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Table 9.1 Number of Schools in Different School Sectors, 2014 School Type Elementary

Government 4009

Church Agency 3138

Authorized, and other Schools 151

Primary

1678

1847

18

3543

Secondary

133

82

4

219

2

114

Vocational 60 52 Source. EMIS, NDOE, 2015 presented in NDOE, 2016, p. 27.

Total 7298

Table 9.2 Number of Students, Number of Teachers by Gender and Average Number of Students Per Teacher for Each Grade, 2014 Sector Elementary Total

Female Students Teachers 855 608 8746

Male Teachers 10727

Total Teachers 19473

Number of Students per Teacher 44

Preparatory

335 257

3377

3598

6975

48

Elem 1/Gd1

274 871

2890

3539

6429

43

Elem 2/Gd 2

245 480

2479

3590

6069

40

Primary total

909 473

12 079

13 258

25 337

36

Grade 3

194 695

2662

1939

4601

42

Grade 4

178 044

2359

2023

4382

41

Grade 5

159 325

2072

2070

4142

38

Grade 6

143 143

1857

2196

4053

35

Grade 7

122 648

1618

2383

4001

31

Grade 8

111 618

1511

2647

4158

27

Secondary Total

155 348

1993

3074

5067

31

Grade 9

61 932

Grade 10

51 031

Grade 11

23 732

Grade 12

18 653

Vocational

41 331

440

749

1189

35

Flexible and Distance Education

11 893

Total

1973653

27 808

27 808

51 066

Source. EMIS, NDOE, 2015, cited in NDOE, 2016, p. 26 Universal Education3 will commence in Preparatory Grade provided with an Early Childhood Education approach to provide a suitable transition to the school system. This year will ensure that all students acquire important foundation skills from the

Universal education had been a goal for the committees chaired by Tololo in 1974 and by Matane in 1986, but had also been an aspiration for Hasluck from the late 1950s (see earlier chapters). 3

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age of six in readiness for further learning from the age of seven. A more innovative approach to providing access to a Universal Education should take account of the challenges facing rural and remote students. Considerable forward planning for building schools will be required at provincial and district levels to reflect local context and to ensure that community needs are met. Although there is significant variation between provinces, many still have schools without enough suitable classrooms or clean water and sanitation facilities to provide students with a suitable learning environment that is inclusive, catering for all children’s needs. As well as the need for additional classrooms, extra housing will be needed for the increased number of teachers required. The goal of quality learning for all considers a more inclusive approach to education. Some children and youths are currently being excluded from the system or are marginalised and almost invisible in society. The groups include girls, children with disabilities, those in remote villages and the very poor. Part of the answer may be to ensure that learners are introduced into regular education while removing barriers hindering maximum participation in education. An in-depth gender analysis will identify trends and the causes of some inequalities in access to education. The elimination of barriers to education and the creation of inclusive, learningenabling schools require a multidimensional approach, with parallel interventions on different fronts. (NDOE, 2016, p. 37) The next NDOE (2019a) corporate plan pointed to a huge increase over the past 40 years in education, suggesting that it was no wonder that it has been difficult to keep apace organisationally. By 2017, there were: 66 789 teachers, 2 157 26 students in 9 400 elementary schools, 4 058 primary schools, 299 secondary and high schools, 148 vocational schools registered and operating in the NES. Work is progressing on the shift from the current 3-6-4 structure to a 1-6-6 school structure through the NEC decision No.315/2016. The decision renamed the elementary sector to pre-school and also reduced it to only 1 year of schooling however, continuing to keep the entry age at 6 years. Primary schooling will remain 6 years but with the inclusion of Grades 1 and 2 which are previously E1 and E2 from elementary school. Secondary schools will now consist of Grade 7 to 12 with junior and senior high school. Different models to bring about the changes are also in place under different arrangements. Models are different but best suited to the geography and the socio-economic structure of the locality. Models may have all sectors in one campus or two sectors in one, while a sector situated in a different location. Implications on the changes are massive but manageable. (NDOE, 2019a, p. 9) However, the accompanying diagram did not reflect this information, and it was difficult to see how children who are in village elementary schools would easily be able to attend a distant primary school. No mention was made of the possibility of a primary school with multiple distant Grades 1 and 2 in villages feeding into the primary school, although this was not impossible. For example, the cooperative school of Madang’s Rai Coast and hinterland had schools in many village elementary schools with two principals, assisting in language work, enthusing the teachers, and providing the logistics for the cooperative school and professional development. It was not funded by the Government generally and not all teachers were qualified elementary school teachers. However, they were all teaching in the vernacular language of the village and seemingly doing well. When the students transferred to Government schools they did very well in literacy. Another structural area of change was the positioning of the various post-school institutions. By 2020 the Technical Vocational Education Training (TVET) Colleges and Primary

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School Teacher Training Colleges (PSTTC) moved to the Department of Higher Education, Research, Science and Technology (DoHERST) with the universities and some other post-secondary tertiary institutions. It is also interesting that the term “training” was being used when “education” was the preferred word if one wished to indicate thoughtful preparation rather than a procedural skill- based approach. Thus, the former two groups were no longer part of the NDOE. Although some educators thought that the NDOE should take responsibility for early childhood, it is not clear that this happened—but the one-year preschools (equivalent to Pre-Elementary or Preparatory) are under the NDOE.

Funding and Change

The other major area of concern was governance and administration. There was no way to guarantee accountability. The leadership was not inspiring those below who felt change was being imposed on them, but there was also silo management. The country needed to be behind the reform but it seemed that it was inadequately explained at all levels. Finally there was no way that sufficient resources and funds were being made available to implement the universal education policy despite its desirable provisions related to equity, moral and educational grounds. At the time of writing, it is not clear that all these issues with respect to the SBE regime have been overcome. It is noted however that there are still silos within the system such as interactions between Curriculum and PNG Institute of Education, Curriculum and Head Office and Teacher Education and Curriculum. The issues of power and the number of dismissals of senior people such as at University of Goroka and school principals, together with a lack of dismissals that might have occurred due to lack of effort in positions, have suggested to some that there has been a lack of “due process.” Misunderstandings about Standards are also evident. However, we are not in a position to critique the revised curriculum at this stage. Only primary school mathematics curricula are currently available. More Papua New Guineans are prepared to write materials for teachers. There were large amounts of money put into resources during the reform period which will still be relevant now. However on a new and good note, overall, the lead for SBE is coming from within the country and not from outside experts. Tuition Fee-Free Education and Accountability In an effort to have as many students as possible attend school and hence work towards the goal of universal education which has been touted on numerous occasions, the Government implemented a Tuition Fee-Free Education (TFFE) policy. This did not mean, as some said free education. The Provincial Board would still set fees, and project fees. However, staff salaries at least would be paid by the Government. NDOE (2016) indicated the areas from which funding for education would come. There were still difficulties for schools with numbers and accessing funding (Paraide, 2015). For accountability of TFFE there is a complex system as pointed out by McLaughlin and McDonough (2019), who provided a diagram to summarise the many policy statements that had been made to provide a structure for funding. The system should lead to accountability but there were too many areas where reporting may not occur. The diagram they prepared is shown in Figure 9.3. Importantly the O’Neil Government provided significant amounts of money for education compared to previous Governments (4 to 5 times as much), but by 2016 monies which were supposed to have gone to schools disappeared. Accountability of school funds and Government funds needed adequate systems (McLaughlin & McDonough, 2019). There was danger of further political patronage occurring with Members of Parliament using discretionary funds to favor some for political reasons rather than ensuring a fair distribution.

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Figure 9.3 Flow of information for accountability of TFFE. Despite this concern, the two corporate plans (NDOE, 2016 and 2019) are remarkably detailed and although there were anticipated gaps in funding they provided an excellent vision for reviving a sound education.

Moving Forward

The most significant aspect of recent education planning and renewal has been an increase in funding. One result has been the provision of attractive guides for teachers to use when teaching mathematics. It is our opinion that much is dependent on the following, as also mentioned to some extent in the plans: • new buildings for the restructure of 1, 6, 6; • clean and safe environments;

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• growth in preservice teacher education; • provision of further inservice professional education such as in the areas of • gender equity, • planning and delivering detailed mathematics inquiry lessons that incorporate ongoing assessment; • incorporating cultural mathematics as intended by goals around integral human education into a syllabus that was developed without this knowledge; and • inservice professional education that allows teachers, especially elementary teachers and teacher educators, to gain qualifications. The new syllabuses and teachers’ guides provide some new approaches, such as the tape or conceptual method (Xin, 2012), although further work on area models for multiplication and division is needed together with professional development in all sectors of education. In addition, the more recent recognition of a sense of size that could build on Papua New Guinean children’s sense of size from walking around and playing could have been incorporated (Möhring, Newcombe, & Frick, 2015; Owens, 2020). Although there is no recognition of mathematics as an important aspect of culture, and no attempt to implement contextual mathematical practices in the teachers’ guides, nevertheless that is possible. However, to do this effectively, vernacular languages need to be kept or revived, as they reflect people’s thinking. Mathematical thinking and reasoning is evident in PNG societies but much of this is visual and spatial, oral and mental. That is not recognised in the curriculum writing to date. The diversity of counting systems—so rich in PNG—and different ways of representing all areas of mathematics have yet to be incorporated in theory and practice which recognizes that there should not be a duality of mathematics (McConaghy, 2000) nor a fracturing of selfidentity and knowledge (Valsiner, 2000). This argument has been well expressed by Paraide (in Paraide & Owens, 2018). An integration of cultural and school mathematics would be likely to strengthen students’ mathematical reasoning and generate stronger mathematical thinking from pre-school to university. References Adler, J. (2002). Teaching mathematics in multilingual classrooms. New York, NY: Kluwer. Agigo, J. (2010). Curriculum and learning in Papua New Guinea schools: A study on the Curriculum Reform Implementation Project—2000 to 2006. Special Publications No. 57. Port Moresby, PNG: National Research Institute. Ascher, M. (2002). Mathematics elsewhere: An exploration of ideas across cultures. Princeton, NJ: Princeton University Press. Barrington-Thomas, E. (1976). Problems of educational provision in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea Education (pp. 3–16). Melbourne, Australia: Oxford University Press. Bondo, J. (2009). The implementation of the outcomes based education concept in an international primary school: A Goroka International Primary School case study. M.Ed thesis, Divine Word University, Madang, PNG. Campbell, S. (2002). The art of Kula. Oxford, England: Berg, Oxford International Publishers. Czuba, J., Homingu, M., Malpo, K., & Tetaga, J. (2013). Report of the task force for the review of outcomes based education in Papua New Guinea. Port Moresby, PNG: National Department of Education. Evans, T., Guy, R., Honan, E., Kippel, L. M., Muspratt, S., Paraide, P., & Tawaiyole, P. (2006). PNG Curriculum Reform Implementation Project—Impact Study 6. http://hdl.handle.net/ 10536/DRO/DU:30010511 Giraure, N. (1976). The need for a cultural programme: Personal reflections. In J. Brammall, R. May, & M.-L. Allen (Eds.), Education in Melanesia (pp. 101–104). Melbourne, Australia: Oxford University Press.

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Guy, R., Paraide, P., Kippel, L., & Reta, M. (2001). Bridging the gaps. Port Moresby, PNG: National Research Institute. Guy, R., Paraide, P., Kippel, L., & Reta, M. (2003). Lower primary curriculum review of edition one materials. Port Moresby, PNG: National Research Institute. Kari, F. (2005). The implemented mathematics curriculum. Contemporary PNG Studies: DWU Research Journal, 2, 24-40. Matane, P. (1976). Education for what? In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 57–60). Melbourne, Australia: Oxford University Press. Matang, R. (2005). Formalising the role of Indigenous counting systems in teaching the formal English arithmetic strategies through local vernaculars: An example from Papua New Guinea. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce & A. Roche (Eds.), 28th Conference of Mathematics Education Research Group of Australasia, Melbourne (pp. 505–512). Adelaide, Australia: MERGA. Matang, R. (2008). Enhancing children’s formal learning of early number knowledge through Indigenous languages and ethnomathematics: The case of Papua New Guinea mathematics curriculum reform experience. Paper presented at the 11th International Congress on Mathematics Education ICME 11, Monterray, Mexico. http://dg.icme11.org/document/ get/322 Matang, R., & Owens, K. (2006). Rich transitions from Indigenous counting systems to English arithmetic strategies: Implications for mathematics education in Papua New Guinea. In F. Favilli (Ed.), Ethnomathematics and mathematics education, Proceedings of the 10th International Congress on Mathematical Education Discussion Group 15 Ethnomathematics. Pisa, Italy: Tipografia Editrice Pisana. Matang, R., & Owens, K. (2014). The role of Indigenous traditional counting systems in children’s development of numerical cognition: Results from a study in Papua New Guinea. Mathematics Education Research Journal, 26(3), 531–553. https://doi.org/10.1007/ s13394-013-0115-2 Matsuura, K. (2008). Why languages matter: Meeting millennium development goals through local languages. SIL International, Texas. Dallas, TX: SIL International. McConaghy, C. (2000). Rethinking indigenous education: Culturalism, colonialism and the politics of knowing. Brisbane, Australia: Post Press. McLaughlin, K., & McDonough, R. (2019). Tuition fee-free education in Papua New Guinea. PNG Insights. https://pnginsight.com/right-of-children-to-free-and-compulsory-education/ Möhring, W., Newcombe, N., & Frick, A. (2015). The relation between spatial thinking and proportional reasoning in preschoolers. Journal of Experimental Child Psychology, 132, 213–220. https://doi.org/10.1016/j.jecp.2015.01.005 Murdoch, K. (2019), Inquiry model. https://static1.squarespace.com/static/55c7efeae4b0f5d 2463be2d1/t/5dcb82551bdcf03f365b0a6f/1573618265386/A+MODEL+FOR+DESIGNI NG+A+JOURNEY+OF+INQUIRY.pdf NDOE, Papua New Guinea. (1998). Lower primary language syllabus, Grades 3–5. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1999). Primary handbook. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2004a). Lower primary language syllabus. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2004b). Lower primary mathematics syllabus. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2016). National Education Plan 2015–2019. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2019a). Department of Education Corporate Plan 2019–2021. Port Moresby, PNG: Author.

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NDOE, Papua New Guinea. (2019b). Mathematics teachers’ guide Grade 6: Standards based. Port Moresby, PNG: Author. Nebas, F. (2008, 9 December). English a problem in schools—Principal. Post-Courier, p. 4. Neofa, Z. (2010). A case study of how primary school teachers in Papua New Guinea understand outcomes-based education. PhD thesis, Queensland University of Technology, Brisbane, Australia. Nongkas, C. M. (2007). Leading educational change in primary teacher education: A Papua New Guinea study. PhD thesis, Australian Catholic University, Australia. http://researchbank.acu.edu.au/cgi/viewcontent.cgi?article=1208&context=theses Owens, K. (2014). The impact of a teacher education culture-based project on identity as a mathematics learner. Asia-Pacific Journal of Teacher Education, 42(2), 186–207. https://doi.org /10.1080/1359866X.2014.892568 Owens, K. (2016). Reforms in mathematics education: The perspective of developing countries on visuospatial reasoning in mathematics education. South Pacific Journal of Pure and Applied Mathematics, 2(1). https://researchoutput.csu.edu.au/ws/portalfiles/ portal/23012697/8991832_Published_article_OA.pdf Owens, K. (2020). Transforming the established perceptions of visuospatial reasoning: Integrating an ecocultural perspective. Special Issue: Mathematics Education Research Journal, 32(2), 257-282 https://doi.org/10.1007/s13394-020-00332-z Owens, K. (2022, in press). Chapter 7. The tapestry of mathematics – Connecting threads: A case study incorporating ecologies, languages and mathematical systems of Papua New Guinea. In R. Pinxten & E. Vandendriessche (Eds.), Indigenous knowledge and ethnomathematics. Cham, Switzerland: Springer. Paraide, P. (2008). Number in Tolai culture. Contemporary PNG Studies: DWU Research Journal, 9, 69–77. Paraide, P. (2010). Integrating Indigenous and western mathematical knowledge in PNG early schooling. Doctoral thesis, Deakin University, Geelong, Australia. Paraide, P. (2015). Challenges with the tuition fee free education policy implementation. Papua New Guinea. Contemporary PNG studies, 23, 47-62. Paraide, P., & Owens, K. (2018). Chapter 12. Integration of Indigenous knowledge in formal learning environments. In K. Owens, G. A. Lean, with P. Paraide, & C. Muke, (Eds.), History of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Patrionis, H. (2013). Education, economics, and public policy. https://hpatrinos.com/2015/11/15/ literacy-rate-in-papua-new-guinea-increases-due-to-early-grade-reading-interventions-2/ Quartermaine, P. (2001). Teacher education in Papua New Guinea: Policy and practice. PhD thesis, University of Tasmania, Hobart, Australia. Rai, H. (2008, 23 May). Plan to introduce vernacular in elementary schools. Post Courier, p. 4. Ralph, R. (1978). Education in Papua and New Guinea to 1950. Unpublished book. Setati Phakeng, M., & Moschkovich, J. (2013). Mathematics education and language diversity: A dialogue across settings. Journal for Research in Mathematics Education, 44(1), 119–128. Smith, G. (1975). Education in Papua New Guinea. Melbourne, Australia: Melbourne University Press. Tololo, A. (1976). A consideration of some likely future trends in education in Papua New Guinea. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 221–225). Melbourne: Oxford University Press. Unage, M. (2007). Reviewing education process: Talking point series. Goroka, PNG: Endeavour Printing. Valsiner, J. (2000). Culture and human development. London, England: Sage. Xin, Y. P. (2012). Conceptual model-based problem solving: Teach students with learning difficulties to solve math problems. Rotterdam, The Netherlands: SensePublishers.

Chapter 10 Mathematics Education and Language*

Abstract: Language of instruction in schools has long been an issue in PNG education—during the many years of colonial administration, and especially in the lead up to Independence in 1975. It is still being debated, almost 50 years later. Policies and politics have played a major role in determining which language should be the language of instruction in PNG schools, and issues of cultural diversity and national values and opportunities have played a significant part in influencing the decisions which have been made. This chapter will consider some of the research undertaken on the impact of language learning on mathematics learning, from students’ and from teachers’ viewpoints.

Key Words: Bilingualism · Language impact on mathematics · Language of instruction · Language policy in PNG · Multilingualism Language is often overlooked because it is in an intangible part of culture and something which is used constantly by people, without them reflecting on it or being conscious of it. Nevertheless, language is one of the most significant aspects of the cultural heritage of any group. McConvell & Thieberger, 2001, p. 1 Introduction In Papua New Guinea (PNG), people identify strongly with their cultures, and language is one of their most important identifiers. PNG has many unique languages and it is not surprising that they are regarded as an important aspect of PNG culture which needs to be preserved, despite the intrusion of Western-type school education. Although McConvell and Thieberger (2001), in the above quotation, were addressing a broader issue, the sentiment they expressed has deep relevance for mathematics education in PNG, particularly because the cultural and language diversity across the nation has resulted in the development of different groups creating their own mathematical ways of thinking and speaking (Owens, 2015; Owens, Lean, with Paraide, & Muke, 2018). Many mathematics teachers believe that mathematical language is just one universal language but PNG shows us a different perspective. Details given in Chapters 2 and 3 of this book revealed that there are many records of different forms of mathematical language in PNG (Edmonds-Wathen, Owens, & Bino, 2019; Owens, Lean with Paraide & Muke, 2018). The ubiquity of language in all its forms in school mathematics classrooms across the nation raises important pedagogical and conceptual issues which arise from the peculiar This chapter draws heavily on the PhD theses of the two Papua New Guinea authors Patricia Paraide and Charly Muke. Their theses are listed in the references. *

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forms of mathematical language in which concepts are embedded Language-related issues arise, for example, to the teaching and learning of fractions in PNG’s multilingual classrooms of PNG. There are many factors which interact in mathematics classrooms which have an impact on students’ learning. We know that some crucial factors, which do not need referencing since they are so much part of the fabric of education, are parental education, a family’s socioeconomic status, and the amount and type of reading material available in a home and the community which surrounds it. These factors cannot easily be altered by what happens in the school, but some significant factors can be changed in classrooms, and one of them is the language used in the teaching of mathematics. Mathematics education research has shown that, in any classroom language is an important factor in the students’ mathematics learning. Although not the first to study this issue, in a pivotal study Clements (1980) used the Newman Analysis to show that in Australian schools students’ reading and comprehension played a vital role in their performance in solving written mathematical problems, particularly for students who were falling behind in their school mathematical learning. Clarkson (1983) showed that these same language variables were important for Papua New Guinean students. Although there have been many scholarly reviews which have looked at this issue (e.g., Ellerton & Clarkson, 1996; Pimm, 1987; Secada, 1992), most studies which were analysed in those reviews assumed that a classroom was a monolingual space (Clarkson & Carter, 2017). However, for most classrooms throughout the world this is not the case, and in Papua New Guinea, in particular, no such assumption should be made with respect to any mathematical classroom. In recent studies and reviews greater account is being taken of the amount of linguistic diversity among students in mathematics classrooms (see, for example, chapters in Barwell, Clarkson, Halai, Kazima, Moschkovich, Planas, Phakeng, Valero & Villavicencio, 2015a). Already in this volume much has been made of language. In a nation in which there are more than 850 languages spoken within a population of less than 10 million people (Worldometer, 2021), how could it be otherwise? We have noted that languages are the most important means of expressing knowledge systems in cultures, and of communicating that knowledge to future generations. A number of examples have been given of how foundational knowledge is still applied in people’s day-to-day activities and how Indigenous languages are used as languages of instruction for the children and young adults in their everyday lives (see Chapters 2 and 3). We have also shown that clearly the Indigenous languages offer the key to beginning to understand the Indigenous cultures, including ritual and kinship, as well as day-to-day activities. We have argued that these bodies of knowledge are just as valuable and practical as Western and other knowledge systems. As part of this examination, we have shown that mathematical activity and thinking threads through many aspects of Indigenous cultures, even though it may not have been chosen for systemic study in its own right. Issues arising from the fact that across the world many students learn mathematics in multilingual classrooms were not carefully studied until in recent times. Various areas of research into this state of affairs have been pursued over the last four decades, including important research conducted in Papua New Guinea, and this has been documented in authored and edited books, conference proceedings, special issues of journals, and doctoral theses. Examples of these are Adler (2002); Barwell (2009); Barwell, Clarkson, Halai, Kazima, Moschkovich, Planas, Phakeng, Valero & Villavicencio (2015b); Clarkson (1991); a special edition of Educational Studies in Mathematics (2007); Muke (2012); Paraide (2010); Setati, Nkambule & Goosen (2011) and many more journal articles. A useful introductory review of the development of this area of research can be found in Barwell, Clarkson, Halai, Kazima, Moschkovich, Planas, Phakeng, Valero and Villavicencio (2015b). For our purposes it will be useful to identify three main foci of these studies; (a) the way the teacher uses verbal and written languages; (b) the competence that students and teachers have in their various languages; and (c) whether the languages and cultures have developed in a direction which allows parallel communication in the students’ languages regarding the mathematical concepts being taught.

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Within the area of multilingual learning and teaching various unwarranted assumptions have often been made. One was that students’ brains could not cope with having to use multiple languages when in school and hence it was best to use one language only. We now know that the human brain is much more versatile than was assumed even 25 years ago, and there is no problem with its capacity to use multiple languages, even when doing mathematics. A second assumption was that the students’ languages were distinct entities. It is clear that when multilingual people are talking to each other it is natural for them to switch between languages. Indeed, many end up in conversation using a hybrid language that is not one or the other. This is almost always the case when using a verbal mode of communication but it is much less the case with writing. Reasons for “code-switching” when doing mathematics are numerous, some of which link to the mathematics but not always (Clarkson, 2007a; Kirsch, 2018)). A third assumption was that if students switch between their languages, as they are apt to do, this can lead to confusion in understanding some of the ideas. In fact, it appears to be the case that there is some truth in that assertion. However, if students are competent in more than one of their languages, this is much less likely to occur. Cummins (1979) showed that students who are competent in their languages actually performed better than monolingual students or students with one dominant language. Dawe (1983) and Clarkson and Dawe (1997), working with multilingual immigrant students, showed that this holds for mathematics. Clarkson also showed that this result held for Papua New Guinean students when doing mathematics as well as in other subject areas (Clarkson, 1992; Clarkson & Clarkson, 1993). Clarkson (1991) suggested that when multilingual students switch between their languages they often gain a deeper, more nuanced understanding of the concept they are learning and that the switching, far from causing confusion, enhances their performance in test situations. However, Clarkson’s work has drawn some criticism. The participating students in his original research work in PNG were reported as bilingual; in fact they were urban students and in any one classroom the students were multilingual and spoke many languages (Barwell, 2016). The languages in which the students were assessed were the teaching language of the classroom (English at that time), and the lingua franca (Tok Pisin) that students used in the playground (even though that was against the school rules; see later in this chapter) and outside of school between themselves. Barwell (2016) also criticized aspects of a teaching model which Clarkson (2009) developed which took seriously the context of bilingual classrooms based on earlier thinking about context (Clarkson, 2007b). Clarkson’s model built on the common advice given to young teachers of mathematics to start their teaching in the everyday worlds of their students, if that is possible, and then move on to the school mathematics. The rise of real world or authentic mathematical problems has been behind this approach. Clarkson’s original model paid attention to the languages which could be involved. Its original formulation is illustrated in Figure 10.1, although in many later presentations the model was extended to show three languages in play and using an expanding arrow down the right-hand side clearly indicating the final teaching aim was the learning of the appropriate mathematical language for all the languages in play. The various figures in Clarkson’s original model have double-headed arrows to convey the reality that students and teachers will switch between languages and registers. The model was later presented by Prediger and colleagues (Prediger, Clarkson, & Bose, 2012; Prediger & Wessel, 2011) and commented on in relation to the PNG context by Owens (2015). Considering Bhabha’s (1995) notion of thirdspace, there is a mental development of meaning between the two languages and a degree of resistance to the hegemony of the dominant language of English. This could also be extended to multiple languages. The merging of languages may form a new hybrid language used by multilingual students in their verbal utterances, and presumably in their thinking, when they are doing mathematics (Barwell, 2016).

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252 Informal communication in L1

Informal communication in English

More structured communication in L1

More structured communication in English

Use of mathematical register in L1

Use of mathematical register in English

Figure 10.1 Clarkson’s original model of language use by teachers of mathematics. Clarkson’s model suggests there are multiple pathways that might be pursued depending on the classroom context in any particular lesson. A model cannot represent fully what actually happens in real life contexts, and models should always be taken as guides not technical manuals (Clarkson & Carter, 2017). It remains to be seen how studying hybridity of languages in the classroom can be captured by modifying this model, or moving to something new (Prediger, Kuzu, Schüler-Meyer and Wagner, 2019). The teacher and the classroom working with culture and language can also develop mathematically acceptable understandings within the community, as a number of individuals’ thinking is influencing the cultural understandings (Owens, 205; Saxe, 2012). Mathematics education research clearly shows that students’ performance will in part depend on language ability. The teacher is the main person in the classroom to orchestrate how and what languages are used in the classroom. Languages are also the carriers of cultures. Hence in PNG there is the possibility of introducing Indigenous mathematics into the classroom if Indigenous languages are used. That issue will be explored more fully later in this chapter and elsewhere in this book in the PNG context. Importantly, research strongly suggests that utilizing a student’s multiple languages should not be an inhibitor to their learning but, in fact, something which enhances it. However, schooling decisions rarely are based solely on what education research has shown. Often politicians make decisions which are not based on the application of logical thinking or the results of research.

Politics of Language in Papua New Guinea

There is no doubt, as has been detailed in previous chapters, that politics defined the type of schooling available in Papua New Guinea from the earliest period of colonization. This should not be surprising. Education is one of the obvious services that governments of any type are duty bound to provide for their citizenry. It is also an area of society for which most people have an opinion. When a new government comes to power, this can and often does usher in changes, normally cloaked in language of improving the outcomes of schooling by change to the school system. However, in Papua New Guinea one policy has existed for virtually all of the past 140 odd years—specifically, the language of instruction in government schools would be that of the colonizer, English (or pre-World War I, German) in their territory of New Guinea. Previous chapters have detailed the nuances of this policy and how for many years most non-government schools used the local Indigenous language as the language of instruction in the

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first few years of schooling, although even they reverted to English (or German) in the final years of primary schooling. Many of the decisions which were made have been seen as the outcome of power by the colonizer wanting to establish an elite, educated enough in Western schooling so they could participate in the economy of the country, which in turn was assumed would serve the interests of the colonizing power. There seems to have been little interest in what that meant for the education of the students so far as their own enrichment was concerned. Of particular interest in this chapter is the fact that the political decisions since the end of World War 1 for most of this period mandated that the language of teaching in Papua New Guinea would be English in Government schools, and in other schools for which funding was provided. However, that policy runs counter to what educational research suggests should be the situation, and indeed is contradicted by the wisdom of practice of teachers and the wisdom of the Elders of PNG education as reflected in both the Tololo Report (Department of Education Papua New Guinea, 1974) and Matane Report (NDOE, 1986). The one exception to this was the postIndependence time of the 1990s through to the early 2000s when the reform structure and curriculum for education was under way (see Chapter 8). The NDOE seemed to be able to develop and implement recommendations which kept the system globally focused. At that time, the vernacular of the children was to be used, especially in the elementary school. Then a new government came to power in 2012 and reverted to English as the language of instruction (see Chapter 9), even in the elementary schools which had been specifically created so that the beginning years of schooling could typically be in the language most commonly used in a village and one that the child was used to hearing and speaking. Some authors have written about the contradictions between what education research suggests should happen in schools regarding the language of teaching, and what actually does happen (Clarkson, 2016; Halai & Muzaffar, 2016). However, teachers do need to work within the parameters that are decided by the government of the day. In the next section we examine the experiences of teachers teaching mathematics in Papua New Guinea and how the issue of language has impinged on their craft. Then we move to a much less examined area—the impact that students’ languages and their language abilities have on their learning of mathematics. Finally, we return to the teachers and look at both their preservice preparation for teaching and their professional learning opportunities once they are in the field.

Students and Learning Mathematics in Papua New Guinea

Two of the authors of this book were students in PNG schools. Their personal recollections of this time provide an interesting introduction to this section. Cameo from Patricia Paraide When I began school, my teachers were allowed to communicate with me in my Tok Ples as this was allowed in Catholic mission schools then. This made it easy for me to understand the simple mathematical concepts that I was being taught. As I advanced to the upper grades in primary school, I was able to grasp most of the mathematical concepts. This is generally attributed to the likelihood that I had understood the basic mathematic concepts when learning them in the lower grades, and because the language of instruction was my own language. I did not speak English competently yet in the upper grades of primary school. I read a lot of English stories—as we were encouraged to do, then. I loved the Western stories I found in the school library because story telling was an important part of my life when growing up. My father entertained us (only his children) with traditional legends and also with traditional stories that taught wisdom and lessons in life. Almost certainly, my reading of a lot of English stories helped me to master the literacy skills, and therefore understand mathematical concepts presented in the

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English language at the upper levels of primary school. I enjoyed learning mathematics in primary school. However, I had difficulties understanding mathematical concepts when I was in high school because I could not discuss them with other students because most of them were expatriate children. The few Papua New Guinean students in my class spoke other languages. I could not use my own language to process the mathematical concepts because I was hindered from expressing them in my own language. Also, using Tok Pisin or Tok Ples for discussions in class was forbidden, and even looked down upon, in this particular high school. I had difficulties with mathematics in Grades 11 and 12, and I believe that was because I had missed the mathematical concept links to advanced mathematical concepts in Grades 7 through 10. I am now of the view that this could have been because the mathematical concepts were not linked or related to their practical applications in everyday life. Later in life I would understand mathematical concepts better when experienced and competent mathematical teachers made practical links and pointed out practical application of mathematics to assist students to understand better the mathematical concepts which were under consideration. I taught English as a subject in Grades 7 to 10 classes in two high schools. At those schools the principals advised the teacher in charge of timetables to timetable me to teach English only. These two principals were first speakers of the English language, and they were of the view that my spoken English was “good.” They convinced me that I was needed to teach English well in their schools. It was emphasized to me that English was a very important subject and therefore students had to be taught well to read and write in it. I did not teach home economics in these two schools even though this was also one of my specialized teaching subjects. Mathematics concepts especially number, measurement, comparison and other mathematical concepts are applied in cooking and sewing. However, I never emphasized any mathematical links between the home economics content and mathematics to the students because it seemed to me that the general view was that mathematics should be taught only during mathematical lessons by specialized mathematics teachers. Therefore, teachers like me, who were designated “language specialists,” were not prepared to think outside the box and to attempt to reinforce the application of mathematical skills in other subject areas. Cameo from Charly Muke I was able to speak three languages before I went to school. I spoke the two local languages, Yu Wooi of Wahgi in Jiwaka Province, Kuman of Simbu Province and Tok Pisin, one of the national languages. I spoke these vernacular languages before I attended first grade in 1977 at the age of seven, in the nearby community school (now called a primary school). At school, the language to be used for learning and teaching was only English. When I arrived at the school for the first time, I was not given any choice of languages, but had to speak only in English within the school premises. As I recall, the first rule in the list of class rules pinned on the wall was that students must always speak English at school. We were punished severely for speaking our own local vernacular, either inside or outside of the classroom. If we were lucky, we would be told to write 100 to 200 times the sentence “I must speak English at all times.” Otherwise, we received smacks on our backsides with a cane in front of the whole school during assembly, or some other corporal punishment during school hours. In the classroom situation I recall some things that went on in my mind. As I was listening I wanted to understand what was going on, and therefore most times I remembered trying to translate what I heard in English into one of the languages I spoke, to match the concept to a context and situation that matched my own experience from my background. By the time I finished checking this in my mind and understood what was going on in one of my languages, I had to then translate what I understood back to English; but the problem was that I could not even say

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a sentence in English. I could not say out loud what I wanted to say in my own language, because if I did that I would have been punished. The best strategy available to me was to be silent during lessons, even though some of the activities related to the lesson I was thinking about and making some sense of in my mind. I only answered in one or two words in English when questioned by the teacher. In most cases, I would bow my head or scratch my head and say nothing. I did have some idea of what was going on in class, but I had no way to say it to the teacher correctly in English. Making mistakes in front of an elder such as a teacher was culturally insulting for him, and this also kept me silent (Muke, 2001). My teachers shouted at me, pulled my ears, gave me a knock on my head, or would tell me to stand on one leg at the back while listening to the lesson’s progress, because I couldn’t answer properly, in English. The classroom was a painful and shameful area to be in for me. But I went through the six years of this education at the Community School with great determination because my parents kept on emphasizing that school was the doorway to the good life. I was determined to finish those six years and make my family and tribe happy. After the six years of Community School, I went to a Provincial High School for four years. By then I had improved a lot in English and this continued at High School, but it was still not good enough to tell a story or start a conversation with anyone. After completing Grade 10 at High School I went to St Benedict’s Teachers College for two years. There I felt that I really improved my English and in 1990 I finally returned as a teacher to a Community School in my area, where I used only English when I was teaching. Cameo from Philip Clarkson Students when they want to learn will use any and all manner of ways to accomplish this goal. As our coauthors have recalled, part of their way to learning in the school setting was to utilize their various languages. Three instances in 1980 shortly after I arrived in PNG started my understanding of the language complexity to be associated with PNG classrooms. The first was a visit to some local urban community schools and I marvelled at the way students at the beginning of their schooling already knew two or three languages and yet they were being taught in English (and the rules that Muke recounts from his classroom were pinned to the wall at the front of the classes in these schools too). The second instance was my first visit to a National High School that taught Grades 11 and 12, the final two grades of schooling. When sitting at the back of a Grade 12 class I heard the teacher explain to the students in a condescending manner that they probably had problems with mathematics because they did not speak English all that well. During the lesson this teacher did little to help with the language of mathematics that he was using although these students spoke multiple languages fluently. The third instance was my first visit to Goroka Teachers College, now the University of Goroka (UOG). I was asked to give a talk to the senior students after their evening meal. A key part of the talk was to introduce the preservice teachers to the Newman Analysis (Clements, 1980; Newman, 1983). During a discussion with a group of the students after the talk I asked them how they had gone about solving the written (in English) mathematical problems that they had been asked to do during the talk. A couple of the students said they could read the problem but had difficulty understanding some of the words and symbols: Comprehension in the Newman Analysis procedure (Flagg, 2014; Newman, 1983). So I asked “What did you do?” expecting them to say they gave up on that problem and moved to the next. But no, that is not what they said they did. They did what they normally did, so I was told; they translated the problem into Tok Pisin and thought about it again. Then to my surprise two of the young men independently said that for a couple of problems with which they had difficulty, they tried to understand the words by translating them into one of their Tok Ples1. I was certainly Tok Ples: In Tok Pisin this means the language of their place that is village or family, their vernacular language or mother tongue. 1

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intrigued to listen to the UOG students say that of course they used their various languages when solving mathematical problems. It did not always help but sometimes it made much more sense to them if they switched languages. I had come across two crucial notions: that students will switch languages no matter what the rules or what teachers say, and at times switching languages can actually help students learn. According to the UOG students, even if there was not an easy translation of the English into Tok Pisin, or even into Tok Ples—and often there was not—the very act of trying to translate when there was no obvious equivalent words made them work hard to understand what the English words actually meant. Clearly the Grade 12 teacher whom I had observed, and who had taught in PNG for many years, never really understood how many of his students were processing the mathematics with which they were engaged. Yarning in Sydney with PNG Nationals One Friday morning in a Sydney Park in September 2018 three of the authors (Kay Owens, Charly Muke and Philip Clarkson) sat with a group of six PNG Nationals for a “yarning circle” about their views and experiences of education in PNG. Some of the Nationals had been teachers in PNG, and the ages ranged from the late twenties through to the sixties. Hence, they spanned a number of decades of experiences of being students in PNG schools, as well as for some, teaching. Two of the group (Dan and Carrie) had attended primary school in the 1960s. Dan recounted a little of what it was like for him starting school and learning mathematics: Dan: I started off in school where the teacher was using Tok Ples you know in maths, counting in my language. Then I progressed to the next level and had a teacher and he was using language from Carrie’s area (his now wife, who had lived in a village on the other side of the large island) and some English, so we were counting in my language and her language. And we were putting those together really easily. And then when I went away to boarding school 400 km from my village the teaching was only in English. Then I had to convert what I knew from her language and my language into English, including counting and so on, and try to understand adding and so on and I had to count in my language or her language depending on what I was thinking in. …. Muke:

What was your experience with your own languages, Dan? Was it benefiting with using English or …? Dan: Yes well I began to drop counting etcetera in my languages in Grade 5. From Grade 5 onwards I forgot about counting in my languages and just using English. But up till then I think it did. Owens: Did you think you had a really good understanding of those basic ideas because you had been using your own languages? Dan: Yes, well you know we used our fingers then our toes and you and Glen Lean got all this and our whole body (counting system) and when you try to … well I found it difficult to do more (that is do operations) than count in my language, and that is when I really started to use English and think in English about this. Owens: What about multiplication? Dan: Oh um. I can’t think of the words in my language for multiplication now. (Dan and Carrie had lived in Sydney for some 30 years for medical reasons.)

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Results from East New Britain Patricia Paraide did well in English and mathematics when she first learnt in school in her vernacular to read and write and do mathematics. By contrast, Charly Muke struggled with both but he thought about mathematics in his head in his vernacular. The following research studies confirm these cameos. Paraide (2002, 2003) had shown the importance of students’ heritage on literacy and mathematics by looking at the results of the early learning in a vernacular language during the 1990s across many schools in three provinces. She found: Some children who did their elementary education in Tok Ples or Tok Pisin were able to reason well. The answers they gave indicated that they could think logically and reason well. Only a few bright children who did their elementary education in English were able to reason well. (p. 61) This was a result that supported Clarkson’s research in Lae, PNG, during the 1980s on the effects of bilingual or multilingual education. As part of her doctoral studies (2010) Paraide also worked with students in schools in New Britain as she sought to understand more fully the effects of language on their mathematical learning. She found that the students better understood the four basic number operations and some measurement ideas in English. That was the case with one elementary school teacher in particular because he built on the number knowledge that the students already had, which for them was coded in their home language of Tinatatuna (Paraide, 2008a). The lower primary teachers involved in her research were originally of the view that students would advance faster in their mathematical learning if their school learning was undertaken in English but started to change their minds after working with Paraide on her research (see later in this chapter). There were two critical factors at play in this context. The elementary teacher knew his own Indigenous number knowledge well and was also very aware that the students applied these skills in their everyday activities. Hence this teacher used the Indigenous knowledge encoded in the Indigenous language as a stepping-stone when he introduced the Western number and measurement ideas in English. The integration of Indigenous and Western knowledge became an established practice in that elementary school site. Some Theoretical Underpinnings In his doctoral thesis, Charly Muke (2012) worked through some of the underpinning theoretical issues that could help explain why in some cases multilingual students can perform better on mathematical tasks than others. Although for many decades, as noted earlier, it was thought that using more than one language for schoolwork was a disadvantage, Muke noted that a good number of researchers have reported data suggesting that under certain conditions, multilingual skills have a positive effect on the learning process, and on the quality and quantity of learning (Baker, 2006b; Cummins, 1981; Cummins & Swain, 1986). Later research has seemed to confirm that multilingual skills can continue to assist learning (Baker, 2016; Uribe & Prediger, 2021). Indeeed, in the case of mathematics, the relationship between multilingual and mathematics learning has long been recognized. Muke (2012), Dawe (1983), Clarkson (1991), Zepp (1989) and Clarkson and Thomas (1993) have all argued that multilingualism does not necessarily impede mathematics learning. These studies have drawn extensively on Cummins’ (1981) work on the relationship between language and cognition. Baker (2006b) described Cummins’ work as the “cognition theory of bilingualism.” Cummins’s theory centered on language proficiency in all languages involved (the “Threshold Hypothesis”) and on linguistic development that takes place as a result of interaction between them while supporting each other within a single communication episode (the “Developmental Interdependent Hypothesis”). These hypotheses can be

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accessed to help explain the language contexts and their impacts on learning in PNG mathematics classrooms. The Threshold Hypothesis. The threshold hypothesis of Cummins centered on the idea of “balanced bilingualism” (Baker, 2006a; Clarkson, 1991), and proposed two thresholds. The hypothesis emphasized that students who are “balanced bilinguals” will experience positive benefits of bilingualism (Cummins, 1984). A balanced bilingual is someone that has appropriate proficiency in both languages for schooling purposes. However, according to Baker (2006a), there are two thresholds that students have to pass through before they reach balanced proficiency, and it is important to understand the impacts of these thresholds on learning. To be in a better position to understand this difference in language proficiency levels and their impacts on learning mathematics, we use Baker’s (2006a) analogy of language proficiency in terms of a house with three floors (see Figure 10.2). In this house analogy, Baker placed the balanced bilingual at the top of the house. Between the first and second floors and then between the second and top floors are the two thresholds. In addition, two language ladders are placed at the sides of the house, indicating that a bilingual child will usually be moving upward (or downward) and will not be stationary on a floor.

Threshold Hypothesis Diagram

BALANCED BILINGUAL Top Floor – Balanced Bilingual At this level, children have age appropriate competence in both languages and there are likely to be positive cognitive advantages.

SECOND THRESHOLD Middle Floor – Less Balanced Bilingual At this level, children have age-appropriate competence in one language. There are unlikely to be positive or negative consequences.

FIRST THRESHOLD First Floor – Limited Bilingual At this level, children have low level of competence in both languages, with likely negative cognitive effect.

First Language

Second Language

Figure 10.2 Different levels of fluency by a bilingual. (Adopted from Baker, 2006a, p. 172.)

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Papua New Guinean students do not start school as balanced bilinguals because of the government’s policy on language of instruction. The Government of PNG has emphasized that English be used as the language of instruction—at present from the beginning of schooling— although as noted elsewhere, during the reform period English was introduced in Grade 3. Although most PNG students speak their local vernacular fluently before they come to school, at that stage most would not be able to read or write in their language. Hence, they meet English for the first time and learn it as an additional language, while also learning the subject content, when starting school. Such a language environment matches Baker’s (2006a) second threshold located at the middle floor of the house. At this level, according to Cummins (1981), children have ageappropriate competence in one language. In this case, PNG students are competent in their local language but not in the language of instruction; hence, there are unlikely to be positive or negative consequences. However, according to Cummins (1981), learning for these students will be slow. This will be the case in the classroom context of PNG because three learning processes take place. The first is the learning of conversational language, and of the school and its classroom protocols and procedures that are conveyed mainly verbally in English, but with some written language in pictures and diagrams presented in wall posters and on the board. The second process is the learning of mathematical content which is expressed in English, again mainly verbally but increasingly in writing as time progresses. The third process is learning to understand the specific mathematical language including the symbols and how to read diagrams, all of which are encoded in English. Baker also suggested that if appropriate linguistic assistance is provided, students will not only learn the mathematical content but develop proficiency in English as well. Applying this to the PNG context, students will travel linguistically up the ladder towards the first floor to become balanced bilinguals and enjoy the positive cognitive benefits as described earlier, as long as they are given the appropriate support in the school. However, when limited or no appropriate assistance is provided, it is possible that PNG students could travel down the ladder to the bottom floor passing across the first threshold. This will promote limited proficiency in both languages. According to Cummins (1984), such a linguistic journey would be likely to have a negative impact on learning. As Clarkson (1991) described it, the above hypothesis on its own suggested, in one sense, that the proficiency in the two languages operated separately (although see above for his change in position) but could have a combined effect on cognition. Cummins (1986) also suggested that there was some form of interaction that took place between the languages, which combined in some way to impact cognition positively. The hypothesis by itself does not explain the interaction mechanism between the two languages, but clearly implies there is a dynamic transfer of meaning between them. This notion is explored through Cummins’ (1986) second hypothesis, called the Developmental Interdependent Hypothesis. The Developmental Interdependent Hypothesis. In order to gain a better understanding of the impact on cognition as a result of the linguistic interaction between Indigenous languages, Tok Pisin and English, we consider the second hypothesis of Cummins, the developmental interdependent hypothesis. As Clarkson (1991) noted, whereas the threshold hypothesis tended to treat Indigenous languages, Tok Pisin and English separately but as having a combined effect on cognition, the second hypothesis suggested that the Indigenous languages, Tok Pisin and English should also be seen as acting on each other, and that the result of that interaction subsequently impacts cognition processes. As indicated earlier, the language of instruction in PNG is English but many students speak their local vernacular(s) and/or Tok Pisin fluently. According to this hypothesis, when instruction takes place in English, there are two results: first the local language plays a role in developing English and hence in understanding the subject content expressed in English; and secondly English also helps in the development of the students’ local language(s), resulting in a deeper conceptual and linguistic proficiency

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and general academic skills (Clarkson, 1991). The following paragraphs examine the possibility that Indigenous languages promote mathematical English, and how English (or lingua francas) has modified (for better or worse) an Indigenous language such as by adopting words rather than using words and grammar with cultural meaning (Meaney, Trinick, & Fairhall, 2012).

For the first of these two processes, the hypothesis suggests that a student’s level of competence in one language is, at least partly, influenced by the student’s competence in the other language (Cummins, 1979). In the case of PNG, in a classroom, students’ competency in their Indigenous language, according to Baker (2006b), should play a role in the learning of English. This should mean that the proficiency possessed by students in their local language should become a resource to assist the students to code-switch and translate most ideas expressed in English, either verbally or mentally. However, this view assumes that the local language possesses compatible linguistic facilities and structures compared to English in relation to the specific ideas under consideration—in particular, specific linguistic facilities and structures, including in our case a mathematics register and mathematical discourse capabilities, for the ideas presented in the classroom. So there are two issues to be considered here. First, that their Indigenous language possesses such capability, and second if their Indigenous language does, the student is also aware of these capabilities. On the second point it is possible that not all members of a village, and hence not all or any children, will be aware of some mathematical possibilities embedded in their Indigenous language, since such knowledge is either rarely needed in everyday life, or the use of English and Tok Pisin over generations, means that they are no longer familiar with certain vocabulary or with complex language structures. However, in all probability there is likely to be some members of the village who will be familiar enough with their language to express these ideas in Tok Ples. When this occurs, at least some students will be well placed to switch between language codes and, as we have seen earlier, this would be likely to increase their levels of understanding (Edmonds-Wathen, Owens, & Bino, 2019). However, it is possible that not all of PNG’s Indigenous languages will have linguistic facilities and structures to match those of English, and in particular those within the English mathematical registers and discourse. First, the local languages may use a verbal or a noun structure primarily (Capell, 1969). However, the main reason is that the cultural history of these languages is different. The motivations to extend and grow languages/cultures are set by the social and physical environments, and clearly PNG cultures are quite different from most of those deriving from European environments. Consequently, knowledge established to make sense of their environment would be different. In this case, the mathematical registers established and embedded in English and in Indigenous languages could be expected to be different. As has been noted already in previous chapters, this is the case. During the elementary school project on mathematics (Owens, Edmonds-Wathen, & Bino, 2015), the possibility of teaching mathematics in Tok Ples was emphasized as part of using cultural knowledge. As a result, in a number of the workshops some time was spent considering the Tok Ples required to express certain school-curriculum-related mathematical ideas. Often robust discussion took place before a considered opinion was reached about the best phrase. In itself, this assisted teachers to understand the meaning behind English words. In a lesson on volume using bottles filled with water, the teacher used a length unit to introduce the meaning of unit. A hand span or a step was taken as defining the unit. This was confusing when associated with volume. Edmonds-Wathen, a linguist, and Owens and Bino (2019) illustrated these issues. Edmonds-Wathen and Bino (2015) showed that it was possible to express Western mathematical terms in Motu language and with meaning. There seemed to be no word initially for this schoolbased word, “unit”. In the Motu village, teachers discussed volume and length (both syllabus topics) together so that a word was found eventually that did not necessarily associate with either length, volume or any other particular attribute. It related to the infinitive “to measure.” Another

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issue was the word for 10. The counting words were established up to nine in which six translates as three pairs, and eight as four pairs with nine being four pairs and one. The word for ten depends on the group of specific items. Hence ten fish is denoted differently from ten shells and so on. In the Waima village in Central Province, a workshop was conducted by Bino and Owens with teachers who were fluent Roro/Waima, Motu (both Austronesian languages) and English speakers. There was extensive discussion about how multiplication could be talked about in Waima in the sense of equal groups, while avoiding a transliteration of the English word “multiplication.” In trying to express the question “How many in eight equal groups of 5?”, the first proposal included the term, bita (how many), but there was discussion regarding the appropriateness of this terms since it is frequently used in bitamo to mean ‘less than a half’, or ‘small parts’ (Beata & Kim, 2014). A second translation without bita was produced after this discussion: Iro’a ikoinai ababani aonai ima ima? NOM-pile

all.of.them

eight

in/while

5 5?

The word iro’a is a nominalisation of the verb “to pile,” so iro’a ikoinai means something like “all the piles” or “all the groups” (Kim & Kim, 1998). As well as a basic meaning of ‘in’, aonai also means “while” or “during” (Kim & Kim 1998). Ima is reduplicated to indicate repeated groups of five. A semi-literal translation of the Waima question might be something like “(How much is) all the groups when (there are) eight (groups of) five?” This example shows a little of how the grammatical structures of Waima affect how mathematical questions or statements can be phrased, which can be different from the phrasing in English. Discussion, multiple attempts, and consensus were used by participants in the workshops to determine phrasing that the participants felt to be both grammatically and mathematically appropriate. In Yu Wooi, there was a discussion about translating north and south with the teachers and elementary school advisers because the word for north being used on the north of the river was source. This was appropriate for those living on that side but not for those living on the south of the east-west Wahgi River where the source of creeks were in the south. There were also discussions about categories and subcategories, and characteristics or attributes. This occurred as we tried to classify different kinds of leaves. Nevertheless, international comparative ethnomathematical studies have shown that all cultures which have been studied throughout the world have shown evidence of engaging in similar universal mathematical activities. According to Bishop (1988), these universal activities are: counting, locating, measuring, designing, playing and explaining. The extent of the development of these ideas varies vastly between cultures and can be expressed quite differently between cultures. In PNG, various cultures have been studied showing that Bishop’s six activities can be identified but there is great variation in the development of each of these activities (see earlier chapters, but see also Owens, 2015; Muke, 2012, 2001; Owens, Lean with Paraide & Muke, 2018). In summary, whether students are able to take advantage of using their Tok Ples to nuance the understanding of mathematical ideas presented in school in English will vary, although it appears that all students should be able to use their own language at least for the beginning of number ideas. Local languages will vary in the degree to which they can give leverage to enhance understandings of measurement and spatial ideas, probability, and logical mathematical argument. It seems to be the case that few local communities are aware of the positive role deep knowledge of their Indigenous language can bestow on students in schooling conducted in English. That draws attention to a further assumption which Cummins (1979) acknowledged; that is, if his hypotheses is to play out in a positive manner for students, then the Indigenous languages of the students not only have to have the technical registers needed, as has been dis-

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cussed, but the immediate society needs to accord these languages the status of being capable of carrying the intellectual thought needed. Hence, there is a social issue in play as well. If students are told that they should only use English, and their own Indigenous language is disparaged as far as schoolwork is concerned, then they may not be taking advantage of a possible beneficial avenue of learning. It is this societal interaction with schools and teaching, both affordances and difficulties, that we will comment on more extensively in the next section, and the possibilities of children not only becoming fully aware of all the possible mathematical ideas embedded in their own language, but also growing their languages to cater for more mathematics.

Teachers and Teaching in Papua New Guinea

Two of the authors, Patricia Paraide and Charly Muke, began their professional lives as teachers in PNG schools. Their personal recollection of this time provides an interesting introduction to this section. A third author, Kay Owens, was a teachers college lecturer and taught in a community school as well as supervising students weekly and during blocks in city and village schools. Cameo from Kay Owens I visited a school at the end of the road out of Lae in 1975. It was a school where children came from villages higher up the mountain and where the local languages, mostly Non-Austronesian (NAN) Nabak2 speakers, were still strong although parents who visited the city could speak some Tok Pisin. Many parents could also speak Kâte, the NAN church language. The well-respected Grade 1 teacher who had had limited professional development was getting the children to read words like “fire.” He had the children say the names of the letters of the words they were to read. It was not the sounds. They spelt it out loudly, and did not write it. There was no discussion of the impact of the final ‘e’ or similar sounding words or the placement of the word in a sentence. Several other words were for objects that they had never seen, such as “train.” If the children spelt words incorrectly they were reprimanded and sometimes hit with a ruler. The teacher would also use Tok Pisin, especially for managing the class, but the children in the playground used Nabak, even though some came from the lower plantation areas, some of whom may have spoken Musom, an Austronesian language. Annie Dommerholt, at Balob Teachers College, was strong on teaching the teachers to say the sounds of letters, to look at similar words with the same sounds and/or spelling, and to write and read the words. She encouraged the teachers to start with sounds that were common in the local languages, and noted the sounds for the vowels “a” and “u,” in particular. Having come from The Netherlands to work in Indonesia and West Papua, and having a good grasp of important language issues, she was far ahead of the PNG teachers taught by monolingual Australians. The student teachers as part of their cultural studies were expected to record mathematical words in language. Both languages had digit tally systems (2, 5, 20) but Musom has a numeral for 3 and Nabak has 4 as ‘two-ly two-ly’, suggesting the language is verb-based (Capell, 1969). They would also try to give words for groups used in arithmetic and measurement. I spent one block in a village higher up the mountain with the students, and some other time in a coastal village with multilingual students, as there was a plantation nearby that employed people from various places. Students went to many different areas of Morobe and into the Eastern Highlands, or to Oro Province3. Our students recorded counting and other mathematical words in the village language as part of their cultural studies. Trans New Guinea Phylum, Finisterre-Huon Stock, Western Huon Family. The college was a Lutheran-Anglican college, and these denominations were situated in these areas after the various mainline denominations were allocated specific areas of the country to avoid conflict in the PNG villages. 2 3

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Cameo from Charly Muke I taught in almost all sectors of schooling starting as a Community School teacher in 1990. After completing further studies I taught at High Schools and then taught as a lecturer at both Primary and Secondary Teachers Colleges. However, this section will describe my teaching experience in Community Schools, since it is this that is most pertinent to the present argument. While teaching at the Community School I believed that my teaching would be most effective if, among other things, I often involved students both verbally and mentally. I knew that if children could actually participate in doing, discussing, arguing, justifying and inventing for themselves during my lessons, then they would be more likely to learn effectively. They would understand more and be creative in their thinking. Therefore, I always prepared my lessons so that I was clearly focusing on the child as an active learner. However, my lessons hardly ever turned out that way. I felt that I never really succeeded in teaching. I would end up talking most of the time, doing almost half of the problems for the children, and would always be guiding them closely to get them to complete the problems correctly in the time available. At the end of the lessons, I would have mixed feelings—happy that I had completed the tasks that I had planned for the lesson, but seriously wondering if the children had learnt anything. They tended to use the skills that I had suggested, rather than the ones they understood best. It seemed that children viewed what I, as their teacher, said and suggested was the only right way to do problems, and did not use their own ways that may have made greater sense to them. There was a mixture of reasons that could have made my lessons unsuccessful if I had attempted to make them more child-centered. Four of these reasons I have discussed previously (Muke, 2001). They related to cultural views of knowledge and learning; the power of existing cultural practices; the influence of teacher beliefs; and the difficulties associated with learning in a second language. The last of those reasons is particularly relevant to this chapter. During my teaching time in community schools, the rule to use only English as the language of instruction was still enforced. Hence my teaching was all in English, and I expected any learning situation I provided for the children would occur while using English, either by speaking, listening, writing or reading in it. Even though I knew that almost every child in the class spoke at least one other language, many of them more than one, and that English was a new language for all of them, I still emphasized that they should participate in English conversations at school. With my own school experience (see earlier), I could see that the children I was teaching tried to learn and understand the English language, and at the same time they were also trying to learn and understand the mathematics. There was nothing wrong with their effort, and indeed my own effort. However, the children had to take in the ideas in English, translate it into their own language, try to understand it, and then translate from their language back to English to respond and participate in the class, just as I had had to do when I was a boy. By the time they had gone through this process in their mind, the lesson had already moved on, just like it had for me back when I was a student. In other words, the processing that involved moving to and fro between two languages needed time. But my lesson moved too fast to allow this to happen, and hence most could not keep up with the flow. Therefore, I ended up doing most of the activities myself, instead of the students actually doing them, and hence my lessons were far from involving students effectively. My concern for the children’s learning prompted me to try and use the local language or Pidgin sometimes, to explain, reformulate important statements or questions, even though the education policy did not allow this practice. I quickly realized that using such practices helped the children a lot, especially it helped them to understand. Unfortunately, I often got into trouble for doing this. It was not only that I got in trouble with the school inspector for using too much local language or Pidgin, but I got into trouble with parents. One incident I remember from 1993 is just one example of many. A father approached me and said that his child must be spoken to

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only in English and not in any other language. The father thought that by listening to the teacher in English quite often, his child would learn English quickly. The father thought that I was inhibiting his child from learning English when I switched languages. As a teacher, I could not argue, but had to balance the demand of the parents and the school’s administration with furthering the understanding of my students. It was hard, but I thought I managed this well, getting the appropriate balance between the demands of these different groups. Cameo from Patricia Paraide While teaching various grades in high school and as a specialist in English and Home Economics, it was the policy that teachers had to use English when teaching all school subjects, and that communication with students and teachers within the school grounds had to be in English. Some of us got into trouble for using other languages to explain or clarify the ideas, thoughts and skills being taught. This was difficult for some of us because we knew that most of the students at this level of their education were not yet competent speakers, readers and writers of the English language. This prompted me to utilize text that was accompanied by pictures to assist students to understand what they were reading. I also brought graded readers for students to read at their leisure and this assisted with their learning of the written text in the English language. In Home Economics, most of the textbooks used to support teaching had illustrations, pictures and diagrams which assisted students to visualize the cooking and sewing tasks which were required. Also, because these tasks involved mostly practical work, students learned from practical demonstrations of the skills, and applied them in practice. They learned a lot through this teaching strategy and some of them were able to be creative with their various tasks in this process of learning. Other Anecdotal Data Some of our colleagues in our yarning circle at the Sydney Park also reflected on some classroom experiences they had encountered in school: Owens: Carrie:

Carrie what about you? Was your education like Dan’s? I was taught generally all English (taught in English through all grades). Oh, it was the priest that drummed everything into us. It started with the kindergarten (what followed was not altogether clear but related to the strict punishment regime and the teaching being in English for all subjects). Owens: Did you like maths? Carrie: Quite a lot. I think we had very good nuns that took us for addition, multiplication, division both short and long, and drummed everything into us. Owens: Did you get punished? Carrie: Yes! For not knowing stuff. Dan: Oh yes if you did not get it right you had to stand on one leg or the other and put your hand on your head. (Laughter all round and chorus of “Yes me too.”) Owens: What year were you going to school when you were getting punished? Sylvester: Oh this lasted up to the early 90s. Clarkson: And you, Dan and Carrie were in school in the early sixties. Dan: Yes. Carol: Well it depended on where you were. In the cities there has not been so much of that type of punishment recently, but out in the country, yes it still happens. Yes,

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and the community would also participate. The community usually believes that the teacher is someone who knows how it should be, you know, what is correct and if this means to punish someone, well they (the teacher) was in the know, so then this means it’s OK. And yes, we had to write out “I must not misbehave” a hundred times or something Muke: Or a thousand times: “I must speak English. I must speak English.” Clarkson: Another question. You have been talking about doing mathematics but it has always been about number. Did you ever do anything else other than number type stuff? Sylvester: Can you give us an example? Carol: Patterns, space. Clarkson: Measurement? Did you ever do anything of that? Michael: Ah no. Carol: Yes, we did some of that but numbers have to come into play there too. Carrie: [stretched out her arms to show measuring non-formal distance.] Dan: Yes we did that and not with a trundle wheel. Owens: But that was done by the teacher, and not coming from your place. Carrie: Yes. Clarkson: Yes, this is school maths Carol: But no, the villagers, they use arm spans too. Clarkson: What for? Carol: Measure like cloths. Measure the boxes they might make for gardens, and in making houses. Clarkson: Was that traditional? Carol: Yes you used … (indistinct “feet”?) Clarkson: You wouldn’t have had a rope, you would have used vines or something? Carol: Yes, you platted vines sort of and used that. Clarkson: What about spatial ideas, like naming shapes, or … ? Muke: We learnt it. But yes triangles, diamonds, squares, rectangles. Carol: And circles. People learnt it in their language and shapes of trees, and … Owens: You had some language words that were used for some shapes? Carol: Yes, we do now after we did some linguistics (They meant that some new words were developed in their language for this purpose of naming shapes which came from the school curriculum.) Clarkson: Yes, that is now. But when you were little at school? Carol: Not so much. Clarkson: (To Dan and Carrie): Was this the same for you where you come from, or was this something that came in, because you obviously did primary school a bit earlier than them? Dan: No. Well you know, “How long would that be?” Well the first thing I would do would be, I would put my hand out like this (arm outstretched and in line with eye and measuring how many hand spans). Or I would perhaps stand up close or stand on something and measure it against my height. And yes, we have some names for shapes. Yes and in Grades 1 and 2 we would be using vine ropes to measure things. We didn’t have any rulers or things to measure with. So the teacher would measure something about a foot long and ask us to guess how long it was. And yes, we would say “Oh it’s about this” (shows hand and then finger lengths), and so on. The teacher had all these lengths cut up made of strips of cane, smaller

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Clarkson:

canes to longer ones all stacked up and you know we would do measuring with those. Oh, OK. So there was some measuring going on—with teacher aids made by the teacher. So you guys were making some connection between the maths that was being taught in school and back into the village. But not all the time. Just some?

(Indistinct, but general agreement) … Muke: Carol:

I want to ask Carol about that counting. That is a very good idea. We did have our own way of counting back at home but because of the English policy at that time it drifted out of use in schools. But many of the teachers used that (vernacular language and counting) anyway. But later I had PNG teachers who were using Tok Pisin. The children came into our classroom when I was a student, from many language groups, so we had to use English and we were not allowed to use anything else. The teachers did not attempt to use a bilingual approach because they stuck to the book (the curriculum guide) that was in front of them. Owens: Dan I think you told me that your teacher did know your language so did they use Tok Pisin as well or just Tok Ples? Dan: Well he was taught in English in his teachers college, but came back and he used Tok Ples.

These anecdotes indicate how teachers would make use of available equipment, experiences and language to try and help their students to learn. Nevertheless, there were issues which arose as a result of the policy which had English as the sole language of instruction.

Indigenous Mathematics and Language

In previous chapters we have given many examples of Indigenous mathematical practices in the various cultures of PNG. Glen Lean (1992), of course, painstakingly catalogued the many counting systems used, and Owens, Lean with Paraide and Muke (2018) have extended his analyses. The study of measurement systems (Owens & Kaleva, 2008a, 2008b) is also significant in showing that a broader range of mathematics than number was evident in these language groups. Owens and Kaleva collated measurement words in over 200 PNG languages. It seemed that most of the languages had words or phrases for the measurement terms used in school mathematics. Owens has outlined and analysed the various spatial abilities across many groups in Papua New Guinea (Owens, 2015). Over a century ago Haddon noted the strength of visual acuity (Haddon, Balfour, Marr, Ray, & Seligmann, 1906), and numerous later studies have also emphasized this (see Chapter 11 for further details). Others have added to this trove, such as Kopamu (2005) who noted counting words to 20 in Enga, and that there are other words, rarely used, for higher numbers. He then showed that in Enga there are many other mathematical ideas that can be conveyed: the mathematical operations of addition, subtraction, multiplication; some fractions (at least 1/2, 1/4 and 1/8); the concept of zero; negative numbers; and some comparisons such as equals and not equals, greater than and less than, and belonging to or not belonging to. Kopamu argued that although some of these ideas are not used in everyday life, nevertheless his language is rich enough to carry the needed meanings if there is a need. An implication that flows from this is that it could be used in classrooms in the teaching of mathematics. And this is the crux of a key issue that needs discussing—can Indigenous mathematical concepts and principles in general be used in what are essentially Western-style schools and classroom? Besides Bishop’s (1988) six fundamental aspects of mathematical activity (counting, locating, measuring, designing, playing, and explaining) other fundamental mathematical activities include the related processes of problem solving, investigating, inquiring with inferring, reason-

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ing, logic, comparing, and modelling (D’Ambrosio, 2006). All of these can be found in all cultures, albeit in some cases for comparing size with little emphasis on numeral tags and by using other language structures. All cultures have developed their own mathematics which meet their cultural needs and have also developed commonly agreed mathematical practices (de Abreu, 2002; Saxe, 2012). Bishop gave a range of examples that bolstered his argument and so did Joseph (1987, see also 1991). Chapters 2 and 3 of this book provide a canon of these mathematical activities in PNG cultures. Although certain mathematical generalisations that might be germane to patterns and relations are discussed in the various languages, there is not a term called “mathematics” that associates an identifiable body of knowledge as we find with Western mathematics. It seems that the need never arose in the various Papua New Guinea cultures although some people were identified as specialists in certain areas like trading or sailing that are highly mathematical. The other point to note is that the various mathematical ideas that have arisen in Papua New Guinean cultures are embedded in the various Indigenous languages. De Abreu and Cline (1998), Zavlavsky (1998), DÀmbrosio, (2001), Ascher (2002) and Beach (2003a,b) have documented similarities between Indigenous and Western mathematical concepts in other cultures, thus bolstering Bishop’s (1988) claim about the existence of universal forms of mathematics. Nevertheless, those who have contributed to relevant literature on this matter have usually noted that schools tend to ignore the mathematical knowledge that the students bring into their classrooms. What students already know is not used as a stepping-stone when introducing similar or new Western-style mathematical knowledge in formal school learning situations nor developed within the classroom (de Abreu, Bishop, & Presmeg, 2002). However, such a divide between home and school knowledge is not just a function of multilingual contexts. The same exists for nominally monolingual communities when students solve authentic or real problems outside of the school but not used inside school (Clarkson & Carter, 2017). A push for Indigenous mathematics to be used as a pathway for the teaching of Western mathematics was a key feature of the reform schooling. This was to occur throughout school but was of particular importance for students beginning elementary school. Paraide (2008b) has argued that vernacular languages are currently seen only as a tool to teach Western knowledge, and not as a rich source of Indigenous knowledge. Quoting Geoffrey Smith (1975) she noted that PNG’s Indigenous education, Indigenous knowledge and Indigenous languages were suppressed in school learning situations before, during and after the colonial era. Consequently, English is still often viewed as the most effective language for schooling in PNG. Paraide (2002) had earlier found that, as a consequence of the implementation of English as the main language of instruction, at all levels of schooling during the colonial era, the use of English simply became an established practice in schools and an expectation of communities. As a consequence, Indigenous languages are still viewed by many as less-effective teaching languages in schools. This was reflected in a teachers’ focus group discussion in Paraide’s (2010) study: We were taught in English when we went to school and we learned well, … this was okay. … We were trained to teach in English. … We are used to this (that is) teaching subjects in English … Teaching in vernacular is new to us and maybe it is okay, but we are not sure. … But the students must learn English well so they can learn better. … They must understand English well because they have to sit the Grade 8 National Examination which is in English. … With vernacular and bilingual education we really do not know if they can learn English well enough to understand this exam. It is not surprising that teaching in English means that children are expected to learn Western knowledge. This form of schooling has dominated past and current forms of the implemented curriculum in PNG, and it is now generally viewed by Papua New Guineans as being more valuable than their Indigenous knowledge—at least for the purpose of progressing society toward a cash economy. Indigenous mathematics is not always observed as valuable, as a step-

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ping stone device in society to assist the development of Western mathematics, as valuable to communities, and as valuable in its own right, and that was true during and after the period of reform. Evans, Guy, Honan, Kippel, Muspratte, Paraide and Tawailoye (2006), in evaluating the reform, noted that the forms of Indigenous mathematics which are embedded in vernacular languages, were generally not integrated with formal or Western mathematical knowledge in PNG’s elementary schools, despite that aim having been one of the crucial tenets of the reform (see Chapter 8). Paraide (2009) found that the lower primary teachers did not use the students’ Indigenous number ideas when teaching group counting in English. However, she also found that when students were being taught in the language they knew best (either their Tok Ples, Tok Pisin, or Motu), they showed, on the whole, commendable understanding of the mathematics that they were being taught (Paraide, 2003). Whether the mathematical concepts which exist in the Indigenous cultures are identical to those in Western mathematics, or not, difference usually becomes an affordance to students if understanding and application are important in mathematics. In Western mathematics a counting number is an abstract notion that can be applied to any quantity, no matter what is being counted: for example “three” tells us about a quantity of anything, whether it be tomatoes, trees, tigers, steps, or members of a team. In PNG Indigenous cultures, the number words used, and at times the number system used, may be dependent on what is being counted but it is usually abstract. Even the systems using classifiers, or specified groups of 10, or even alternative counting systems for counting different objects which may be borrowed from a neighboring trade partner, seem to have some kind of abstraction. There are systems in place which are recognized by community. Smith (1981) also noted that counting has been used as a measurement device. The number of objects, which may only approximate “being the same,” was a unit of measure for many transactions. In this vein, one author (Clarkson) can recall examining carefully the pyramidal piles of four tomatoes at the Saturday morning market in Lae, all selling at the same price, but choosing the pile with what appeared to him to have the largest-sized tomatoes, much to the amusement of the stall keepers. Clarkson and the stall keepers were clearly working from two different, but intersecting, measurement/number systems. Display systems for exchanges may also involve counting with showmanship, or visually. Although there are similarities between the two, Indigenous and Western, there are differences. Both have served more than adequately the needs that the different cultures have developed. Both require cognitive abilities that need to be learnt within the culture. The question then becomes whether using the Indigenous mathematical notions as an introductory device in schools to begin the teaching of Western concepts, given that each differs from the other in essential ways, will be confusing for the students or an advantage for them. That is a question to which we will return to later in this chapter. An Earlier Exception: The Tok Ples Skul Movement It has been shown above that the debate about the language of instruction continued during the years leading up to Independence and beyond. There was a concern that an English-only school system was alienating children from their own language and culture (Clarkson, 1983; Litteral, 2001). Several of the 20 new decentralized provincial governments heard this concern expressed by parents and took steps to introduce vernacular education approaches outside the formal schooling system. One of these provinces was the North Solomons Province (now Autonomous Region of Bougainville). There, the Bougainville Islanders proposed giving their children two years of preschool education in their own language before the first grade of primary schooling, and then moving to English as the language of instruction in primary school. The Viles Tok Ples Skul (VTPS) (Village language school) schemes thus emerge as a non-formal community-based, vernacular language preschool education option. Later, it became known as Tok Ples Pri Skul (TPPS) (Vernacular preschool). By the early 1980s this movement was gradu-

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ally spreading, first to Chimbu province, and then to a number of other provinces. The Morobe/ Madang Lutheran Tok Ples schools also continued. A review of the VTPS program found that not only did children who attended a village vernacular preschool before entering the first grade gain distinct educational advantages, but their communities also enjoyed social and cultural benefits (Delpit & Kemelfield, 1985). Primary school teachers noticed that the transition to the English-only classroom was much easier for children who had attended the vernacular language preschool compared with those with no previous school experience. Such results were becoming clear even by the early- and mid-1980s when Clarkson and a colleague (Bryn Roberts) from the NDOE visited schools on Buka Island on two occasions. In particular, teachers were introducing mathematics in the local vernacular by first working with the children on the mathematics they had already learnt in the village in their own language when doing the various tasks and playing the different games they enjoyed. When it seemed appropriate the teacher would then branch out from the local mathematics across to beginning Western mathematical ideas. It seemed to Clarkson that the teaching which was occurring in these schools, undertaken by teachers who were from the local village, using the local language and, I was told, very knowledgeable of local customs, could be usefully studied with the possibility of developing a set of guidelines that other teachers could use. In 2015 as part of the elementary mathematics project, Owens and Bino, with local organizers, ran a workshop for some teachers from the Madang area. One teacher had completed his own schooling in Tok Ples and was now teaching in Tok Ples without a good oral understanding of either Tok Pisin or English. We were told he was an excellent teacher. The skills of the Tok Ples Village Schools conducted under the auspices of the Lutheran church in Madang and Morobe Provinces in the 1970s had a lasting impact.

Valuing Vernacular Languages in Education

As has been documented in an earlier chapter, a major assessment was completed across the whole education system in Papua New Guinea in the early 1980s by a group chaired by Sir Paulias Matane. This generated the influential report called the Matane Report (NDOE, 1986). A major influence on this Report was the success of the VTPS program, and became, as we have seen, the basis for recommendations of major education reform throughout PNG—although a number of the recommendations echoed what had been written in the earlier Tololo Report of 1974 (NDOE, 2002). One critical recommendation was to change the English-only policy. This Report recommended that local vernaculars could be used within the formal education system. The following section describes the change which occurred with language policy in the 1990s. At this time one of our colleagues began her preschool years in English at an international preschool on Bougainville. She knew her own language well. Subsequently, as an influential early childhood teacher without specialist training she has influenced governments to run preschools and schools in English because she herself did well in this system. She is also fluent in her husband’s language from Central Province. It is not uncommon to hear villagers who went on to high school say they think English should be taught from an early age as was the case with what they experienced. Two things tend to be forgotten in these contexts: first the smartness of the persons involved; and second, the differences in teachers’ English experience over time. Teachers Use of Indigenous Language When Teaching Mathematics In examining the model depicted in Figure 10.1 there is no issue if there are well-known and developed mathematical registers available within the linguistic structures of each language. If there are, then the teacher and students have many options of what languages can be used. However, if the mathematical register is restricted in a local Indigenous language, then the pathways available are restricted. The key point is that how the teacher uses language in the classroom cannot be assumed to be the same in different teaching situations or with different language

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contexts. Variations will always occur, and these will depend on the extent of the mathematical registers. In South Africa there are multiple Indigenous languages as well as English and Afrikaan. Setati and Alder (2001) found that South African teachers could not shift directly between the formal Indigenous mathematical language to the formal English mathematical language in most African languages because the mathematical registers in the African languages were often not sufficiently well developed in order that the Western mathematical ideas could be clearly expressed. (Setati and Alder illustrated this using their own model, not Clarkson’s.) Instead, they found teachers normally started with informal African language and moved to informal English, and then down the English column to formal mathematical English, using part of Clarkson’s model (see Figure 10.1). Muke (2012) conducted an extensive investigation of the bridging by Grade 3 teachers teaching mathematics during the mid-2000s, when Grade 3 was designated as the bridging year— with the bridging in this case being from the vernacular languages, Wahgi and Tok Pisin to English. The vernacular languages were used in the elementary schools, and English was used as the language of instruction in the upper grades of the primary school. (Primary schools started with Grade 3 under the reform changes, and continued to Grade 8.) The study was conducted in several primary schools in the Wahgi Valley in the Western Highlands, Muke’s home area, and hence he spoke fluently all languages being used. The teachers encouraged understanding and discussion of mathematical concepts by using multiple languages while teaching mathematics, and this approach was encouraged at the time, unlike the classroom situation Muke had experienced when he was a boy (see his cameo earlier in this chapter). His key focus was how the teachers used these multiple languages, and what hybridity eventuated as they code-switched between Wahgi, Tok Pisin and English, and between the intra-languages (informal, structured, mathematical register; see Figure 10.1) within each of these. In his analysis, the middle row of more structured language mainly disappeared, and Muke primarily discussed language moves between the informal languages and the more formal languages of the mathematical registers. According to Skiba (1997), there are two identifiable forms of code-switching: between sentences (inter-sentential), and within a single sentence (intra-sentential). The purpose of inter sentential switching varies, but according to Setati and Adler (2001) it includes questioning, repeating, paraphrasing, explaining etc., which all help with ensuring the conversation is understood by both speakers. Similarly, code-switching that occurs within a sentence (intra-sentential) can be broken down into two processes. If the practice involves a single word, this is described as “code borrowing.” But when a number of words are switched this is referred to as “code mixing.” The most common reason for a speaker to borrow a single or group of words is the absence of such a term in the language, or it is deemed that such a term is known to the speaker and listener, and is embedded into that language, being used regularly in dialogue. However, the core purpose of code-switching, no matter which type is used, is to try and ensure the clarity of shared meaning between the people speaking. Classroom discourse is more than just mathematical, and Muke analysed the different kinds in terms of code-switching. Mathematical Classroom Discourse Muke analysed the language used in classrooms of a number of teachers and identified when they used code-switching in their teaching between Waghi, Tok Pisin and English. From the context of the moment and a detailed examination of the transcription he hypothesized why and how the teacher had used code-switching. Then viewing the video tapes of the lessons with individual teachers, he asked them to reflect on why they had code-switched.

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Both types of intra-sentential switching were commonly observed in sentences constructed to ask questions, to give explanations or definitions. When the teacher asked a question, it was normally about the English mathematics register. In most cases, the question was asking for the definition or explanation of the English term, which had been embedded within the sentence. A similar pattern was observed when the teacher asked for an explanation. Similarly, when the teacher gave a definition, the sentence was constructed using one of the local languages (Wahgi or Tok Pisin), but using the appropriate English mathematical term. The aim of the sentence constructions using intra-sentential switching indicated that the local languages were used as a resource to guide the learning of mathematical English and the related mathematical content. The other type of code-switching this study examined within mathematics discourses was described as inter-sentential switching. It appeared that the overall aim of shifting languages between completed sentences was again to guide effective learning. In this way, teachers nuanced their questioning, explaining and repeating using this type of code-switching. For example, when examining teacher verbal communication that involved questioning, it became apparent to Muke that it was mainly for purposes of eliciting. Eliciting involved teachers asking students about something that they already knew but checking if the students shared with the teacher the same understanding of the concept. What was unique about eliciting in this study was that the teachers alternated between languages to ask questions as part of eliciting. This study did not target how it impacted students in their learning, but the teachers said when interviewed that they thought that this process would help their students’ understanding. It appeared to the researcher in observing classes that due to the language shifts, the students were comfortable in responding, and it seemed to him that they participated more in these lessons when this language use was employed (unlike his own experience in grade 3). When examining the verbal interchanges that involved explaining, Muke found that it was normally terms and phrases from the formal mathematics registers which needed elaboration, particularly expressions from the English mathematics register, but also some in Wahgi and Tok Pisin. In the main inter sentential switching used the local language to elaborate, paraphrase and summarize the major concept under discussion. Again, it appears that using code-switching in this context increased the potential for teachers to effectively guide the learning of the students. Inter-sentential switching between languages also occurred in the context of repeating. Repeating occurred commonly in three contexts: repeating questions, repeating explanations, and repeating students’ responses. Repeating questions and repeating explanations served the same purposes as discussed above, but the practices were in line with how cognition processes take place for unbalanced multilingual learners (Clarkson, 2006). When a learner with an unbalanced language background is presented with a concept in a new language, the learner constantly switches into the fluently spoken language trying to repeat as accurately as possible what was said to make connections with what is already possessed as background knowledge to enhance understanding. They do this either mentally or verbally. As the researcher recalls from his own experience as a young learner, such a language process was purely a mental process for him, as English as a foreign language was only allowed to be used for teaching, and there was no verbal articulation in a local language. In such a situation, the learning was slow because the students were undertaking such mental processes, but in the new situation without realizing this, the teacher had sped ahead in his/her teaching (as Muke had done when he was teaching). When a teacher repeated a term or phrase, the students followed the pattern of the teacher’s thinking and the speed of the lesson was regulated more to the students’ progress of cognition. Hence, in this context it is likely that when a teacher repeated a question in the local language, the students were given the opportunity to understand and respond not only accurately but in a relatively short time after they were asked. Similarly, when a teacher repeated an explanation in the local language, the students were assisted to understand what was being explained in the lesson through the switching of languages. Therefore, the teachers’ practice of repeating questions or explanations

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in different languages probably helped these unbalanced multilingual students to process ideas mentally and keep up with the speed of the lesson. Thus, hopefully they were guided far more effectively in their learning than the researcher had been as a young learner of mathematics. Another strategy that teachers used in their teaching was repeating students’ responses. The purpose of this was to confirm or reject what the students publicly shared in the classroom. The responses of the students were mainly in the local language, but as part of the teachers repeating the students’ responses, the teachers not only confirmed what the suitable mathematical expressions were in the local language, but also led students to the formal mathematical expressions in English. Hence, it appears that the result of teachers repeating students’ responses was not only that students received feedback on their attempts to answer questions, which was likely to grow their knowledge, but it also confirmed and incorporated what students said in the flow of the discourse, often elaborated with a shift to English. Non-Mathematical Classroom Discourse Effective mathematics discourse in a classroom does not happen in isolation. There are always elements of non-mathematics discourse in each lesson and these often play a major supportive role in enabling the mathematics discourse to be effective. Analysis of the non-mathematics discourses showed that code-switching was evident. In summary, the ways teachers talked had the aim of regulating both the physical and mental behaviour of students. For this to occur, teachers used strategies which involved questioning, instructing, and repeating instructions. The questions asked, often as part of code-switching in non-mathematics discourse, were not for eliciting explanations or answers as was found within mathematics discourse, but were aimed at managing classroom behavior. The questions were commonly asked in English, but in some cases they were also asked in the local languages. The questions in English promoted the authority of the teacher, and thus helped manage the behaviour of the students. The few non-mathematical questions that were asked in the local languages were also for regulatory purposes, but required students to understand what they were required to do. Thus, these questions played an important part in the progress of the lesson. Similarly, the overall aim of instructing and repeating instructions after inter sentential switching were also for the purpose of regulating behavior. Using these strategies was one way that the teachers managed the flow of the lesson, and hence facilitated the success of the related mathematics discourse. Teachers’ Reasons for Using Code-switching Teachers were interviewed about their reasons for code-switching. On one hand, teachers had very strong views that English, in this case mathematical English, was important. The main reasons that the teachers gave to support this view were related to using English for future employment and communicating with the rest of the world. The teachers believed that the students had to have access to English, as early as Grade 3, if they were going to learn enough English to enter good jobs or obtain entry to tertiary institutions, and this meant passing the Grade 8 examination, which was in English. On the other hand, the teachers said that using English only in teaching did not promote deep understanding at Grade 3 because at that stage students were being introduced to the language for the first time. Therefore, to help students understand the mathematics being taught, the teachers strongly believed that they should use the fluently-spoken local languages as a teaching resource. In this way the students would more easily come to understand the mathematics concepts being taught, and this would help them to learn successfully. However, they expressed, often quite forcefully, what the students really needed to understand were mathematical concepts expressed in English.

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Summary Comments Muke found that the teachers did not shift straight from informal language in Wahgi or Tok Pisin, to the formal language in English. Instead, the teachers aimed to continue using the local language as the predominant language but when necessary adopted the language practice of code-switching. That is when they used terms from the English mathematics register and embedded that language into the sentence structure constructed in Wahgi or Tok Pisin. In particular such borrowings were in the form of a noun phrase and the local language was used to form the rest of the sentence as a verb phrase. There were two main reasons that teachers gave for using the language practice of code borrowing. The first was that there was no appropriate language in the mathematics register in the local language that they could use, and therefore they borrowed from English. The second reason was that even if there was an appropriate local mathematical language available, they still code borrowed since their main aim was to promote mathematical English. They saw no reason in this school context to enhance the local mathematical register, since that was the business of the village, not the business of the school. Muke also noted in this study that the mathematical register of Waghi and Tok Pisin, did not mirror the English mathematical register although there were easier links for mathematics taught in Grade 3. In other words, the grammatical structures of the two languages and concepts were different although it was possible to translate one to the other after due consideration. Other studies carried out within Papua New Guinea, and mentioned earlier in this chapter and in this book, reinforced this idea. Hence this becomes one of the main obstacles for the local languages to be seen as useful resources in schooling. Most teachers and local speakers do not know how to deal with the partial incompatibility of the registers, and this hampers the way forward. It is no surprise, then, that many communities do not believe that the use of local languages will be of benefit to teaching and learning in PNG. However, English could be used to help in the development and maintenance of local languages by developing the mathematics registers in local languages to a degree similar to that of the mathematics register in English (Clarkson, 1991), which could enhance mathematical discourse both in local languages as well as in English. Muke noted that to elaborate the mathematics registers for each local language would be a big, time-consuming but worthwhile task. It would assist each local language to develop and survive for generations to come. Such an approach was taken for the Maori language in New Zealand (Meaney, Trinick & Fairhall, 2012). Muke suggested that the elaborated mathematics registers must use new local terms (avoiding code borrowing from English if at all possible), or reuse existing terms to have double meanings, and then embed these terms within the linguistic structures of the local language for successful mathematics discourse. For this to happen there would need to be action both at the academic level and at the policy level, but in close consultation with the Elders and members of each culture. Such a process may be challenging at the beginning, but it would be worthwhile in the long run for the local languages not only to become compatible with English, but also to allow local languages to be updated, developed, and hence survive for future generations.

Teacher Education in Papua New Guinea

During our yarning circle in the Park, we discussed some of the teacher-education issues that might have occurred. Owens:

Does anyone remember having based 10 blocks or Dienes blocks or other base blocks in their primary school? [No-one answered.]

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Clarkson: Well Dan and Carrie wouldn’t, because they were not introduced until the mid- to late-sixties, after students had completed Grade 6. But what about you others? [No-one answered.] Clarkson: Yes. well I don’t think they got very far out from Port Moresby, Lae and maybe Hagen and Rabaul

Three or four of the participants would have attended primary school at some time in the late-1960s to mid-1970s, so if the Dienes’ learning materials were in any of their schools, maybe they would have remembered them (see Chapter 5 for more details on Dienes’ work). But they did not. Owens saw these blocks together with pattern blocks (which included a parallelogram as well as a rhombus) and attribute blocks in community schools in the suburbs and surrounding schools of Lae in the 1970s, but the blocks had gathered dust in the storeroom. Teachers at the time could explain how the different base blocks could be used but did not see any point in using them. Representing numbers in different bases with blocks and symbols, and adding and subtracting in different bases was then part of the primary school curriculum. It was not easy for the children. In her detailed review study, Southwell (1974) showed that one of the reasons the Dienes material failed was the lack of appropriate, and indeed the number of, in-service sessions held for teachers. She also noted the importance of the verbal language that could be generated by the interesting material, as Dienes (1969) intended—as Clarkson (1979) concluded was the case for students in Australian schools. All agreed it was the conversations around mathematical ideas that counted more than the blocks themselves. Southwell noted that if such discussions in PNG schools were in English, following the policy of the time, then students would also have been hampered by not being allowed to use their own Indigenous language. There had been a curriculum, and much support material, set up which followed the logical ideas of Dienes but there was still a lack of adequate professional development of teachers and teacher educators, despite Dienes’ attempts to encourage this. Cameo from Philip Clarkson Language for mathematics was an issue expressed by Alkan Tololo4 to Clarkson and Bryn Roberts (from NDOE at the time) while they waited for the ferry from Buka Island to Bougainville Island in 1983. One of the recommendations of the Tololo Report (Department of Education Papua New Guinea, 1974) was that teaching in the first three grades of schooling should be in the functional language of each community, that is teachers should be using the vernacular of the children (NDOE, 2002, p. 7). Tololo believed that none of the recommendations in his report were implemented, since all of them were opposed by the established business interests and the expatriate advisers (including at UPNG) that were still so influential in the years immediately before Independence in 1975. Both groups insisted on English as the language of instruction for all grades because they were only interested in children finishing school with a good understanding of written and verbal English, so they could go straight into the cash work force. It is worth noting that in 2012 when the new Prime Minister told the nation that English was to be used from the beginning of schooling, that Paraide had a paper published in a local business journal (Paraide, 2008b). On hearing that we were both working in mathematics education, Tololo also told us that although the Dienes material had had great potential, very few of the teachers really knew how to use it in classrooms, largely because of the paucity of hands-on experiences in professional learning sessions with any of the materials. Tololo was the Chancellor of PNG University of Technology and also, as Director of Education, was chair of the committee which wrote the so-called Tololo Report (Department of Education Papua New Guinea, 1974) 4

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Professor Solon, former Vice Chancellor of the University of Goroka, told a similar tale to me in 2003 when I visited the University as part of the evaluation of the PASTEP Project (Clarkson, Hamadi, Kaleva, Owens & Toomey, 2004). Solon had attended a one day-long briefing on how to use the Dienes material in the late 1960s, and he remembered being more confused than convinced that the blocks would assist teaching and learning. I saw the result of a similar story regarding Dienes’ materials in primary schools in Fiji in the 1990s; again there had been a lack of prolonged and worthwhile professional learning with teachers, and little work with the materials in preservice education programs. As an aside, at PNG University of Technology and at Balob Teachers College, both in Lae, Dienes’ multi-based arithmetic blocks have been used to introduce students in a physical way to the multiple counting systems of the country along with associated language for different places. This could offer important conceptual understanding of number systems for any mathematics student, anywhere in the world. Introducing Innovations The introduction of any new innovation in teaching is fraught with difficulty. Most teachers seem to be reasonably conservative in their approach to teaching, and unless they become convinced that the new innovation will benefit their students, they are likely to continue their normal ways of teaching which, they will tend to say, have been successful for them. So, if innovations are going to prove fruitful, they need to grip the teachers’ imaginations. No matter what new curriculum documents are written or what powerful speakers advise that the new way is better, teachers will not be convinced unless it seems to them it works in their own classrooms. That means that a long school-based process is needed to embed the innovation. This did not happen when the moves were made to have teachers use Indigenous language as the language of teaching in elementary schools and at least in Grade 3 in primary schools in the early 1990s. Like the Dienes reform many years earlier, this reform was not given the necessary support. Teachers were fortunate if they received inservice training in child development and learning and literacy (not bilingual per se) for a couple of weeks, with follow-up work in schools from the head teacher, or an assignment on which they would be expected to submit a report. During several visits which Kay Owens made to the PNG Institute of Education between 2000 to 2015 it became clear to her that people at the Institute who did the training were not particularly familiar with the diversity of PNG’s counting systems, and there was a tendency to teach teachers to teach mathematics with a Western focus. This was despite there being some good materials for teachers from the curriculum unit, including one on patterns that possibly needed to go further in terms of counting. Paraide (2010) conducted a professional learning program for teachers in their schools aimed at revealing the difficulty of embedding the new innovation of using an appropriate Indigenous language as the principal language of teaching for the beginning grades of schooling. The difficulties which arose in the preservice programs will be illustrated by drawing on the results of the evaluation of the PASTEP, which had an impact on the design and structure of future preservice education programs for primary school teachers. Professional Learning for Teachers The implementation of vernacular instruction at the elementary level of education and bilingual instruction at lower primary level began in the early 1990s under the reform. The change was made based on empirical research which was intended to assist the formulation of the new language policy (NDOE, 1990, 1991, 1995, 1996a, 1996b, 1997, 1999a, 1999b). However, the resistance to vernacular and bilingual instruction was powerfully prevalent in many communities during the implementation of the reform, and it was this resistance that Paraide’s (2010) study set out to investigate. The study focused on both elementary school teachers who were to

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teach in a local vernacular, and lower primary grade teachers who were being asked to implement the Grade 3 bridging program as students were bridging from experiencing teaching in their vernacular to teaching in English. Elementary schools. At the elementary school site of the study in the Gazelle Peninsula of East New Britain Province, the teachers had varying acceptance of the proposed language reform by which Tinatatuna would be used as the language of instruction and the integration of Indigenous and Western knowledge in the formal learning environment. One teacher seemed comfortable with these teaching strategies, but the other teachers accepted teaching in Tinatatuna, but had difficulties with the integration of Indigenous and Western number knowledge (discussed in Paraide, 2003, 2015). The acceptance of the vernacular teaching strategy was attributed to the fact that these teachers had recently undertaken the elementary teacher-training program, which had as its main focus the use of vernacular teaching through the integration of Indigenouscommunity and Western-formal knowledge. Primary schools. Paraide found that in the three schools in which she worked the lower primary grade teachers were resistant to the new language policy. This was primarily because these reasonably experienced teachers had all undertaken their initial teacher education courses in English, with the expectation that they would be teaching in English. They were unable to embrace a radical change away from their tried and true manner of teaching. These teachers, then, took the professional learning program aimed at introducing the reforms, but they were not yet happy, or confident, to apply the bilingual teaching strategy; nor were they convinced that bilingual instruction was an effective teaching strategy for teaching English language literacy skills. They still believed that the bilingual teaching strategy would not prepare primary school students in English reading comprehension and writing competency to a level that would ensure a high performance on the Grade 8 National Examination—the results on which would determine whether their students would be able to progress to high school. Therefore, a general reluctance was observed in relation to teaching in two languages. The teachers still had the view that English instruction, beginning in Elementary Prep, was the most appropriate teaching strategy. The professional learning program on bilingual teaching that these teachers experienced encouraged the use of pictorial charts that had both Tinatatuna and English labels, but these were noticeably absent from their classrooms. However, an abundance of English-labelled pictorial charts were noted in all classrooms. None of the students’ written work showed activities that were related to bridging. Curriculum documentation. The primary school teachers had in their possession the Lower Primary Syllabus documents developed for the reform (NDOE, 2004). These documents provided justifications for the bilingual teaching strategy. It was found that the teachers had not studied the curriculum documents in any detail. The documents were generally hidden away and rarely used by the teachers. The mathematics syllabus and teachers’ guide embedded within the curriculum documents were, in any case, not sufficient to establish support for the integration of Indigenous and Western mathematical knowledge. There was only one indicator each for the number and measurement learning areas which related to the possible integration of Indigenous and Western measurement and number knowledge. This suggests that the actual integration of Indigenous and Western number and measurement knowledge was given low priority during the development of the documents. The mathematics syllabus, as part of the curriculum documents for the elementary schools, was little better (NDOE, 2003). There was hardly any emphasis in the material on the integration

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of Indigenous and Western mathematical ideas, leaving an overwhelming impression that it was the teaching of the Western ideas which was crucial. It was no surprise, then, that the teachers who embraced the idea of teaching mathematics in English only, had difficulty teaching anything but the kind of formal content that characterizes Western mathematics. Professional learning programs. The professional development in bilingual education to prepare primary school teachers to apply this teaching strategy effectively was minimal before and during the implementation process, and hence did little to balance out the inadequate curriculum document. No wonder Paraide (2010) reported that the primary school teachers appeared to have had a limited understanding of how to embrace the bilingual teaching strategy. In fact, Paraide (1998) and Evans et al. (2006) had argued that the success of the implementation of the new teaching approaches promoted by the reform would be largely dependent on the quality of the professional learning programs conducted both before and at its introduction, and that they would need to continue for some time after that time. They hypothesized that if the teachers had been better informed about the policy, and if they had had mastered sufficient skills in the vernacular and bilingual teaching strategies, then they would have been more willing to implement them. Evans et al. (2006) investigated the success of the “clusters support strategy,” by which selected teachers from a cluster of schools worked together for professional learning and then in turn trained teachers in their own school. This approach was used at the start of the reform. However, Evans found that its success was heavily dependent on committed and effective cluster team leaders, on the ability of the teachers in the school clusters to function as a team, and on the teachers’ view of the quality of professional support provided. He reported that he had found all of these aspects wanting. In her study, Paraide found that hardly any of the teachers who had attended such training trained other teachers in their schools. Amazingly, many of them did not teach at the lower primary level, and, in fact if they did attempt to train their colleagues, they were not sufficiently confident in what they presented to train the other teachers successfully. Not surprisingly the lower primary teachers, as has been noted above, were reluctant to use this approach. In a later study by Paraide (2010), in which participating teachers were involved in their own classrooms, she demonstrated the teaching strategies involved in bridging from Tinatatuna to English. The bilingual teaching strategy she used resulted in the teachers becoming willing to try bilingual teaching, and this resulted in a slight increase in the teachers’ communication with the students in Tinatatuna. By the end of the fieldwork, the lower primary teachers were beginning to see the value of bilingual teaching and how the integration of Indigenous and Western knowledge was possible. The teachers began to acknowledge that the students advanced in their learning faster when they were already competent in the language of instruction—in this particular case, Tinatatuna. The lower primary teachers’ observations of their own students’ positive progress seemed to convince them that, contrary to what they had always generally assumed, using a language which the students knew best as the language of instruction did not hinder the students’ learning, or hold back their cognitive development. Preservice Education Two large Australian Aid projects, the Curriculum Reform Implementation Project (CRIP) and the Primary and Secondary Teacher Education Project (PASTEP) were undertaken at the end of the 1990s and early 2000s (see Chapter 7 and 8 for more details). Both were intended to introduce the reform agenda with its emphasis on vernacular languages and culture into practice. However, an evaluation of PASTEP (Clarkson et al. 2004) suggested otherwise. The relevant result from this evaluation was that mathematics education staff with the responsibility of train-

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ing PNG’s teachers of mathematics saw no reason to deal with language issues at all, let alone the notions embedded in the bridging ideas for Grade 3. For them the main issue was making sure their students understood the mathematics their students would teach, and then dealing with a some of the pedagogical issues associated with teaching mathematics. With respect to understanding mathematics they never considered using Indigenous mathematical ideas or techniques but confined their attention to Western mathematics. An inspection of curriculum guidelines produced for the teachers colleges would not have provided much input for the lecturers (Baing, 1998). The guidelines had a gestation period of four years starting in 1994 and were written when the new syllabuses for schools were also being prepared. However, that hardly got a mention. The one place language and bridging were mentioned was in the mathematics-science strand, was where it was relegated to one of nine possible topics from which lecturers could choose in the one mathematics methodology unit (Baing, 1998, p. 84). The lecturers’ approach was confirmed by the evaluation of PASTEP’s survey results from college students and interviews—two small samples of first-year teachers in schools within a short drive of the colleges were consulted. Neither of the two groups saw the need to consider language issues or Indigenous mathematics in the lecture on this topic in the main mathematics subject. All that would be offered was an elective on ethnomathematical approaches. Indeed, on the question of who should teach the notion of bridging and related issues in the colleges the opinion of the whole staff of the colleges was sought, and not just the mathematics lecturers. The recommendation was that it should be a matter for the Language Department, although there was some support that perhaps those teaching Social Science should also be involved. Most of the college lecturers indicated that any idea that these notions should involve all subject areas, and especially mathematics and science—since they should be applied to all areas of teaching in schools—was far-fetched. There were a couple of exceptions, specifically those who had attended Balob and Madang Teachers Colleges who noted the importance of community and cultural knowledge for school mathematics. In fact, one of these lecturers had designed posters showing her own counting system and other mathematical ideas from her language group. The lecturers in the Language Departments had their own concerns. Their feeling of unease revolved around how to fit more into what was already an overloaded syllabus. Clearly there was no deep understanding of what the notion of bridging implied. There was no evidence that other notions which should have been regarded as applying across the whole of the college curriculum, and hence regarded as important by lecturers in all subject areas, were given due consideration. For example, little attention was given to gender issues. The other key result that concerns the discussion in this book was the poor performance on mathematics tests by the college students. Items on these tests assessed the mathematics of the primary school curriculum. The results suggested that graduating students would not be competent in their Western mathematical knowledge of the curriculum material that they had to teach. The PASTEP and CRIP advisers were developing guidelines and materials similar to those used in Australia. A follow-up professional learning Master of Education for some of the teachers college lecturers, usually leaders, in which Kay Owens was involved in 2003, was based around research methodologies and did not consider the ideas of the reform as a priority, as it was then assumed the lecturers were aware of the reform goals and implementing them. Perhaps this was another missed opportunity although the one paper on women and child abuse, in one province and another on language issues, were used as exemplars.

Concluding Comments

The Government may have again changed the policy back to using English only for schooling purposes after moving away from the reform package, but this will not change the reality of the language environment in PNG classrooms and more specifically, the impacts of local lan-

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guage on learning. Many students will always be speaking their home language fluently before they come to school and meet English for the first time at school as an additional language that they have to learn. The current policy may not allow students to use their local language officially, but the reality is that students will use their fluently spoken local language to think, and, with their peers, to clarify understanding, as they learn English at the same time. The original problems and their possible resolutions that both the Tololo and Matane Reports recognized are still issues that need solving. One of the interesting features of the research reported in this chapter is the comparison between the teachers that Muke worked with, those that Paraide worked with, and the results from the evaluation of PASTEP. Muke’s study in many ways was a natural investigation where he went into classrooms and tried to understand through observation and discussion how the teachers were using language to teach mathematics, and to find out why they were using those particular strategies. Few of these teachers said they were teaching in this way because of reform expectations. They knew they could teach effectively this way—it came naturally. Paraide was more interested in whether the reform as mandated was being undertaken and in the end why it was not influencing the teachers she observed and with whom she talked. In this case, the teachers needed encouragement in terms of professional development to see how mathematics as well as literacy could be taught in vernacular, and indeed how rich the mathematics would then be. Again in the PASTEP Evaluation, Clarkson and colleagues were trying to ascertain why particular reform issues were not at the forefront of the college teaching, even though there was in progress a huge effort to think through with the staff the deeper meaning of the reform. In Paraide’s and in Clarkson’s studies, little was made of what many teachers in the lower grades of schooling in PNG actually did, which in fact was the central point of Muke’s study. Muke found that teachers switched languages when they deemed it was likely to help their students’ understanding. This may point to a bigger issue—it may be that teaching techniques that emerged in Western education do not fit PNG schools as they exist in the developed Westernstyle classrooms. It may be that an Indigenous style of teaching needs to be grown for PNG, borrowing from outside from time to time, but only to enhance those ideas that were PNG Indigenous in informal education and in classrooms with teachers knowing the languages of the students. The commentary during activity, the use of words to assist (not direct) the problem solver learning a new skill, the use of visuals and visual memory, and working alongside the learner. Deconstructing and constructing are also common in everyday gatherings when issues for the community are being discussed. It is likely that many of the approaches used with Australian Indigenous communities (Yunkaporta & McGinty, 2009; Stronger Smarter Leadership Institute, 2017) and Maori communities (Meaney, Trinick, & Fairhall, 2012) will be appropriate for PNG communities. The other interesting issue that has threaded its way through this chapter is the alignment of the various Indigenous mathematics that exist in PNG, with the form of Western mathematics that is taught in schools. There is a general agreement from many studies that using the local Indigenous mathematics to branch out into school Western mathematics is a useful approach. It may be in the lower grades. However, there has been little research on how the differences can be used to strengthen school mathematics except by Owens (2000; Owens et al., 2018), Matang (2005; Matang & Owens, 2006, 2014), and, earlier, by Kemerfield (1983), and recently with the Australian Research Development Award study by Owens and her colleagues. The findings of another study in theWestern areas of Gulf Province (Petterson, 2013), revealed that students who learnt to read in their Tok Ples did best, while those trying to learn in three languages remained illiterate. The differences in meanings of number, as just one example, in the different mathematical systems, seems to have escaped any NDOE Department taking the research seriously. It is probably not enough to say that Indigenous words can just be used, and that they will suffice when in some Indigenous languages the words for number is a real issue. It

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also assumes that teachers are fully familiar with the mathematics embedded in their language, avoiding the real situation that in many PNG cultures not all people are, or need to be, fully aware of the mathematics that is available in their own culture. These are issues still to be resolved. Muke suggests one possible way forward, citing what the New Zealand Maori have done. Elders need to sit down with the mathematics teachers and develop their mathematical dictionaries by first selecting key concepts in both languages and, for those that are not obvious, to establish them in terms of phrases or clauses. The discussion in the Park reverted to discussing teachers and then how for one participant English became virtually his main (only) language for a time, which caused difficulty when he moved back to his village. This raised questions about what it means to have properly learned a language, and the need to code-switch when discussing issues with the community. The following transcript is presented in its unedited form. Michael:

No. Well the teachers that I had were also [English speakers]. I remember my Grade 3, 4, 5, 6 at least. Well in 6 and 5, I had nuns but in 3 and 4 I had Papua New Guinean teachers and one of them, Grade 4, is still there [in my village]. I met him three weeks ago, he has changed of course now and is in a successful business, but he got trained in English, … in teachers college where … I mean he spoke in English and he still speaks in English and it was very good English as far as I could make out at that time. And looking back at him now and having spoken to him not so long ago he speaks excellent English. Owens: Where was he from? Michael: He was from Hagen, and from our community as well. So we knew his family and the community from where he came, but he had come on to teach us. But then we had teachers that came from say the coast or from Madang or New Ireland or Tolai, and they came over to the Highlands as well and they engaged with us and they were very, very articulate in English. And so they taught us, and were also enforced by the fact that we all had to learn English. So, many of us eventually, well not eventually, in school started to cringe when we had to speak in our language in the classroom. We couldn’t. You know we … we really struggled to get our own language out in the classroom. But in the village and outside the classroom we rattled on and rattled on in our own language. And we … eventually after we went to uni, when using my own language, I really struggled. Whereas in English I really—I think got so used to it, more and more and I found it easier to work in English. But then when you are in the village you are expected to communicate in the local language. And when I couldn’t do that it was quite embarrassing when you weren’t fluent in your own language. Owens: That embarrassed you? Michael: Yes. Clarkson: Can I follow that up because there is a bit of a division then starting to happen here in the languages you are using and in your thinking? Michael: Yes. Clarkson: Now, then, I remember in Charly’s (Muke) thesis—and we talked about this quite a bit—where there seemed to be in the end that this was the mathematics that you did at school and this was the mathematics that we did in the village. And there wasn’t all that much overlap. Oh, there was some, but when it came to it, and I can remember we talked about examples that the kids should be using. Should they be using examples in maths that were very much village-based or not? And I think that when Charly was talking to the teachers he was observing, there was an understanding that there were certain things that we teach in the village to the kids; you know our cul-

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ture. But there are certain things that get taught in the school. And there is not necessarily a coming together of all those things for examples in the classroom. Would that be right in your experiences? [Murmurs of agreement from a number of people]. Michael: Yes, well you could say basic things. Yes, but in formal settings you’d be asked to say things in front of a community or sometimes with just some members of the community listening, you would say a few things (in Tok Ples) and then you would … code-switch into … Pidgin which is quite handy. Ah, then, of course you get some rebuking from members of the family saying “You know you’re supposed to speak in Tok Ples so what you doing rattling on in Pidgin?’ [Indistinct but with laughter: “Yes why not in Tok Ples?”] Michael: That’s right. So, you know, you were admonished for not speaking your language properly as you were expected to. So, later on, I found that I had to learn it. So, I made my point now of going back and learning all the languages. So, learning my language fluently so that I can now code-switch at any time I want and I, I really make it my point to speak every word in language. Muke: Depending on the audience you’re speaking to? Michael: Yes, absolutely and deliberately so. Sometimes not code-switching at all. Clarkson: But is it, not just the audience, but, and [emphasized], so this was another reason allied to what Muke and Michael had said, what you are talking about, so there are certain things you would only talk about in language? Michael: Yes, yes, the subject matter. Muke: Yes, it gives a little more meaning and flavor to it than saying it in English. Clarkson: Or than even saying it in Tok Pisin? Muke: Yes. See Phil, yes that is a very good point. But something more. Papua New Guineans, we need to decide what type of education we want. We need to recognize what is in our culture, and therefore what is not there. We have to think about how we can bridge it. We bridge it or we don’t bridge it? What level? This is what we need to do. And the other side of the things is language … is going to die and culture is going to die and disappear. So, this is one of the things we are going to say in this book as much as possible. Michael: Yes, well in my culture it is base 5 [number system] you know (counts in language), and I use that as a basis for my art sometimes in terms of making art you know, and my defence has been that we, from the bilum to many other things we make and do, and how we see and value beauty and then connect it with life and all the rest of it. There is a different way in which this operation happens, and you know. It can’t be just simply an object where there are formal qualities that we have to look at in terms of design and layout and color and contrast and all the rest of it, as happens in Western art. Sure, that operates in that frame of reference, but in another frame of reference the ways in which that’s discussed and deliberated upon is very different. And I am a strong advocate of that now. That is why in education, later on in university, I support it. Like ethnomathematics—the rise of it, and the research support of it—because I could see the value of what we were doing. Then what’s being done by this kind of thing? We need to protect our systems if you like, and we need to articulate and advocate it more. Carol: The challenge now would be for the population—well a lot of them are at least bilingual speaking, Tok Pisin and English now, and you would not find a lot of them speaking in their mother tongue because we have a lot of mixed marriages as well.

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And many children will be coming into the classroom speaking with … their Tok Pisin and English. And this is where we need to find a space for the connection, to connect what they know in their own environment, which is basically bilingual. Clarkson: Are you talking about villages out there or urban areas? Carol: Urban areas. Clarkson: Yes, that has been so for a long time in urban areas, but it is still an important issue. Yes, but what about out in the villages? Carol: Well, a lot of them are speaking in Tok Pisin too. Clarkson: OK. Owens: Yes the Eastern Highlands has people from so many provinces it tends to have, well Tok Pisin has really taken over the languages in Eastern Highlands. Probably around Madang as well since there are so many languages. But everyone still speaks Hagen in Hagen. Michael: Yes, but they code-switch with Pidgin. We were out in the region three weeks ago. Accessibility out there was great 30 to 40 years ago but now there are no roads, no nothing. You’ve got to get in by MAF [Christian sponsored airline, flying one-propeller aircraft into remote communities] once a week if you are lucky—If the weather’s good, and if there are passengers going in or coming out. Anyway, we went there and we stayed there for a week and … then we went to other communities. Interestingly, there is a big contrast between the two communities in how language operates. You can code-switch very quickly between my Melpa dialect and Pidgin, but mostly 75% of the time the talking was in Pidgin whereas in the first place 75% of the time the talk was in language. And there it is only in a certain proportion of the population, that was either say 25 years and down, that were code-switching. But 30 and over were preferring to operate in their own language, their own vernacular. And you see that there was a very stark contrast, whereas in urban areas with Melpa where everyone speaks predominately in Pidgin but they can code-switch back into language although most of the time it is in Pidgin. And so, I made the point in the meetings when I said you know they have four or five species of bananas there. And I said OK what’s this one? And in each of their languages they could name each one of the species, and they can tell you why that is like that and why and how. But in Pidgin it is only a banana, banana, banana, …

It seems that the use of Tok Ples and Tok Pisin is alive and well in the villages, for village life. Teaching literacy was not going to be easy as the local language and Tok Pisin were usually used for oral communication, with English being valued and taught often alongside other languages for literacy—although in elementary schools the Tok Ples should have been paramount. A new phenomenon for literacy for villages without books is text messaging (Schneider, 2016). Even though the Government’s policy has now changed back to an “English only” policy for the language of teaching, the students will always speak their local language and it will have an impact on their learning. The students will use their local language(s) as a learning resource for all their subjects, including mathematics, whether it is allowed at the policy level or not. Good teachers know this. Hopefully they will continue to utilize this fact for the enhancement of the learning environments of their students. Perhaps it is a real loss that it took the country 20 years to adopt Tololo’s goals for education in terms of beginning school in local language. In that time, many had lost their language and many others succumbed to using Tok Pisin in the place of Tok Ples.

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Moving Forward

Despite the loss of languages, the degree of usage of Tok Pisin, the need for time and discussion to talk about mathematical concepts in Tok Ples, there will be a time when the country will appreciate not only its own ways of thinking mathematically but also its ways of expressing the concepts in a language that is more akin to the people’s cultural and social and ecological ways of thinking. The identity that is strengthened by language will also assist with valuing mathematics, both Western and Indigenous. By recognizing the language of mathematics, there may be a greater impetus to record and use the more complex Indigenous mathematical knowledges. It is realized that some of the ways of thinking might not be easily expressed in Tok Ples, and in that respect the use of visuospatial reasoning has a role to play, as will be discussed in the next chapter. References Adler, J. (2002). Teaching mathematics in multilingual classrooms. Dordrecht, The Netherlands: Springer. Ascher, M. (2002). Mathematics elsewhere: An exploration of ideas across cultures. Natural History, 112(7), 2–9. Baing, S. (Ed.). (1998). National curriculum guidelines for the Diploma in Teaching (Primary). Port Moresby, PNG: Staff Development and Training Division, NDOE, Papua New Guinea. Baker, C. (2006a). Cognitive theories of bilingualism and the curriculum. In C. Baker (Ed.), Foundations of bilingual education and bilingualism (4th ed.) (pp. 167–186). Clevedon, England: Multilingual Matters. Baker, C. (Ed.) (2006b). Foundations of bilingual education and bilingualism (4th ed.). Clevedon, England: Multilingual Matters. Baker, W. (2016). English as an academic lingua franca and intercultural awareness: Student mobility in the transcultural university. Language and Intercultural Communication, 16(3), 437–451. https://doi.org/10.1080/14708477.2016.1168053 Barwell, R. (Ed.) (2009). Multilingualism in mathematics classrooms: Global perspectives. Clevedon, England: Multilingual Matters. Barwell, R. (2016). Mathematics education, language and superdiversity. In A. Halai & P. Clarkson (Eds.), Teaching and learning mathematics in multilingual classrooms: Issues for policy, practice and teacher education (pp. 25–39). Rotterdam, The Netherlands: Sense Publications. Barwell, R., Clarkson, P., Halai, A., Kazima, M., Moschkovich, J., Planas, N., Phakeng, M., Valero, P., & Villavicencio, M. (Eds.) (2015a). Mathematics education and language diversity (The 21st ICMI Study). Dordrecht, The Netherlands: Springer. Barwell, R., Clarkson, P., Halai, A., Kazima, M., Moschkovich, J., Planas, N., Phakeng, M., Valero, P., & Villavicencio, M. (2015b). Introduction: An ICMI study on language diversity in mathematics education. In R. Barwell, P. Clarkson, A. Halai, M. Kazima, J. Moschkovich, N. Planas, M. Phakeng, P. Valero, & M. Villavicencio (Eds.), Mathematics education and language diversity (The 21st ICMI Study: pp. 1–22). Dordrecht, The Netherlands: Springer. Beach, D. (2003a). Mathematics goes to market. In D. Beach, T. Gordon, & E. Lahelma (Eds.), Democratic education ethnographic challenges (pp. 99–122). London, England: Tufnell Press. Beach, D. (2003b), The politics, policy and ideology of school mathematics. In G. Walford (Ed.), Investigating educational policy through ethnography (Studies in Educational Ethnography, Vol. 8, pp. 1–15). Bingley, England: Emerald Group Publishing. Beata, U., & Kim, N. (2014). Tentative grammar description for the Waima language. [Working paper]. Ukarumpa, Papua New Guinea: Summer Institute of Linguistics. Bhabha, H. (1995). The location of culture. London, England: Routledge.

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DÀmbrosio, U. (2001). What is ethno-mathematics, and how can it help children in school? Teaching Children Mathematics, 7(6), 308–310. D'Ambrosio, U. (2006). Ethnomathematics: Link between traditions and modernity (M. Borba, Trans.). Rotterdam, The Netherlands: Sense Publishers. Dawe, L. (1983). Bilingualism and mathematical reasoning in English as a second language. Educational Studies in Mathematics, 14, 325–353. de Abreu, G. (2002). Towards a cultural psychology perspective on transitions between contexts of mathematical practices. In G. de Abreu, A. Bishop & N. Presmeg (Eds.), Transitions between contexts of mathematical practices (pp. 170–189). Dordrecht, The Netherlands: Kluwer. de Abreu, G., Bishop, A., & Presmeg, N. (Eds.). (2002). Transitions between contexts of mathematical practices. Dordrecht, The Netherlands: Kluwer. de Abreu, G. & Cline, T. (1998). Studying social representations of mathematics learning in multiethnic primary schools: Work in progress. Papers on Social Representations, 7(1–2), 1–20. Delpit, L., & Kemelfield, G. (1985). An evaluation of the Viles Tok Ples Skul scheme in the North Solomons Province. ERU Report No. 51. Port Moresby, PNG: University of PNG. Department of Education Papua New Guinea. (1974). Report of the Five-Year Education Plan Committee, September, 1974 (The Tololo Report). Port Moresby, PNG: Author. Dienes, Z. P. (1969). Building up mathematics. London, England: Hutchinson Educational. Edmonds-Wathen, C., & Bino, V. (2015, July). Changes in expression when translating arithmetic word questions. Paper presented at the Annual conference of International Group for the Psychology of Mathematics Education, Hobart, Australia. Edmonds-Wathen, C., Owens, K., & Bino, V. (2019). Identifying vernacular language to use in mathematics teaching. Language and Education, 33(1), 1–17. https://doi.org/10.1080/095 00782.2018.1488863 Educational Studies in Mathematics. (2007). Special Issue on Language and Mathematics. 64(2). Ellerton, N., & Clarkson, P. (1996). Language factors in mathematics teaching and learning. In A. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp.987–1034). Dordrecht, The Netherlands: Kluwer. Evans, T., Guy, R., Honan, E., Kippel, L. M., Muspratt, S., Paraide, P., & Tawaiyole, P. (2006a). PNG Curriculum Reform Implementation Project—Impact Study 6. http://hdl.handle. net/10536/DRO/DU:30010511 Flagg, V. (2014). Newman's error analysis and mathematical language: Diagnosing mathematical errors on word problems made by my 4th graders who attend a low SES school. Ph.D. thesis, Mercer University, Macon, GA, USA. Haddon, A. C., Balfour, H., Marr, J., Ray, S., & Seligmann, C. (1906). Anthropogeographical investigations in British New Guinea: Discussion. The Geographical Journal, 27(4), 365– 369. https://doi.org/10.2307/1776236 Halai, A., & Muzaffar, I. (2016). Language of instruction and learners’ participation in mathematics: Dynamics of distributive justice in the classroom. In A. Halai, & P. Clarkson, (Eds.), Teaching and learning mathematics in multilingual classrooms: Issues for policy, practice and teacher education (pp. 57–70). Rotterdam, The Netherlands: Sense Publications. Joseph, G. (1987). Foundations of Eurocentrism in mathematics. Race and Class, 28(3), 13–28. Joseph, G. (1991). The crest of the peacock: Non-European roots of mathematics. Princeton, NJ: Princeton University Press. Kemerfield, G. (1983). Innovation and training in policy-oriented research: The North Solomons education research project. In G. Guthrie & T. Martin (Eds.), Directions for education research (pp. 173–185). Port Moresby, PNG: Faculty of Education, University of Papua New Guinea. Kim, D., & Kim, N. (1998). Waima grammar essentials. Ukarumpa, Papua New Guinea: Summer Institute of Linguistics.

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Kirsch, C. (2018). Young children capitalising on their entire language repertoire for language learning at school. Language, Culture, and Curriculum, 31(1), 39. Kopamu, S. (2005). Mathematical concepts in Enga Tokples. Papua New Guinea Journal of Mathematics, Computing and Education, (7)1, 13–18. 55. https://doi.org/10.1080/079083 18.2017.1304954 Lean, G. A. (1992). Counting systems of Papua New Guinea and Oceania. PhD thesis, Papua New Guinea University of Technology, Lae, PNG. Litteral, R. (2001). Language development in Papua New Guinea. Radical Pedagogy, 3(1), http:// radicalpedagogy.icaap.org/content/issue3_1/01Litteral002.html. Matang, R. (2005). Formalising the role of Indigenous counting systems in teaching the formal English arithmetic strategies through local vernaculars: An example from Papua New Guinea. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce & A. Roche (Eds.), 28th Conference of Mathematics Education Research Group of Australasia, Melbourne (pp. 505–512). Adelaide, Australia: MERGA. Matang, R., & Owens, K. (2006). Rich transitions from Indigenous counting systems to English arithmetic strategies: Implications for mathematics education in Papua New Guinea. In F. Favilli (Ed.), Ethnomathematics and mathematics education, Proceedings of the 10th International Congress on Mathematical Education Discussion Group 15 Ethnomathematics. Pisa, Italy: Tipografia Editrice Pisana. Matang, R., & Owens, K. (2014). The role of Indigenous traditional counting systems in children’s development of numerical cognition: Results from a study in Papua New Guinea. Mathematics Education Research Journal, 26(3), 531–553. https://doi.org/10.1007/ s13394-013-0115-2 McConvell, P., & Thieberger, N. (2001). State of Indigenous languages in Australia—2001 (Australia State of the Environment Second Technical Paper Series—Natural and Cultural Heritage). Canberra, Australia: Department of the Environment and Heritage. Meaney, T., Trinick, T., & Fairhall, U. (2012). Collaborating to meet language challenges in indigenous mathematics classrooms. Gewerbestr, Switzerland: Springer. Muke, C. (2000). Ethnomathematics: Mid-Wahgi counting practices in Papua New Guinea. M.Ed thesis, University of Waikato, Hamilton, NZ. Muke, C. (2012). The role of students’ language in learning mathematics when teachers change the teaching language from a vernacular to English: The Papua New Guinea experience. PhD thesis, Australian Catholic University. NDOE, Papua New Guinea. (1986). Ministerial committee report: A philosophy of education for Papua New Guinea (chairperson P. Matane). Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2002). The state of education in Papua New Guinea. Port Moresby, PNG: Education Reform Facilitating and Monitoring Unit, NDOE, Papua New Guinea. Accessed on 1/7/2020 at http://www.paddle.usp.ac.fj/collect/paddle/index/assoc/ png038.dir/doc.pdf NDOE, Papua New Guinea. (2003). Cultural mathematics: Elementary syllabus (and teaches’ guide). Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2004). Mathematics: Lower primary syllabus (and teacher guide). Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1990). PNG strategic plan. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1991). Education sector study. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1995). Education resource study. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1996a). Papua New Guinea national education plan, 1995–2004, Volume a. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1996b). Papua New Guinea national education plan, 1995–2004, Volume b. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1997). Elementary handbook. Port Moresby, PNG: Author.

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NDOE Papua New Guinea. (1999a). Language policy in all schools, Education Circular. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1999b). National education plan (1995–2004). Update 1. Port Moresby, PNG: Author. Newman, A. (1983). The Newman language of mathematics kit. Sydney, Australia: Harcourt, Brace, & Jovanivich. Owens, K. (2000). Traditional counting systems and their relevance for elementary schools in Papua New Guinea. PNG Journal of Education, 36(1 & 2), 62–72. Owens, K. (2015). Visuospatial reasoning: An ecocultural perspective for space, geometry and measurement education. New York, NY: Springer. Owens, K., Edmonds-Wathen, C., & Bino, V. (2015). Bringing ethnomathematics to elementary teachers in Papua New Guinea: A design-based research project. Revista Latinoamericana de Etnomatematica, 8(2), 32–52. http://www.revista.etnomatematica.org/index.php/RLE/ article/view/204 Owens, K., & Kaleva, W. (2008a). Case studies of mathematical thinking about area in Papua New Guinea. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Annual Conference of the International Group for the Psychology of Mathematics Education (PME) and North America chapter of PME, PME32—PMENAXXX (Vol. 4, pp. 73–80). Morelia, Mexico: PME. Owens, K., & Kaleva, W. (2008b). Indigenous Papua New Guinea knowledges related to volume and mass. Paper presented at the International Congress on Mathematics Education ICME 11, Discussion Group 11 on The Role of Ethnomathematics in Mathematics Education, Monterrey, Mexico. https://researchoutput.csu.edu.au/en/publications/ indigenous-papua-new-guinea-knowledges-related-to-volume-and-mass Owens, K., Lean, G. A., with Paraide, P., & Muke, C. (2018). The history of number: Evidence from Papua New Guinea and Oceania. New York, NY: Springer. Paraide, P. (1998). Elementary education: The foundation of education reform in Papua New Guinea. Papua New Guinea Journal of Education, 34(1), 1–18. Paraide, P. (2002). Rediscovering our heritage: An early assessment of lower primary learning in the reform curriculum (DER Report number 76). Boroko, Papua New Guinea: National Research Institute. Paraide, P. (2003). What skills have they mastered? Paper presented at the conference on National Education Reform: Where Now, Where to? Goroka, Papua New Guinea. Paraide, P. (2008a). Number in the Tolai culture. Contemporary PNG Studies: DWU Research Journal, 9, 69–77. Paraide, P. (2008b). The progress of vernacular and bilingual instruction on formal schooling, Spotlight with NRI, 2(4). Paraide, P. (2009). Vernacular languages and the systems of knowledge embedded in them. Port Moresby, PNG: National Research Institute. Paraide, P. (2010). Integrating Indigenous and Western mathematical knowledge in PNG early schooling. Doctoral thesis, Deakin University. Petterson, R. (2013). The vernacular factor in literacy in West Gulf Province schools in PNG. Paper presented at AUSAID Workshop Conference on Language and Literacy Work in Papua New Guinea. Digital Resources SIL Org Electronic Working Paper 2013–001. https:// www.researchgate.net/publication/269702665_The_Vernacular_Factor_in_Literacy_in_ West_Gulf_Province_Schools_of_PNG Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. London, England: Routledge. Prediger, S., Clarkson, P., & Bose, A. (2012, August). A way forward for teaching in multilingual contexts: Purposefully relating multilingual registers. Paper presented to Topic Group 30: Language and Communication in the Mathematics Classroom, International Congress on Mathematics Education (ICME12), Seoul, Korea.

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Chapter 11 Visuospatial Reasoning, Calculators and Computers

Abstract: Visuospatial reasoning has become a well-researched theme within mathematics and mathematics education, and there is evidence that it is often used in mathematical contexts in Papua New Guinea (PNG). This chapter provides a review of related research in Papua New Guinea which has been conducted over the past 40 years, and discusses the diversity of contexts aspects, which points to the importance of visuospatial reasoning including its relationships to some language aspects within the nation. The chapter also considers the position of information communication technologies (ICTs) within PNG. In fact, in the 1980s the PNG University of Technology situated itself so far as ICT was concerned, ahead of many universities around the world. In more recent times however, it has been hard for the nation to keep pace with advances elsewhere, stemming from the cost of internet services from advances in teaching with technology. Nevertheless, despite the nation’s relatively small population, and the impact of that on the cost of internet services there have been numerous projects and other advances in terms of ICT within the nation. This chapter will focus particularly, on PNG research on visuospatial reasoning in mathematics, mathematics education and computer education. Key Words: Alan Bishop · One laptop per child · One laptop per teacher · Spatial thinking · String figures · Teacher professional development in remote areas · Visualization · Visuospatial reasoning · Teaching mathematics and statistics with computers

Technology, computers and calculators, engage students in problem solving and visualising. Students receive, process, and retain knowledge during learning. Cognitive, emotional, and environmental influences, as well as prior experience, all play a part in how understanding, or a world view, is acquired or changed and knowledge and skills retained. Computers and technology assist students to construct knowledge, concepts and physical objects. The parts make up the whole like a kit of Lego. The role of the teacher is to create the conditions for invention rather than provide ready-made knowledge. Computer programming is the most powerful medium of developing the sophisticated and rigorous thinking needed for mathematics, for grammar, for physics, for statistics, for all the "hard" subjects. ... In short, I believe more than ever that programming should be a key part of the intellectual development of people growing up. Paraphrasing Seymour Papert

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Introduction Although it is well known that language factors affect the learning of mathematics, there has been less attention given to how visuospatial reasoning has an impact on mathematical thinking. In this chapter the term “visuospatial reasoning” includes spatial thinking, spatial capabilities, physical and mental visual representations, and types of visualising. In Chapters 2 and 3 we discussed foundational mathematics, that is to say, the mathematics which was part of cultures from earliest times until the present, and intimated that visuospatial reasoning was always important in that regard. For example, there have been a number of studies associated with the creation of string figures and artefacts, and with how people located places and found their ways across the seas or jungle. This chapter will relate some of the first contact commentaries to research conducted in the 1970s which followed a particular interest in spatial abilities around the world, and to the more recent worldwide interest in spatial capabilities and reasoning, especially in recent PNG research. Some of that research has particularly focused on geometry, but visuospatial reasoning will be seen to be relevant to other aspects of mathematics as well. In particular, it will be argued that mathematical thinking associated with ecocultural contexts is significantly influenced by visuospatial and other kinds of reasoning.

Highlights of Foundational Mathematics Visuospatial Reasoning

Mikloucho-Maclay (1975) recorded extraordinary details of the artefacts of the Madang groups with whom he lived in the late 19th century, as well as many other aspects of their lives. He also studied skills such as navigating. Mennis (2014) used old records and experience to continue the studies into navigation skills as well as trade, especially with pottery made in Madang. The Haddons (A. C. Haddon, 1900, 1904; K. Haddon, 1930, reprinted in 1979) studied and wrote about the visual acuity and string-figure abilities of Papua New Guineans as well as Torres Strait Islanders (Australia) in the first decades of the 20th century. However, given that these mathematical activities in many cases have continued into the present (some modifications and loss has occurred as a result of colonization), it is not surprising that there are modern studies on this subject by Papua New Guineans themselves as well as by others (see, e.g., Vandendreissche, 2007, 2015). Some activities require a physical representation of what is in a person’s mind, what is being visualized, and how the thoughts can be associated with visual images, words and actions. These images, words and actions give rise to spatial representations of objects, places and activities. The extensive degree of creativity, not necessarily featuring written words, has been particularly noticeable within various PNG communities. This creativity has required a high level of visuospatial reasoning, and the first studies in PNG in this direction were undertaken by expatriates well versed in educational psychology studies related to visuospatial reasoning—in particular, the research reports of Alan Bishop, Glen Lean, Ken Clements and Kay Owens were based on research carried out while those researchers were based within the PNG University of Technology (Unitech). Their openness to observing and learning from participants in studies was in marked contrast to the comparative studies of others, especially those that were focusing on Piagetian stage theories—such as those undertaken by the Education Research Unit at UPNG (see, for example, Price, 1978; Shayer, 2003; Shea, 1978). Studies Conducted in the 1970s which Focused on Papua New Guinea Students’ Learning Educational psychology in the 1970s was influenced by formal testing of abilities, creativity, and various aspects of intelligence. Theories put forward by Guildford (1966), Thurstone and Thurstone (1941), and others, were popular, and widely referenced around the world, as was Piaget’s stage theory and associated clinical interviews, which incorporated testing using care-

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fully designed items based on a theoretical model. There were also suggestions that some people might have strong abilities in verbal or visual learning and memory. Piaget developed models in relation to spatial abilities (Piaget & Inhelder, 1956) and mental imagery (Piaget & Inhelder, 1971). Information processing theorists such as Paivio (1986) were divided on the issue of whether ideas were stored in memory in verbal or visual forms, or both, no matter whether the initial input was verbal or visual. In PNG, Rosanne Smith (1973) carried out a study of pictorial depth perception with 28 male and 23 female students from Grades 3 to 6; 10 boys and 9 girls in Forms 1 and 2 from high schools; and 3 schooled male adults and 4 schooled female adult Government workers, in the Ialibu District of the Southern Highlands Province, which was quite remote at that time. Smith modified a test developed for use in Africa so that it presented representations of PNG pigs, a man and a cow. A series of questions were asked about each picture to identify the relationship of the animals and man, with mountain slopes and size being relevant. Two other tests, Kohs Block-Design Test (1920) and a WISC block design test (Weschler, nd), used models and pictures of models with blocks carefully positioned so that only one or two colored sides were showing when they were first presented. T-test comparisons on all aspects were made with 48 Tasmanian subjects, and some statistically significant differences were reported, with the Tasmanian subjects doing better on all items compared to all three PNG groups—although the schooled adults obtained similar results. That analysis suggested that schooling could play a significant role in learning in relation to the visualization of three-dimensional objects. However, the sample sizes for this study were small. Around the same time, Saunderson (1973) was interested in whether the use of concrete materials with post-secondary students in Papua New Guinea would have an impact on their spatial abilities. In his training study he used two-dimensional (2D) and three-dimensional (3D) activities like tangrams, pentominoes, and tile shapes; the tests covered both 2D and 3D areas. Analysis of the Form Board test data and of performance on the activities that Saunderson devised suggested that students improved their spatial skills by improving their analytical skills. The Mathematics Learning Project and the Mathematics Education Centre at the PNG University of Technology brought a number of mathematics lecturers from overseas to prepare appropriate materials for the students and also to research PNG students’ learning. There was an interest in identifying the best ways to teach PNG students the mathematics they would need to know if they were to be better placed to assist others in a new independent country. Importantly, the University put research into practice by attaining appropriate equipment such as tetris 3D cubes, equipment to simulate harmonic motion, vectors (using pulleys and weights hanging from a pinboard), binomial theorem equipment, Dienes multi-attribute and multibase arithmetic blocks, measuring equipment, and programs which drew graphs and represented situations found in real life. Figure 11.1 illustrates how the research was linked into the university course development, and also indicates the areas of research being undertaken. Glen Lean and Lesley Booth (Booth, 1975; Lean, 1975a) undertook an extensive study of spatial abilities of school students, since it was thought that this was one area of cognition for which PNG students might outperform students in Western countries. After modifying seven educational psychology spatial tests with connections to mathematics and visualisation1 and after converting them to a group test appropriate for PNG, they conducted trials at Lae High School and at the University (Unitech). They also administered the tests over the course of a week to 489 students (239 males and 150 females) in high schools located in five New Guinea Islands Region’s high schools (Forms 1 to 4, that is Grades 7–10). They found that on all tests there were significant differences between male and female students, always favouring the males. Test results correlated fairly well, but correlated with age for the males only on one test (predicting geometric Cross-sections of 3D objects, nets of 3D objects, cut and rearrange cloth to make a square, directions on a grid, comparing areas of 2D shapes when made with same perimeter, comparing 2D shapes by sight, walks that give different loci. 1

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Source. Lean (1975b). Table 1, p. 5 Figure 11.1. The Mathematics Learning Project sections); for the females on three of the tests (predicting geometric sections, nets of surface of solids, and loci of points). Data from the test on predicting geometric sections correlated with data from each of the other tests, and Lean and Booth considered it to be the best test for measuring spatial abilities. A study on conservation of area (involving the comparison of areas of non-congruent rectangles which had the same perimeter) tried to assess Piagetian levels. Data suggested that in New Guinea there was more diversity of performance and not such a clear developmental trend as there was in Geneva. Rearranging parts of a square and asking about area for both arrangements suggested that PNG students, like many non-Western students, were not as advanced as Genevan

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students so far as conservation of area was concerned. It was suggested that the results pointed to a need to emphasize spatial thinking in the school curriculum, and that teachers needed to be made aware of the problem (see David Shields’ cameo in Chapter 7). Fredericks (1981) particularly recognized the difficulties with some items on examinations in which diagonal or oblique lines were to be considered equal in length to a horizontal line (in 2D representations of 3D objects) even though this was not the case for a representation of 2D figures—he also pointed to the importance of language issues. Visualisation and Spatial Abilities Research In 1979 Alan Bishop visited PNG from Cambridge University, and was based in the Department of Mathematics for more than six months. In interviews with Unitech students Bishop found that although they struggled when asked to interpret images in photographs, it did not take long for the students to learn to “understand the photographs.” That did not surprise Bishop because, after all, very few of the students owned a camera, and most of them had not had much access to magazines and books which contained photographic images. He concluded that the students were not used to interpreting pictures or interpreting photographs of people and objects, and conjectured that there were two different visualizing capabilities: interpreting figural representations, and visualisation (Bishop, 1979). Later, Glen Lean would review studies on spatial abilities involving 3D objects and on the impact of training (Lean, 1984). Like most people who have taught in PNG, Bishop was challenged by his experiences in PNG. Over the next decade he would write his ground-breaking book, Mathematical Enculturation (Bishop, 1988). However, he was not happy with the idea that this book might suggest that Western school mathematics was expected to be the pinnacle of everyone’s mathematical learning. Collaborating with Guida de Abreu and Norma Presmeg, he was able to write and edit a book which included examples from many places showing how students could maintain their own culture and cultural mathematical knowledge and how these could be integrated with Western mathematics (de Abreu, Bishop, & Presmeg, 2002). Mathematics educators who refer to his first work in which he noted six mathematical activities are strongly recommended to read the second book which discusses the importance of ethnomathematics along with school mathematics. There should not be a duality or a simplifying of ethnomathematics to fit the concept of school mathematics. Enculturation into Western school mathematics is not the goal but rather an understanding of ones’ own culture and the mathematics of both—with a synergy that comes from growing and learning about both. Part of the research and discussion around information processing related to whether people’s problem solving and mathematics were related to whether they were processing stimuli visually or analytically/verbally. Krutetskii (1976) had suggested analytical thinking was necessary for good problem-solving abilities. This was the background to Ken Clements’ visit in 1980 and he, working with Glen Lean, provided an extensive critical review of the literature on whether students could be classified as visualizers, analytic thinkers (verbalizers), or “harmonic” thinkers (that is to say, those who might choose to use either, depending on their reaction to the item context (Clements, 1983; Krutetskii, 1976). Lean and Clements critiqued the interpretation of research by Moses (1977), who argued that verbal/analytic approaches were best. Clements (1983) noted that spatial problems might provide diagrams, or ask for diagrams to be drawn. Clarkson (1993) suggested that Australian students often ignored diagrams in questions, even when questions specifically directed their attention to them. Others suggested visualizing was helpful, but the research was not systematic and relied somewhat on what the students tried to do. Clements’s Indonesian student, Stephanus Suwarsono (1982), however, developed a problem-solving test that also tested for mental processing and generated a “visuality score,” which measured the extent to which a student processed items using visual rather than verbal

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analysis. At Monash University, Nongnuch Wattanawaha2, working with Ken Clements, developed a way of classifying spatial tasks, and used that classification scheme to analyze responses to spatial items (Wattanawaha & Clements, 1982). Webb (1979) later indicated that a pictorial representation accounted for a sizeable proportion of variance in scores when pretest scores were taken into account. Another of Clements’ students, Thomas Lowrie3 (1994), found that students, especially good students, would choose the most appropriate way of thinking about a particular problem or item, and often that required visual reasoning. Some items required a pictorial visual image to be drawn as a place to start (Paivio, 1971, 1986). Though these studies were carried out in many different countries, the PNG research on spatial abilities and visuality provided worldwide leadership for studies in this area of cognition. Lean and Clements (1980, 1981) investigated the extent to which 116 Engineering entry students at PNG University of Technology preferred to process mathematics tasks by using visual methods (as opposed to verbal/analytic methods). Independent variables were measured by four different spatial tests, a drawing test, and an adaptation of Suwarsono’s problem-solving processing questionnaire, and dependent variables were the students’ university mathematics tests (one routine pure mathematics test involving algebra, trigonometry and vectors, and a second test on applied elementary mechanics). The validity of the analytical-visuality questionnaire for this current group of students was checked against a group of Australian Grade 7 students (and a correlation of 0.9 was obtained) and by interviewing 10 students. The levels of success on the pure mathematics test was best predicted by the analytical-visuality processing score, the only independent variable that provided a significant prediction. Visuality processing emerged as a unique factor, separated from other spatial factors, in a factor analysis, but all of them correlated. None correlated significantly, however, with the applied mathematics test. Lean and Clements’ explanation for lack of predictability was that the dependent variable questions were fairly straightforward word problems, unlike those used in other studies. Further studies assessed students in senior high schools, with an additional check of adding a bonus mark for correct solutions being applied. However, this did not change results (Clarkson, 1981, 1982). Clements and Lean (1981) also undertook a study of relationships between mathematical thinking, visual memory, and spatial reasoning with school children from three grades, with three groups of children with different backgrounds. Two schools in the Highlands Region and two in the New Guinea Islands Region were involved—altogether, there were 45 children in Grade 2, 33 in Grade 4, and 36 in Grade 6. The second group were nationals who were attending an International school4 (7 in Grade 2, 5 in Grade 4, and 5 in Grade 6). A third group, comprised expatriate students in the same international school (n = 42, 48, 26 respectively). Some students were interviewed and given additional tasks. The tests covered visual memory, an understanding of spatial conventions, coordination of perspectives, simple mathematical problems, conservation tasks, and estimations of length, area and quantity. Tests included remembering the positions of objects on a grid which had been covered, recognition of a face, and identifying 3D objects (which could be inserted in a hollow, toy plastic ball (from Tupperware) in which the differentshaped blocks could be fitted through appropriately-shaped holes), modelling from a line figure, photographs of 3D shapes, and viewing an object from different perspectives. The grade levels of children in all groups correlated with their performances, but there was little difference in the performances between the groups. Performance on a memory-for-design task was better for the group with less Western influence, but that group did not do so well on the Clements’ experiences in PNG, India, and other parts of Asia enabled him to attract many graduate students from Asian nations. Many of them are now leaders in mathematics education in their own countries. 3 Lowrie, like Owens, has continued to carry out research in visualisation. He is now Professor of the STEM Education Research Centre, at the University of Canberra. 4 It is expensive to send children to this school. It is likely that only parents with careers requiring educational qualifications could afford the books which the students were required to have. 2

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auditory tasks (measured by digit span). Results of conservation tasks were quite varied for all groups, but overall the students did well, except that one-third of one community school group found a conservation of substance task quite difficult. Nearly all students could select a pentagon as the odd shape when it was presented with quadrilaterals of various kinds. However, their explanations generally favoured reference to its shape rather than the number of sides. Only 27% of the community school subjects, compared to 46% of the international school subjects, mentioned the abstract idea of “number of sides.” Although the community school students in lower grades did not do so well on the fittingshapes task, by the end of school they performed at a similar level to the international school students on that task—but note that their ages were also, on average, higher. Interestingly, in interpreting the line drawings, most of the community school subjects in all grades produced two-dimensional responses when asked to “make what they saw” in the diagrams. The international primary school students in all grades and in both groups produced three-dimensional responses, which supported Bishop’s findings on an “Interpreting Figural Representations” variable. Although all interviewed students were able to select a perspective, the younger students in all groups and the community school students in higher grades, did not find it easy to place photographs of a complex model in the order in which the photographs might have been taken given that the photographer had moved systematically around the model. In one larger group interviewed at one of the community schools, the girls did not do as well as the boys on that task, but that was also the case for the mathematics test suggesting that the girls may have been at school to equalize the number of female students rather than by selection of the brighter girls. With the group tests, it was clear that the language of mathematics was the main challenge for the students particularly, with polarized comparative language terms like “more” and “less,” and items on time (for which terms like “hours before,” “later,” or “between given hours” caused difficulty, especially with Grade 4 students—although Grade 6 students did better. One major difference between community school children and international school children was the amount of homework the students would have done—this could have amounted to 1000 hours over the six years including much more reading at home. A lack of opportunity to see pictures in any format or creative art works was evident for the community school children, and most of them only heard English spoken in the classroom. All of them indicated that mathematics to them was related to computation, although a couple mentioned blocks (Dienes’ attribute blocks were available in both kinds of schools). All groups did better on remembering objects than numerals or symbols, and even then they gave the objects names to remember. The researchers suggested results indicated no cultural differences in basic cognitive processing although the community school children had better memory of shapes. Visual memory was an advantage for the younger children, and could continue to be strong as Bishop had found with PNG university students who had English as a second language. Clements and Lean also suggested that the mathematics curriculum should take account of the students’ backgrounds and not be directed to high language processing or diagram reading without earlier experiences in mathematics with appropriate problem solving having been provided. Both the teachers’ and students’ confidence with higher mathematics needs to start with appropriate levels, to avoid the rote learning of facts and procedures. Except at one community school, none of the children were aware that their own results in the examination would help select them for high school—they just assumed they were unlikely to go to high school, reflecting a social phenomenon which undoubtedly affected student motivation with respect to their schooling. The issue of knowing and interpreting representations of mathematical concepts was also evident in a study by Ausburn and Ausburn (1980), who noted the number of children who made a two-dimensional representation for a three-dimensional object and were unaware of the conventions of perspective and scale. In addition, they found that students were unable to select an image slightly different from another before they were trained to analyse diagrams systemati-

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cally—and then they could do well. In everyday life in the rural areas, this kind of detail may not be missed. A follow-up study on a range of tests which they linked to learning styles but some of which were generally used for spatial reasoning studies generated some interesting results. Results on a test assessing recognition of embedded figures indicated that university students were able to complete these at a similar level to Australian trade apprentices, but not as well as American college students. In a remote village where children did not generally draw, the embedded figures and matching familiar figures were completed equally well by three community school children and three children who did not attend the school but had had some other minimal schooling. However, the school children were more concerned about whether they were correct or not. Another extensive study on spatial reasoning was carried out by Lean (1982) with 389 students from five secondary schools in the New Guinea Islands Region. Following a thorough critical analysis of the studies in the area, Lean administered seven tests to the students. He particularly considered the development of surfaces of solids in an 8-item test requiring students to draw the surfaces of the eight solids. He found the results were similar to Faw’s (1977) results in USA rather than Piaget’s Belgium results based on data gained in individual interviews. Over the grades, students improved on all shapes gradually but not with any significant jump to the next grade results. The boys did better than girls in all grades but without any one significant jump. Lean (1982) also made conjectures about levels or stages. Since the results were dependent on rating, high inter-rater reliability was achieved, through careful definitions of levels of responses. Overall, the majority of students were in transition stages. It could be said that there was no distinct jump in capability at any one grade or stage, a finding that was not uncommon in other studies. The one solid mastered by the younger group of 12- to 13-year olds was the square pyramid, followed by the cube and triangular prism, then the tetrahedron (triangular pyramid), the cylinder, cone, and finally the parallelepiped, which few could do at a reasonable level. The study was a good development of Mitchelmore’s (1976) study in Jamaica, which made use of a rectangular prism. Data from most of the other spatial tests which were given correlated with the results on this test except for those of females on a perceptual maze test and for males on the visualization/spatial memory test. Results from this latter test did not correlate with those from a form board test (involving finding shapes to complete a shape). Lean also carried out an extensive review of training studies related to spatial reasoning. However, unlike what transpired in many other studies, he related them back to the two categories that Bishop (1979) had identified earlier: Interpreting Figural Information (IFI) and Visual Processing (VP). The latter category tended to be where spatial abilities testing and most training studies had occurred. He noted that all the VP training programs lacked a “dynamic display of various transformations of spatial representations” (p. 51), and that a relatively short training period by Bishop assisted IFI. Clements and Jones (1983) discussed how Atawe Koigiri, a mathematics lecturer at Unitech, had, as a young child, been moved from a very remote village to another area with Western schooling, and after gaining a secondary and University education, had become a mathematics lecturer at Unitech. According to Clements, risk-taking, and an ability to understand numbers visually were two key components to Atawe’s success. This resonated with Kay Owens’ many experiences of students in PNG. Atawe later became a parliamentarian and returned to lecturing and supporting open learning at PNG University of Technology (Unitech) before he died after a short illness in 2011. Later Research in PNG on Visuospatial Reasoning In 1997, Kay Owens noted that Architecture students were using their cultural and ecological backgrounds when designing beautiful sculptures from paper and cardboard (without glue). Her analysis of interview data pointed to certain strengths having been developed from their

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PNG heritage (Owens, 1999). The synthesis of results indicated the role of visualizing in their problem solving as well as the impact of their cultural identities on their ways of noticing and bringing ideas together to solve problems. Building on the work of visualizing through diagrams to represent algebraic information, and the role that visualizing played in PNG societies, Owens provided an account of significant creative designs through history that were both aesthetically pleasing as well as practically useful. This was the case when designs were enlarged without due consideration given to balance and to forces (Owens, 1998). In 2014, at the first International Conference on Pure and Applied Mathematics, a history of mathematics was presented. Despite making reference to other cultures, there was no mention of the role of visualization or history of mathematics in PNG. The presentation drew a quick response from Owens, who suggested that ethnomathematics encompassed school mathematics, a theme on which she would elaborate in later papers, basing her argument on her research in PNG (Owens, 2016a) (see Figure 12.1). During Wilfred Kaleva and Owens’ measurement study in PNG (Owens & Kaleva, 2008a, 2008b), Kay Sinebare was a primary school teacher in North Goroka when the researchers were looking at children’s spatial thinking in mapping. Being an excellent teacher, Sinebare recognized that the students needed to have a further lesson on drawing a map and she was able to share the strengths of her students’ drawings with others in the class (see Figure 11.2). Interestingly, she noted that some students made mirror images of the school buildings and the oval, and others mapped their house without walls or a boundary; but otherwise they designated areas inside their houses (which were small, sometimes round, with between one and five small rooms) accurately. She also used local clay to enable her students to carry out 3D modelling. As mentioned above, representing 3D space on a 2D page was not a common experience for PNG children. However, there was much use of visuospatial reasoning during measurement activities, especially the activities observed in villages. That observation inspired the writing of a book (Owens, 2015). Owens began her initial studies on visualizing and spatial reasoning by assessing

Figure 11.2. Teaching mapping at primary school.

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spatial abilities through an innovative test (Owens, 1992) which included items which assessed spatial abilities mentioned in the psychology literature (Eliot & McFarlane-Smith, 1983). For her grounded-theory research, her series of learning experiences in classrooms, two of which were in PNG, and the others in low-SES suburbs of Sydney, showed that spatial abilities could be improved through training. Analyses of classroom activities indicated that the amount of responsiveness during problem solving was a key. The immediate classroom environment, such as having open-ended questions, the activities in which the materials were used, and peer and teacher comments, influenced the learners’ thinking. The cognitive processing of heuristics, visualising, adding words, feelings and attitudes were expressed in their responsiveness that in turn led to modification of the materials, or peer or teacher responses, and so the cycle continued throughout the problem-solving learning experiences. Owens’ continuing work on the ecocultural aspects of visualizing has been influential (see for example, Owens, 2007, 2008, 2012, 2013, 2020a, 2020b, 2020c; Owens & Paraide, 2019). Arguing from a critical theory perspective, Owens maintained that cognitive processing was influenced by culture and the ecology surrounding a person. A particularly interesting area of investigation has been the noticing and the visuospatial reasoning about ratios of lengths and areas. For example, villagers would carve a canoe using memory and observations of other examples of canoes of different lengths. They would build a rectangular house that has a length “half as much again,” but knowing how much more floor, wall and roof materials would be needed for this larger shape. Similarly, for a round house, they intuitively knew the proportional increase in the volume of a round house for an increased radius, how much more wood would be needed to warm the house, how many more saplings for the building, how much more wall coverings, how much kunai grass would be needed for the roof, and the extent of additional gardens which would have to be planted to support the relatives who assisted with building the house. Owens’ PNG research on visuospatial reasoning has continued, based on reports and books such as those recognizing the breadth of connection between the spatial representations and cultural beliefs (Owens, 2016b, 2022 in press). Other research has been reported particularly by Majewski (1997, 1998) and others interested in the visual representations of 3D objects using computer packages (see below). Owens (2016c) delivered a paper at the 2nd International Conference on Pure and Applied Mathematics on the use of the freely available GeoGebra in developing countries, and on why it was beneficial to make use of visuospatial reasoning, which seemed to be a strength of PNG students. The points made by Owens were immediately taken up by the ICT manager, Russell Deka, at the University of Goroka, a case of being unable to easily access reliable information on programs in PNG until this conference presentation, an issue raised by Chris Wilkins at PNG University of Technology (see below for further comments).

Language for Location and Direction

PNG and the Pacific has also been an important area for the study of how language can be related to locating and directing, which is an important aspect of spatial geometry. Many studies have looked at how language is used for giving directions from the place where a person is standing and how they are facing. Cardinal directions (East-West, North-South) and local landmarks or topology (e.g., coast or mountain) are often used (Edmonds-Wathen, 2012). Senft (1997) recognized the diversity of ways to refer to space without the use of prepositions such as “on,” “in,” and “before.” Spatial deictic references may be prepositions or postpositions; locatives (local or place adverbs or local nouns); directional; verbal; presentative like “there,” or demonstratives like “that” (Senft, 2004). Senft (2004) noted the use of body parts and elements in the environment for designating position was common among Papuan (Non-Austronesian), Austronesian and Australian languages. Position and dimension are involved. Often familiarisation with the local area was

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important for wayfinding, and for referencing positions. However, this is not simplistic or non- abstract. The Austronesian Oceanic languages mainly refer to the Sea and Land. Quoting Mosel (1982), Senft, noted that Tolai, an Oceanic language of East New Britain, had a complex deixis5 for describing relationships in terms of space. Tolai not only can describe “here” and “there,” but there are generally two classes: (a) that regarding the speaker’s position and some other place; and (b) 15 hierarchical subclasses related to remoteness, action, location, goal and source with a specific hierarchy. Words indicate: 1. The level at which the indicated place is located relative to the speaker’s position 2. Whether the indicated place: (a) is a place at which an action takes place …; (b) is a place where something or somebody is found …; (c) the goal of an action …; (d) the source of an action. 3. Whether or not the place pointed at is known to the hearer (Mosel, 1982, p. 111 cited in Senft, 1997, p. 20) Takia, an Oceanic language influenced by the language of a Papuan neighbor on the same island, has post-positionals attached to the noun, and nouns for positions such as “underside” as well as numerous directional and positional verbs (Ross, 2004). Ross noted that “up” may refer to “up the volcanic mountain” or “out to sea towards the horizon.” However, the order of verbs becomes important in other utterances. For example, the second verb indicates direction or position of a movement. The Takia language also refers to sun rising and sun setting. For Kilivila of the Trobriand Islands, “here,” “there” and a “distant there”6 are often accompanied by a physical indication with a finger, eyes, tipping of head or movement of the lips. Positions are also described by linkage to a noun such as “corner of a garden.” “Left,” “right” “middle,” and “in front of” are also used together with a number of nouns derived from body parts such as “face” or “in front of,” and the post-positionals can also carry a numerical indicator. However, the Yupno, who have a non-Austronesian language in the mountains of Madang, divided their valley into four quadrants, but they also describe routes by actions and what people know of the area (Wassmann, 1997). Many of the Highland areas referred to “up” or “down” the river or valley. Muke (2012) noted that in mid-Wahgi, people referred to North as the source of the little rivers without due regard for which side of the big Wahgi they were on. Referencing position often involves the connection of the space to the person’s current view, and particular areas (Owens, 2015). Senft’s (1997) research also took account of Papuan as well as Oceanic languages outside of PNG. These linguistic studies are likely to be ongoing especially as Ross and others investigate proto languages.

Technologies and Mathematics in Papua New Guinea

In earlier chapters, we referred to some of the tools for teaching, such as Dienes’ attribute blocks and multi-base arithmetic blocks. However, PNG was also early in adapting calculators at the university level. At the PNG University of Technology, there was a calculator laboratory with large desktop calculators in the mid-1970s. These were particularly useful for building the confidence of entry-level students in learning mathematics which required computations.

Deixis is a linguistic term relating to indicating. This might include the speaker, participants in the communication, their location or orientation in space, whatever indexing acts they perform, or the time of the utterance in relation to these. For a full discussion see Senft (2004). The studies covered here are especially about spatial deixis. There have been a number of cognitive psychology studies linked to linguistic studies (see, for example, Edmonds-Wathen, 2012). 6 This is common also in many Australian Aboriginal languages such as Wiradjuri. 5

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Like the physical equipment used through the Mathematics Education Centre for teaching Unitech students, and the calculators, the University also made early use of computers. When desktop computers became available, so did discs with particular programs (now they would be “apps”) such as the one for linear programming. All students learned to use EXCEL proficiently and the accounting students could use MYOB. Lecturers shared their research with staff and students. For example, in one seminar, Landau discussed the position and stability of a chair with three or four legs on a curved surface—picture a small doll’s chair and an orange for a demonstration. Fairclough discussed the random pattern of raindrop splashes on different sloping grounds and linked this to erosion (appropriately Lae has a very high rainfall each year). One Open Day at Unitech, the enterprising Mathematics Department decided to set up a casino where students could spin the chocolate wheel and the computer recorded how many turns and how many guesses equalled the result on the wheel. A table and a graph on the computer screen provided details. This was really intended to show how little chance someone had of winning in gambling. It worked very well until an astute student noticed a slight bias in the homemade chocolate wheel, so he guessed correctly! Research on Calculators The use of calculators also has been an area of research. Alan Edwards, the first Director of Unitech’s Mathematics Education Centre, led many workshops on the use of calculators in technical colleges and schools. In particular, he noted the advantages for primary school students who were likely to remain in the village with a trade store or other enterprise, and also for those undertaking technical courses. Edwards (1981) noted a number of issues that could be highlighted in the use of calculators by PNG adults. In particular, he recognized the difficulties of the decimal point with money calculations and with shop assistants’ tills which automatically inserted the decimal point. Another issue was the use and interpretations of negative numbers which appeared in some subtractions. “Mark Ups,” for which selling prices were given, also required the use of the memory—if a calculator had one—and therefore needed greater explanation. Measurement units could be an issue but Edwards chose to use decimal fractions of a metre for calculations. He also simulated random numbers by using ÷ 7 = = = . Beth Southwell returned to PNG on a number of occasions and in 1984, as a visiting scholar in the Mathematics Education Centre on study leave from the Nepean College of Advanced Education (now the University of Western Sydney), she carried out a study on the use of calculators with all students in Grade 7 at two Lae high schools (Southwell, 1984). One of the schools was a boys’ school with mainly boarding students, so many students came from more remote areas of Morobe; the other school had day students who lived in town or nearby, but came from many different parts of PNG. Two classes were introduced to using calculators to solve both vocational and non-routine problems. A series of six lessons were based on an overseas approach to using calculators. Among the variables were scores on the questions attempted, the number of responses showing working, and the number of correct answers given. Several analyses of variance and t-tests were applied to compare control and calculator classes; gender; and scores on pre-, post- and retention tests were also used as variables. The most significant results were gain scores on post and retention tests over scores on the pre-test for the students using calculators, compared to control students, with many more items being attempted, working shown, and more correct responses on the retention test. The author noted that the reason for the improvement, especially in the confidence to attempt questions, could also have been due to the researcher/ instructor difference or the teaching of simple strategies for problem solving like making estimates and turning the word problem into mathematical code. There were also game-like problem-solving activities for which the calculator had advantages for speed. However, learning how the calculator works, and, for example, the constant operation of addition for multiplication may also have developed students’ confidence.

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Musawe Sinebare (1985), who would become Head of the Department of Mathematics, then Pro-Vice Chancellor at the University of Goroka, Acting Head of National Research Institute, later,Secretary of Education in Port Moresby, and then Vice Chancellor at the University of Goroka, wrote an early article on the advantages of calculators for problem solving.

Computers in Mathematics

Papua New Guinea’s universities were early users of computers7. By 1979, PNG University of Technology had a mainframe computer which was used for university administration and also by most departments. Each Faculty team at the University had an air-conditioned computer room of terminals and, later, desk-top computers. There were terminal rooms for teaching students in most Faculties. Its first two managers were incredibly dedicated to their work. Excellent statisticians and teachers like Peter Jones, Phil Sheridan, Chris Wilkins and Don Lewis made extensive use of Minitab on the DOS computer system, when teaching probability and statistics. Lewis (1985) wrote of the value for simulation of real-life examples for looking at yields of plants in agriculture on different blocks of land, modifying impact of catalyst, pressure, and temperature on engineering production, and the issue arising from missing data in an examination question also requiring Latin squares (blocking). One disadvantage seemed to be that the computers and programs would not be available to students after they had left university. The issue of recognizing natural variation was also raised. Lewis emphasized the importance of bringing field experience into the “design of an experiment “class in a more direct way through having the students use Minitab when carrying out a joint project. Phil Sheridan (1985) wrote a DOS script so that when students were sitting their examinations they could move between MS Word spaces reading questions, giving answers and using the Minitab program to do their calculations. All their work was available for marking. Meanwhile, Don Lewis taught staff the syntax for SPSS and the details of statistical methodologies—all of which was made possible through the computer. The University used tape for entering code for only a short period of time, since the terminals by 1982 provided the entry. The Mathematics Department had its own computer laboratory as well as an air-conditioned staff room with computer terminals. In another building the students had access to their own computer laboratory for work outside of lectures and tutorials. The students’ risk-taking and willingness to work on the unknown computers meant they did not develop a fear of this technology. After their first lecture, they were learning how to break into another student’s computer and send messages. However, email messages were slow to arrive from outside the University, there being just one dedicated desktop computer and its cover was propped open to keep it cool. Messages were printed and distributed to staff. One of the computer uses was to allow students to teach themselves school mathematics by selecting a set of items on a topic that they needed to understand better. Although they could skip a set, they had access to instant feedback, so they could decide if they were correct and hence thinking correctly about an item. It was assumed that if they were wrong then they would be able to work out why, and move on to the next item. One particular set were questions involving multiplying decimal numbers by 100 or 1 000; another was dividing two decimal numbers with the same digits but of different values to get 100 or 1 000 (Shield, 1982). These were run and controlled with simple DOS commands. Staff in various institutions were writing their own programs to assist learning (Fredericks, 1983), but as Wilkins (1983) pointed out, the use of computers needed to be engaging and not just directed at rote learning. Toward that end, soon after the advent of microcomputers, Rod Selden (1983) illustrated how his graphics program on the Apple II encouraged students to See Kay Owens’ cameo in chapter 6

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explore features of graphs, to find new curves for different equations, and to recognize features of these graphs. Students used Visual BASIC as a programming language at PNG University of Technology, but Selden introduced programming with Logo which was gaining prominence overseas (Papert, 1993). He also supported the International schools with microcomputers and educational software. Wilkins (1983) pointed out the difficulties of knowing what software to buy when in PNG. The PNG computer scientists and mathematicians were sharing as best they could on the usefulness, or otherwise, of available software from overseas. As Wilkins noted, there was no shop to go and try things out and the programs were very expensive for PNG to take the risk in purchasing. Some came as a result of expatriates coming to teach or visiting their home countries, some were purchased in the hope that they would work. Nevertheless, programs did come and get used. By 1995, Majewski was exploring computers for representing complex ideas such as fractals using Pascal programming language and POV-Ray. Majewski introduced Scientific Notebook as a tool in mathematics education, and also compared the effectiveness of Maple, POV-Ray and Mathcad (Majewski, 1997). By 1997, spreadsheets which had been used from the start were being used in more sophisticated ways in a range of modelling applications in agriculture (Arganbright, 1997), architecture and other areas where recursive computations were needed. By 1998, C++, Visual Basic, HTML, Cobal, were all being taught to students. To assist with mathematics, LaTeX and Java applets were also used (Majewski, 1998). As an aside, Kopamu provided training in LaTeX to his staff at the University of Goroka in 2014, and prior to computers, secretaries like Lenore Adams (now Burton) and Betty Agum, at Unitech, and Anna Johansson, in Goroka, had become expert at the use of the “golf balls” on electric typewriters for producing high quality mathematics handouts. Betty and Anna remained as secretaries for many years providing high quality material on computers. Phythian (1998b), Head of Mathematics and Computer Science at PNG University of Technology, emphasized that computers were becoming part of the fabric of education for life and living, for earning and for fulfilling opportunities to innovate. Importantly, computer modelling could facilitate imaginative abstraction beyond immediate real-life boundaries, and could be used to predict. It was useful for “queuing theory,” and encouraged creativity. Phythian (1998a) saw potential in expanding PNG human technological knowledge so that it could boost export potential beyond that of the current mineral, forest, and sea resources to other countries. He provided examples of designs from equations and designs using POVRan and then simulation modelling using SIMIAN, which was particularly useful for queuing theory (initially taught by Owens to architecture students, using basic formulae and available tools like a calculator, EXCEL and Minitab but connected to other tables of requirements for building design). Phythian’s examples were concerned with banking, computer use, and stadium entry— queues being frequently experienced in PNG. The Mathematics and Computing Department was burnt down in 1999, with a firewall protecting the next section of the building. The Department went from four network Linus classrooms to one together with a small staff room of computers. Wamil and Hanspeter (2002) indicated that they were using Derive and Mathematica for their Engineering Mathematics classes, and Minitab was still used for Forestry and Agriculture. The students were also bringing other packages with them to try to share but, when the enthusiastic Len Raj left, Derive did not continue to be used in mathematics classrooms. Mathematica also proved to be inappropriate—it took up a lot of computer space and there were difficulties in using it. Although Minitab was not particularly user-friendly, many students liked to use it. Networks in PNG needed constant monitoring to prevent user and viral issues (Roberts, 1998). During the PASTEP project, as an additional development of the project, teachers colleges were provided with secure, air-conditioned computer laboratories. In general, material, such as a World Encyclopaedia, was made available on disc. Students were expected to use Microsoft

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Office at least for their assignments. At first, staff were also able to use the internet to source extra resources and in some cases research related information. Staff also had access to email. However, soon viruses crept in. Some of the staff, like Neville Undaki and Sorengke Sondo, who were selftaught to run computer laboratories, developed really creative educational materials and staff were supported as they attempted to cope with a full range of computer difficulties. However, the computers only made external connections when the TELIKOM (telephone company) bill was paid and lines were working. Sondo purchased, out of his own money, an updated virus protector to reset the computers for use. The One Laptop per Child program, which involved internet connections and video streaming of quality materials, seconded the PNG Institute of Education’s computer room and when the program officially finished, the centre was unfortunately not used by the Institute for many years. Computers in Schools In 1995, a conference of mathematics teachers discussed whether there should be a computer course for schools (Phythian, 1996). All agreed there should be one at the junior secondary level, as an appreciation course for teaching keyboard skills, word processing and filing, and EXCEL should be taught in mathematics, and databases in the social sciences. All hoped for internet connections, especially in the senior high school. Only a few schools managed to be able to do that, however, although two new private day schools (a national high school and a Grade 1 to 12 school in Port Moresby) did so, despite the very high costs of data download needed for quality computer education and education with computers. The high schools also needed a more advanced computer-related subject in Grade 12 since some students would not gain a place at a university, and businesses in the community looked with favor on such skills. The list of topics for this included graphics, computer systems management, computer algebra, network management, school records, computer architecture, social effects of computers, using spreadsheets for accounting, and statistics by computer. It was agreed that the first course could produce a certificate of attainment, and guidelines for competencies should be made available together with information on software and possible hardware. It was not expected that it would be examined. There were some issues, such as at some schools, teachers and students would be using the same computers (a data security issue), and some teachers felt uncomfortable about teaching computing in their substantive subject—like word processing in English. By 2006, some students who had attended high school were able to become research assistants since they could demonstrate exceptionally good word processing and spreadsheet skills, often being more competent than Australian students at that time. Interestingly, some of these basic computing skills were still being taught at the University of Goroka in 2014. Sinebare, as Head of Mathematics, encouraged the University of Goroka administrators to make computers available to all students, and especially to give beginning secondary teachers some computing experience. Japanese aid would make this possible. When changes are made to senior personnel in the Universities, the websites of the Universities get changed—but they are often not finished and materials are lost as they are often reliant on overseas volunteers. That is largely because of expectations put on the computing sections within the universities. Some of the One Laptop per Child schools such as at Oksapmin and Wewak continued to maintain their computers and solar panels and undoubtedly that assisted with the education of their students. The growth in intraschool networks continues. A Rotary Club in Australia has spearheaded some of this in technical schools in PNG, with the intention that it would be taken over by Unitech graduates. They networked, with very small Chinese computers attached to the rear of each student screen, together with solar power. Waimo preschool, at the initiative of one of the community members who had previously worked for AirNiugini, had a computer with free education games, but only one that the teachers could use.

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Sinebare’s doctoral studies (1999) at University of Wollongong, in Australia, was titled Private Computer Training in PNG: From Chaos to Order. Although by the end of the 1990s universities and a few post-secondary institutions were providing computer education, these were mainly for the elite students who gained entry to such institutions. Hence many commercial organisations began offering computer training to meet the needs of the country. In 1999, Sinebare wrote, in his doctoral thesis: Private computer training in PNG is chaotic … There is neither a specific policy formulated nor a curriculum guideline provided to cater for IT education and training in the country.... Several recommendations based on the research findings, are put forward for implementation by various existing authorities as well as recommendations for a new authority and new structures under which IT education and training can be facilitated. … because of the potential in using IT in many socio-economic aspects of the country, and more importantly because of its potential to create a knowledge industry within the country. In order for this to occur, the Government must play the pivotal role in formulating a national IT Policy for implementation by both public and private sectors. Further, necessary administrative and physical infrastructure should be established as a matter of priority in order for IT education and training to be promoted, applied, and implemented. (abstract) Sinebare summarized what had happened in other developing countries, and also reported information relating to the situation in PNG that he had gathered through responses to a questionnaire from many organizers of ad hoc training programs. It seems that he was successful in persuading government to establish regulations for training programs. Despite the fact that university computer graduates often run businesses outside their own work their clients being private individuals and companies—for whom they manage websites, carry out maintenance and attend to issues associated with viruses—some university computer graduates find it difficult to gain jobs. There have been some good successes with dedicated staff at the University of Goroka, such as Smith Wendell, who has used his expertise for the Flexible and Distance Learning Division. Unitech provided computer laboratory space for the International Education Agency (IEA), and works closely with Microsoft and others to provide courses. Don Bosco high school in Port Moresby began with computers available off campus and continues its service. Port Moresby Grammar, a private school, also has good computer facilities. However, overall, Unitech and others have found it difficult to attract suitably qualified computer specialists to lead a push to upgrade computer education at the University or across the nation. Several notable people have remained faithful to the cause, sometimes moving from one university to the other. One such person is Lakoa Finia who served Divine Word University for many years before transferring elsewhere. He is now professor at UPNG. Peter Anderson has been at DWU for many years, first as a lecturer but then as the Head of Department and Professor, and has worked hard to extend education in this area. DWU continues to have excellent online library and internet facilities for staff and opportunities for postgraduate training. Chris Wilkins, who married a Papua New Guinean, and Benson Mirou, deserve much praise for their untiring efforts—despite having to cope with a decade without a mathematics building after the fire and the difficulties experienced by Benson as he sought to be given an opportunity for doctoral studies overseas. He is currently working on programs for other faculties of the University as part of his local doctoral studies. Russell Deka, who came as a Japanese volunteer, married a Papua New Guinean and has continued to assist and develop the University of Goroka computers, and has sometimes assisted in other places.

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During the elementary schools mathematics project (Owens, Edmonds-Wathen, & Bino, 2015), teachers in remote areas were provided with a solar panel, battery, and a laptop computer with a professional development program on the teaching of elementary school mathematics. However, many teachers in remote areas missed out on receiving computers because of the high import duty (which should not have occurred for education or aid), with the extra cost per computer reducing the number available. Nevertheless, the elementary teachers learnt to use a computer and the solar panels and batteries were used at least for their mobile charging although many continued to use the computers as well as study the professional development. One of the team, Susie Daino put together a sizeable range of free software, including some on mathematics, and encouraged the use of these in early childhood settings to extend young children’s positive use of screens. Attracting overseas and locally trained staff and then providing them with opportunities to extend their knowledge, has proved difficult for this still young nation where monies do not stretch far enough for extensive computer facilities at any level, and where humid hot environments affect electricity supply and make computers short lived. In 2014, when Kay Owens was supplying computers to teachers in rural areas and teaching them how to use them, she began to realize that although carrying a computer laptop bag was a status symbol, laptops might need to be hidden to avoid them being stolen. However, in 2021, Unitech is struggling to attract the best or even enough students into Computer Science. Despite all these difficulties, when opportunities exist, some incredibly good computer and information technology work has taken place and that has served the nation well. References Arganbright, D. (1997). Modeling growth and harvesting on a spreadsheet. PNG Journal of Mathematics, Computing and Education, 3, 7–18. Ausburn, F., & Ausburn, L. (1980). Perception, imagery, and education in developing countries. Mathematics Education Centre, PNG University of Technology Report, 17. Lae, PNG. Bishop, A. (1979). Visualising and mathematics in a pre-technological culture. Educational Studies in Mathematics, 10(2), 135–146. Bishop, A. (1988). Mathematical enculturation: A cultural perspective on mathematics education. Dordrecht, The Netherlands: Kluwer. Booth, L. (1975). Investigation of teaching methods and materials I: Investigation of a 'learning for mastery' approach in elementary coordinate geometry Progress Report 1973–1975 (pp. 49–54). Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Cajori, F. (1907). A history of elementary mathematics with hints on methods of teaching. New York, NY: Macmillan. Clarkson, P. (1981). A study in visual ability with Papua New Guinea students. In J. Baxter & A. Larkin (Eds.), Research in mathematics education in Australia (pp. 30–48). Adelaide, Australia: Mathematics Education Research Group of Australasia. Clarkson, P. (1982). Mathematics learning in PNG National High Schools: A comparison of years 11 and 12 across schools. Retrieved from Lae, PNG. Mathematics Education Centre, PNG University of Technology. Clarkson, P. (1993). Gender, ethnicity and textbooks. Australian Mathematics Teacher, 49(2), 20–22. Clements, M. A. (1983). The question of how spatial ability is defined, and its relevance to mathematics education. Zentralblatt fur Didaktik der Mathematik, 1(1), 8–20.

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Clements, M. A., & Jones, P. (1983a). The education of Atawe. In I. Palmer (Ed.), Melbourne studies in education 1983 (pp. 112–144). Melbourne, Australia: Melbourne University Press. Clements, M. A., & Lean, G. A. (1981). Influences on mathematical learning in Papua New Guinea: Some cross-cultural perspectives (Report No. 13). Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. de Abreu, G., Bishop, A., & Presmeg, N. (Eds.). (2002). Transitions between contexts of mathematical practices. Dordrecht, The Netherlands: Kluwer. Edmonds-Wathen, C. (2012). Frames of reference in Iwaidja: Towards a culturally responsive early years mathematics program. PhD Thesis, RMIT, Melbourne, Australia. Edwards, A. (1981). Adult numeracy with a calculator: Some problems. In P. Clarkson (Ed.), Research in mathematics education in Papua New Guinea 1981 (pp. 43–50). Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology. Eliot, J., & McFarlane-Smith, I. (1983). International directory of spatial tests. Windsor, England: NFER-Nelson. Faw, P. (1977). A study of the development of the ability of selected students to visualize the rotation and development of surfaces. PhD dissertation, University of Oklahoma, Oklahoma. Fredericks, H. (1981). A test of measurement—Or is it? In P. Clarkson (Ed.), Research in mathematics education in PNG 1981 (pp. 52–57). Lae, Papua New Guinea: PNG University of Technology Mathematics Education Centre. Fredericks, H. (1983). Instructional computing—Microcomputing capabilities for computer presented lessons. In P. Clarkson (Ed.), Mathematics education research in Papua New Guinea (pp. 1–6). Lae, Papua New Guinea: PNG University of Technology Mathematics Education Centre. Guilford, J. (1966). Intelligence: 1965 Model. American Psychologist, 21, 20–26. Haddon, A. C. (1900). Studies in the anthropogeography of British New Guinea (Continued). The Geographical Journal, 16(4), 414–440. https://doi.org/10.2307/1774323 Haddon, A. C. (1904). 21. Drawings by natives of British New Guinea. MAN, 4, 33–36. https:// doi.org/10.2307/2839985 Haddon, K. (1930, reprinted in 1979). Artists in strings. New York, NY: AMS. Kohs, S. (1920). The Block-Design Tests. Journal of Experimental Psychology, General, 3(5), 357–376. https://doi.org/10.1037/h0074466 Krutetskii, V. (1976). The psychology of mathematical abilities in schoolchildren. In J. Kilpatrick & I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics. Survey of recent East European mathematical literature (Vol. II: The structure of mathematical abilities, pp. 5–58). Chicago, IL: University of Chicago. Lean, G. A. (1975a). An investigation of spatial ability among Papua New Guinean students. Mathematics Learning Project: Progress Report, 26–47, and Appendix 22. Lean, G. A. (1975b). The Mathematics Learning Project 1973–1975 Progress Report (pp. 1–13). Lae, Papua New Guinea: PNG University of Technology. Lean, G. A. (1982). The rotation and development of surfaces—A study of a Piagetian spatial task with PNG secondary school pupils. In P. Clarkson (Ed.), Research in mathematics education in Papua New Guinea (pp. 117–159). Lae, Papua New Guinea: Mathematics Education Centre, PNG University of Technology, Lae, PNG. Lean, G. A. (1984). The conquest of space: A review of the research literatures pertaining to the development of spatial abilities underlying an understanding of 3-D geometry. Paper presented at the Fifth International Congress on Mathematical Education, Adelaide, Australia. Lean, G. A., & Clements, M. A. (1980). Spatial ability, visual imagery and mathematics performance. Mathematics Education Centre, PNG University of Technology, Lae, PNG. Lean, G. A., & Clements, M. A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12, 267–299.

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Lewis, D. (1985). The use of simulation programs in teaching experimental design concepts. In N. Wilkins (Ed.), Research in mathematics education in Papua New Guinea (pp. 36–42). Mathematics Education Centre, PNG University of Technology: Lae, Papua New Guinea. Lowrie, T. (1994). Horses for courses: Ways in which children solve mathematical problems. Australian Association for Educational Research. https://www.aare.edu.au/publications/ aare-conference-papers/show/1158/horses-for-courses-ways-in-which-children-solve- mathematical-problems Majewski, M. (1997). Evaluation of Scientific Notebook as a tool in mathematics education: Realistic visualizationof 3D Maple plots PNG Journal of Mathematics, Computing and Education, 3, 25-38, 39–48. Majewski, M. (1998). HTML approach in publishing mathematics on the WWW and Publishing live mathematics on the WWW. PNG Journal of Mathematics, Computing and Education, 4, 53–62, 63–72. Mennis, M. (2014). Sailing for survival. University of Otago Working Papers in Anthropology, 2. http://hdl.handle.net/10523/5935 Mikloucho-Maclay, N. (1975). New Guinea diaries 1871–1883 (C. L. Sentinella, Trans.). Madang, Papua New Guinea: Kristen Press. Mitchelmore, M. (1976). Cross-cultural research in concepts of space and geometry. In J. Martin (Ed.), Space and geometry: Papers from a research workshop. Ohio State University: ERIC Center for Science, Mathematics and Environmental Education. Mosel, U. (1982). Local deixis in Tolai. In J. K. Weissenborn, Wolfgang (Ed.), Here and there: Cross-linguistic studies on deixis and demonstraion (pp. 111–132). Amsterdam, The Netherlands: John Benjamins. Moses, B. (1977). The nature of spatial ability and its relationship to mathematical problem solving. PhD dissertation, Indiana University. Muke, C. (2012). Role of local language in teaching mathematics in PNG. PhD thesis, Australian Catholic University, Melbourne, Australia. Owens, K. (1992). Spatial mathematics: A group test for primary school students. In W. M. Stephens & J. Izard (Eds.), Reshaping assessment practices: Assessment in the mathematical sciences under challenge. Melbourne, Australia: Australian Council for Education Research. Owens, K. (1998). Creating design using mathematics. PNG Journal of Mathematics, Computing and Education, 4, 79–98. Owens, K. (1999). The role of culture and mathematics in a creative design activity in Papua New Guinea. In E. Ogena & E. Golla (Eds.), 8th South-East Asia Conference on Mathematics Education: Technical papers, (pp. 289–302). Manila, The Philippines: Southeast Asian Mathematical Society. Owens, K. (2007). Changing our perspective on measurement: A cultural case study. In J. Watson & K. Beswick (Eds.), Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 563–573). Hobart, Australia: Mathematics Education Research Group of Australasia (MERGA). Owens, K. (2008). Culturality in mathematics education: A comparative study. Nordic Studies in Mathematics Education, 13(4), 7–28. Owens, K. (2012). Papua New Guinea Indigenous knowledges about mathematical concepts. Journal of Mathematics and Culture (on-line), 6(1), 15–50. Owens, K. (2013). Diversifying our perspectives on mathematics about space and geometry: An ecocultural approach. International Journal for Science and Mathematics Education. http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10763-0139441-9

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Owens, K. (2015). Visuospatial reasoning: An ecocultural perspective for space, geometry and measurement education. New York, NY: Springer. Owens, K. (2016a). Culture at the forefront of mathematics research at the University of Goroka: The Glen Lean Ethnomathematics Centre. South Pacific Journal of Pure and Applied Mathematics, 2(3), 117–130. http://www.unitech.ac.pg/sites/default/files/ICPAM_ PAPERS_final_3%20%281%29.pdf or https://researchoutput.csu.edu.au/ws/portalfiles/ portal/12488842/8991526_Published_paper_OA.pdf Owens, K. (2016b). The line and the number are not naked in Papua New Guinea. International Journal for Research in Mathematics Education. (Special issue: Ethnomathematics: Walking the mystical path with practical feet), 6(1), 244–260. http://sbem.iuri0094.hospedagemdesites.ws/ojs3_old/index.php/ripem/issue/view/99 Owens, K. (2016c). Powerful reforms in mathematics education: The perspective of developing countries on visuospatial reasoning in mathematics education. South Pacific Journal of Pure and Applied Mathematics, 2(3), 104-116. https://researchoutput.csu.edu.au/ws/ portalfiles/portal/23012697/8991832_Published_article_OA.pdf Owens, K. (2020a). Indigenous knowledges: Re-evaluating mathematics and mathematics education Quaderni di Ricerca in Didattica (Mathematics), Quaderno numero speciale 7-Palermo, 28–44. http://math.unipa.it/~grim/4%20CIEAEM%2071_Pproceedings_ QRDM_Special_Issue%207_Plenaries.pdf Owens, K. (2020b). Noticing and visuospatial reasoning. Australian Primary Mathematics Classrooms, 25(1), 12–15. Owens, K. (2020c). Transforming the perceptions of visuospatial reasoning: Integrating an ecocultural perspective. Mathematics Education Research Journal (Special Issue on The relation of mathematics achievement and spatial reasoning), 32, 257–283. Owens, K. (2022, in press). Chapter 7. The tapestry of mathematics – Connecting threads: A case study incorporating ecologies, languages and mathematical systems of Papua New Guinea. In R. Pinxten & E. Vandendriessche (Eds.), Indigenous knowledge and ethnomathematics. Cham, Switzerland: Springer. Owens, K., Edmonds-Wathen, C., & Bino, V. (2015). Bringing ethnomathematics to elementary teachers in Papua New Guinea: A design-based research project. Revista Latinoamericana de Etnomatematica, 8(2), 32-52. http://www.revista.etnomatematica.org/index.php/RLE/ article/view/204 Owens, K., & Kaleva, W. (2008a). Case studies of mathematical thinking about area in Papua New Guinea. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Annual conference of the International Group for the Psychology of Mathematics Education (PME) and North America chapter of PME, PME32—PMENAXXX (Vol. 4, pp. 73–80). Morelia, Mexico: PME. Owens, K., & Kaleva, W. (2008b). Indigenous Papua New Guinea knowledges related to volume and mass. Paper presented at the International Congress on Mathematics Education (ICME 11), Discussion Group 11 on The Role of Ethnomathematics in Mathematics Education, Monterrey, Mexico. https://researchoutput.csu.edu.au/en/publications/ indigenous-papua-new-guinea-knowledges-related-to-volume-and-mass Owens, K., & Paraide, P. (2019). The jigsaw for rewriting the history of number from the Indigenous knowledges of the Pacific. Open Panel 106 Indigenous mathematical knowledge and practices: (crossed-) perspectives from anthropology and ethnomathematics In A. Arante et al. (Eds.), Annals of 18th World Congress of United Anthropological and Ethnological Societies IUAES18 Worlds (of) encounters: The past, present and future of anthropological knowledge, Anais 18 Congresso Mundial de Antropologia (pp. 3468– 3487). Florianópolis, Brazil: IUAES18. Paivio, A. (1971). Imagery and verbal processing. New York. NY: Holt, Reinhart, & Winston.

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Southwell, B. (1984). Problem solving with calculators for grade 10 students. Retrieved from Mathematics Education Centre, PNG University of Technology: Lae, PNG. Suwarsono, S. (1982). Visual imagery in the mathematical thinking of seventh grade students. PhD thesis, Monash University, Melbourne, Australia. Thurstone, L., & Thurstone, T. (1941). Factor studies of intelligence. Psychological Monographs, 2. Vandendriessche, E. (2007). Les jeux de ficelle: Une activité mathématique dans certainess sociétés traditionnelles (String figures: a mathematical activity in some traditional societies). Revue d'Histoire des Mathématiques, 13(1), 7–84. Vandendriessche, E. (2015). String figures as mathematics: An anthropological approach to string figure-making in oral traditional societies. Dortrecht, The Netherlands: Springer. Wamil, J., & Hanspeter, I. (2002). Teaching mathematics through computers, PNG University of Technology: A case study. Journal of Mathematics, Computing and Education, 6(1), 15–23. Wassmann, J. (1997). Finding the right path. The route knowledge of the Yupno of Papua New Guinea. In G. Senft (Ed.), Referring to space: Studies in Austronesian and Papuan languages. Oxford, England: Oxford University Press. Wattanawaha, N., & Clements, M. A. (1982). Qualitative aspects of sex-related differences in performance on pencil-and-paper spatial questions, grades 7–9. Journal of Educational Psychology, 74, 878–887. Webb, N. (1979). Processes, conceptual knowledge and mathematical problem-solving ability. Journal for Research in Mathematics Education, 10, 83–93. Weschler, D. (nd). Weschler Intelligence Scale for Children (WISC) (5th ed.). Pearson. Wilkins, C. (1983). Microcomputers in education—Surely not just a rote practice and audiovisual machine. In P. Clarkson (Ed.), Research in mathematics education in Papua New Guinea (pp. 7–13). Mathematics Education Centre, PNG University of Technology: Lae, Papua New Guinea.

Chapter 12 The Impact of Globalization, Colonialism and Neocolonialism on Education in Papua New Guine

Abstract: Globalization has had, and is having, an impact on colonized developing countries in a variety of ways, one of which in education is the adherence to such movements as standards and outcomes-based education. At the same time in postcolonial Papua New Guinea, neocolonial tendencies continue the influence of Australia and other countries with education aid projects influencing, in significant ways, curricula and teacher education. The neocolonial phase is also perpetuated with the internal support it receives from the educated PNG elite who succeeded with the colonial style of education and see little reason to deviate from it. One of the crucial results from the various impacts of neocolonialism and globalization has been the loss of traditional languages and cultures from schools and the education system as a whole. In this chapter we look at the result of neocolonialism, as well as at a number of other critical issues within education, and in particular in mathematics education. Most of these have already been described in earlier chapters within their historical settings but here we try to bring the lenses of globalization, colonialism and neocolonialism to bear so they can be viewed from different perspectives. Issues related to policy, the content of mathematics education, consideration of quality education, how international standards have been used, how aid projects have influenced the wider syllabus, teacher education, and the issue of gender equity will all be addressed.

Key Words: Aid donors in developing countries · Education funding · Epistemic hegemony · Gender equity · Globalization · Neocolonialism · Translocal WHITEMAN What do you need from me? You have turned my land into a desolate place I stumble along with a half-white mind What is it you want, whiteman? Where am I? What am I? Not a recognised race! There is wasteland ahead and wasteland behind. Wangu, ~1981 Introduction Papua New Guinea has seen its fair share of international aid and companies. The disparity between these organizations and the local people’s circumstances, together with the cargo that can come one’s way through them, has resulted in both cooperation and corruption since colonial times. Either way, neocolonialism exists. In most cases, the word “corruption” does not adequately reflect the cultural mores within which various people work. The import is usually more © Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9_12

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subtle, but sometimes the level of corruption is blatant and the benefits to the outsider can be illustrated by the following case: A company owned by Australian brothers Mamdouh and Ibrahim Elomar— patriarchs of a notorious Sydney family linked to ISIS, and themselves formerly convicted of international bribery—have been found by a court in Singapore to have improperly paid more than $6m to a senior PNG official and his wife to buy a PNG timber company at an estimated discount of more than 90%. The purchased PNG company, Cloudy Bay Sustainable Forestry, was set up using revenue from the massive Ok Tedi gold and copper mine to provide “a model for responsible forestry development” in a sector notorious for rapacious foreign loggers: its profits were intended to fund the construction of roads, schools and health clinics for some of PNG’s most marginalized rural communities. Instead, Cloudy Bay’s assets were allegedly stripped and sold, while its owners, past and present, fought each other through the courts: those who it existed to benefit, forgotten. (Lassett & Doherty, 2021) The exploration of the copper mountain with a gold cap, Ok Tedi, in the far west of Papua New Guinea began before Independence. The first gold was mined about 10 years later, in 1981, by BHP Billiton, the largest mining company in the world with the BHP precursor company having been founded in Australia. For years, as was the story at the earlier Panguna mine in Bougainville, now known as the Autonomous Region of Bougainville (ARoB), little was done to ensure the environment was protected and just a trickle of the wealth made its way to the villagers who had been adversely affected by the mining operation. With the effective nationalization of the mine in 2002, far more wealth theoretically should have gone to the villages. But as Lassett and Doherty report, much of the wealth never reached its assigned destination. Such has been the continued story in the period of colonization, and now neocolonization, of Papua New Guinea. Philip Foster in his conclusion to the UPNG 1974 Education in Melanesia seminar at which he was a key speaker, reflected “I had a profound sense of déja vu as I listened to speakers raising issues that I heard expressed in almost identical manner in Africa over a decade ago” (Foster, 1975, p. 516). People in PNG were no different to others, their perception was that the foreigners’ schooling c ontributed to economic advantages. Foster (1975), however expressed reservations about the terms “modernisation” and “development” and about how much western schooling was a source of “new” values. He was speaking about the strength of the values and teaching implicit in the village society. Another speaker at the same seminar, Hiap Salaiau, a Manus woman university graduate, suggested that both traditional and colonial education was restrictive of girls and that colonial schooling made the mistake of not (overtly) building on to what the children already knew. Her advice was that “While there is a difference, the former can be a prerequisite of the latter in order to keep Papua New Guinea's own identity” (1975, p. 336). It was clear that by the 1990s school access for girls and women was improving although retention rates were poor (Wormald & Crossley, 1988; Yeoman, 1987a, 1987b). But what was not learned, or accepted, were new methods of birth control. “Too many children, too early, too late and too close together, pose a threat to the health and wellbeing of the mother, her children and indeed the village, province and nation.” (Gillett, 1990, Quoted in Quartermaine, 2001, p. 2) Pam Quartermaine (2001) maintained that PNG had not fully learnt from other colonized countries. In particular, it had not and still does not value the role of culture in terms of education, and of mathematics education in particular, or the importance of women in society and education. The colonial powers, the national leaders, and the neocolonial powers have a way to go. The fact that there is still a continuing expatriate population at university level, mostly from former

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British colonies in Asia and Africa, and the fact that so little of cultural mathematics has been learned in schools at any time, has meant that a neocolonial perspective on mathematics has been maintained. Even in 2014, a mathematics lecturer at the International Conference on Pure and Applied Mathematics—held at Unitech and organized by the Mathematics Association of PNG— gave a lecture on the history of mathematics which began with an outline of the history of Indian and Egyptian civilisations, and moved on to an argument which suggested that the Englishspeaking world should not dominate school mathematics. Unfortunately, there was no recognition of the mathematics of the older civilizations of PNG, or that he was referring to the Western forms of mathematics in schools and academia, as that is known overseas. Attempts to redress this issue by Owens, in discussing the elementary school project, resulted in her creating Figure 12.1 on ethnomathematics. An acceptance of such a recognition of mathematics and its origins and developments might change the way curriculum writers and educators view mathematics. Privilege could be given to Indigenous cultural mathematics and to that of subcultures (Hunter, Civil, HerbelEisenmann, Planas, & Wagner, 2018) A chief aim of colonizers is to extinguish all other stories so their own story can be privileged. At the heart of this process is the assumption and belief that the colonizer’s culture is superior. The colonizers force an acceptance of this belief on those subjugated, and in the process relegate the subjugated culture to a much inferior status (McConaghy, 2000). In Papua and New Guinea that was the case with early mission education and with later Government education (see Chapter 4). This continued well into the 1980s and has not yet been fully redressed by the colonizers. For the colonizers, interactions with the colonized were not carried out as with a people who were equal in any way to that of the colonizers. The colonized could not be trusted to have a civilized discourse between equals, because clearly, they were not equal. Even if they had had their own ancient languages, social and legal systems, they could not be equal with the colonizer. Both beliefs and power determined these factors. Of course, colonial portrayals and decrees have nearly always been resisted by the colonized, who, over the years and in many places, attempted in a variety of ways to retain their own languages and cultures, sometimes through overt actions, but given the normal power imbalance, more often through covert behaviour. Over time, the colonizers and the colonized often developed a strategy of accommodation. So, PNG teachers in the early grades of schooling who used traditional languages were more often than not just ignored by the colonial powers. Traditional cultures continued to be taught in the villages and these beliefs and ways of living became quite

Ethnomathematics Indigenous Cultures

Subcultures

Ancient Greece Renaissance Europe Some Middle East

School and academic mathemacs

Modern Academies Scientific Revolutions Warfare; Space explorations; Climate change; Medicine; Economics

Figure 12.1. Kay Owens’s perspective on ethnomathematics and school mathematics.

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isolated from official school education. The reports of committees such as those chaired by Tololo and Matane were attempts to resist colonization and to express the voice of the people. However, this was not enough to stem the impact of colonization. In the prelude to and after Independence, Papua New Guinea gradually moved from the state of being colonized to that of being in a period of neocolonization. The difference between the two is instructive. Under colonization the financially stronger nation, Australia in this case, has the power and authority over the weaker nation, Papua New Guinea. The unequal relationship between the colonized and colonizer leads more often than not to exploitation of both human and natural resources for the benefit of the colonizer. Thus, the profit distribution is skewed in favor of the colonizer and profits gained from the resources of the other are not used to the benefit and the development of the colonized, but are transmitted “back home,” that is to the colonizer. We have noted at various points in this book examples of this process. In a period of neocolonization the power imbalance between the former colony and the colonizing country is still being played out. Since the former colony was not developed to an adequate level to become equal in the relationship, after Independence there was inevitably a dependence of the former colony on the stronger nation. Although in Papua New Guinea’s case it has been blessed with massive natural resources in mining, agriculture, forestry and fishing, the expensive technologies needed to exploit these resources in the modern world could not be paid for by the national Government. Hence the Government’s need to rely on substantial ongoing grants from Australia (of the order of 25% of the PNG Government’s annual budget) after Independence, and its willingness to negotiate with other powerful countries (e.g. Japan and China) and multinational companies to exploit their natural resources (The Guardian, June, 2021). However, this is not only an economic process. In general, an original colonizer, using its undoubted power, continues to export colonial ideologies thus keeping at a deeper level “their” colony in a state of dependency. Later in this chapter we will indicate how this is the case for Papua New Guinea. We will analyse a number of education projects funded by the Australian Government when the advisors took little to no notice of their PNG counterparts and insisted that their knowledge of what is a worthwhile curriculum and what are appropriate teaching practices, including in mathematics, is what was needed for Papua New Guinea. Globalization Globalization is inevitably interwoven within the processes of colonizing and neocolonialism in the modern world. There are many definitions of globalization. For this chapter the key components of this many faceted notion is that as a concept it conceives of the world as being compressed and hence the world can be talked of as being one. Hence, the many differences and constraints of geography, cultures, operations within societies, are glossed over or disregarded, and the people involved know that these various constrictions are happening. Hence globalization is not just a multinational company or an international body such as the IMF or OECD operating across various countries. Their key aims are global in outlook and their aims will only pay real heed to the local contexts when their global aims are impeded by the local situation (Atweh & Clarkson, 2010). As globalizing forces entrench their presence at the local level, often opposing reactions set in to promote the interests of the local and impede what is normally seen as an overwhelming globalization force (Atweh, Clarkson & Nebres, 2003). These fundamental societal processes have had a huge impact on virtually all countries during the modern era and, in particular, have had an impact on intra-country societal processes, including education. Within education, science and mathematics have been influenced along with languages, syllabuses, and assessment regimes (see Carter, 2015; Carter & Walker, 2010; Clarkson, 2011). In the previous chapters of this book, we have mainly taken a historical approach trying to tell something of the story of education, and especially of mathematics education, in Papua New Guinea throughout the period of colonization up to Independence, and then during the time of neocolonisation, post-Independence. At various times we have also noted the impact

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of globalization—for example, the Australian Government’s awareness and actions with regards to the reports of the UN Trustee Council in the 1960s, in Chapter 5, as well as the spread of textbooks published by an international company supplanting those developed by the NDOE in Chapter 6. In this chapter we use the intersecting notions of globalization, colonialism and neocolonialism to emphasize their impact on this story which may in turn reinforce some of the contentions we have already made, but also help chart a useful path forward in Chapter 13. The term “globalization” can evoke opposite reactions among Indigenous communities. For some it promises to provide opportunities, but for others it raises the injustices of neocolonial practices and attitudes (O'Sullivan, 2012). Some claim globalization results in sovereignty transcending state-imposed situations. For others, state implementations of education are fraught with prejudices, domination, and the lack of self-determination. In addition, O’Sullivan (2012) suggested that it is difficult to implement the Declaration of Human Rights in a country where the Indigenous people are not a minority, as is the case in Papua New Guinea—although racism and elitism may infringe on people’s rights to culture and education appropriate for them in the spaces in which they reside. Decentralization One particular belief regarding education that has been affected by the onset of globalization is the notion that a good education system must be one that is decentralized. Globally, this looks different in every state or jurisdiction, and although decentralization aims to ensure education meets the needs of the local, inevitably central governments retain control over defining aspects of the system, thus muddling the whole notion. Thus, for example, in PNG, school syllabuses for mathematics are nationally determined, but it is the provinces which manage their implementation, and indeed are responsible for professionalism in all of its aspects. Thus, a rhetoric emerges for implementing a curriculum, one which has been determined centrally, of the need to take account of the needs of individual children in their place, and with their language and culture being given due attention. However, that message is largely unrealizable, given the small amount of relevant professional development made available to teachers and given the level of resources available at the periphery—the funds needed for such “extras” are mainly held in the center. Furthermore, high-stakes examinations, set at the center, tend to dominate conceptions of practice, with a well-greased national assessment system overwhelming any initiatives that a province on the periphery may wish to advance. Pre-service education for teachers is another key component that is managed from the center. Management even within provinces is difficult, given the lack of the internet due to high charges for lines for PNG’s relatively small population (9 million in 2021) and other limited infrastructure. Curriculum Given the central Government’s determination to maintain global standards for education, it is not surprising that it is not only interested in curriculum content and the outcomes of schooling, but it has also invested in the organisation as a whole—its structure, internal processes and general environment. However, the Government’s dilemma is that the standards need to reflect local, political and socioeconomic circumstances as well as educational needs of society as a whole. After Independence, PNG has needed external financial aid for carrying out improvements in education aimed at meeting the perceived global standards. However, efforts to improve education need not start with top-down measures but can start with teacher education and community involvement in developing education curricula—as has been suggested in Chapters 7 and 8. Such an approached is not unknown elsewhere in the world (e.g., for the Maori in New Zealand, Navajo in USA, and Sámi in Nordic countries), but is often hard to achieve especially if funding is required. For this to happen, foremost is a recognition of village languages and the value of maintaining these languages through the village and schools. The argument throughout this book has been for multilingual and multicultural education.

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Despite the intentions of the reform of education which was intended to provide “education for all,” the centrally devised curriculum and associated syllabuses were a mandated set of outcomes with a lack of detail, implementation materials, and strategies. As a result, with poor to non-existent teacher education for the elementary teachers, the reform did not achieve its intention of providing an education appropriate for all children in their own places, particularly in the first years of schooling. With management difficulties in a country with so much remoteness and cultural ownership of land, despite the goodwill and intention of inspectors and teachers, a lack of funding made it difficult to achieve the desired objectives, especially given the lack of a sufficiently mature approach to teaching reading and a lack of understanding of mathematics as portrayed by the findings of ethnomathematics research (Matang & Owens, 2014; Owens, 2015). Examinations The Measurement and Assessment Unit in the Department of Education seemed to be managed effectively with expatriate support for many years. Year 12 Examinations were constructed by teams of experts like Clarkson and Owens and were carefully designed to measure the syllabus objectives or outcomes. Criteria were set for different levels. However, as mentioned in earlier chapters, these examinations also prevented students capable of further education from continuing, not because the examinations themselves were defective, but because they were used as sieving devices to fill quotas determined by the few places in secondary schools and tertiary institutions. Although during the reform period it was possible to increase numbers when Grades 7 and 8 were removed to primary schools leaving space for more students in the secondary system, an increasing number of students were seeking places. This was evident by the huge number of students who tried to enrol in secondary schools up to Grade 12 when fee free education started. From the perspective of globalization, the centralized notion of examinations attuned to standards echoed practices used in Western countries; that notion was encouraged by UNESCO and the OECD, and supported by donor countries such as Australia and Japan. Thus, formal measurement of learning achievement can be regarded as a practice inherited from the colonial era. Foucault (1987) and Fellingham (1993) criticized strategies, such as externally-set examinations and assessments which are used to set people apart from each other. Before the implementation of the educational reforms, the primary aim of the Grade 6 National Examination was to assist the selection of students for the limited places available in provincial high schools. It was also used for the purpose of monitoring standards at the primary level. The education reform called for an education system that focused on preparing students for formal employment as well as for possible return to their communities (NDOE, 1996a, 1996b, 1999b, 1999c; 2004a). However, examinations were still used to identify the successful students who would go on to secondary school and further training, and the unsuccessful ones would return to their communities. The former Grade 6 National Examination was replaced by the Grade 8 National Examination which was developed for selection purposes only (NDOE, 2003c). CRIP and the Australian Council for Educational Research (ACER) have also developed new assessment tools to measure students’ learning achievements in Grades 3, 5 and 8 at the primary level. These types of assessment may not adequately measure other out-of-school variables and individual differences in terms of skills, talent, and other achievements that may contribute to how students perform on these assessment tasks. The assessment tasks may incorrectly label some Papua New Guinean students in comparison with other students. For example, if someone does poorly in mathematics that person may be labelled as intellectually less competent than others; however, that poor performance may have been related to language or teaching issues which were not be captured in the assessment tasks. Foucault (1987), Lather (1991), Davis (1994), Fellingham (1993), Pennycook (1998) and Said (1994) have all documented how “voiceless” people are treated by influential others. Mechanisms are generally established to categorize people, and attitudes towards them are created over time and become established practices. Similarly the national examinations in PNG

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have contributed to the emergence of two groups of Indigenous Papua New Guineans—the influential and the less influential. The less influential people do not usually have a voice concerning development issues in their rural communities and self-help town settlements. From our observations as researchers, the influential people generally have power over these groups of people and in most cases make decisions that they think are better for them. Similar to Lather’s (1991) discussions concerning long-term practices which create labels for groups of people, these formal assessments provide the basis for “defining” successful and unsuccessful students in the Papua New Guinean school system. The examination systems also created the label “drop-out” for the majority of young people. Paraide’s people used a more degrading term, tabururu, which means, that these groups of young people are useless and all they are good for is to destroy. Paraide suggests that her people have felt that this was a fair description, as these young people were deprived of their communities’ Indigenous education, and so did not learn the skills needed to survive when they returned to the village. They had received some formal schooling, but that was geared towards formal employment. Consequently, when they were forced out of the education system through their performances on the examinations, the implication was that they did not gain sufficient skills from their formal schooling to survive in the Western world, and had not gained adequate skills from their Elders to be able to survive in their own communities. It is worth saying here that in the recent times with the reform change of education valuing culture and the focus in preparing young people for self-employment through technical and vocational programs, the word tabururu seems to have disappeared from the vocabulary for students who fail to gain a place in secondary school at the end of primary. Local Focus for Education We noted in the introduction to this chapter that with the force of globalization, local resistance can grow. So, in contrast to the globalization of Western education for Papua New Guinea it is important to set out alternative possible worldviews, especially those held by the Indigenous peoples. Foundational educational processes in Papua New Guinea and elsewhere have been carefully crafted around observing natural processes, adapting modes of survival, obtaining sustenance from the plant and animal worlds, and using natural materials to make tools and implements (see Chapters 2 and 3). Paraide (2010) noted that having English as the language of instruction in formal education has been, and still is, favored over having vernacular languages as languages of instruction. Papua New Guineans and others have yet to realize fully that Indigenous bodies of knowledge are embedded within these vernacular languages. As discussed in Chapter 10, languages are the most important means of expressing knowledge systems in cultures and communicating that knowledge to the next generation (McConvell &Thieberger, 2001). Knowledge is built into the day-today activities of people, and the choice of languages to be used as languages of instruction carry messages about what is important in everyday life. In that sense, the key to an Indigenous culture with its ritual and kinship is its Indigenous language. These cultures embody forms of knowledge that are just as valuable and practical as that found in Western and other knowledge systems. The National Department of Education, through the reform curriculum, recognized and encouraged local decision makers to have Tok Ples as the language of instruction, not only in the early years of schooling but also, when needed, in higher grades. Their aspiration was for this change in policy to become over time a powerful bonding agent between Indigenous education and schooling (National Department of Education Papua New Guinea (NDOE), 1999a, 1999b, 2000, 2003a, 2003b). If the Tololo vision had been implemented at Independence and not 10 years later (when Matane attempted to introduce it) or in fact 20 years later when the reform reached schools, the loss of language and culture would have been significantly reduced and the aspirations of these committees would have been more easily implemented. However, significant lack of funding for elementary schools and for preparing orthographies and early readers in the local languages

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meant the reform aspiration were not realized. The policy changed in 2013 reverting to the use of English as the language of instruction once again. Paraide (2008) had shown, vernacular languages had long been seen only as a tool to teach Western knowledge, and were not a rich source of Indigenous knowledge. Much earlier, Smith (1975) had maintained that PNG’s Indigenous education, Indigenous knowledge and languages had been suppressed in formal schooling situations since the colonial era. It is not surprising, really, given this history that English is still viewed as the necessary language of instruction for formal schooling in PNG. Western knowledge still dominates current PNG curricula, including in mathematics, because it is generally viewed by Papua New Guineans, particularly the Western educated elite and power brokers within the wider society, as being more “valuable” than their Indigenous knowledge. Given this situation, in recent times, vernacular instruction at the lower levels of formal education has again been abandoned and the formal language of instruction at the lower levels of formal education has reverted back to English.

Colonialism and Neocolonialism

Beginning in the 1400s through to the 1900s European colonizers from Italy, Spain, Portugal, The Netherlands, United Kingdom, France, Russia and later Germany and the United States of America sought to acquire land in various parts of the world. These colonizers invaded populations, and used their superior weaponry and organizational structures to assert control, in various ways, over many countries across the world. Of course, the Western cultural expansion was nothing new. Many groups have expanded throughout recorded history and almost certainly much earlier than that. Japan and China, for example, at various times also extended their borders or were forced to contract due to outside forces. Following World War 1 some European countries withdrew from their colonies, principally Germany, as they were required to do so by the victors. Others, such as Japan established themselves in new territories, namely Korea and in parts of China. However, after World War II, sometimes by agreement (e.g. India), and often by armed force (e.g. Algeria) many former colonies became independent. To assist the process, a number of international organisations were established which became avenues for globalization—and, interestingly, the International Monetary Fund (IMF), the World Trade Organisation (WTO), and even more importantly the United Nations (UN), were all substantially underwritten by the United States of America. This new world code was promoted by the UN, and developed and implemented by the Western powers, and in particular the United States of America. Various countries, such as the states in the Pacific including Papua New Guinea, the many African states, South and Central America, and some in South East Asia were expected to adhere to the decisions which were made. It was claimed that the decisions promoted liberty through free, fair and liberal elections, a free media with civil society and private sector involvement; through fighting corruption and mismanagement, providing transparency and accountability of public affairs and service delivery; through protection of human rights, especially of women and children; and through working to eradicate poverty (the latest version of which is to be found in the Millennium Goals, 2000). Thus, the core values and structures of the recent colonizers were expected to be embraced by their recent colonies, even though the colonies were ostensibly now independent. This was neocolonization. Wickens and Sandlin (2007) insightfully commented on the new state of affairs: Neocolonialism describes a situation wherein although many formerly colonized countries have gained geographical and political independence, “cultural and economic independence was never really, if at all, won. The colonial systems of domination continue ... as the former colonizers continue to economically, culturally, financially, militarily and ideologically dominate what constitutes the so-called developing world.” (p. 276)

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Smith (1996) noted that “neocolonialism operate[s] according to a number of distinct processes: through the terms of trade; the need for aid; the repatriation of profits; and technological dependency.” For example, Kenya’s approach was to have an NGO Co-ordination Act (1990) (Kenya Gazette Supplementary Acts, 1991). Lang’at (2008) studied the enactment of this approach in Kenya and elsewhere and noted that colonialism restructured not only the economy of the countries but also the education systems which were in place. Such restructuring continued after independence, this being a key element of neocolonization. In restructuring the education systems to align with Western ideas, transgenerational knowledge sharing that had occurred for at least centuries was affected significantly by schooling and particularly during high school, when students were often absent from their villages for long periods of time making such enculturation difficult. The same has been occurring in PNG. Neocolonialism is in some ways even more pernicious than colonialism. Another feature of this process is that racist attitudes of those who were colonized persist, at least among the Western educated elite, and they in turn can begin the process anew of intra-colonizing. Such a process is enhanced when “better” education means the elite is “given” the power to colonize (Wickens & Sandlin, 2007). Wickens and Sandlin point out that neocolonialism maintains the myth that literacy will bring prosperity, in the case of Papua New Guinea, literacy in English. But despite schooling, many Papua New Guineans remain illiterate due to a lack of reading material in any of their languages, and the teaching of literacy occurs in a language not spoken from birth and rarely heard in many villages even today (see various comments from those interviewed in Chapter 10). Even social conversations between fluent English speakers, such as teachers, are not normally in English. For most of PNG’s history, right up to the present, most knowledge is communicated orally, remembered, then used in local political situations and in economic exchange systems, albeit with discussion and sometimes with conflict. Rarely are written records kept. According to the classification of Lassou, Hopper, Tsamenyi and Murinde (2019) the situation in Papua New Guinea is termed “soft neocolonialism,” that is to say there is no forced or armed intervention. However, like many of the African states, Papua New Guinea is still driven by international monetary systems governed by Western approaches through global companies so that much money returns to the colonizing power (Australia), or other external powers (Japan, China) and others such as international global companies. In PNG the accounting systems have often been managed by overseas agents in conjunction with locally trained professionals (initially graduates from PNG University of Technology but now from all the universities including some who graduated from overseas universities). Despite this, the spending of aid money has had systemic issues that still persist. Many aid personnel and advisers, mainly expatriates, and their counterparts, particularly employed in the private sector although some on Government projects, receive high salaries and allowances, stay in expensive hotels for security reasons, and fly or drive 4-wheel drive vehicles; they often have armed security guards and satellite systems as a necessity, due to poor road conditions and major personal safety issues. Other aid personnel, employed by missions and at times Government, are volunteers living on local salaries which are topped up by health and travel insurance and/or the provision of free education for their families. Even with the latter “extras” the disparity of conditions has been obvious to those with whom they work, and at times an impediment to driving forward the projects on which they are involved. For nationals in some places, attaining respect and safety also requires similar conditions, although it is often obtained at cheaper rates through local acquisition. Lack of engagement from local people has been commonplace, if there was no perceived immediate benefit to the participant. This can be seen within villages but also within institutions. In the Evaluation of PASTEP project (see Chapter 7), the team convinced the funding body (AusAID) to extend the project by six months so a preliminary visit could be made to the teachers colleges that were involved, and to the NDOE, to ensure that they were not only aware of what the project involved, but also that full cooperation would be forthcoming when data were being gathered. Subsequently, each individual visit to each college was extended by one day so there

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would be time not only to talk to the principal and staff, but also to the senior students regarding the Project. During the initial visit and on every subsequent occasion, one question that was almost always asked was “How will we benefit?’ Colleges were not interested in participating in a project for which the outcomes may have some systemic value but would be irrelevant to their own immediate situations. Money has become important for the status of “big persons” in Papua New Guinea, as it is in most other countries. Aid money for the funding of various projects has often been syphoned off in order to supplement the salary of a big person if that person’s salary alone was not deemed sufficient to enhance the person’s status. At other times, some held on to resources in the hope of receiving large sums for the sale of the work achieved, even if funding was provided to develop the resources. Intellectual property rights were recognized in the country, but often guidelines were either ill-defined or potentially crippling with respect to the progress of a project. During the colonial era, linking classroom mathematics with the students’ ethnomathematics was generally discouraged. There was little attempt to show the similarities and differences between the mathematical systems in order to assist students to gain a better understanding of either, or both. Before the 1970s similar concepts such as number and measurement in both knowledge systems were rarely considered, and in the formal teaching of mathematics there was no effort to let them complement each other. Parallel teaching was not expected, and integration of mathematics in both knowledge systems, was not included in the primary and secondary mathematics curricula. This resulted in many Indigenous mathematical knowledges being lost because in the formal school learning environment the major focus was on Western mathematical knowledge. This issue was exacerbated in secondary schools since most of the students in those schools had been removed from their Indigenous communities in order to learn the Western knowledge systems. Therefore, only a small amount of curriculum time and space, if any, was devoted to learning and applying Indigenous knowledge to situations which would be likely to arise in the students’ everyday lives. As a well-educated Papua New Guinean and as a researcher, Paraide (2010) observed that during the colonial pre-Independence period, and some years after Independence, PNG’s social organisations and Indigenous knowledge systems and practices were viewed and categorized by the colonizers and some post-Independence expatriates, as “not normal human behaviour.” The traditional social organisations, knowledges and practices were different from those of the expatriates’ own practices and ways of life and, therefore, the expatriates deployed strategies and practices to suppress them (Paraide, 2010). Pennycook (1998) and Said (1993), in their discussions of colonial influence, also highlighted the fact that colonizers promoted their own languages, knowledge systems, and practices in the colonies. As noted a number of times in earlier chapters, Australian officers working in PNG during the colonial era, and for some years after Independence, had a great deal of influence on the type of services that were provided to the “native” (sic) population.

Language as a Tool in Colonial and Neocolonial Approaches to Education

Although there were some Indigenous languages used for trade negotiations prior to Western contacts and colonization, the main focus on language after colonial times, and something which is still evident today, has been what might be called “the underlying language ideologies.” According to Makihari and Schieffelin (2010), these are cultural representations, whether explicit or implicit, of the intersection of language and human beings in a social world. Mediating between social structures and forms of talk, such ideologies do not just concern language—rather, they link language to identity, power, aesthetics, morality, and epistemology in terms of cultural and historical specificities. Through such linkages, language ideologies influence not only linguistic form and use, but also affect workings within the social institutions and are used to define fundamental notions of persons and community. (p.14)

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As noted a number of times in earlier chapters, from the beginning of colonization for both Germany and Australia, the issue of language of instruction has been a matter for debate. During the colonial period the decision on what would be that language in schools was made by the colonizing power, although leaders within many mission schools and other schools, defied the official position by teaching at least the first years of schooling in Tok Ples. Brown (1908), Murphy and Moynihan (1936), Mennis (1972), Smith (1975), Barrington-Thomas (1976), Dickson (1976) and Giraure (1976) all addressed the language-in-school issue as it existed before Independence (which was achieved in September, 1975). There was also a body of anecdotal and at least of descriptive research to draw on. Toward the end of the reform period, projects previously implemented by SIL in hundreds of language groups across the country, were being recognized and used to train more teachers from elementary schools in the use of phonics beginning with local familiar sounds and words (local languages) and then new sounds and new English words with both literacy in the local language and English. In addition, projects have been implemented aimed at revitalizing the use of shellbook—that is to say, picture books in which the story of the pictures can be written in local language. Culturally relevant books have been introduced into elementary schools through a literacy project (Simoncini, Smith & Gray, 2020), through the mathematics for elementary schools project (Owens, Bino, Edmonds-Wathen & Sakopa, 2014) and the Buk Bilong Pikinni libraries (sponsored by the Australian Ambassador’s wife). A New Zealand aid project has provided school magazines and early readers to schools. Furthermore, a preliminary workshop aimed at defining “appropriate books” was held with the support of Australian Aid—it was chaired by Kay Owens and organized by Philippa Darius from the Elementary Schools Curriculum Development Unit. It followed a one-day workshop which pointed to the importance of replacing English with vernacular languages in the schools. In particular, Simoncini et al. (2020) showed that the culturally-relevant books were most read by the children. With the transition to the neocolonization period post the reform under O’Neil’s government, the issue of what language of instruction should be used was revisited. It was reported that not only was there vast anecdotal evidence for the use of the language that students came to school with, but the research evidence was also building supporting the use of Tok Ples, especially in lower grades (Clarkson, 1992) and by teachers (Muke, 2012). The literature supporting that conclusion was reviewed in Chapter 10 of this book. However, it will be useful here to draw together the continued discussion on language use in the classroom from a different point of view, one which has only been hinted at in previous chapters. Paraide’s (2010) classroom observations as a former National Monitor and currently as a researcher and education consultant showed that schools since 2000 continue unquestioningly to adopt a Western-style system of education, with English as the language of instruction, including in the lower levels of schooling. Teachers use English as the language of instruction, despite the introduction of the language policy in the late 1990s with the reform which prescribed the use of vernacular instruction in elementary schools and bilingual instruction in the lower primary level (Guy, Paraide, Kippel, & Reta, 2003; Paraide, 1998, 2002). McKinnon’s (1976) argument supporting English to be the formal language of instruction was buttressed by the view that the use of PNG’s vernacular languages had little support in the literature. He claimed that it would have been too expensive to embark on translation programs. McKinnon did not seem to envisage that appropriate textbooks could be written by nationals in their own languages, including in Tok Pisin; but that happens now with the on-line news website of the Australian Broadcasting Commission (ABC)1. Teaching English is a rather modern invention developed in India during the mid-1800s. It was a deliberate colonial strategy implemented by the English to teach the “natives” how to use “proper English,” and one which sidelined thousands of years of Indian oral and written literary traditions (Somerville, 2021). O’Sullivan (2012) https://www.abc.net.au/news/tok-pisin/

1

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argued that a similar process had sidelined Māori literature in New Zealand as well as many of their other indigenous traditions. Such an argument throws into relief the continued justifications offered for why English had to be the language of instruction in Papua New Guinea, and perhaps why the use of traditional languages have been suppressed, and why traditional mathematical ideas are rarely recognized in schooling in Papua New Guinea. McKinnon (1976) also argued that English would promote national unity for a new nation formed from a huge number of different linguistic and cultural groups. This was a clear marker of neocolonialism. Interestingly, Laycock (1976), writing around the same time as McKinnon, argued that the lingua franca Tok Pisin had been a widely spoken lingua franca in PNG all through the colonial era and would continue to be spoken after Independence. Interestingly, to this day Tok Pisin is often used in the national parliament, and has been since Independence. In fact, Laycock’s prediction proved to be true, with 44% of PNG’s population being Tok Pisin literate in the early 2000s (Rannells & Matatier, 2005), even though it officially played little role in schooling. Clearly it could have been chosen as the unifying national language. Laycock (1976) argued that: An essential factor in a democratic government is the ability to communicate with all of the people, to hear their grievances, to disseminate information of national importance. Pidgin is fulfilling this function now through the Pidgin radio broadcasts. (p. 186) Today, there is very little English radio. As a well-established Creole with grammar and vocabulary, Tok Pisin could have been chosen as the formal language of instruction. However, as Laycock (1976) observed Tok Pisin: is hampered in its bid for recognition, at least as far as the non-Indigenous population is concerned, by the unsuitable name. The name ‘Pidgin’ [a name denigrading the Creole and not used today] tends to rouse feelings of hostility among Europeans, especially those who know nothing of the language and regard it solely as a form of broken English. (p. 185) Childs and Williams (1997) cautioned that when groups of people are not treated equally over a prolonged period of time it can create superiority and inferiority myths and develop powerful and less powerful divides that lead to serious misunderstandings. Fellingham (1993) and Said (1999) also argued that it is inevitable that the dominant partner’s perceptions of the world, how the world is organized, and those practices which are viewed as “better,” are forced upon the less powerful partner. A further consequence in such cases is that the less powerful partners have minimal influence, contribution, and participation in any form of development. The dominant partner, with sometimes a lesser contribution from the weaker partner, identifies problems and possible solutions and then drives decisions on such issues affecting both parties. Clearly this happened with the language-of-instruction question at Independence, and has continued to happen since then, with the interruption in the late 1990s during the period of the reform curriculum. Matane (1976) and Tololo (1976), members of the weaker party in a neocolonial political structure, showed that PNG Indigenous knowledge, teaching strategies, and learning styles were made virtually invisible by both missionaries and Government education policies through the colonial and new Independence periods—although many of the missionaries did support the use of traditional languages in the early years of schooling. As noted in earlier chapters, in fact Tololo in particular set out a position that would have led education in Papua New Guinea down quite a different path, particularly his emphasis on the use of Indigenous languages in education. However, that was not to be. The colonialism, so ingrained in the way the country operated, and then morphing gradually to a neocolonial phase, was not prepared to risk such a strategy.

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We believe that the whole issue with respect to the official language of instruction remains a vital key for further movement in Papua New Guinea education. As Makihara and Shieffelin (2010) pointed out: Though some expect language to be transparent—a list of vocabulary items containing meanings with largely functional equivalencies across code boundaries—this is not the case. It is often not only difference in codes and problems in translating between them that make understanding difficult, but ideas about the nature of language itself or its functions which, when taken for granted on one side and unimagined or even unimaginable on the other, lead to the misrecognition of meaning and even intentions. (pp. 13–14) The languages are fast changing, especially for groups that are small or scattered. For example, May and Loeweke needed to revise the New Testament in Fasu after 20 years because the difference in current language was like the centuries of difference with modern and old English. With further oil and gas explorations in the area, more changes have occurred. Indonesian has affected the counting languages for large numbers in the base 6 Kanum language (Donohue, 2008). Changes had occurred in the past that led in some cases to truncated body-tally systems, but this continued with the introduction of pounds and shillings and later decimal currency (Saxe, 2012). However, English particularly affected the language for operations like multiplication, so that transliteration was used in Motu and Roro with teachers needing to discuss how to talk about multiplication and other arithmetic operations in their own languages (Edmonds-Wathen, Owens, & Bino, 2019, based on 2014 and 2016 field work). Nevertheless, conversations around translation have “heightened awareness of code boundaries and differences between codes” (Makihara & Shieffelin, 2010, p. 21). A willingness to vary the way of combining frame words to refer to a particular number may be more flexible for many languages than for English (Owens, Lean & Muke, 2018). Differences are both cognitive and social; power plays of colonization and neocolonization are still prevalent. Particularly important is the Melanesian understanding of language itself as an identifier in a much more socially significant way than in most Western countries. The use of words like wantok from Tok Pisin or poro from the Highlands (Hagen words, incorporated into Tok Pisin) provides a case in point. Using English identifies a person’s status in some cases, like the laptop case. Code-switching is used for more than for explanatory or immediate linguistic purposes. “Reciprocal multilingualism” (p. 24) also maintains balance and egalitarian relations in the social system. It plays an important social role in the relationship of speakers within a situation. We will return to the issue of language later in this chapter, and again in Chapter 13.

Reform Period and Neocolonialism

As with language policy, school curriculum reforms were dominated by outsiders, as would be expected in a country’s neocolonial phase. The question arises, however, how long will this phase last? A newspaper article entitled, “OBE (outcomes-based education) is a recipe for disaster” (Maima, 2008) argued that Australian donor financial support shaped, if not determined, the then reforms being implemented in PNG school curricula—with the Curriculum Reform Implementation Program (CRIP), in particular. At the same time the World Bank recommended (dictated?) that the structure of the school system would have to align with an aspirational universal education format, and that would be one of the conditions for providing World Bank loans for reforms. The dominance of the past colonial power, and the involvement of the World Bank, were markers of the interlinking of neocolonial and globalization forces working in tandem. During the late 1990s through to the 2000s Australia became involved in three large aid projects and in so doing provided many of the personnel, resources and much of the cash, thereby maintaining Australian influence. One was the Primary and Secondary Teacher Education Project

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(PASTEP), discussed briefly in Chapter 7. The second was the Elementary School Project (ESP), which built on earlier work carried out in Papua New Guinea and held considerable influence within the Department. However, with ineffectual teacher education policy at its grassroots, ESP had little lasting impact. The third and most enlightening part relevant to this discussion was the Curriculum Reform Implementation Program (CRIP). The structure, and underlying philosophy put forward to support the CRIP’s outcomes-based (OBE) curriculum documents, as in Australia (ACARA, 2009, 2017; Ellerton & Clements, 1994), dominated the ontology and hence epistemology of the curricula that were to be introduced. It is no surprise that assessment, under the guise of standards, was the central plank of the product. CRIP aimed to develop and produce written documentation which would stimulate curriculum reform. To enable this to happen effectively, the project was tasked to build an internet backbone so that electronic files could easily be shared across the country. In the early 2000s, that tactic was being used in many nations, across the world. However, with respect to the critical process of developing the curriculum documentation, at the start there were no meetings with the staff of the Curriculum Unit of the NDOE for over a year. Hence at the end of the first year of the project, the project’s Curriculum Management Plan was launched without Curriculum Unit representation. The power exerted by the Aid organisation, which was a clear marker of neocolonialism in practice, was shameful. There seemed to be a lack of vision and of understanding by the CRIP’s leadership. As time moved on there was finally forceful representation by PNG’s Curriculum Unit management (Ryan, 2008) which resulted in the Unit’s curriculum advisers beginning to work alongside the expatriate CRIP staff. However, even then monies were spent on CRIP while the Curriculum Unit was starved of funds. For example, little money was provided even for travel by Unit members so they could work with the CRIP Advisory Committees. This was despite the Unit actually providing materials for schools based on their PNG experience. Sadly, many Curriculum Unit personnel reported (personal communications) that their suggestions and work were often dismissed in an off-handed manner. It took another protest from the teachers colleges before their staff were belatedly brought into the CRIP process. After all, they were supposed to be preparing teachers for this reform, but they had not been kept informed or consulted on the outcomes-based education philosophy on which CRIP was premised. The CRIP process was perhaps the most blatant of neocolonial processes adopted at the time of the reform period. The grassroots were, at best, only partly involved. Advice from professionals, already in curriculum positions within the existing Papua New Guinea structure, was not sought, and if it was offered anyway it was often regarded as a hindrance and was virtually ignored. Expatriates used their Western management systems and provided the associated list of products and timelines to be achieved. “Progress” (assuming OBE premises to be appropriate) was ticked off on completion of set tasks. Within this reform, those developing the “new’ science and mathematics curricula unashamedly took on the Western science and mathematics assumptions being adopted by curriculum developers in Western nations, despite the fact that sometimes these were sharply criticized (see, for example, Ellerton and Clements’ (1994) critiques of what was being done in Australia). The only acknowledgement that these subjects were developed for Papua New Guinea students was the inclusion at appropriate points, of some cultural objects. There was no evidence that the expatriate developers understood that Indigenous science and mathematics were integral to people’s ways of understanding existence. The notion did not seem to occur to these expatriate education writers who had grown up in very different worlds, that the many years of cultural development which allowed these cultural entities and ways of knowing to emerge. Nor did they recognize the possibility that such knowledges were worthy of comparison with Western science and mathematics, and at times overlapped with their own understandings (Ryan, 2008; authors' personal experiences). It is difficult to escape from the conclusion, therefore, that the reform period, epitomized by CRIP, lacked the advantage of having received grassroots advice from Papua New Guinean educators on curriculum and schooling reforms. There were some saving graces, how-

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ever—particularly, in the training of counterparts, in the framing of other projects, and in the establishment of School Boards. Language issues were often at the forefront of views fuelled by an emphasis on achievement. Apart from the predominance given to the ideas of the elite, who mainly sided with an English/Western culture bias for Papua New Guinea education, another compelling issue was associated with the acceptance of attitudes and influences that international donors brought to policy decisions. Development and progress standards in education are generally set by donor countries (Hickling-Hudson, Mathews, & Woods, 2004). World standards of education are used to measure progress, or non-progress, in education, in developing countries such as PNG; in mathematics education, international league table in performance on language and mathematics and science tests, especially TIMSS and PISA results, are often used as the basis for such comparisons. In PNG, a study of Pacific nations suggested PNG was not achieving as well as these other nations although it could be argued that many of these nations had a common local language used in instruction, fewer languages and cultural differences, and indeed remoteness although these were aspects of the other nations. Many of the Papua New Guinean critics of vernacular and bilingual education argue that high standards are crucial for Papua New Guinean progress, and that this might best be achieved via funding from external funding bodies—particularly if that is used to develop greater competency in English among students, because English is the “language of progress.” In 2013 that kind of thinking was used to justify the move back to the introduction of using English as the language of instruction from the beginning of formal schooling in PNG. Those who supported this argument maintained it would ensure better mastery of the English language and other literacy skills, and would generate higher educational standards indicated by improvements in performance, perhaps even comparable to those achieved by developing countries across the global community. They were not convinced that vernacular and bilingual teaching strategies could ever achieve these goals. It would seem that often the politicians and their advisors were not even aware of the counter-arguments based on educational research that could be, and were being, used (see, e.g., Ellerton & Clements, 1994; Halai & Clarkson, 2016).

Revising the Curriculum and Structure of Education

At Independence, there was hope that an education policy might be put in place in PNG that would be suitable for all students. Despite attempts to improve local-language instruction in preschools, and the naming of schools as community schools in 1974, there was insufficient detail in curriculum documents and a preponderance of expatriate advisers to education who failed to interpret appropriately the intentions expressed in the Tololo Report. Although implementation was likely to be difficult in such a rugged country with limited teacher education facilities, there was not the necessary knowledge to appreciate both the mathematics and the mathematical language of the various societies, or how they might link to school mathematics. In the following decades, work around the world on ethnomathematics would indicate how Indigenous mathematical knowledge could be valued and used to encourage equity in education. In many respects, PNG provided early input through the work of Alan Bishop and his PNG students, through Glen Lean, Geoffrey Saxe and other cognitive cultural psychologists, and through the work of those who followed—like Rex Matang, Patricia Paraide, Charly Muke, Philip Clarkson and Kay Owens, and the many teacher education students who made connections in their elective research on mathematics, language and culture (Owens, 2014). The area of language was a place in which PNG was rich and could offer to the world different perspectives on education. Arguably, there were too many advisers without the vision, and the knowledge of PNG’s history, which was needed. Nevertheless, implementation of the reform and instruction in vernacular including Tok Pisin in elementary schools, looked promising. There were hopes it could generate desirable reforms in education and in mathematics education. However, its implementation was far from

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satisfactory. The 2012 announcement returned the structure of schooling to an earlier form where only one year might be likely at elementary school in the village. The organisation of this change took another 6 years and is still not well implemented. Grades 1 and 2 could be taught in the village, or students could travel to the nearest primary school. However, this might have reduced the opportunity to improve even the basics of education. The naming of schools also changed over the years from “community school” to “elementary school and primary school” and then to “primary schools” with Grades 1 through 6 which may include the start of “high school” in Grades 7 and 8 depending on availability of classrooms. There was a continuing Western curriculum, albeit for mathematics that was interpreted by the Japanese advisers who seemed to have little to no regard for PNG local knowledges (see Chapters 2 and 3). The syllabus was dominated by assessment ideas. Education was seen, not as something to enrich village life but to assist individuals and groups to gain money—just as had been the case with earlier African projects. This did not happen and resulted in there being students roaming the streets and villages, lacking a sense of cultural belonging, as there were no paid jobs for them. Students had been educated to believe that secondary and tertiary education would help them get jobs, and the reality was that often this did not happen (in Owens’ last visit to PNG in 2016, she met several men and women making that statement—see Yombu Selden’s cameo in Chapter 9). In the years following Prime Minister O’Neill’s announcement of free education in 2012, it became clear that tuition fee free education would not necessarily provide the answers to PNG’s education problems—fee free education had occurred in 1982 (Bray, 1987) and 2002. Although the O’Neill Government did provide considerable funding to education to manage tuition fee free education, the numbers of students in schools became overwhelming. Schools and teachers colleges had enormous problems placed on them, especially the teachers colleges which struggled to produce enough teachers for schools with rapidly increasing numbers of students. With English as the language of instruction, Western values began to dominate curriculum implementation. Neocolonialism, despite high nationalisation, was again evident.

The Role of Tertiary Education in Neocolonialism

In this section we look again at the tertiary sector, not so much as with the historical detail of Chapter 7, but through a colonial/neocolonial lens. Tertiary education was established late in the colonial period of Papua New Guinea. The movement for this tier of education was really only contemplated in the early 1960s when it began to dawn on Australia that like so many other colonies throughout the world, Papua New Guinea would one day become independent. So again, circumstances dictated this change of thinking rather than any forward-looking idealistic motives. Coincidently, change was afoot at the tertiary level in Australia at this time as well. New universities were being opened such as Macquarie in Sydney and Monash in Melbourne. It had been many decades since additions had been made to the original six universities, one in each state. Hence UPNG fitted this movement and for the years up to Independence was regarded as one of these new Australian universities, albeit a very small one. So structurally the new university, UPNG, was folded into the system run by and servicing the colonial master. Interestingly, the upgrading of the original Institute of Technology to a university (Unitech) predated the same movement of a number of well-known Australian senior technical tertiary institutions to university status (e.g. RMIT in Melbourne, which only occurred in 1990). Furthermore, this university established teaching with technology before many of the amalgamated universities in Australia such as Western Sydney University. Maybe this case was one of the few instances (Chapter 11) in which the wants of the less powerful country over-rode normal procedure of the more powerful, thanks to visionary expatriates and a willing workforce. The two universities in PNG (UPNG in Port Moresby and Unitech in Lae) were seen at Independence as necessary to produce professionals to run their own country at an Australian level of achievement—another marker of colonial/neocolonial thinking. This was not difficult to

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achieve because student entry was in the main after completing Grade 12 or completing an equivalent Preliminary Year at university. With only the two universities the number of entry students was severely limited due to the number of school examinations embedded in the school system from Grade 6, with only the highest-performing students making it through to Grades 11 and 12, let alone to tertiary studies. As was noted in Chapter 7, UPNG did not always fit in with Government plans both before and after Independence. Being modelled on Australian liberal arts universities which guarded their independence for thinking, researching and speaking truth to power, this is not a surprise (see Chapter 7 for more detail). Indeed, as has been noted earlier in this book, many of the founders and early parliamentarians of Papua New Guinea learnt their craft when attending UPNG or Unitech. This model of university was strengthened by the Australian and New Zealand personnel recruited to the new universities in PNG who recognized from firsthand experience, the need for independence of universities and appropriate courses for the country as well as high standards for the professions2. This was a far cry from the universities of India and Africa that at first mimicked British ones, and for many years after their independence still regarded education as a service to the imported, colonial professional organizations and systems (Ashby, 1966). For example, the PNG University of Technology (Unitech) recruited its first Vice- Chancellor and its early Heads of Department—of the calibre of Professors Woodward, Dale, Wong, and Greenwood—as persons who would seek to educate future PNG professionals to the highest possible standards. They all sought to have their own academic freedom, to make courses applicable for the PNG context, to recruit staff who were well qualified professionally, and yet were high-quality teachers so their courses might be regarded as having world standing. In testament to these initial and continuing aims, the engineering graduates were able to gain registration with the Institute of Engineers, in Australia. Was this yet another marker of neocolonialism? Whatever it meant, it is clear that many graduate students were risk-takers and capable, even if they did not have everyday Western experiences of diverse engineering systems other than those that the Australian Government had instituted in PNG (concerning areas like water supplies, sewerage, electricity grids, and telephone systems). Early vice-chancellors were themselves Papua New Guineans, capable but gradually limited by lack of funding that held back their work, so it is a credit to them and their other senior PNG university staff for what would be achieved. If one crucial goal for Papua New Guinea’s universities is now to create an environment in which everyone can succeed regardless of their social backgrounds, there must be a concerted effort to incorporate alternative ways of coming to know the world which better reflects the backgrounds, experiences and languages of the diverse student bodies within the universities. It was noted above that the school system, with its examinations, sieves many students out leaving only a very few who graduate from Year 12 and hence can progress to university. The high fees charged to tertiary students exclude many other possible students. There are also racial and other forms of discrimination which impinge on tertiary students’ and lecturers’ lives. For example, it is well known that women who study at university still often face sexual abuse, and hence females are less likely to attend than males, an issue that has plagued all PNG universities from their inceptions (see Naomi Wilkins’s MPhil research, personal communication, 1997). If potential teachers opt out of a university course because of such non-educational reasons, the education for the whole nation suffers. Also, due to racial bullying some staff and students have left tertiary institutions (personal communications with a number of ex staff). Such practices are a long way from the thinking of those who worked to create the initial tertiary institutions and Senior High Schools—almost all of whom wanted their institutions to be fully co-educational, and drawing their intakes from all (or at least many) Provinces to create

They were willing to stand up to the ex-Africa colonial administrators and demanded greater rights for women staff and other PNG staff. 2

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Table 12.1 Number (Percentage) of Male and Female Teachers and Head Teachers in Primary Schools, and Lecturers in Primary Teachers Colleges (in 2009) Gender Teachers Male 26 904 (61%) Female 16 934 (39%) Source. Waninga, 2011, pp. 4–5.

Head Teachers 6228 (74%) 2139 (26%)

Lecturers 151 (68%) 72 (32%)

Table 12.2 Reasons Given for Why Girls Left Schools Reason for Girls Leaving Schools Non-payment of school fees by parents Lack of support from parents who gave preference to boys Early pregnancy and forced marriages Teachers’ negative attitude towards girls Laziness and inability to cope with school work Peer pressure and involvement in social activities Lack of facilities to cater for older girls’ needs Conflict of interest with opposite sex Tribal fights and fear of threats and rape Basic needs not met by relatives Rapid maturity and ashamed of being over-aged Threatened because of drug abuse by boys Family problems Withheld by parents to assist in family obligations Disease/sicknesses Accusing women and young girls of witchcraft Source. Waninga, 2011, p. 6

No. of Schools 29 25

Percent 100 86

23 19 17 16

79 66 59 55

13 11 8 8 7 5 5 5

45 38 28 28 24 17 17 17

1 1

3 3

learning spaces where students would come to see themselves first and foremost as Papua New Guineans. That was, perhaps, another colonial dream which was not well thought through. Interestingly, university qualifications and in particular doctorates have been merged into prestige status within the cultures. In 2009, there were less than 40% of primary school teachers who were female and less than a quarter of the head teachers were female. Similar percentages occur for lecturers at primary teachers colleges (Waninga, 2011 based on 2009 UNESCO project figures—see Table 12.1). Waninga, Yoko, Tieba and Apingi (2007) reported, in their study of child friendly schools, that they had asked head teachers of 33 schools (from which there were 29 replies) why girls left school (see Table 12.2). Fees were an issue, but there were also other reasons given. At the time of the study 2004, HIVAIDS was still taking its toll of parents’ lives, so students were cared for

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by relatives, but general poverty was also an issue for larger families3. It should be noted that girls also did not continue if they did not receive sufficiently high grades to continue into the next level of schooling, so the numbers of female students continued to fall throughout school. Noticeably, after a Gender Equity in Education Policy (GEEP) and a Gender Equity Strategic Plan (GESP) were developed and put in place in 2002, there was little progress made so far as girls’ achievement was concerned. Perhaps that was because the Policy and Plan were drawn up with external advice that did not take account sufficiently of local attitudes and social situations. Komhiol4 (2020) considered preservice teacher education institutions, one in a matrilineal area and the other in a patrilineal area although both could take students and staff from anywhere in PNG, to analyse why no or little progress was occurring in the gender area. Primarily the majority of interviewees from Port Moresby Head Offices to those in the two Teachers Colleges did not know about the Policy or Plan, and had not seen any documentation about them. This lack of awareness and education about the importance of these documents for a fair society that meets international human rights, that adds value to the society, and increases women’s ability to give of their best for education and in leadership was well documented by Komhiol: The findings also indicated that foreign ideologies and gender principles conflicted with traditional governance structures, belief systems and practices. Dominance, hegemony, suppression, exclusion, leadership, religious beliefs and practices, cultural maintenance and social status were identified as major impediments to the effective adoption and implementation of the GEEP and the GESP. Overall, the findings indicated a lack of congruence between policy development and implementation practices. The promotion of gender equity and implementation of the GEEP and the GESP remain ineffective and need urgent attention from all respective stakeholders. (Komhiol, 2020, abstract) Women do not apply for leadership positions as they believe that they will be given to men, who won’t “give away” their status allowing females to lead. Such would be a sign of weakness and loss of power for men. Females responsible for gender equity have little to no say, are allocated little funding, and have no power to ensure that policies are implemented. With women being kept out of the business sector, and often being financially dependent, they are unable to establish their positions. In short, they are vulnerable. Traditional beliefs make it especially difficult for officers and teachers to promote gender equity in rural areas. Clearly, there needed to be more understanding and recognition that both men and women need to be agents of change. There needed to be greater awareness, promotion and professional development to take account of gender equity issues arising in education policy and for their dissemination and implementation (Komhiol, 2020). Nevertheless, in the 2020s there are now more females attending secondary and tertiary level education institutions, although this may not yet be a 50-50 male-female ratio. That ratio may already be the case in some secondary schools in urban areas. DWU has recently indicated that it has slightly more females than male students enrolled (49% male, 51% females). However, a crucial factor will relate to graduation levels of female students, and to whether there will be provision of continuing employment and leadership roles for the female graduates. It maybe that the very way universities are administered may need to be rethought. At present the universities follow Western lines with Councils, Chancellors, Vice-Chancellors (or Presidents), various Heads of Units within the university, academics who are expected to have

It is ironic that one of the first Government Departments to be axed after Independence was Population Planning. Since that time the population has doubled, mostly due to more births with some West Papua refugees. 4 The author is Teng Waninga Komhiol 3

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doctoral degrees, and other teaching staff who have graduate degrees and relevant experience. However, since Papua New Guinea is still under the sway of neocolonial ways of doing things, it is hard to envisage radical change in universities happening any time soon. It would appear to be the case that there are few who can imagine what such a change might entail5. What will it take for the universities to find alternative ways of coming to know the world from a Papua New Guinean perspective? For example, Australian universities now have a senior position and many policies and plans in place to ensure Indigenous representation. Certainly, as we have argued earlier, when it comes to mathematics education much more attention needs to be paid to the forms of Indigenous mathematics already in PNG cultures, and the structures of the languages which embody them. Cultural technology, mathematics, and science should be a part of all student degrees as well as language and cultural studies.

Hegemony of Neocolonialism

Neocolonial forms of thinking evident in PNG has resulted in the once hegemony of the colonial masters being in part transferred to that of an educated elite of whom many are strongly influenced by the past’s ruling colonials who still exert power within the country. In his classic book, Fanon (1952) notes that the elites and the educated could often carry the same attitudes as colonial “masters.” Fanon asserted that the “collective unconscious is not dependent on cerebral heredity; it is the result of what I shall call the unreflected imposition of a culture” (p.135). “I read white books and little by little I take into myself the prejudices, the myths, the folklore that have come to me from Europe” (p. 136). Yet there is more to this. It is a feeling of not being like the others with the same skin colour, of failure to be culturally capable, and at the same time failing the white man’s myths (of lacking civility, capability, and concern) and white man’s standards. Reconciling, the loss of your own culture, the unconscious acceptance of black is not good, not capable, and the unconscious desire to have a standard equal to the white man, has consequences of fulfilling the negative dream of character as having a useless culture, an uneducated culture or denying this perspective and claiming exaggerated standards beyond the white man’s expectations. This is the case for many PNG men, and women, located in different regions within the nation. The language of instruction being set as English has meant that those with an English and Western mathematics education are often seen as able to achieve more than those without English and school mathematics. This was evident in questionnaires we sent to educated Papua New Guineans. Whether their initial schooling was in English, or their local languages, they had achieved due to their mastery of English and their “completion” of education. They were the elite, who learned, despite not always understanding their initial schooling if it were in English. In those cases, employed parents or community leaders assisted by giving them an opportunity to engage in schooling, and the community was prepared to recognize their natural intelligence and supported them. Whichever route was taken, the myth of literacy in English as the means for progress was maintained. Thus, an educated elite emerged ready and willing to take over the mantle left by colonialism in this neocolonial phase. Epistemic hegemony is also evident in the difficulties faced by women to reach higher positions in the education system, both within schools and in other arms of education administration. Such a difficulty was certainly true in colonial times and, arguably, only minimal progress has been made with respect to it since Independence. The same ideology applies. There have been no

The situation is not helped by the government’s dismissal and appointment of Vice-Chancellors and Pro-Vice Chancellors at the University of Goroka with apparent lack of due process or transparency. Only recently Komhiol and team marched into Sinebare’s office with a government minister’s dismissal of him and his team (four leadership teams since we started to write). 5

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equity movements, such as affirmative action, to provide more opportunities for women, with Australian Aid emphasizing equality of opportunity rather than equity. Even some male advisers did not encourage equity for women. Post-Independence, as in colonial times, there have been pockets of encouragement for more girls and women to go to school and continue to higher levels of education. However, there have been few role models for females to emulate. Excellent female Acting Principals have been replaced by males who do not value new initiatives nor negotiation, and with limited experience in assessing issues and solving the myriad of problems faced by schools. Some of these men were poor role models in their personal lives. Increasingly, women have had to deferentially make suggestions, eat separately, and not necessarily be part of decision-making conversations. However, there have been a few notable male principals whose encouragement for women could not be denied. There have been systemic issues in the Department stemming more from PNG culture than from external agencies. Leaders and advisers might ensure that women were represented on committees, etc., but often the men ignored the women’s advice (personal communication, Head Office employee, ~1981). As far as University students are concerned there have been marked changes. When Kay and Chris Owens went to PNG in 1973, smart but penniless men and women were receiving a tertiary education, wearing their one and only, often second-hand, T-shirt and shorts or skirt but having a living allowance which they were often able to stretch to support their younger siblings’ education. Ten years after Independence things began to change. University fees became the norm, but universities struggled to get funds from the Government. Universities continue to have student unrest from time to time, and staff strikes occur when money, even for salaries or to support scholarships for further studies, are not provided. Nevertheless, during the 45 years since Independence, there have been more paid jobs and increasing amounts of money flowing within the community which has made more money available for luxuries such as mobile phones, DVD players and computers, batteries and lights, cakes and fast food, more clothes and longer-lasting home materials. Not surprisingly, some university graduates are beginning to wonder whether the benefits they are receiving from their superior education are worth the efforts they had to put in to get there. Thus, they too, whether consciously or not, perpetuate the hegemony of neocolonialism. With second and third generations of tertiary educated students, there is another educational phenomenon in PNG that too feeds into this hegemony. Eminike and Plowright (2017) describe this as Third Culture Indigenous Kids (TCIKs). Third Culture Kids (TCKs) was often a term used referring to missionary and expatriate children in PNG. Many completed their primary and some their secondary education in Papua New Guinea, with a few returning after completing tertiary education in the home country of their parents. However, by the 1990s the majority of students in International Schools, originally begun for children of expatriates, were actually Papua New Guinean. So, a remnant from the colonial past changed its character in the neocolonial phase, with the acronym also changing with the inclusion of the “I” in TCIK. The staff profile in the schools also changed. Once all staff had been expatriates, but by the 1990s most were Papua New Guinean. So, by the 1990s these schools dealt with global issues but from points of view rooted in a society dominated by neocolonial ideologies. They were trying to provide a PNG-focused education as well as an international education. A minority of other students, initially expatriates and then more and more from well-to-do Papua New Guinean families, attended years of church-based boarding schools, normally in Australia or less often in New Zealand. Such schooling often overlooked discussions on issues of relevance to PNG. Indeed, such issues did not even register in the thinking of most students and teachers in those schools. It is reasonable to use the TCIK term for both the recent (since the 1990s) Papua New Guinean school graduates, both from the international schools and from boarding schools overseas, and indeed from well-educated parents who may have taken them overseas during their own studies or work abroad.

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Eminike and Plowright’s (2017), who analysed responses of students in international schools in Nigeria, commented that: Students appeared to idealise Western cultures and denigrate their own indigenous customary practices in order to validate their desire to emulate Western attitudes and behaviour. …Their comments bestowed an inferior status on (their own) indigenous cultures whilst attributing superiority to cultures and values of Western origin. (p. 9) Significant in this argument is that these TCIK students’ identities are borderline and often represent variants of Western perspectives. PNG TCIKs may still regard peers who attended community or primary schools as lacking in “real education.” It is becoming easier for them to believe that these peers do not share suitable democratic values or worldviews which align with the progress of the country—or, in general, the peers who did not attend an International or a boarding school are less able. Such a divide is often seen in Western societies in which those who attended well-endowed private schools think of themselves as superior to peers who “only went to Government schools.” These TCIKs tend to become more individualistic, a characteristic of Western cultures, and often begin to feel quite at variance with central markers of Papua New Guinean cultures. At the same time, as they feel a loss of their original cultures and positions of influence in their local societies, they often seek to compensate by attempting to buy back into their culture by bringing monies, opportunities, advice, or goods to their villages. This may bring them closer to their own cultures and at times give them some status in their societies. Nevertheless, they become part of the elite that perpetuates the hegemony within neocolonialism. Another interesting outcome was similar to that found in Kenya, seen both as a result but also a perpetuating aspect of a similar hegemony in Kenya. Lang'at (2008) noted the “haves,” those who kept their land and attained education, lord it over the others. However, it has been rare in PNG that an individual has taken over colonial land holdings in which original land rights were being claimed (Ballard, 2010; Connolly, 2005). However, there are other “haves.” Giving something small for any service in government offices was endemic in Kenya (Lang'at, 2008). Gift exchange in PNG was universal but there were always cultural rules that governed such exchanges. Gift exchange for services grew after colonization in PNG, but in this context, there were different cultural rules that governed such a practice, and hence it grew into something else entirely and became more akin to bribery (Walton, 2013; Wickberg, 2013). This form of bribery became important for establishing positions as “big men” or “big women” affecting relations between employees and indeed external change agents, whether from another tribal group or an aid organisation. Importantly, neocolonial discourse continues around transparency, given the strength of the “big man” mentality in decision-making. Nakata (2012) expressed a similar fate for Torres Strait Islanders (Indigenous Australians who live in the strait between Australia and PNG), a term in itself which does not recognize the tribal groups and languages of the people. Nakata’s great grandfather, grandfather, father and himself had all tried to provide a better life for their family through education, despite great personal and cultural loss. This is not uncommon in PNG. Importantly colonialism and neocolonialism, has continued. As Nakata noted (2012): The symbols and representations of ideology (in this case, Western) effectively turn (our) history into … ‘exotic’; we are ‘natural’, that is, we are devoid of any history which we name for ourselves; and most importantly to westerners, we are culturally marginal … which legitimates colonial interventions. Educational institutions, policies, research, pedagogies and curricula will all continue to be developed unabated with regard to a people who have been relegated to a marginal space and who are culturally deficient. In Australia, there have been times and places where some of us have become the best representatives for ongoing colonial presence and dominance in our lives. So often, so many of our own people too, continue content with recent liberal versions of our cultural history, rather than with a more emancipatory agenda.

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But why should we be surprised by this? Do we in fact know any more what the colonial presence looks like in our lives? Do we know when we are speaking outside colonial discourses? Colonial discourses and their narratives are now so dense that it is very hard to make out whether one speaks from within them, or whether one can speak outside them, or whether one can speak at all without them. (p. 83) Significantly neocolonialism continues based on perceived “lacks.” Many who would be regarded as being in the PNG elite, despite their achievements, still feel as Nakata (2012) does, that he and the system need to do better. We have, at various times, lacked intellect, language and education. We have lacked health, hearing and nutrition. We have lacked control over alcohol, finances, land and sea. We have lacked as fathers and mothers. We have lacked as children. We have lacked as students. We have lacked so-called mainstream experiences. This was noted and written at the time of the first anthropological expedition by Haddon in the 1880s and more than 100 years later, Western experts can still name it and write it, and so we still lack it. (p. 88) Haddon, who visited PNG in the late 1800s, noted the visual acuity of Torres Strait Islanders but then went on to say they “lacked” the intellectual capacities of higher-order thinking—what a terrible colonial interpretation. Reading this, we reflect back on Chapter 11 which dealt with visuospatial reasoning and computer technology in PNG, and on an earlier book by Kay Owens (2015) which showed that these were strengths, and types of armour for their intellectual capabilities. Chapters 2 and 3 of this book also point out that Papua New Guineans had no lack of technology, agriculture, economics, education, and social welfare which pre-dated colonization by centuries, if not by millennia. However, neocolonialism is not just about external soft neocolonialism of Australia and England, which definitely occurred through the rejections of the main recommendations of the Tololo committee, especially its guidelines and its acceptance of an Australian academic’s view of self-government and Independence. Neocolonial thinking was also to be found in later projects governed by external, albeit well-meaning (which in a way makes them less identifiable) governments, such as CRIP, PASTEP, and the UK VSO projects especially for teaching English. Although virtually all the external government projects had an impact on mathematics, the later Japanese aid for mathematics in PNG was dictated by global trends such as standards and assessment regimes rather than learning regimes which took seriously Papua New Guinean mathematical cultures. The neocolonial impact of the elite Papua New Guineans, with their sometimes sub-conscious values and attitudes aligned to the external intervening countries, also continue to play a part. Unlike Nakata (2012), who is fighting against perceived lacks and deficit perspectives that have become instilled and institutionalized in Australia, the PNG elite may not be aware that they could also be doing the same. PNG cultures and remote communities in particular may not be well served by Papua New Guinean elites promoting Western or global views of education, including assessment methods that will inevitably result in remote children being seen as having deficits which need to be changed. This perspective does not take into account their identities, particularly their mathematical identities (Owens, 2015), their ethnomathematical knowledge and their ways of learning and solving problems mathematically.

Teaching Perspectives

Guy et al. (2002) maintained that formal examination results indicated that there is a general weakness in Papua New Guinean students’ mastery of basic Western mathematical skills. Some scholars have queried whether such results have been brought about by intellectual weaknesses, language difficulties, or poor teaching. Comments by some western educators, such as, “Papua New Guinean students do not grasp or process mathematical concepts well” (Dickson,

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1976, p. 36) may indicate that they believe a weak mastery of mathematical concepts is a consequence of a lack of intellectual capability. As stated by Kaleva (1998) and Paraide (2008), Papua New Guinean students interact with sophisticated reasoning and well-developed skills when placed in in practical situations in their everyday lives. It follows, therefore, that an understanding of basic mathematical concepts is not beyond the students’ intellectual capacity or capabilities. Similar to de Abreu, Bishop, and Presmeg’s (2002) findings with migrant students in Europe, students in PNG often do not link abstract representations that they were taught in formal learning situations to their practical applications of mathematics in, say, village contexts. Although the well-intentioned CRIP team members thought they were supporting the kind of education reform needed in Papua New Guinea, they were predominantly non-Papua New Guineans who had little opportunity or time to learn about cultural mathematics or Indigenous mathematics. Even Papua New Guinean officers working on curriculum development projects have not necessarily had the opportunity to make conscious the kind of Indigenous knowledge which fosters cultural bonding. That kind of learning rarely comes from being instructed about, or reading about, Western teaching strategies which emphasize “outcomes.” After all, those in previous generations who were instructed along Western lines had virtually no opportunity to recognize cultural mathematics including the recognition of the influence of language forms in relation to mathematics learning (Ellerton & Clarkson, 1996). Western teaching strategies can be a hindrance, if students are not familiar or comfortable with them. For example, PNG students are more familiar with learning through observation and participation in meaningful activities, as these are common learning strategies in their home environment (Department of Education Papua New Guinea, 1974; NDOE, 1986). Although the reform curriculum materials encouraged this approach, it did not correlate well with how lessons were often being taught in classrooms. From Paraide’s (1998, 2002) classroom observations, a considerable amount of the curriculum was still taught by rote learning, and in many cases in isolation from the students’ practical application of the knowledge. Her classroom observations also indicated that students’ Indigenous mathematical knowledge was not usually used as a stepping stone, when introducing similar or new Western mathematical concepts in formal learning environments. For example, the lower primary teachers did not use the students’ Indigenous number when teaching formal number—like, for example, how group counting of coconuts, for example, can be used to introduce and give meaning to multiplication. Vernacular languages are currently used only to teach prescribed Western mathematical content in elementary and lower primary grades. Paraide’s classroom observations mirror those of the other authors of this book. Evans et al. (2006) also noted that the Indigenous mathematics which is embedded in vernacular languages was generally not integrated with formal or Western mathematical knowledge in many elementary schools in PNG. As Ascher (2002) stated, the practical use of mathematical concepts is similar worldwide, but is expressed in different forms. Accepting similarities between Western and Indigenous mathematical knowledge, and not only tolerating but sensitivity recognizing the differences, was and is currently minimal in PNG classroom practices. Also, adapting strengths from each knowledge source, such as the effective teaching of mathematical concepts and skills, was given minimal attention even during the reform period. This is because Western teaching strategies have been elevated over supporting Indigenous ways of learning. Consequently, Indigenous teaching strategies have been viewed as less effective. Battiste (2002) in her discussion about placing Indigenous knowledge at the same level as Western knowledge, stated: To date, Eurocentric scholars have taken three main approaches to Indigenous knowledge. First, they have tried to reduce it to taxonomic categories that are static, over time. Second, they have tried to reduce it to its quantifiably observable empirical elements. And third, they have assumed that Indigenous knowledge has no validity, except in the spiritual realm. None of these approaches, however, adequately explains

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the holistic nature of Indigenous knowledge or its fundamental importance to Aboriginal people. (p. 10) In PNG, Indigenous teaching and learning strategies have been largely ignored in formal curriculum (Matane, 1976; Tololo, 1976). Past and present curriculum documents show that Western textbooks and educators have substantially influenced mathematical teaching and learning in the past, and still do so in the present.

“Look North”: To Neocolonialism Asian Styles

In the early 1990s, PNG’s Prime Minister told the nation to forget Australia since its aid was tied to projects and most money went back to Australians and not to Papua New Guineans. This was true in large part but there was somewhat more to the story. Australian Aid for some years immediately after Independence had been given as untied government money to the Government of Papua New Guinea to spend. It was soon realized by the Australian Government that there was little monitoring of how the money was actually being spent. A significant percentage was being pocketed by “middle-men,” and in the end with little project accountability the monetary aid was not making any difference in areas on which the Australian Government had intended it to be spent. Gradually, through the 1980s, Australian aid began to be targeted specifically to specific projects rather than given as untied aid. Whether it was realized or not, Australian Aid, both untied and then tied, initiated a phase of neocolonialism in Papua New Guinea. In an attempt to break this neocolonialism, the Prime Minister went on to direct that the country should look North rather than south, to Asia rather than Australia, for assistance. Even so Australian aid continues to flow and is still by far the greatest source of support for Papua New Guinea. There was an immediate response to this statement. Chinese and Korean companies moved to fund many large infrastructure projects in PNG, the latest of which seems to be a city, airport and wharf to be built on the island of Daru, very close to the Australian border. Chinese businessmen have discussed projects with politicians over dinner. Large trade stores were set up or taken over by Chinese owners based in Indonesia, and previously locally-owned small trade stores have gradually been taken over by Chinese owners employing mainly Chinese, but also Filipinos, and Indonesians. Elsewhere a large nickel mine, Ramu Nico, with majority ownership by Metallurgical Corporation of China (MCC), was set up with armed guards keeping locals away and dumping waste close inshore6. In the education sector, the dormitories at the University of Goroka were built by Chinese companies which employed Chinese professionals and labourers with the help of some Papua New Guineans; a number of plumbing issues resulted from a lack of familiarity with the available town infrastructure and cultural ideas. PNG University of Technology accepted the rebuilding of the Mathematics and Computing Department ten years after it burnt down, although updating its facilities, rather than rebuilding, might have been better. Neocolonialism was bolstered by China using soft power to influence policy of the weaker power, and to gain access to its resources. Another Asian country which has provided significant amounts of aid to PNG, especially since Independence, is Japan—which, it ought to be remembered, invaded it during World War II. Japanese aid has provided funding and expensive resources to Papua New Guinea, but one could argue that this has contributed to the perpetuation of neocolonialism. Japan paid for a number of public buildings for various Government departments, including buildings for Education.

There was a closure in 2019 when the toxic slurry spilled into the Basamuk Gulf near Madang. The locals did not want this mine but the lawsuit against it was nullified (Harriman, 2019). The Owens were on the same boat as the so-called government advisers on the impact of the mine on the environment before it started. It was clear that even before the mine started. the advisers did not have the knowledge or equipment needed to carry out effectively the environmental impact study. 6

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The University of Goroka funded its own library, designed and managed by a PNG architect company and builders, and it reflects the cultures of PNG. Japanese aid supplied the bookshelves, each marked with a Japanese flag, and a large room of computers for students at the University of Goroka, which otherwise would have been beyond the reach of the average student around 2005. Importantly, Japanese JICA aid workers have been important in setting up computer systems and keeping them running for many years. This aid has been important for the University. Japanese has been taught by Japanese volunteer teachers at Sogeri National High School and at the University of Goroka for many years. In fact, this enabled a PNG University of Technology mining student to complete a PhD in Japan. The new mathematics curriculum for the Standards-Based Curriculum was developed with aid, advice and training coming from Japan. The key Japanese educator in this process was able to speak English well, had produced textbooks, and communicated relatively well with his Papua New Guinea colleagues. However, as with some of the previously Australian-funded education projects, much of the writing was by Japanese advisers, situated in Japan, rather than by PNG curriculum officers, and this clearly influenced the textbooks and curriculum documents that were produced as parts of the Project. As might be expected the Japanese brought a wealth of mathematics education knowledge, but their knowledge was situated in a Japanese context, not a Papua New Guinean context. It appears to have been the case that they conceived of school mathematics as transcending cultural contexts. The PNG curriculum officers involved in the development of the project have been taken to Japan for training and writing. Nevertheless, one such officer struggled to understand the core draft document (personal communication, 2016). There was limited PNG input as there was no mention of an advisory committee helping to prepare the curriculum—compare this with what took place in the 1970s and 1980s, as mentioned by Quartermaine and Owens in Chapter 6. The local vernacular language and culture was mentioned in the teachers’ guides, but not as the key for developing new concepts. The older syllabuses are still available to teachers. While drafts of the new curriculum provided much detail in relation to assessment, this is reduced in later forms, but clearly positioned for each lesson in the teachers’ resource. The lessons are directed by the new textbook materials so that PNG teachers may lose ownership and find it difficult to adapt to their students’ cultural background and needs. The colonial background for teachers suggests that they will follow the guide fairly closely without thinking about whether it is sufficiently aligned to cultural considerations. If something is deemed, by a teacher to be too hard to implement it may be ignored. This is quite different to the professional learning offered in the elementary school project where teachers started with their cultural knowledge, and prepared their own inquiry lessons (Owens, Bino, Edmonds-Wathen and colleagues mentioned previously). An integration of the two approaches to curriculum development would have been valuable. There is no doubt that Papua New Guinea will continue to diversify its efforts to obtain aid monies from alternative sources, even though PNG can be fairly confident that aid will continue to flow for education from Australia and, to a lesser extent, from New Zealand. However, there is an urgent need for protocols to be developed which will ensure that Papua New Guinean expertise is privileged in making decisions regarding education matters, no matter from where aid monies are obtained. In some ways that is the relatively easy part of a necessary two-way process. The hard part will be ensuring the adherence to such protocols once the money is flowing for a specific project. Whether external aid agencies will be prepared to donate in such a context and relinquish control even if only partly, given a history of monies disappearing from the intended programs in Papua New Guinea, is hard to predict. In a push to standardize the whole of the curriculum with English as the language of instruction, the link to culture in the name for elementary school was discarded. Now the elementary schools teach mathematics with some use of Indigenous languages rather than beginning with cultural practices and language. Elementary teacher education programmes in the colleges still

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do not give due attention to the importance of cultural mathematics, how to promote it, and how to link it to school mathematics. Little has changed in 20 years since the time we were evaluating the PASTEP Project (see Chapter 7). Reflecting and analysing her own and other expatriates’ roles in developing science curriculum in PNG, which can also apply to mathematics, Ryan (2008) wrote of a need to: Explore how and why local knowledge has been successfully repressed and replaced by forms of Western privileged knowledge and understandings; a process of repression and replacement that can be seen in the neocolonial activities of education consultants. A discussion of the activities of global aid agencies working in the National Department of Education highlights the issues. The nature of neocolonial activities in the education context, and the associated promotion of sameness rather than differences [also need to be investigated]. (p. 665) During the colonial period in PNG, schools were agents of change bringing Australian values, histories, and knowledge to the people, but this process of colonial education fractured long-standing cultural knowledges. The mythologies and ontologies of the cultures that tell of the very existence and being of the people were not part of this landscape of colonial school science, mathematics or even geography, although those provide the landscape which establishes relations between people and knowledge of the ancestors and their values and beliefs. For PNG to move forward this fracture must be healed and space found for those who dissent from the current practices, who see in the now marginalized knowledge a way forward, and who wish to experiment with alternatives (Dawson, 2020; Stein, Andreotte, Bruce, & Susa, 2016). There is, however, some hope. Currently there is a closer alignment of the outcomes-based education (OBE) and the new standards-based education. The newly developing curricula present outcomes together with performance indicators which both outline what are progressions for achieving the outcomes and ways of assessing with standards and criteria. However, there is no recognition of language or culture in the syllabus outcomes, or in the details presented in the syllabus except for statements that were already there in older syllabuses regarding language and culture, gender and other equity. How these are actually implemented in schools would take a revised teacher education course and considerable professional development of teachers in schools.

Moving Forward

We have noted in a number of places that a combined aspect of neocolonialism and globalization is the continuing input of overseas, mostly Western approaches to mathematics education resulting from the continuing influence of overseas aid and overseas experts, bolstered and supported by privileged Papua New Guineans who have achieved within the present neocolonial system. However, there needs to be not only a recognition of neocolonialism, and how it is influencing PNG thought and policy making, in general, but also an academic study of how it is having an impact on mathematics education, in particular. The balance between neocolonial forces in education and the cultural and social changes that are occurring is worthy of investigation and research. We need to know the extent in Papua New Guinea that educational changes being made by diffusion or coercion, or by independent innovation parallel to global development, or by mere transfer from one system to another without due regard for culture. Examples from overseas, Papua New Guinea and Pacific Indigenous communities that may help in such investigations can be found in many books and papers (e.g., Elementary Mathematics Manual by Owens and colleagues; Bino, Owens, Tau, Avosa & Kull, 2013; Burarrwanga et al., 2013; Dawson, 2013; Furuto, 2014; Hunter et al., 2018; Hutchins, 1983; Lipka, Adams, Wong, Koester & François, 2019; Lipka, Wong & Andrew-Ihrke, 2013; Owens, Cherinda & Jawahir, 2015; Pinxten & François, 2012; Pinxten & Vandendriessche, 2023 in press; Rosa & Orey, 2017, 2020; Rosa, Shirley, Gavarrete & Alangui, 2017). There are many other potentially useful

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transformations in Pacific Societies: Oxford, England: University Press. https://oxford. universitypressscholarship.com/view/10.1093/acprof:oso/9780195324983.001.0001/ acprof-9780195324983-chapter-1 Matane, P. (1976). Education for what? In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 57-60). Melbourne, Australia: Oxford University Press. Matang, R., & Owens, K. (2014). The role of Indigenous traditional counting systems in children’s development of numerical cognition: Results from a study in Papua New Guinea. Mathematics Education Research Journal, 26(3), 531–553. https://doi.org/10.1007/ s13394-013-0115-2 McConaghy, C. (2000). Rethinking indigenous education: Culturalism, colonialism and the politics of knowing. Brisbane, Australia: Post Press. McConvell, P., & Thieberger, N. (2001). State of Indigenous languages in Australia. Australia State of the Environment, Second Technical Paper Series. Canberra, Australia. McKinnon, K. (1976). Development in primary education: The Papua New Guinea experience. In E. Barrington-Thomas (Ed.), Papua New Guinea education (pp. 226–251). Melbourne, Australia: Oxford University Press. Mennis, M. (1972). They came to Matupit. Vunapope, Rabaul, East New Britain Province, PNG: Mission Press. Muke, C. (2012). Role of local language in teaching mathematics in PNG. (PhD), Australian Catholic University, Melbourne, Australia. Murphy, J. M., & Moynihan, F. (1936). The National Eucharistic Congress. Melbourne, Australia: The Advocate Press. Nakata, M. (2012). Chapter 6. Better: A Torres Strait Islander's story of the struggle for a better education. In K. Price (Ed.), Aboriginal and Torres Strait Islander education: An introduction to the teaching profession (pp. 81–93). Melbourne, Australia: Cambridge University Press. NDOE, Papua New Guinea. (1986). A Philosophy of Education for Papua New Guinea. (The Matane Report- Report prepared by a committee chaired by P. Matane). Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1996a). Papua New Guinea National Education Plan, 1995 –2004, Volume a. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1996b). Papua New Guinea National Education Plan, 1995 –2004, Volume b. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1999a). Elementary education: Making your own community curriculum. A guide for elementary trainters and teachers. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1999b). Language policy for all schools—Education circular. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (1999c). Language policy in all schools. Education Circular. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2000). Lower primary mathematics: Bridging to English resource book. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2003a). Elementary teachers’ guide. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2003b). Implementation support booklet for head teachers of elementary schools: Supporting the implementation of the Elementary Curriculum. Port Moresby, PNG: Author. NDOE, Papua New Guinea. (2004) A National Plan for Education 2005 –2014. Port Moresby,PNG: Author. NDOE, Papua New Guinea. (2009). Gender Equity Strategic Plan 2009–2014. Port Moresby, PNG: Author. O'Sullivan, D. (2012). Globalisation and the politics of indigeneity. Globalisation, 9(5), 637– 650. https://doi.org/10.1080/14747731.2012.732424

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Chapter 13 Moving Forward: Overcoming Neocolonialism in Education in Papua New Guinea

Abstract: This chapter provides an epilogue to the book. It contains the authors’ deliberations in looking back to what has transpired and looking forward to a possible future, bringing arguments from alternative perspectives. Three approaches that could overcome in-bred neocolonial attitudes in education for Papua New Guinea (PNG) are summarized and discussed. The first is the valuing of differences and, in particular language differences, associated with different cultural groups. This would demand a concerted effort to maintain or re-establish local languages. In tandem with language, differences in foundational mathematical ways of understanding and working with the diverse ecologies. A second focus would be gender equity. Although it has been addressed in many policies and externally funded projects, effective progress is yet to take place. The third matter would be to recognize the entrenched hegemony and neocolonial thinking of the elite, in particular, as they balance the need for globalization in the economy and in education with a recognition of the importance of applying the concept of “education for all,” and the importance of encouraging the development of identity based upon language, culture, relationships, and guardianship of the land and embedded mathematics. Other areas of difference are capabilities of individual students, socioeconomic status, and place. Some comparisons are made with other post-colonial countries and First Nations people, especially with respect to establishing an Indigenous focus in education. Key Words: Cultural identity · Epistemic hegemony · Gender equity · Globalization · Global-local education · Glocalization · Neocolonialism · Neoliberalism · Overseas monies and aid.

The most successful in learning … often know the least about the traditional technologies … due to their formative years in the school classroom. The second reason for the lack of awareness of traditional technologies among the newly educated in PNG is more serious. In spite of … [their] prevalence throughout the country, there is the tendency to consider them to be primitive and of little value. … This not only denigrates traditional culture but also discourages the study, and thus understanding, of the technical and practical viability of traditional techniques. Warpeha, 1994, p. vi

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Introduction The previous chapter summarized the argument for recognizing the impact of neocolonialism on mathematics education in Papua New Guinea (PNG). The concept of neocolonialism embraces the idea that some powerful Papua New Guineans maintain aspects of neocolonial approaches to education and the economy. In particular, we noted the impact on mathematics education of overseas donors who provided expertise and funds for various projects but were working from neocolonial perspectives. Papua New Guinea cultural groups have foundational forms of mathematics that exhibit sophisticated designs for activities such as string-figure making (Vandendriessche, 2014, 2015) and in visuospatial reasoning in construction, travel, and location (Owens, 2015b). Kinship systems (Shaw, 1974), provide some variation, but have not been fully analyzed mathematically, and foundational counting systems are sometimes quite intricate mathematically (Owens, Lean, with Paraide, & Muke, 2018). Chapters 2 and 3 in this book offer the first known attempt at analyzing and summarizing the mathematics involved in many foundational activities in many different PNG cultural groups. The forces of colonialism and neocolonialism have meant that the depth of the mathematics within Indigenous communities has largely gone under-recognized. It is ironic, perhaps, that before and after Independence a number of colonial “experts” were brought from Africa, from other mainly British-colonized states, as well as from the colonizing country Australia, to PNG to advise on education in PNG. Although the states from which they came were almost always lacking in knowledge of Melanesian ways and associated mathematics, they were thought to have sufficiently enlightened attitudes to culture and education that they would be in strong positions to advise usefully on PNG education. Although it was the case that some of these advisers acknowledged and valued Papua New Guineans’ input into their way of thinking, nevertheless, the colonial systems prevailed. One might say, “white” privilege prevailed. We begin this chapter by making a comparison with other colonized countries and the Autonomous Region of Bougainville to consider how other places have fared with neocolonialism and to garner if they offer a way forward to improve education in PNG in terms of building on the funds of knowledge (Ewing, 2014; González, Moll, & Amanti, 2005) of the community relevant to school mathematics education.

Comparative Studies on Ethnomathematics in Schools

Papua New Guinea is not the only country where neocolonialism is an issue for curriculum development. For example, around 1990, for many school children around the world fractals was a recently conceived area of mathematics. However, in many parts of Africa, fractal mathematics had been used for thousands of years in many different activities—in, for example, settlement patterns, buildings, hair styles, and textiles (Eglash, 1999)1. Northern Australia There is no doubt that policies in Australia were intended to privilege English culture and were implemented, often cruelly, and that attempts for greater autonomy of school curricula were not easy to attain. Nevertheless, some teachers managed to introduce mathematics of the community in various ways and recognised the importance of language in this respect (Roberts, 1998). At one point, there was an emphasis on bilingual education. This generated teachers who speak, read and write in both their home language and English well. In some cases, these Elders A video of this work is available - https://www.ted.com/talks/ron_eglash_the_fractals_at_the_heart_of_african_ designs?language=en 1

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and teachers are still influencing learning in schools (e.g., FM in Walpiri, Harrison, personal communication, 2019). For the Yolngu, in the Northern Territory of Australia, strong examples of recursive and spatial mathematics have existed for thousands of years. At Yirrkala, decisive action was taken to incorporate this mathematics into their school curriculum. In 1985, the Yolngu teachers at Yirrkala Community School, led by the then Principal, Mandawuy Yunupingu, banned from the School the teaching of maths, because it was doing so much harm to the students. The complete ban on Western Maths held for two years. (Thornton & Watson-Verran, 1996, p. 3) A small book was prepared for Yolngu on teaching mathematics through kinship (WatsonVerran, 1992). Yolngu kinship is recursive with a set of basic names, a set of rules for devising further names from any one name, and a set of rules for using the system within the material world. It has a commutative rule, and also unitary reciprocal relations: that is “if you are my son, then I am your father,” but not dual reciprocal relations as in English “if you are my son, I am your father or mother.” David Shield (personal communication, 2021)—mathematician and linguist—has pointed out that clan system relationships can be described in terms of what might be called “algebraic group structure, modulo 4” in its recursive nature. More recently, place and other relationships have been described by Burarrwanga et al. (2013). As in many PNG cultures, the kinship relationships are well-known with “ideal” marriage patterns recognized within communities. Importantly, the Yolngu curriculum had its origin with a group from each moiety. The mathematics comes “from already existing social and intellectual contexts which are constituted by the languages, the conceptual structures and community ideals of Yolngu society” (Watson-Verran, 1992, p. 9). The English curriculum emphasizes individual value and separateness, so that eight is the value of eight ones. In Yolngu, relatedness is the ideal rather than the individual value. Generalization and abstraction are the basis of these relationships. Although kinship relationships in PNG cultures and languages can be identified and the “rules” and ideals associated with them (Shaw, 1974), to the best of our knowledge a similar mathematical study has not been undertaken or encouraged within the PNG languages. The Northern Territory government has not continued to support such initiatives in areas where teaching was culture-based and in language. Some recent initiatives and government- funded research have explored the importance of oral home languages in schools (personal communication, Edmonds-Wathen, 2018). An excellent study of bilingual education in the Northern Territory is provided by Devlin, Disbray, and Friedman Devlin (2017). It seems Australian Indigenous communities have struggled to gain autonomy as have the PNG peoples in education, especially mathematics education. Autonomous Region of Bougainville In 2007, following a ten year war between the PNG Government and the people of the Autonomous Region of Bougainville, the United Nations Development Program sponsored a Human Development Report in which the people of Bougainville were highly involved. Education was seen as a key for moving forward. A large number of surveys were carried out. Basic Skills test results revealed that although students were mostly in the advanced categories a significant number had had no schooling related to these kinds of test items. Significantly, the report recommended that education should be provided which “blended modern systems with customary values and wisdom” (p. v). The UNDP (2009) report asserted: The bottom line is that children will get the education they need if communities are fully engaged. We especially applaud the decision by Bougainville women to take a lead role in life-long learning—A goal that will go a long way towards realizing basic

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education for all in the Autonomous Region. (United Nations Development Programme, 2009, p. v) The report encouraged Elders to continue teaching in the schools along with teachers, and that in return the students provide some assistance to them as they would do in a cultural setting. The report noted the resilience, problem solving, self-direction and emotional intelligence of the people during the crisis. “These skills and the customary methods that have sustained and developed them during the crisis need to be emphasized in formal and informal curricula” (p. 39). It maybe that this approach is working well in this region given that this region was the site of an early Tok Ples Skul initiative in the 1980s (see Chapter 10). Nevertheless, there are some clever privileged people in early childhood education today who prefer English as the language of discussion and teaching in early childhood settings despite knowing the research evidence because of their own personal experience where their own home language was strong and their preschool education was in English (Dinah Ope, personal communication, 2015). Other South Pacific Countries The South Pacific countries considered here are 11 island countries2 that were protectorates (one remains a protectorate) of a range of other countries, with some experiencing a number of invaders like New Guinea. Samoa was first under Germany and then, after World War I, it was handed over to New Zealand control by the League of Nations (Begg, Bakalevu, & Havea, 2019). In general, these islands are within Melanesia, Micronesia or Polynesia. All of the countries are much smaller than PNG, many with populations of about one million. Half the people of one of the nations, Fiji, are of Indian origin because the British brought indentured labourers from India to Fiji. The Solomon Islands is closest in diversity and cultures to PNG and has a population of about 610 000. All of these countries, as far as education at least is concerned, are affected by neocolonialism. They are sourcing textbooks and aid mainly from the colonizing country. Some have struggled to find the money needed for adequate professional development and the provision of local resources. The old approach to teaching with the teacher telling and students copying is still common. Their curricula is partitioned into subjects rather than being holistic and they have drawn from “Australia, New Zealand and other Western countries for guidance rather than to their own cultures for content, concepts, contexts, and teaching approaches” (Begg et al., 2019, p. 13). As elsewhere, “curriculum documents emphasize assessable subject content knowledge rather than aims” (p. 17). Logical thinking is emphasized by mathematics but Begg et al. (2019), suggest school curricula in the region should emphasize different types of thinking: Empirical (sensory/experience-based) thinking, creative thinking, meta-cognitive thinking, caring thinking, cultural thinking, contemplative thinking, subconscious thinking, and systems thinking. Using all these forms of thinking broadens one’s view of mathematics. (p. 17) Such an approach, together with an emphasis on vernacular languages and culture, can assist countries to break the neocolonial mould. In the Pacific islands, the language of instruction in most schools is a local language, and then there is a transfer to English (or French) or other languages. For example, in Solomon Islands and Vanuatu, Pijin and Bislama respectively are frequently used. We are referring to Fiji, Cook Islands, Kiribati (Gilbert, Phoenix & Line Islands), Marshall Islands, Nauru, Niue (Savage Island), Western Samoa, Solomon Islands (British Solomon Islands), Tokelau (Union Islands), Tonga (Friendly Islands), Tuvalu (Ellice Islands). There are other islands within the region under control of other countries—France, USA, Ecuador and Peru. 2

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Tonga, though not colonized, also failed to recognize the value of ethnomathematics in its teacher education and school curricula. As in the Solomon Islands, the material tends to follow overseas curricular approaches. However, it is in the teaching approaches that neocolonial approaches are most evident. Solomon Islands The Solomon Islands faced the same issue with students not being allowed to speak Pijin (similar to Tok Pisin in PNG) or their home languages in schools. As Jourdan (2013) made clear, students learned wrongly, often from expatriates, that Pijin was like bad English without a grammar: The Solomon Islands faced such issues when they gained independence from Britain in 1978. Repeatedly, the question of the language of instruction surfaced: educators and Government officials, aware of the 1953 UNESCO recommendation of the use of mother tongues as a medium of instruction, were concerned to facilitate literacy and numeracy for the children. The role that Pijin, the lingua franca of the country, could play in education was never seriously considered. Regularly, the Government implemented policies that favored English as a medium of instruction, save for the first two years of primary schooling. (Jourdan, 2013, p. 270) … Many teachers’ views on Pijin coincided with that of the colonial administration. They transmitted their prejudice against Pijin to their students, many of whom went on to become members of the current post-colonial intellectual, social, economic, and political elite. Not surprisingly, when the time came to establish a post-colonial education system, English remained the language of instruction and in the process, as we shall see in what follows, the language of social discrimination as well as the language of social promotion. For years, Solomon Islanders were told that Pijin was not a real language, and many of them came to believe it and embraced an ideology that demeans them. Our recent research shows that many young Solomon Islanders, who were educated in post-independence Solomon Islands schools and taught exclusively by Solomon Islands teachers, still believe today, as they told us, that Pijin is not a real language and that it has no grammar. In many cases, Pijin was their mother tongue and the only language they knew until they picked up English at school (Angeli, 2008; Jourdan & Angeli, 2008, February). Inherited from colonial times, the prejudice against Pijin lives on. (Jourdan, 2013, p. 274) It is interesting to those of us who have taught some of these Solomon Island students in PNG, that their English was often very good compared to the English of many of our PNG students. Jourdan (2013) continued that in the Solomon Islands: the education system favored the urban children, particularly those of Honiara. Educational resources are better and more numerous in Honiara; parents have more opportunity to earn the money necessary to pay the tuition fees (skulfi in Pijin); and the greater proximity to English through popular culture facilitated the acquisition of the medium of instruction. But the education system worked in a way that the students’ ability to progress through it depended on their proficiency in English and the availability of seats in schools, rather than on the student’s real mastery of the curriculum and real ability to succeed in higher grades. Competition was, and still is, fierce and attrition rates from one grade to the next are such that many deserving children found themselves pushed out of the system. (p. 276)

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This same story can be heard within PNG, even to this day. Reducing school fees did not necessarily improve education as class sizes skyrocketted after 2014 but it did provide a lost opportunity. However, there are still costs. Many times, opportunities have been lost due to a lack of valuing of difference. Hence, the student from the coast in a Highland school might be discriminated against or bullied. Self and family interests are commonplace in PNG. The 2018 Pacific Regional Initiative for Delivery of Education (PRIDE) (Zhang, Chan & Teasdale, 2018) has focussed on identifying the learning processes operating in Oceanic countries. It became particularly relevant after fighting broke out in the Solomons between those who had power and those who did not, at a time when a divide existed in Honiara based on racial allegiances. Pijin was seen as a possible language for reconciliation and for restoring peace. However, there were issues including the hegemony of the status of English that had developed among the elite as well as the choice of which Pijin to select (should it be that of the local streets, or the more classical village Pijin, or that of Bible translation?) Since the restructure associated with the introduction of standards-based curriculum in PNG, the language of instruction in schools, from Preparatory to Grade 12, has been English despite it being used infrequently by teachers and in the village or community. There continues to be inequities for children in remote areas and from less educated families where English is not spoken or not heard. Solomon Islands provides a good example of a quality literacy education program built on Australian and New Zealand aid with books being made available for the students to read which are about cultural events in their lives. Detailed teachers guides and textbooks for all the schools were available although a few schools did not collect them from the centers due to local disagreements. However, although the literacy examples are from the country, the curriculum follows the structured learning expectations of materials that were prepared elsewhere. There are no really new approaches—such as the tape (box, conceptual) diagrams for word problems involving arithmetic operations (Xin, 2012) which can be found in the latest PNG curriculum documents. The division operation particularly is poorly explained and developed in the teachers’ guides. Nevertheless, there was a good degree of local input into the curriculum. Group work was expected. All that said, schooling is not compulsory and not all children complete Grade 6. West Papua An interesting comparison for neocolonialism in PNG can be made by considering the history of education in West Papua (Ipenburg, 2009). Besides some islands like Timor further to the west, and countries in Oceania (Solomon Islands, Vanuatu, Fiji, New Calendonia), this is the other area with Melanesians. Although Dutch schools through the missions (first Protestant and then Catholic) started earlier than in PNG, and in local languages, few pupils attended. Those that seemed to benefit had been freed from slavery (the Netherlands also had missions in Indonesia and Malaysia where the Islamic regimes had slaves but some European business enterprises may have treated workers like slaves). The emphasis was on preparing students for cathecism in the church. The vernacular and then Malay were the languages of instruction, although in the 1960s there were a number of schools for Dutch and Chinese students at which Papuans could attend if they spoke Dutch at home. It is interesting to compare with PNG schools which catered for immigrants and for the colonialists’ children in schools and to compare with Paraide’s introduction to school where her teacher was amazed at her progress in English despite Tolai being spoken at home and with her early schooling being in Tolai. The schools offered three years of instruction for quite a few students but it was likely that these students lost their literacy skills once schooling finished with little need for school literacy in community. In PNG, a similar situation occurred. However, in PNG at least students, especially girls, discontinued school because of personal and family reasons after two or three years.

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The next three years of school, to which only about one in a 100 could continue, included an attempt to localize to the Papuan situation, which was considered better than what occurred in Indonesia at the time. It covered Middle Age history, then Papuan history (or at least the Dutch explorers and colonisation), and then a history of China. One assumes the mathematics was similar to that being taught in The Netherlands at the time. At the end of the 1950s and into the 1960s there was a flurry of education, now in Dutch although some language groups had literacy classes in their own languages where the church had established a fairly sizeable recording of materials. Many expatriate Dutch-speaking teachers came. Most, but not all of them were trained teachers and some, like Annie Dommerholtz, had a strong grasp of teaching reading using sounds and strong approaches to literacy through whole language and grammar including functional grammar. (Later, Annie would bring this knowledge to Balob Teachers College in Lae, PNG). There were some West Papuan students who went on to be teachers, pastors and administrators. The Netherlands handed the country to the United Nations in 1962, which quickly moved to have it administered by Indonesia in 1963 and finally being incorporated into Indonesia proper in 1969 after a ‘refererendum’. This act of “freewill” was not really free as the selection of voters, not the whole population, was carried out under Indonesian military control. The Indonesian military with pro-Indonesian supporters imprisoned and shot dead most of the up-and-coming leaders of West Papua during the 1960s. During this time transmigration began mainly from Java with “outsiders” being given plots of land—none went to the Papuans whose land it might rightfully have been. It was not long before half the population were transmigrants, schooling was all in Malay or Bahasa Indonesia. Positions in education and other jobs and businesses were given to the transmigration settlers, and West Papuans were not even allowed to sell their food inside the markets. However, the number of schools increased rapidly and a university was established. About a quarter of the population (local and immigrant) gained a full primary school education, and further education opportunities were made available to some. In the 1980s there were some ten thousand West Papuan refugees in PNG and this number doubled in the following years. Displacement from traditionally owned lands and government brutality were the main reasons. Many West Papuan residents (the population doubled from the three-quarters of a million to 1.5 million in 30 years through transmigration) had schooling, and by 2001 over 1600 had received a tertiary education. So-called “autonomy” was granted to the region in 2001 but most of the laws for this have not been implemented. Under the Indonesian approach there was no allowance for local practices in mathematics. A West Papuan refugee from the 1960s went to Papua New Guinea and became a teachers college principal in PNG and trained with many others in Australia. He was then an Education Administrator before he returned to West Papua to take up a Senior Education Administration role, some 40 years after he had fled. It is likely he then had some influence on the direction of education. Indonesia Unlike Papua New Guinea, Indonesia had been colonized by different groups for thousands of years (Patahuddin, Suwarsono, & Johar, 2019). Initially migrants came from different parts of Asia, India, Vietnam and Cambodia in numbers that swamped the original inhabitants. The first colonizers were Indian traders who brought Hinduism, around the first century of the Common Era (CE), and then Buddhism to the country, and there are still many examples of their influences to consider in terms of ethnomathematics. By the seventh century CE, a great Indonesian Kingdom rose in Sumatra. Sriwijaya was famous for its quality education with religion but also mathematics and science. Another Hindu empire emerged in East Java during the 13th century with schools being established around the temples. Then the Muslims came and set up schools around the 14th century. Clearly the schools were influencial in spreading Islam. Next to come

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were the Portuguese with Catholic missions, and the Portuguese language was popular along with Melayu (the origins of Bahasa Indonesian). Portuguese influence declined in 1605 when the first Netherland traders and Protestant missionaries arrived. Their first school was in Ambon and others followed teaching a Christian life but also reading and writing; but by the mid-1700s the Dutch influence on education had declined. England arrived very briefly between 1811 and 1816 but no changes to education were introduced at that time. Then the Dutch resumed control, and established a composite primary and high school for the Dutch and wealthy Indonesians. 1848 saw the start of primary schools for Indonesian students in the local language or Bahasa Melayu with reading, writing, language and arithmetic. During the next period there were two levels of schooling for locals and a school for the Chinese. The gap between the primary school and the current high school led to an intermediary school. The achievement level was strong and covered arithmetic and geometry. The high school was mostly staffed by Dutch—teachers required a doctoral qualification—and successful students could then attend a Dutch doctoral program in the Netherlands. Upper-class Indonesians performed as well as the Dutch students. After the Japanese invasion, Indonesia was made Independent in 1945. The Government looked like it was setting up a system for Indonesia but the mathematics books at the time were very much like the old Dutch books. Indonesia was trying to provide one education system for the whole country. The curricula was renewed with little change three times until 1964 with the only modification being to have students placed in either a language and culture stream or a science and mathematics stream in the senior high school (Patahuddin et al., 2019). A new-order curriculum was established between 1966 and 1998, with the 1975 mathematics curriculum following the curriculum as developed elsewhere in incorporating sets (so-called Boubaki “New Maths”) but also probability and statistics, logic and calculus at the senior high school. The writings of Piaget, Dienes, Bruner, Gagne, and Ausubel were discussed at seminars. In 1999 there was a Reformation in thinking about education, with nine years of basic education being followed by three years of senior secondary (general or technical/vocational) education. Indonesia, like PNG, introduced education ideas from Africa, and “modern mathematics” was introduced in the 1980s, with calculators and computers in 1984. Senior mathematics textbooks were written for junior high school students by Stephanus Suwarsono, who completed his PhD at Monash University in Australia where he created a test for spatial reasoning (see Chapter 11), which would become widely used around the world. Schools were often obliged to use specific books. Ideas such as student active learning, amd realistic mathematics education (RME), which became much used around the world but originated with Indonesia’s last colonizer, The Netherlands. There were textbooks for Grades 1 and 2 with forward approaches to teaching mathematics. Surabaya University continued to push for contextual teaching and learning. By 2000, Indonesia Indonesia tried lesson study (which had originated in Japan). Experiences, Language, Pictures, Symbols, Application and Assessment components were considered in planning (in what resembled an Australian approach), students’ performances were improving, but were nevertheless low in the TIMSS and PISA international performance “ladders.” PNG did not participate in any of the TIMSS and PISA investigations. Then came a very large Australian Aid program for professional development for all mathematics teachers by which Facebook was used to contact teachers. National examinations were scrapped, due to corruption. A few Indonesian professionals continued to be influential worldwide as well as at home. In the years immediately after 2016, Australian Aid supported a huge teacher-education program across Indonesia. There is now a strong ethnomathematics group in Indonesia and much interest in their local languages (Papuan Linguists’ Conference, held in West Papua, 2006). Many ethnomathematics papers were written and published in the Journal of Physics Conference Series between 2018 and

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2021 including one which incorporated technology. Most look at the mathematics of structures, others everyday activities. It should be noted that Adam (2010), in Malaysia, prepared an excellent thesis on the mathematics of weaving of food covers. Other Colonies Students from many ex-British colonies in Africa and on the Indian subcontinent still complete external exams, like the Cambridge (British) international examinations. Nuffield Foundation scholarships and British Commonwealth scholarships continue to attract postgraduate students as well as European Union scholarships (to the United Kingdom and other European countries). Australia itself was not exempt from neocolonialism. This was particularly evident between 1980 and 2000 when there was much reference in Australia to how the Cockcroft Report, which had been prepared in England, encouraged new approaches to learning mathematics and to the assessment of mathematics learning. Cockcroft argued that assessment would drive change. National curriculum, assessment and outcomes-based education were imported, particularly from the United Kingdom. Ideas emphasizing group work with concrete materials came from the United Kingdom through people like Richard Skemp. Australia followed the United Kingdom in bringing in calculators as did PNG. Graphing calculators and computer algebra systems (CAS) also flowed in from other European countries and were well adopted by experts in Australia— although they had virtually no impact in PNG and Indonesia due to costs. Even so, PNG Universities adopted computer-aided instruction (see Chapter 7) before many Australian universities. In each of the countries mentioned in this chapter, it is possible to see the elite, the educated leaders taking on neocolonial attitudes. However, in Indonesia and in Australia there were strong research cultures, and Australia has become a respected contributor to research in mathematics education. Australia also has a strong emphasis on quality teacher education and research, with many universities focussing on this area. There is a beginning recognition of the mathematics of Indigenous cultures (Burarrwanga et al., 2013; Mousley & Matthews, 2019; Owens, 2012a; Owens et al., 2012). Indonesia has encouraged ethnomathematics along with other South-East Asian countries like Nepal (Luitel, 2009, 2013; Pradhan, Sharma, & Sharma, 2021) and the Phillippines (Alangui, 2017, 2021; Alangui et al., in press). Nepal In Nepal, with its 139 languages, there are interesting differences in mathematical practices. Chundaras’ “teaching and learning activity involved participatory and cooperative approaches in which they learn with the help of their parents” (p. 7), but the Nepalese have created many games which could usefully be incorporated into school mathematics. In addition, there are many unique activities and buildings which have been used in relation to the teaching and learning of mathematics (Pradhan et al., 2021). First Nations in America Several First Nations groups in the United States of America have fostered these approaches which can bring cultural values into mathematics and leadership. For example, leaders of the Cherokee bring values of family to self-determination and problem solving to the benefit all (Smith, 2013). The Yu’pik incorporate mathematical activity into curriculum implementation and note the central importance of symmetry in cultural beliefs as well as mathematics (Lipka, Adams, Wong, Koester, & François, 2019; Lipka, with Mohatt, & the Ciulistet Group, 1998; Lipka, Wong, & Andrew-Ihrke, 2013). Navajo recognize the capability of noticing and visuospatial reasoning that is part of intergenerational cultural knowledge and incorporated this into

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school mathematics especially geometry (Pinxten & François, 2011; Pinxten, van Dooren, & Harvey, 1983). These forms of assessment and teaching are needed in the curriculum of PNG. In other words, assessment and teaching need to reflect the values that PNG societies bring to education.

Privilege and Equity

It is not without significance that some authors of this book have white privilege, and neocolonial beliefs and attitudes are therefore, despite our personal critical reflections and collaboration, inevitably embedded in parts of what has been written in this book (McConaghy, 2000). The length of time that these Australians have spent in PNG, and their knowledge of PNG mathematics, though limited, have encouraged co-investigation and sharing of this knowledge. Yet it is recognized that just as Papua New Guineans can critique from their own identities, backgrounds, and ownership, as Nakata (2004) did for himself and his family and community, our team of coauthors, with the assistance of other knowledgeable colleagues in PNG, can do the same. The privilege has included knowing how to gain access to written documents, many of them written by non-Papua New Guineans, most of which express colonial or neocolonial ideas. We acknowledge that even the Papua New Guinean authors of this book cannot represent or know the mathematics of all the different cultural groups in their nation—nevertheless, they have offered a useful foundation for the writing team’s reflections on the history of mathematics, and of mathematics education in PNG. Much of the direction of this book came from the theses of authors Paraide and Muke, whose pride and knowledge of their cultures and their cultural identities provide book-ends for what has been written in this book—but we acknowledge that they are among the elite and privileged in PNG. Nevertheless, we now make recommendations based on research that might assist those working to overcome vestiges of neocolonialism still present in PNG. Many of the thoughts with respect to significant Indigenous and/or other multicultural groups which are expressed are based on experiences and thoughts from other countries. We first examine underlying assumptions and ways of identifying and overcoming deficit attitudes, paternalistic attitudes, arrogance, and other neocolonial expressions.

Re-Examining Dialogue on Attitudes and Purpose

We tell a story which might help to set the scene to understand that people are often judged through neocolonial lenses. What is work? This was a question posed by the anthropologist Michael Smith (1994) when working with the Kairiru on an island off the coast of Wewak, East Sepik Province. Responses to the question indicated that work or time spent working was not equated with making money. Sitting in an office or the cabin of a mining vehicle or transport truck was not considered as work, but was regarded as like being in a prison cell. That kind of paid “work” meant that “the sun was moving slowly,” that is to say, time was passing too slowly. They craved that they would be able to find the time for the “active work” of the village, whether preparing a canoe from a tree, building a house, fishing, or talking and discussing before making important decisions. When the men walked into the bush to prepare a tree, time passed in waiting while some took detours to see people or attend to different matters—but at the tree they all worked together to complete the job at hand. When this approach to time and work is met with the view that education might bring money through a paid job, or it might bring a more influential voice in the community, then the response can engender deep conflicting feelings and attitudes. The neocolonial emphasis on education sits uncomfortably with many villagers who see little relevance in it

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to their situations. Mathematics, like English, is seen as providing a pathway to a higher level of schooling through the competition of examinations. Any other relevance for formal mathematics education is not always clear to students. However, the neocolonial attitude is one of thinking that the students are lacking, are apathetic, incapable, not interested in progressing themselves, or trouble-makers (“rascals” in Tok Pisin). Little thought is given to the students’ situations in terms of their health or hunger, their feelings of inadequacy or uncomfortableness in being made to sit inside for a long time, or their feelings that it is all pointless given their life circumstances. Little thought is given to the relevance of the curriculum to the students or to the quality of the teaching that they receive. However, these attitudes to students can be changed by focusing on the child through the lenses of family and Elders in the community. In Australia, Elders from the community are welcomed in schools and are encouraged to participate in sharing their cultures and languages with the students. The child, family and school need the Elders as leaders, decision-makers, and sources of knowledge and teaching (Owens, 2015a; Owens, et al., 2012; Thornton & WatsonVerran, 1996). In PNG, there have been occasions when we have met Elders, often chairs of School Councils, in schools, sharing with the teachers and children; but usually they are discussing finances or organizing work around the school. Their role in facilitating intergenerational knowledge sharing could be extended and other Elders involved. However, the demands of their subsistence lives may mean that the Elders cannot spend much time in classrooms or schools so there needs to be rewards for participating. Furthermore, the attitude seems to be that the teacher is the fountain of school mathematics, “not me.” There is yet to be the necessary motivation gained from recognizing the value of their foundational mathematics as well as school mathematics. Before and after Independence, there was an emphasis on adult education that tended to have a school mathematics focus and vocational interest in terms of attainment. The programs3 “aimed at instilling attitudes of good citizenship in Indigenous people” (McConaghy, 2000, p. 162). Some such programs exist today at the University of Goroka by which vocational and technical education is made available to teachers. It is indeed an expatriate attitude that many Indigenous people lack motivation or are apathetic if schools or vocational centers are not well attended. Little thought goes into considering real reasons behind a lack of attendance—such as an irrelevant curriculum, or teaching that does not show respect to the child and family. For both adults and children, there needs to be more PNG context, purpose, values, and mathematical thought and not just the use of examples that are “localized” in terms of words and situations. Although that is a good start4. However, it can be teaching styles that lead to meaningless, rote learning, that are the most damaging factors in the current neocolonial situations. It has been common practice to teach via a standard approach—there is the Initiate (usually a teacher’s question), a Response (from a pupil), and the Follow-Up/Evaluate by the teacher. This approach suggests that the teacher is the fount of all knowledge and does not encourage the student to use their own knowledge and inquiry. This has been found to be detrimental in Australia and it is certainly so in PNG because of its neocolonial implications (Pickford, 2008). The need to limit the use of, and even overcome, this approach was emphasized in the United Nations Development Programme (UNDP) (2009) for the Autonomous Region of Bougainville by recognizing the collaborative discussion approaches of PNG cultures and the capability of collaborating in problem solving. The Aotearoa Maori approach was also recognized as potentially a significant way forward in education. Thus, a change of attitude to vernacular knowledges and ways of learning needs to be reflected in schools and curriculum. The PNG programs became role models for Australian Northern Territory programs. Some schools and parents buy books, from book stores, written by authors in places like USA.

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With the loss of language which occurred with the delay, for 15 or more years, of the implementiation of the report of the committee chaired by Tololo at self-government, it is important for the nation to realize how much of their cultures is tied up in language and in moving toward maintaining their local languages and encouraging them to survive. This may include taking time to prepare proper grassroots mathematics, and identifying language appropriate for the real meanings of Western, school mathematics. Australian First Nations have been sorry to see their languages fragment under colonization but are now relishing opportunities to reinvigorate them, so that their importance will be realized. Now every group is endeavoring to revive its language and to ensure that all children (Indigenous and otherwise) learn some of the culture and language of the people where the school is placed. Herein is a message to PNG before more of their languages are lost. With language goes aspects of culture and in particular for this book, mathematical cultural practices. It is time to reclaim these before they are lost. Concern for Identity and Valuing Identity One of the amazing accomplishments of Papua New Guinea is that it has remained a unified country even though it is home to the most widely varying occurrence of different languages and cultures found anywhere in the world. In many other places either there is a fracturing between language groups, or within groups, or alliances of small groups become dominant and harmful. Neither of these scenarios has played out in Papua New Guinea. Once Australia started to take a real interest in its “colony” after World War II, it considered that the creation of a unified country was essential. It did not want micro-states emerging on its border. The unity has remained since Independence to a large degree, although events unfolding in the Autonomous Region of Bougainville5 may bring a new twist to this story. The colonial decision to work toward unity seeped into thinking about education. This was especially the case with respect to the Australian approach which was adopted toward curriculum before a national curriculum was achieved some time after Independence. English was to be the language of instruction. The emphasis in education, at least, has always been a recognition of diversity but this has been perceived as a disruptive element if the aim is to achieve an efficient and effective education system. The notion of disruption came originally from colonialists whose experiences were mainly of sameness, and hence they assumed that diversity was to be somehow managed so that a common approach to schooling was employed. A need for “sameness” has remained an emphasis of the external/internal forces of neocolonialism, and particularly the external forces of globalization. From a neocolonial perspective, the country’s huge diversity in education has been deemed a negative for the progress of the nation. A return to the education visions of Tololo and Matane is needed and new ways must be found to implement these so that diversity becomes more valued. Besides the one brief paragraph in the syllabus, a recognition of human rights and peace that comes from promoting multilingual inclusive education and gender equity needs strengthening. In particular, as D’Ambrosio argued in his contribution to the plenary panel of the International Congress for Mathematics Education in Copenhagen in 2004, ethnomathematics is “the backA 10-year war between freedom fighters who wanted sovereignty of their land and rights was fought with the PNG Government. An escalation by the Government bringing in the Sandline mercinaries in 1997 was averted by the then Head of the Defence Force, Saniok. who from his conscience broke the contract, costing himself and the country dearly. However, his decision was welcomed by grassroots people, especially those who remembered the second World War. It saved lives in that it led to reconciliation forces (such as Australian unarmed forces, negotiators, counsellors) securing peace, although mainlanders and outsiders were still held to be suspicious. The whole issue arose partly from the colonial approach to the Bougainville Gold and Copper Mine, the big hole, destruction of the land, river and sea with the tailings, and a lack of negotiation in the first instance with the women in this island of matriarchal lineage. During the War, the people’s resilience and reliance on cultural practices was strengthened but school education was left to those who could provide it in an unstructured way. 5

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bone of mathematics”. Time, money and expertise will all need to be sufficent for enlivening small group discussions in every culture in order to provide an equitable cultural and language approach to mathematics education in PNG.

Funding Crisis and More Neocolonialism

By the 1990s, either economic development in PNG stalled or monies were mismanaged and education was starved of funds. In the succeeding years the World Bank, an instrument of globalization, provided funds as repayable loans that would finance various recommendations that would reform education in ways that its advisors had formulated. At the same time, Australian Aid, an instrument of neocolonialism, also provided funds for three large projects (Curriculum Reform Implementation Project, Primary and Secondary Teacher Education Project, and Elementary Schools Project), which we have discussed earlier in some detail noting that the authorisation for expending these funds remained mainly in the hands of Australian employees. Global companies such as Oxford University Press, another marker of globalization, were employed to publish textbooks. However, these expatriate writers offered little emphasis on the context of Papua New Guinea, let alone using any notions of ethnomathematics to guide the process (see Chapter 10). In some of the World Bank and Australian-funded projects PNG counterparts were appointed and although they did receive good salaries and gained much experience, in comparison to how the bulk of the funds was expended this was a minor amount. However, as Yoko, (2002) revealed, when quoting the Acting Assistant Secretary of the Curriculum Development Division: My personal fear, shall we say, is that there is insufficient dialogue between the various agencies which are responsible for implementing the philosophy and strategies that go with it. … I am afraid but there is no avenue for mutual exchange of ideas. (p. 8) For effective change, there needed to be mutual planning of purpose and implementation with all stakeholders—from students, lecturers, and teachers to senior officials—having a way to communicate their ideas but also with increased capabilities to implement new approaches. Financing Education Several suggestions have been put forward in African countries for moving away from neocolonial approaches to funding education. First, fees for attending educational institutions should be much less than they are now or better there should be no fees. Second, outsourcing to aid organisations should be much less, with the added advantage of often making education more cost effective. Third, global companies aiming to make profits from education should not be supported (Lassou, Hopper, Tsamenyi, & Murinde, 2019). An implication from the second theme for Papua New Guinea would suggest that national advisory committees for curriculum and other education policy might usefully be reinstated and composed of nationals who are well steeped in ethnomathematics, and aware of their own neoliberal, neocolonial points of view. “Advisers” should not be invited to advise unless they are demonstrably quite familiar with the country. No outside organisations, either private or external government agencies, should be involved. Extending these ideas, monies for addressing education projects requested by PNG should not be tied to grants meeting expectations of the donating country. Tied grants should only be accepted if they will clearly promote the goals of such committees, like those chaired by Tololo or by Matane, and more recent ones dealing with social justice and equity. Indeed, a core premise of any grants should be that there is no profit element embedded in the costing that will be sent out of the country, and payment to aid workers and advisors should be such that they are not exces-

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sive so that excess monies will not be siphoned off and/or transmitted back to the donor countries. A fundamental principle should be that ultimately “aid” money ought not be accepted unless its use will clearly profit Papua New Guinea. For example, for the elementary project in which two authors were involved (Muke and Owens), only the purchase of equipment (solar powered computers with the professional development package) that was left with teachers in PNG villages and some printed manuals and games, were purchased in Australia and not with the intention of promoting the manufacturers of the materials. It was purchased at discount. They were pilots requiring time to source and connect compatible equipment. No salaries were paid to expatriates, national lecturers or their universities from the aid money although funds for release from teaching commitments and for the employment of national research assistants were available. The aid money went into the professional development and resources for the teachers and for necessary transport sourced as cheaply as possible, usually village buses, cars and dinghies. The personnel used low-cost accommodation such as village and transit houses. Food was sourced locally usually from the villages, trade stores, or student mess and not from hotels unless the Provincial Education Office selected to do this.

Overcoming Colonialism/Neocolonialism Through Vernacular Languages

Our use of the term “neocolonialism” comes not just from the continuing presence of Australians and New Zealanders as key players in aid projects in PNG, but also in the continuing attitudes and expectations of many of the Papua New Guinean power brokers who were aligned with them. Despite statements of intention to give greater responsibility for education to those closer to the sites of implementation, by decentralizing control to the provinces, there has still been a centralized curriculum and policy roll-out as noted in Chapter 12. This mix of political power between the center and the periphery has at times led to political interference from the center and has resulted in a lack of ownership by both central and provincial senior bureaucrats. Thus, a sense of uncertainty and a lack of ownership in schools and the various administrative groups has often prevailed, since both groups have had insufficient professional training to cope with such an important change initiative. At the school level there is a lack of materials, “inspection”, and initial and inservice teacher education. Elementary teachers’ salaries are dependent on school education, teacher education and inspection, and so salaries are often not paid. The salaries, professional education, and number of teachers all need to increase. Within this turmoil, just earning enough money for survival, meeting wider family expectations, and ensuring that one keeps one’s position seems of paramount importance for most teachers and administrators. They rarely have the energy, money, professional training, or Government support that might assist them to educate themselves further. At the present time the extent and quality of professional development is not sufficient to support current teachers who wish to implement new curriculum/education initiatives. Some project officers have assumed that teachers will just read the information in the syllabuses and teachers' guides that they produce and implement them accordingly. However, often teachers are not well informed of the reasons for curriculum change, so they may not understand well the value of it and therefore not enthusiastically support moves to implement it. This can be a contributing factor to curriculum not being accepted or implemented well. In other words, teachers may not feel any ownership of curricular proposals, or decisions, or indeed the whole process, because they generally have had little or no input into their development. The few teachers who are usually selected to represent the teachers’ views and to make inputs during the curriculum development stage often do not represent all teachers' views and concerns about subject content and the best teaching strategies for PNG children and students. Externally funded projects have to be completed and reports presented by specific dates and

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therefore the consultation processes in country may not be achieved. Procedures, including the selection of key personnel, need to be reviewed in future curriculum development projects in order to get a much wider input from teachers during development phases. For Papua New Guinea, legacies from the colonial era are using English as the language of instruction with its inevitable focus on Western knowledge and teaching strategies and aiming to prepare students to move into formal employment. These are also clear markers of a neocolonial perspective. The idea of having English as the sole language of instruction has become so embedded in Papua New Guinea’s educational curricula and classroom practices that even educators who are Indigenous Papua New Guineans have accepted this as normal—which, again, is a marker of neocolonialism. Many educated Papua New Guineans believe that being competent in the English language, and having completed a Western-style education, is the key to progress and development. Western educational practices are normally viewed as beneficial by those Indigenous Papua New Guineans who have succeeded within that context. On the other hand, some have viewed them as a hindrance for those who have not been successful (see, e.g., Tololo, 1976). The welleducated become more powerful and make decisions for the majority, including the “less welleducated.”6 The well-educated tend to control which educational issues are discussed and have more influence when decisions are made concerning education changes that might need to be made in order to facilitate national progress and development. An important part of the decision-making process relates to the continuing debate on the advantages and disadvantages of vernacular and bilingual education. Those who have experienced success being taught with reasonably good vernacular and bilingual teaching strategies usually acknowledge their value and would like to see them adopted across the nation. However, these are outnumbered by those who have had a reasonably high level of “English education”. The reader will have noticed that throughout this book we have argued that a move toward vernacular instruction would be one way of overcoming deficit views of Papua New Guinean students’ achievement due to their language and their ethnomathematics. Grote, Oliver and Rochecouste (2014) pointed out that encouraging “code-switching” between local languages and Standard English is likely to help students strongly identify with their foundational culture. Although the same three authors also noted that education involving code-switching may remain a form of colonization if it is mainly seen as a strategy for assisting students to become competent in English, they argued that it could be employed to the benefit of both teacher and students. Paraide (2010; Paraide & Owens, 2018) and Owens, EdmondsWathen, Bino and Muke found that it only took a short intervention to show teachers that they could easily achieve competence in code-switching techniques, and indeed many instances of code-switching occur quite naturally in many classrooms, and it is precipitated by both teachers and students. Muke also showed that early-year teachers often use code-switching in various situations in classrooms, and some of them do not even realize that they are doing it (Muke & Clarkson, 2011a, 2011b). Clarkson (1983, 1992; Clarkson & Clarkson, 1993) showed that students who were competent in their various languages and employed code-switching performed better in mathematics and in other subjects than comparable students who did not. There is no doubt that sociocultural environments within PNG should have an impact on what languages, registers and representations students and teachers use. Even so, how multilingual students use the various languages, registers, and representations of mathematics will vary and that will need further study. However, according to Uribe and Prediger (2021): For multilingual mathematics classrooms, the instructional approach of relating languages, registers, and representations needs to be applied more flexibly, taking into account students’ different starting points. When doing so, students’ connection proThis is in terms of Western schooling.

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cesses should be supported and explicated more systematically in order to fully exploit the students’ repertoires. (p. 1) In PNG, teachers are likely to share at least one of the students’ languages especially in the village schools, so that the strengths of the students’ languages can be built on. Another support for the students is the approach to assist with reading and in particular pronouncing words in two languages. There have been well-tried methods of introducing phonemic awareness starting with local sounds and words. Many, but not all, of the languages of PNG have an accepted orthography (i.e., ways of writing down the sounds of a language), thanks to the efforts of many linguists and Bible translators in the country. On the other hand, no real effort has been made to ensure that all languages have an orthography and no real effort has been made to teach major mathematical concepts from an Indigenous or oral (hearing, listening, and visualizing) approach. Cameo from Kay Owens I frequently rejoiced in the wonderful opportunities of viewing, hearing and occasionally at church participating in PNG dancing and singing (singsing). The very first occasion was in 1973, at Bukawa7, in Morobe Province, when the villagers were celebrating the arrival of Lutheran missionaries in their village 90 years earlier. Other occasions were associated with self-government, Independence, shows in Lae, Goroka and Mt Hagen, ad hoc village fund-raisers and celebrations, the opening of the dormitories at the University of Goroka, and the cultural welcome and display at the second International Conference on Pure and Applied Mathematics (in 2016). The diversity of dance, song, music, and “bilas”8 was always extraordinary. When I was at the Western Sydney University and teaching in the block-release Aboriginal Rural Education Program, I met a number of rural Aboriginal teachers who have moved on to be leaders in their communities along with those we had met in the 1960s. One of these students, Diane McNaboe, became my teacher of the Wiradjuri language and culture at Western TAFE when I moved to Charles Sturt University, Dubbo, NSW Australia. The other student-come-teachers in the area continue to influence my thinking as my friends and colleagues. On one occasion, I organized a small three-day conference at Charles Sturt University for our staff and local community and Indigenous friends from overseas: we had a Sámi (Ylva Jannok Nutti) and other Swedish academics, Patricia Paraide from PNG, and Dominic O’Sullivan from Aotearoa New Zealand. A small group from the local Buninyong school put on a dance segment for us—Tears ran down my face as I observed the visual pride and identity of these children and their teachers, something that connected with my many PNG experiences. This was nearly a first in New South Wales, a school dance group using local song and dance after many years of language loss. I had previously seen a small group at Western Sydney University and the famous Bangarra Theatre Company, but these were my local school children. The Wiradjuri had virtually lost their language—colonization had not allowed them to speak the language for fear of children being taken away from their parents, or of their parents thinking that English would assist them to break out of the cycle of poverty and discrimination. Two missionaries had decided to write down as many words as possible of the language. Now it is recognized that there are at least two dialects, but Stan Grant senior, with the assistance of other linguists (especially John Rudder), was able to prepare a dictionary and grammar and with Elders still speaking words or more of the language, they have been able to revitalize their language and culture. Every child (Aboriginal and all others too) in NSW is expected to learn something of the language and culture of the Aboriginal people on whose land their school is situated, We were walking along the coast from Finschaffen to Lae, a three day walk. Bodies were painted and exquisite decorations of bird plumes and other items were added to their bodies, the beat of kundu drums, small drums, pipes, and large bamboo pipes enhanced the singing and diverse dance steps. 7 8

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no matter whether they are in Sydney or in the bush. How I wished that Papua New Guineans would value their languages and make every effort to keep them, since language is the essence of culture. I have heard that it is the second generation who most commonly realize what they have lost with their language and identity, and want to revive it. My PNG friends who have married across different PNG cultures have usually encouraged their children at least to understand one if not both languages, and in most cases to speak at least one of them. However, some struggle to learn to do this with Tok Pisin and English being the main languages of communication in the home and with friends. When visiting Sorengke Sondo’s, Vagi Bino’s homes and villages, Wilfred and Roa Kaleva’s home, and Charly Muke’s village, it is lovely to listen to their conversations in their Tok Ples languages especially as the Elders discuss important issues. In 1973, we witnessed a local court case and over many years, various village cultural celebrations. At one, the arrival of books donated in Australia were delivered to a school with much “wailing,” speeches, and song to Charly who had organized and delivered them. Michael Mel9 told me that after all his schooling (in English), he had to go back and learn his home language properly so he could speak with his community in the language that they would expect of a person with authority—his home language had to match his worldwide, educated, and senior leadership experiences. Gairo Onagi10 had taken time to write down his language, Magi, as a dictionary with some grammar. The neighboring language group whose language was swamped by his and Motu, wanted him to spend time to do the same for them before it was lost. Many Bible Translators throughout PNG have recorded “their” languages, prepared materials for literacy and often numeracy, as well as translated much of the Bible. In Ngaing, Madang, Ruki Kalaing was one of the few sent to Australia for agriculture training at Warwick, in Queensland, in preparation for Independence. He then worked in agriculture and a bank before returning to help in his village. He has tackled the translation of the book of Revelation and of the Old Testament into his vernacular language, which is no mean feat. His leadership in the translation team is outstanding, involving much problem solving, and he has done that despite failing hearing and eyesight, and living a day’s walk away from the coast. How precious language can be in enabling people to appreciate their culture, meaning of life, and identity. Responding to the challenge of acquiring one’s vernacular language can enhance one’s self-valuing, and one’s concern for others in the community. It is time to engage the knowledge of Elders before more is lost, and to bring language and cultural mathematics and technology into schools. Recognizing the Cognitive Advantage of Bilingual Education Students who are competent in several languages know that this gives them a cognitive advantage. Hence, we do not argue against the use of English as the language of teaching— rather, we believe that it should be taught alongside a student’s foundational language. The additional advantage of bilingual education, with the encouragement of language strategies such as code switching, will be the interplay of the cultures embedded within each language. The English inevitably “carries” Western culture, and likewise an Indigenous language carries the associated Indigenous culture. For example, earlier in this book we noted some of the similarities between Western and Indigenous mathematical systems, like the counting in sets in the Tolai people’s language (Paraide, 2008). Other aspects of mathematics that have similarities in Western and Indigenous cultures, taking cognizance of Bishop’s (1988) list of activities and processes that define mathematical Michael Mel was Pro Vice Chancellor at the University of Goroka Gairo Onagi was Vice Chancellor at University of Goroka.

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systems, have been described in earlier chapters. However, there are also important differences when Western mathematics and foundational mathematics systems are compared—such as in logical systems that are used (Lancy, 1983). Given such similarities and differences we have also argued that appropriate comparisons and contrasts between the knowledges should be a core planning principle for curriculum developers. One language and associated culture should not be privileged over another. The Indigenous pathway might be used as a way of introducing or accessing Western ideas, but ultimately the best solution could be for them to sit side by side in their own social and historical contexts. Such an approach would achieve greater equity. Equity Equity has been particularly difficult to achieve with respect to education in PNG. Language policies have raised, and continue to raise, major concerns with those competent in English finding it easier to be placed in and keep their places in privileged positions. Cultural differences continue to exhibit a lack of equity. For instance, those who are competent orators, and are perceived as having much influence within their cultures, are likely to be able to overwhelm those in smaller less assertive groups. Wealth has also been an issue associated with accesss to educational advantage. Encouraging gender equity, language difference, cultural recognition and development, and economic equity would break neocolonial approaches. Gender Equity Gender equity has been noticeably absent in many societal contexts. We have posited that giving equity due attention is a key to reducing neocolonialism. During colonial times, some teachers made a stand to have girls at school but the roles of the girls within the village communities often prevented their enrolment or regular attendance at school. Owens observed an increasing pressure on women to take a secondary role in schools even though they were often more capable. Nevertheless, especially in matrilineal societies, a number of outstanding women emerged with high levels of education. In some areas this has continued, but it is still the case that there are fewer women than men in parliament, and in many communities, even in families, women continue to be regarded as second-rate citizens in general who should have minimal or no power with respect to education. During the many neocolonial overseas aid projects, gender equality (not equity) was seen as an important focus—that was the case with the Primary and Secondary Teacher Education Project, for which workshops on gender equality were provided for teachers, and there was a focus on females having a say in decision-making. The United Nations Development Programme (2009) was guarded in talking about “gender awareness,” and in discussing the role of women in teaching during the crisis in the Solomon Islands, even with respect to the matrilineal societies of the Autonomous Region of Bougainville. The number of female representatives in many areas of the Government, including departments relating to education, is much smaller than the number of male representatives. However, there does continue to be a female presence. Colonialism, Globalization and Gender Issues However, more indirect influences are occurring. For example, the availability of royalties and other payments for extracting oil and gas, has raised issues associated with the identification of land ownership. Among the Fasu clan, who in colonial times were brought into villages rather than left as scattered communities, there has been a return to recognizing the male lineage of land in the kepo11 for establishing the Incorporated Land Group (ILG) system, in accord with the Male head and male children.

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Lands Group Incorporation Act established by the national (pre Independence) Government in 1974 which allowed local groups to register for royalty payments. The kepo consists of the male head and his adult male sons. Due to the adaptable approach of the Fasu, the number of ILGs has increased from 54 to 88 with no increase in total payments (Gilberthorpe, 2007). Interestingly, the Onabasulu, the Fasu’s neighbors, developed 17 legally fixed clans based on the cosmogonic and cultural importance of that number for them (Ernst, 1999). However, the females’ marriages are now even more controlled by the males. The sago palms remain for exchange but cash and other goods are now related to the wealth attainable through the cash economy. As a result, women are tied to subsistence, and are less likely to continue with education as they are expected to tend to sago making and child rearing. With the payment system, relations are based on land rather than exchange (Strathern & Stewart, 1998), bringing more division between clans and genders. In addition, men might desire to marry to reach into the wider cash world (of Port Moresby and other larger towns) leaving a number of unmarried Fasu women. The need for the staple crop sago for the kepo and other exchanges are causing a “feminisation of subsistence” in the context of so-called developments driven by the global economy (Moore, 1988). Many families have also left for the cities or cannot return from the cities either because their land has been taken over or for fear of sorcery and fighting. Furthermore, in the latest reports on family violence, one in every three women has been affected (Oxfam 2021 brochures), sometimes very seriously, and that has also traumatized many of the children. All these factors have had an impact on education. It reduces the amount of positive recognition given to women as teachers and leaders in education and it reduces and negatively skews children’s education. Overall, the original balancing of the economy through exchanges based on clan reciprocity has broken down. Men and women often marry outside of the reciprocal clan or moeity and many families may no longer recognize land ownership in the same way as before. Marrying outside of the expectations, divorce and remarriage has led to a blend of families with the various exchange relationships and land ownership in future doubt. In these cases, the place of women and their identities have often been further eroded and the need for a woman and her children to survive often leads to further non-exchange relationships (Bettina, 2015). Has the loss of exchange relationships been partly due to the lack of respect accorded to Indigenous customs, including associated mathematics within formal education? In PNG there are a variety of kinship systems so there are likely to be reciprocals, commutativity and modulo arithmetic to be recognized which might strengthen the system and people’s understanding of mathematics. For instance, birth-order in a range of languages recycles terms as found in the Papuan languages of the Huon Peninsula, Morobe Province12 (Hannah, 2013; Holzknecht, 1992), and in Madang and Sepik Provinces, and in nearby Austronesian languages. Probably that has been due to marital relationships (Hannah, 2013; personal communications with a number of Sepik women living in Australia, 2021). In general it could be a relatively local innovation to avoid using a person’s name in order to show respect. One wonders if it is too late to recognize the value of kinship relationships by recognizing the mathematics of these systems as the Yolngu have done in Australia (Thornton & Watson-Verran, 1996). However, we recognize the possibility that our perspectives on language and gender may be regarded as “cultural relativism” (McConaghy, 2000). By this we mean that they may seem to be the product of a form of liberalism that has underlying connotations for conserving culture and languages. However, it still might have arisen from the structured perspective of Western schooling. It is only when cultural perspectives and structures are in place that such decisions regarding culture, languages, and education should be made.

For example, Kutong, Irumu, Ma Manda, and Nek.

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A New Approach to Curriculum

The expatriate authors and to a large extent the PNG authors of this book have looked at mathematics through a Western lens. Our use of terms like “patterns,” “symbols,” “reasoning,” “problem-solving” and “inquiry” indicates a Western perspective that ought not to be associated with a mathematics of community, since the term for “mathematics” is hard to describe in PNG languages. Systematic ways of thinking, designing, and associated patterns and language, purpose and transgenerational sharing of knowledge might best establish the meaning of mathematics when it is trimmed of most Western words. Focusing on the bigger processes of mathematics such as dispositions and reasoning, and in particular on visuospatial reasoning, it is likely that a new curriculum could be established based on cultural mathematics. Mathematical reasoning develops concepts and procedures. There are multiple starting points and pathways to achieve mathematical reasoning. Nevertheless, the learning progressions can be based on research, and in particular on observational research. Furthermore, they should be seen as temporal and subject to modification through use and analysis. They also need to be useful for driving instruction, and local cultural contexts offer suggestions pointing to different approaches for effective instruction with respect to each concept. The progressions will involve visual, descriptive, analytical and relational reasoning. A progressions approach can be associated with various concepts, and with topics that become progressively more difficult to analyse and achieve. No matter the culture, visualizing, thinking and talking are interrelated cognitive processes since visual perception is dependent on cognitive assumptions that can be associated with the visualized objects, and language that is employed around those objects. Keywords, visual mediators, narratives and routines are involved in the mathematical reasoning (Sfard, 2008). Each develops in a cultural way, and developments follow specific cultural ways of representing mathematical reasoning carried out during contextualized problem solving. Thus, for example, the mathematics behind sailing covers a well-developed approach to learning how to sail and how to modify reasoning about various problems that arise while sailing. Visualizing and visual representations are also contextualized and need to be recognized for spatial activities involving objects, paths, and spaces. In order to link with Western knowledge, there is a need to encourage different forms of representation of mathematical concepts and the connections between them. Learning to communicate mathematical reasoning and representations will encourage a greater recognition of mathematics embedded in cuture and the possible links between different aspects of mathematics. In particular visuospatial reasoning is highly developed in creative activities and measurement activities in the cultural contexts of PNG (Owens, 2014, 2015b, 2022 in press). After Independence, the second prime minister was purported to say that Western education was vital since “you can’t fly an airplane in a Melanesian way”13. However soon after, the Melanesian way was redefined by Narakobi (1980)—later to be Minister for Justice among other cabinet positions—who said that the core of the Melanesian way valued self-sacrifice, reciprocity and obligations to others past, present, and future, with a belief in the ancestors and their links to the land. Narakobi was pointing back to the five National Goals that had been identified at Independence: 1. Integral human development, 2. Equality and participation, 3. National sovereignty and self-reliance, 4. National resources and environment, and 5. Papua New Guinea ways. An amateur “Can’t Get Much Worse Theatre Company” based at Unitech and comprised mainly of expatriates created a clever skit by Garth Bond in which a pilot described to his co-pilot a take-off in Tok Pisin. Most of the audience including all the expatriate lecturers and others (who had all made an effort to learn Tok Pisin) followed this with delight, as did the students—that was, perhaps, an indication of the quality of expatriate staff in the 10 years following Independence and the power of Tok Pisin as a rich and unifying language. 13

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Although always given lip service, in the mid 1980s these started to become crucial in national debates. So, drawing on these five Goals and the earlier Tololo Report (Department of Education Papua New Guinea, 1974), Matane published the Philosophy of Education (Matane’s Committee Report) (NDOE, 1986), which worked out what the National Goals implied for education. This Report was finally accepted as the way forward in 1986. In current times, PNG men and women are in the teams of pilots for the two national airlines in PNG, and they carry their various Melanesian roots and ways into the cockpits. They transport Melanesian people throughout the country and overseas. It can be argued that they are providing air service “the Melanesian way,” an integration of foundational cultural ways of thinking and Western knowledge of relevance for their roles. For their learning and performance, there is a strength in this marriage. In particular, their visuospatial reasoning is their strength (Owens, 2015b).

Learning and Teaching

Colonial and neocolonial education focused on one key issue, that of identity, particularly national identity without due regard for difference. The curriculum, teacher educators, teacher education students, teachers, and administrators, all needed to show an understanding of difference and to offer appropriate respect. Although there were some efforts in the Social Sciences to reveal differences and to discuss them (see Yombu Selden’s cameo on Goroka Teachers College students’ “writing” information for Social Sciences), there was little emphasis on differences in the school curriculum (Pickford, 2002). There needs to be, as is developing in Australia (see, for example, the NSW Aboriginal Education Consultative Group and the Reconciliation Action Plans of many organisations), an awareness of the role of community and Elders in developing appropriate curriculum and respect in schools based on local culture. As Narakobi (1991) noted: The real challenge to institutions of learning … is to give the possibility to teachers and scholars to explore knowledge in the same way that they would physically explore a mountain, a jungle or sea. I believe that if we can instil this idea we will be planting in people’s minds the possibility of creating knowledge, of creating thought which will assist in addressing human needs and problems. … The task of liberation and freedom is not a political task in the sense that you chase away the colonizing power and take over political control of your country. The process … is a process of liberation … of thoughts which impede and obstruct our construction of a better society and better people. (pp. 24–25) This is not an easy task as it depends on the knowledge sharing of communities, especially Elders, in ways that the teacher can understand and at times when Elders are available. It may be difficult, too, as students have migrated to towns. Yet the idea of “exploring knowledge” means that ownership of the teaching will lie with the teacher and students, and not be dominated by a top-down authoritative education system. This, of course, is not easy given the emphasis in the new curriculum on assessment of progress in a predetermined, centrally-controlled curriculum. Furthermore, this goes well beyond the use of simple cultural examples for teaching mathematics education. Kaleva (2001) has pointed out that although there are other approaches that are more difficult to achieve, the mathematical cultural examples would at least provide a start. This assimilation approach would be a sensible starting point for the national curriculum. Ethnomathematics, and in particular ethnomodelling, is a way of identifying the structures of the cultural mathematics. This approach is accommodation, for which the learners’ cultures influence education (Narakobi, 1991). However, more could occur with cultured adults (Elders) being able to share significantly in educational decisions and provisions. This would not just be the idea of a school council that would concern itself with housing, classrooms, teacher selection (in the

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case of elementary schools) and attendance. It would mean that Elders would need to spend considerable time with teachers and students in order that the curriculum includes forms of cultural mathematics, some of which might link to school mathematics. Time and place are needed for such discussions. Both villagers and teachers can be short on these commodities unless there is strong planning, and strong support from the central Government, from education departments (national and provincial), and from personnel who support teachers in the field. Teachers need time to work with other teachers to support and develop ideas. Resource centers for education were established in the late 1990s but they have ceased. However, teachers will need time to meet and communicate. Our experiences indicate that language is a rich source of meaning but time is needed for discussion to ensure the meaning is adequately identified and developed. The Yolngu community in the Northern Territory managed this with mathematical philosophers, teachers, and the community at Yirrkala to develop a mathematics curriculum based on models for their kinship and location (Thornton & WatsonVerran, 1996). Can teachers as researchers do the same, having gained some mathematical examples of ethnomodelling (Orey & Rosa, 2021a)? The skill of noticing is a key component for researchers learning to understand cultural mathematics. Dominguez (2021) emphasized the link between sensing, in particular, from movement and noticing, and the sense-making of mathematics. Teachers need to notice in community when movement and activities are taking place as well as noticing features on artefacts. Through discussion and trial of model-making aimed at understanding cultural mathematics (mathematizing or thinking mathematically), teachers and their students are likely to develop their mathematical thinking and sense making. Importantly, mathematical thinking is fluid and not fixed in “schemas, constructions, representations, or conceptions” (Roth, 2015). Students may be expected to think mathematically if they are encouraged to notice change of objects, movements, their environment, and thinking in what is called the “zone of promoted action” (Valsiner, 2000). They need not be directed to think in a particular way if they notice what is occurring and patterns which occur. Furthermore, the reciprocity of an action or comment (vocalized or not) from the student or peers moves the activity and agency along towards the solving of a problem through mathematical thinking. Through trusting the student to make sense through their sensing, students establish conceptual mathematical knowledge—such as what fraction of a soft drink is sugar and how much that actually is in terms of their carbohydrate needs (hence a real-life situation that motivates their movement and sense making) (Dominguez, 2021). Thus, in PNG, by participating in community activities and observing skilled specialists, students should notice the mathematics of the movements and activities. Discussion may occur among the Elders that will also hint at the factors which are in a relationship. In learning cultural mathematics, students observe and note important aspects, often prompted by parents’ comments. Thus the idea of sensing and noticing objects, movements, environment and thinking, can be built into school mathematics. This was the approach we took for the elementary teachers’ project that linked culture and language to early mathematical thinking (Edmonds-Wathen, Owens, Bino, & Muke, 2018) especially during times spent with village Elders talking and demonstrating their everyday activities. To be effective, such approaches need to be ongoing and purposeful over long periods of time. Furthermore, teachers need to be the learners, to be the researchers, to think differently, and to plan with Elders. Guidelines for this can be provided. For example, Yunkaporta and McGinty (2009) provided an “8-ways” approach that can guide teachers in their planning for Aboriginal Australian students and their schools and teachers. Some of their ideas were: • • • •

using stories for learning, deconstructing and reconstructing knowledge with the children, clarifying learning pathways, linking to the land and culture,

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• recognizing that learning is not generally linear but ideas come from various sources as the knowledge is built up, • non-verbal ways of learning are important, • symbols identify critical ideas, and • linking with community for education. Each of these is further linked through values, protocols, systems, and processes. Progress requires “inner deep listening and quiet still awareness” (Stronger Smarter Indigenous Education Leadership Institute, 2017). Teachers need to be intentional about inclusivity of culture and take into account the needs and strengths of children with different cultures. The Western philosophy with its emphasis on the importance of an individual’s rights has become a central plank of the PNG education curriculum. Assessment of individuals is crucial to the West’s understanding of education and the system, but that is certainly not what was envisaged in the original interpretation of the National Goals by the committee chaired by Tololo (Department of Education Papua New Guinea, 1974) or the later Committee chaired by Matane (NDOE, 1986). Nevertheless, the neocolonial baton has been passed on to the PNG elite. They now have the knowledge to promulgate these Western philosophies, reinterpreting the National Goals and the Eight Point Plan for education in such a way that there is a seamless knit. The backdrop to this enterprise has been a huge loss of local languages and a loss of strong cultural values due to the invasion of Western views, foreign languages, and an acceptance that there is a universality of one scientific and mathematical system. Translocal Maybe there is another possibility. What would happen if one was to take seriously the internal local social forces which grow in concert with those external forces of globalization, as noted in an earlier section of this chapter. Maybe some of those forces can be viewed as generating positives among such diversity. The notions of “translocal” and “glocalization” (both implying that the global can profitably be interpeted in terms of the values and knowledge of the local) capture some of these possibilities and hence may point to another useful ingredient in plotting a way forward for Papua New Guinea education. The concept of translocal seems to have been first developed in the 1970s. It speaks to the life space of people—where and how they interact, influences of jobs (or schooling), leisure pursuits, their family, residence, and such like. A critical application of translocal has been to do with migrants relocating to new countries but seeking to keep language and some cultural ties with their homeland (Smith, DeMeo, & Widmann, 2011). It speaks to the migrants lived practices as they make their new place, their place, but keeping reciprocal links open between their new place and their place of origin. The term has more recently been applied to those who move between locations but do not necessarily cross national boundaries, as was the case in its original use. For example, Fataar (2012) used the concept to help understand students living in a depressed area of Cape Town in South Africa but who crossed boundaries to attend school in a “better” area and hence mixed with students who were not from where they locate their out-of-school living. The concept of “translocal” has also been found useful in Vanuatu for describing those who have relocated to the capital city from outer islands but still wish to keep cultural and language ties to their villages (Petrou & Connell, 2017). Kudo, Allasiw, Omi, and Hansen (2020), in Japan, have reported some learning principles that arose when introducing notions of sustainability into a community. The term “translocal” refers to the linking (“trans-”) of different localities (“local”), and the purpose of this approach is to create an iterative learning space across multiple communities. Specific conditions of individual communities, such as their socioeconomic status, climate conditions, and type of governance system, are not considered as limiting factors in the translocal

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learning approach. This is because a locality is treated as a resource for learning instead of a factor that highlights the differences or incomparability of communities. For PNG there is a growing transition from village to town for various reasons so that this notion is particularly apt. There is a need to recognize the original local landowners and local “villages” (settlement places) of the towns, and the cultures from which the students come. Such an approach would fit with our earlier and perhaps main line of argument. The differences between PNG cultures should not be dismissed as deterrents to learning, or be supplanted by an alien Western common learning space for mathematics. Instead, the differences should be embraced, perhaps including the Western approach to education and mathematics when that is appropriate, so that within the diversity, the various ways of doing can be described, experimented with, and contrasted. Thus, a deep appreciation not only of a student’s foundational culture can be developed but understanding of other cultures can also be known. For example, the customary kinship relations have rules for each culture but in PNG these are the same and different from those in other cultures. There could be a comparison of rules, and hence of mathematics. The same has occurred when comparing the various counting systems of PNG. To a large extent these can be classified into various types, but it would also be educative to recognize the diversity within those types and the similarities—such as the common use of 5 as a bridge to numbers up to 10 and the common use of selected cycle numbers upon which other numbers are named. For other counting systems, the use of classifiers, body-part tallies, or the cycles of the systems link to cultural practices, and are worthy of careful investigation (Lean, 1992; Owens et al., 2018). More specifically, using the students’ own cultures and languages, and their essential localities, as a primary resource for their learning fits neatly with research already discussed. We have presented arguments for the use of the students’ first languages to be used at least in the early years of schooling, although the upper limit of when this should cease is probably in university years. In particular, the importance of multilingual students gaining and maintaining competence in their first language(s), as well as in the language of teaching has been noted already (Uribe & Prediger, 2021). Students immersed in such an approach could gain a significant advantage in mathematics performance. A model (Clarkson, 2009) which encouraged the use of all students’ languages when learning in school, including when doing mathematics, has also been described. Both those approaches can be extended to the cultures in which a student’s language is embedded. This would include mathematical practices and thinking found in such cultures which we and others have begun to outline in earlier chapters of this book—but see also in Owens (2015b) and Owens et al. (2018). Clearly, the Western culture within which English is embedded, already dominates the school curriculum. If the concept of “translocal” were to be taken seriously by schools, particularly in urban areas, then it would follow that the multiple foundational languages and the multiple foundational mathematical ideas would also become part of the curriculum. The common, but nevertheless fallacious, belief that students do not have the mental capacity to work in multiple languages at school, and if they do they are likely to get confused between their languages and the different mathematical systems, still needs to be addressed. The first point is that it is normal behaviour for students (and adults) to switch between languages as the situation demands. If care is taken in comparing and contrasting the different mathematical systems, then confusion will be avoided. For the past 50 years, languages like Vietnamese were used in Australian schools for teaching students mathematics and literacy alongside English (Cabramatta Public School in Sydney is a case in point). In other schools, Japanese- and Chinese-background students were permitted to use their home languages in the classroom. Now, in the 2020s, most students in NSW are learning the cultures and languages—although many were nearly lost—of the First Nations (Aboriginal) people on whose lands their schools are situated. This is spearheaded by the local Elders and the Aboriginal Education Consultative Group, which is consulted by schools and principals. Change is possible.

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Beyond just privileging students’ first languages and selecting aspects of their mathematical cultures to teach in school, there needs to be built an ecology of knowledges of different groups (Jacob, Gonzales, Chappell Belcher, Ruef, & RunningHawk Johnson, 2021). The translocal approach could provide the theoretical base for this broader approach to the diverse ethnic groups of PNG, as well as the Western approach to learning, teaching and structural organisations. In Aotearoa, New Zealand, and in universities in Australia there are strong changes to curriculum following the recognition of Maori and Australian First Nations and their knowledges respectively. It is now the case that all beginning teachers during their own schooling or at university meet and learn from First Nations people and are being educated in the diversity of First Nations cultures and languages. Sadly, as noted in an earlier chapter this is not occurring to any extent in Papua New Guinea, where diversity of cultures, languages and local knowledges are yet to be recognized in the same way at the tertiary level. In some cases, this notion is regressing from earlier progress. For example, it does not seem to be taught often at the Teachers Colleges and with changes in the Director of the Glen Lean Ethnomathematics Centre at the University of Goroka, continuing research and funding has stagnated. However the University still teaches the course (subject) Mathematics, Language and Culture as an elective in its education degree. It appears, however, that little use is being made of the Centre’s library of materials in teaching this subject. The authors of this book believe that Mathematics, Language and Culture should be a compulsory subject in all of PNG’s teacher education institutions. Glocalization Closely related to the concept of “translocal” is that of “glocalization,” which is the way in which global features are interpreted by locals, for locals, and with local values. Importantly, there is the global external perspective and local perspectives, and “glocal” emphasizes the dialogue between both. Importantly, it recognizes that the product needs to be of worth for the local as well as for the global. It takes a social justice stance into account, sociopolitically. For this, ethnomodelling provides a way forward. In this approach ethnomathematics is portrayed in terms of the models which are used. Orey and Rosa (2021a) explain that this dialogue is carried out for mathematical modelling through “ethnomodelling.” By discussing the mathematics of culture and developing a mathematical model for the practices, it is likely to generate culturally appropriate mathematical modelling (Orey & Rosa, 2021a). It is a dialogue that permits the outsider to work alongside the insider; the mathematician and syllabus writers to interpret mathematics for local schools in conjunction with teachers and students to assist them to develop their own culturally appropriate mathematical models. This approach is best epitomized by the work of Geddes in Mozambique, Lipka with the Yup’ik in Alaska, and Pinxten with Navajo Native, First Nation Americans. Others have achieved this with students in Brazil, Israel, Australia, PNG, Aotearoa New Zealand, The Philippinnes, Nepal, and Indonesia (for examples, see chapters in the "International Conference on Ethnomathematics 6," 2018; Pinxten & Vandendriessche, 2023 in press).

The Value of Ethnomathematics and Ethnomodelling

It is time for Papua New Guinea to properly fund the early years of schooling with priority going to each language establishing its own literacy program incorporating its own mathematical knowledge and feeling free to adopt any relevant and related Western mathematical concepts (Edmonds-Wathen, Sakopa, Owens, & Bino, 2014). The first two chapters of this book, albeit based partially on Western perspectives, could offer a rich foundation for cultural mathematics.

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Such a revitalisation of early schooling will only be sustained in the long term if analogous changes are made in the teachers colleges. At present, teachers education courses on ethnomathematics are rarely to be found, and if they are taught at all, they are only seen as an introduction in some way to the “proper” mathematics, Western mathematics. The few ethnomathematics subjects focus mainly on counting systems. The approach needs to be broadened with, for example, explorations of the longevity, cultural context, and complexity of some relational systems, and foundational spatial ideas found in traditional cultures (Owens, 2012b, 2015b), and the various games, some of which are described in Chapter 2 of this book that embody various pattern learning (Owens et al., 2018). Such important ideas should be considered seriously by all teacher education students for a substantial number of weeks, and not located to an optional category, as occurs now. There are also higher order mathematical explanations related to group theory, polynomes and other systems, some based in the kinship systems, some in games such as string figures (the cat’s cradle), others in complex art, craft, building systems, navigation, and many other foundational knowledges (Campbell, 2002; Crawford, 1981; Owens, 2015b, 2016, 2022 in press; Vandendriessche, 2015). The preparation and publication of foundational mathematics research by Papua New Guinean scholars is well overdue—although the student reports written as part of Mathematics, Language, and Culture units at the University of Goroka could provide a good start (Owens, 2014). In addition, alternative assessments are needed in education. For example, in Aotearoa New Zealand, there have been efforts to ensure that assessment caters for the well-being of the Maori students and provides contextual approaches such as manaakitanga (care, respect, hospitality), wānanga (a forum, a sharing of knowledge, a place of learning) and culturally sustaining pedagogy (Kerr & Averill, 2021). Pasifika students’ cultures, foundational and current, together with their translocation context are recognized in professional development of teachers (Hunter & Hunter, 2018). Increasingly around the world, it is acknowledged that school mathematics is only a part of mathematics. Ethnomathematics contains all mathematical developments over time and place. Diversity and history in mathematical practices abound in PNG. Brian Greer (2021), when expounding on Jens Høyrup’s writings, noted “history is the future” and “creativity and consolidation” are both needed in mathematics. In that sense PNG’s diversity of mathematics provides for much creativity in developing and sharing in the classrooms of PNG and in connecting with school mathematics. Greer (2021) continued “diversity is manifest in all families of practices involving mathematics, with the notable, and arguably unfortunate, exception of school mathematics.” We have noted that particular assessment regimes, and the use of English as the language of instruction, reduce diversity and hence social justice for all students (D'Ambrosio, 2006). In terms of diversity, the Elders need to introduce their knowledges and relationships into schools. Ethnomathematics ought to be the focus from the start at all education levels. This is more than the mere inclusion of examples from culture, but a full recognition of the mathematical ways of thinking and language associated with cultures. This should apply to more than just counting systems. Such a focus will prioritize the more recent realization that learning about number through comparing, noticing differences in measuring, and then associating numbers is a sounder way to develop number sense than just through counting (Möhring, Newcombe, & Frick, 2015). From this respect, it should be significant that PNG abounds in visuospatial reasoning (Owens, 2015b, see Chapter 11). Ethnomathematics has a role to play in social justice by giving voice to minority cultures. Multicultural approaches are essential for growth and for harnessing the gifts of multicultural and multilingual schools (Barwell, et al., 2015). The development of creative school-based curricula calls for more time, money and expertise.

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All development is not just to be guided or controlled by those with land but in consultation with others including experts who may be from another PNG culture or from overseas. Importantly, overseas experts should not be given dominating, and well-funded, power but should nevertheless be free to offer insights into how ethnomathematics can be the core to effective curricula and learning. It is important, in establishing diverse ways of knowing and doing things differently, to recognize and value why groups carry out mathematical practices. Providing a voice and using dialogue in the community and the classroom, and providing suitable recognition and privilege, can help to bring the mathematics of the community into schools. D’Ambrosio (1990) pointed out that all cultures take part in mathematical practices such as observing, evaluating, comparing, classifying, organizing, measuring, quantifying, inferring, and modelling. Bishop, heavily influenced by his sabbatical in PNG, suggested mathematical activities included counting, locating, measuring, designing, playing, and explaining (Bishop, 1988). Both D’Ambrosio and Bishop noted that problem posing, problem solving and investigating should be crucial elements of mathematics education. From an ethnomodelling and ethnomathematical perspective, the construction of mathematical knowledge should incorporate the reality, daily life, and context of members of distinct cultural groups. The ethnomodelling process should begin by introducing new situations and problems to learners, for them to master within their own context and their unique cultural and experiential reality. Helping them answer the question: Why is this so? (Orey & Rosa, 2018). This process enables new forms of mathematical knowledge and experience to be learned; as modelers construct or deconstruct their knowledge they link to a larger mathematical universe. In accordance with Rosa (2010), it is with a sight (perspective) towards both historical, social, cultural, as well a diverse, or alternative forms of understanding, comprehending, and resolving problems that these diverse uses of mathematics become relevant and meaningful to students. (Orey & Rosa, 2021b, p. 158) Nevertheless, ultimately problem solving should be based on problems arising from the students’ local, natural environment. As a preparation to developing appropriate mathematical skills, D’Ambrosio suggested posing the following three questions: 1. How are ad hoc practices, problems, and solutions developed into methods? 2. How are methods developed into theories? 3. How are theories developed into scientific invention? (Quoted from Orey & Rosa, 2021b. p. 155) Ethnomodelling provides a means of developing the curriculum of the classroom through culturally sustaining dialogue. Such dialogue would respect the individual and community, their identity, knowledge and beliefs. In many ways, this would provide a way forward for the complexities of learning situations (Davis & Sumara, 2006) where cultures are in flux as in PNG and where there are local as well as global expectations. While that is often theorized, we suggest the theory needs to be operationalized for classrooms and communities. Teachers, students and community need to realize that there are diverse ways of understanding and representing mathematical thinking, that there is not linearity in mathematics or mathematics learning, or a duality of culture and school mathematics. Rather there are emerging unpredictable ideas for open classrooms for trialling and discussing concepts, ideas and problems. There can be networks of thought that will self-organize through dialogue, feedback and relationship building.

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A Warning to Those Intending to Develop a Translocal, Glocalized Curriculum

There are at least two critiques to make of the above discussion. The first is that it is portraying a somewhat deficit perspective for the colonized. There is a lack of money for education. Aid is still flowing into the country generally but with undeniably considerable influence from overseas advisers. There has been resistance to this but the push back to date has been ineffectual in the sense of not being replaced by energetic, accepted alternatives. The second critique concerns duality. Duality does not benefit a decolonizing of the curriculum or of education in general. It is evident in categorizing: colonizers and the colonized; the elite and the less advantaged; the rich and the poor; the good English speaker and those who are not so good; National High School or International School students and other students; Government officials or teachers and village Elders, and School Councils; men and women; Highlander and coastal. In developing education and providing a voice for the majority and for difference, there needs to be a breakdown in duality. Power is likely to come from one side of the duality. Importantly, the elite often bring neocolonial attitudes into the discussion and into decision-making. Resistance from the other side may not necessarily achieve its purpose. It needs to be recognized, from the outset, that structures which are in place may not be changed, and discontent, lack of progress, or poor implementation may result. Good practices and good people can be lost when political advantage or power take over from steady well-advised change. The rift which has developed between home mathematics and school mathematics needs to be bridged. This may bring peace within people and between people in terms of valuing the mathematics they use, and the mathematics which others use (D'Ambrosio, 2006; de Abreu, Bishop, & Presmeg, 2002). Establishing a mathematical identity from a social identity strengthens not only the person but the country (Owens, 2014, 2015b). Conclusions The early chapters in this book described a diverse range of mathematics used by different cultural groups within Papua New Guinea, from before first contact until now. Recognizing that the foundational forms of mathematics need to be understood by both expatriates and nationals, in order that they will learn to appreciate that ethnomathematics offers foundations for mathematics education, can enable the nation to move forward from neocolonial perspectives of mathematics and mathematics education. Throughout the middle chapters of the book it was argued that the historical records indicate that the colonialists looked for control and sameness, and stigmatized difference and cultural knowledge. Although there were indications that language was an important aspect of culture, identity of self and group, local languages were seen as somehow inferior to English. For missionaries, the vernacular was the language of education but for the colonial administrators English (or initially in the north, German) was selected as the language of instruction. The importance of vernacular languages and cultures were again recognised about 120 years later during the reform era with attempts to maintain and promote them as the basis of education. Unfortunately, colonialism and neocolonialism had taken control of the thinking of too many teachers and educators unfamiliar with their own and other PNG cultures, languages, and ways of learning. There were monetary constraints in education, as well as massive money influences in the fields of mining and forestry. These induced significant changes especially with movements of people within the country, and reduction in exchange relationships and power that was associated with both education and money. People wanted closer control of government services, and thus provinces were given control of education but not of curriculum, which remained under the control of the national government. Attempts to ensure that the curriculum was appropriate for all students and that schooling reached into remote corners, were not successful.

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When the reform was introduced it was 20 years too late due to colonialism and neocolonialism. By that time, the influence of local languages and cultural practices had been significantly reduced especially in schooling as a result of Western influences and internal conflicts, and there was no recognition of a need for a major focus on ethnomathematics or on the peace it might bring between people (D’Ambrosio, 2006). However, even the reform was tainted by neocolonialism with the structure of the curriculum being brought in from overseas. About that time, too, the national advisory committees were no longer as strong in establishing curricula. Although there were similar workshops, they were more ad hoc and more likely to be heavily influenced by the external advisor or those within the country who had the privilege of education and experience overseas, and a stronger grasp of English—the neocolonial elite. Any sense of ownership by local educators of the curriculum was lost (Quartermaine, 2001; Tapo, 2004). Although countries want to know their progress in literacy and numeracy over the years, and although schools and education providers need to make decisions requiring assessment, including the provision of examinations, it should be remembered that testing is not necessarily the best way by which decision-makers can be informed. Many a capable student has been lost to education as a result of examinations. Quantitative analyses such as that of EGRA (Early Grade Reading Assessment) may tell only part of the story. Despite Paraide’s evidence (Paraide, 2003; Paraide & Owens, 2018) and Matang’s evidence (Matang, 2005, 2008; Matang & Owens, 2006, 2014), the reasons for the so-called failure of elementary schools were complex. Foremost among them was the lack of education of the teachers, a point that neither the colonizers (through the Australian Aid project), the politicians, the institutions, or the elite necessarily wanted to admit. The second was the high cost of adequate implementation, but when writers pointed this out at various stages, it was regarded as a neocolonial attitude. The economy was ruling decisions. In addition to an emphasis on ethnomathematics, building gender equity will bring greater creativity, peace and social justice to the country (D'Ambrosio, 2006). Women too may have specific mathematical areas for which they are responsible and the associated forms of mathematics need to be established. Recognizing and valuing language as the carrier of culture will encourage the effort needed to voice mathematical ways of thinking and maintain the languages of PNG (Hunter, Civil, Herbel-Eisenmann, Planas, & Wagner, 2018). Due to the effects of colonization, we hope the first few chapters in this book, and some of our other writings (Owens, 2012c, 2014, 2015b, 2016, 2020, 2022 in press; Owens et al., 2018) will assist in clarifying the richness of mathematics of the diverse communities of Papua New Guinea. We finish with a quotation from the late Rex Matang, former Director of the Glen Lean Ethnomathematics Centre, at the University of Goroka: It is not uncommon to hear many Papua New Guinean students at all levels of the national education system commenting that they have always found mathematics, compared with other prescribed school subjects, a difficult one to learn and understand. Depending on the type of cultural background that one comes from, and how well one attaches to mathematics as a formal school subject, there will surely be a variety of responses on this comment given by different individuals. For some, such comments are expected and they would wholeheartedly welcome it, because it further increases the knowledge gap between what the few privileged individuals know about the subject as an academic discipline and those who are not, particularly in terms of its everyday use as an important cultural tool. For others who show no interest in the comment just ignore it …For those who seriously call themselves mathematics educators, such a comment is unacceptable, because it directly brings into question their professional integrity. (Matang, 2003, p. 1)

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As argued by Resnick (1989, p. 166) (in Masingila, 1993, p. 18), that “schools place too much emphasis on the transmission of syntax (procedures) rather than on the teaching of semantics (meaning) and this discourages children from bringing their intuitions to bear on school learning tasks.” Providing the necessary link between the students’ ethnomathematical knowledge gained in out-of-school situations and the formal mathematics learnt in school is where the role ethnomathematics becomes fundamentally important particularly if students are to make any meaning of the abstract mathematical concepts taught in school mathematics. In other words, ethnomathematics complements the efforts of both the teacher and students in the learning of formal school mathematics in providing the relevant contextual meaning to many abstract mathematical ideas which otherwise would be difficult for students to learn and understand (Boaler, 1993; D’Ambrosio, 1990). What is required of the classroom teacher is to build upon the ethnomathematical knowledge that students bring into the mathematics classroom from their everyday experiences. One way to achieve this is to encourage students to make relevant connections between these two worlds whereby the role of the teacher is to facilitate mathematical activities in which students identify and use relevant everyday cultural practices of mathematics to explain the conceptual meanings associated with the abstract mathematical ideas found in school mathematics. Such a teaching approach will also formalize the students’ ethnomathematical knowledge gained through practical experiences in which students also develop sense of ownership to that knowledge. This way it encourages the students to learn mathematics in a more meaningful and relevant way. Teaching of school mathematics, where ethnomathematics plays a very central role, can be more effective because it has the potential to “yield more equal opportunities provided it starts from and feeds on the cultural knowledge or cognitive background of students” (Pinxten, 1989, p. 28, in Masingila, 1993, p. 20). (Matang, 2003, p. 5) Matang’s vision deserves serious consideration. References Adam, N. A. (2010). Mutual interrogation: A methodological process in ethnomathematical research. Procedia—Social and Behavioral Sciences, 8, 700–707. Alangui, W. (2017). Ethnomathematics and culturally relevant mathematics education in the Philippines. In M. Rosa, L. Shirley, M. Gavarette, & W. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education. Cham, Switzerland: Springer International. Alangui, W. (2021). Building stone walls: A case study from the Philippines. In A. Rogers, B. Street, K. Yasukawa, & K. Jackson (Eds.), Numeracy as a social practice: Global and local perspectives. Abingdon, England: Routledge. Barwell, R., Clarkson, P., Halai, A., Kazima, M., Moschkovich, J., Planas, N., Phakeng, M., Valero, P., & Villavicencio, M. (Eds.) (2015). Mathematics education and language diversity (The 21st ICMI Study). Dordrecht, The Netherlands: Springer. Begg, A., Bakalevu, S., & Havea, R. (2019). South Pacific: Mathematics education in the South Pacific. In J. Mack & B. Vogeli (Eds.), Mathematics and its teaching in the Asia-Pacific region (pp. 1–34). Singapore: World Scientific. Bettina, B. (2015). Cross-sex siblingship and marriage: Transformations of kinship relations among the Wampar, Papua New Guinea. Anthropologica (Ottawa), 57(1), 211–224. Bishop, A. (1988). Mathematical enculturation: A cultural perspective on mathematics education. Dordrecht, The Netherlands: Kluwer.

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Appendix 1: Brief View of History of Early Contact

It took about four centuries for European navigators from Portugal, Spain, France, England, the Netherlands, and Germany to map the coastline of the mainland of New Guinea and the islands off the coast (see Table A1.1). Their sense of longitude was not strong, but there was a possibility that land masses were positioned depending on which side they fell of the longitudinal line which divided the area with west claimed by Spain and east by Portugal. The distance sailing across the Pacific from the Straits of Magellan by the Portuguese was particularly short of reality. Most of the ships are not recorded as stopping. The cartography was often passed on to future sailors from different countries, often with inaccuracies made by mapmakers like Dalrymple. Colonies were not generally started. Mendana in 1595 failed—disease killed most who attempted to stay (Wentley, 2007). Although some of the sailing ships came close enough for various reasons such as a safe harbour and the need for fresh water, these encounters were not always recorded. However, there are a number of recorded encounters. One gets the impression that some of “intruders” responded in criminal ways or were more interested in profiteering than in people. Some of these encounters are summarized in Table A1.2.

Early Settlers

When, in the late 1800s people from Europe settled in what is now called “Papua New Guinea,” they were usually armed, or were interested in profiteering, or had a scientific or moralising purpose. Although Mikloucho-Maclay seems to have had respectful relations, the opposite can be said for D’Albertis. Documentation of the Royal Anthropological Society Expeditions recorded interesting information, especially in terms of language. Some of the church and government men were reasonably enlightened in terms of respecting all people. Some considered their purpose was to work alongside their fellow humans, to learn the local languages, and to bring medicine, education and Christianity to them.

Mainland Patrols

Many of the patrols were for government information such as the one led by MacGregor, mission work (such as Flierl), or seeking gold (such as Leahy). Some combined both document ing government information and seeking gold (such as Taylor and Black). These routes are marked on the map in Figure A1.2. The earlier routes were up the two main rivers from the south and north—the Fly River and the Sepik River, respectively. Until the end of World War 1 their routes also made use of rivers and valleys as well as crossing from one to the other. After WWI, Australia administered both territories but with different administrations and mandates. That continued until WWII at which time the administration was combined. For further information, © Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9

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382

see the documents and books cited in the next paragraph and also Gammage (1998) and Leahy (1994). These patrols were generally in uncontrolled areas. Figure A1.2 indicates where the patrols went with the number referring to the information in Table A1.3.

Detailed Accounts

Detailed accounts are available in D'Entrecasteaux (2001), The Rattlesnake (Goodman, 2005), Mikloucho-Maclay (1975), Chinnery (1924, 1926), the Lutheran Church (Wagner & Reiner, 1986), and the United Church (Threlfall, 1975, 2012), Whittaker, Gash, Hookey, and Lacey (1975), Torres (Hilder, 1980), Murray (West, 1968), and the many accounts by Ray (e.g., Ray, 1891, 1895, 1919, 1929, 1938). Roberts (1996), and others, also provide interesting accounts. Table A1.1 European Sailing Ships 1500-1850 Attempting to Explore Papua New Guinea Date < 1500

Person/Ship

1526

Jorges de Meneses

1528

Alvaro de Saavedra

Country of Origin Malaya, Indonesia, China, Molluccas

Area of PNG West Papua

Comment Mostly visitors but settlements were established in Vogelkop (Bird’s Head).

Portugal

North coast

From Molucca. There is no evidence of a landing on the mainland, but there is for western islands of West Papua

Spain

North coast, Manus

1537, 1545 Ynigo Ortiz de Retes

Spain

North coast

From the east

1567

Alvaro de Mendaňa

Spain

Solomon Islands

From the east.

1605

Jan Lodewijkszoon The Roosengin and Netherlands willem Jansz

South coast of Papua and West coast of Australia’s Cape York but not the passage between

Ship Duyfken from the East Indies; followed by Pera that returned to Ambon, and Arnhem that saw tip of Australia’s Arnhem land but not much more than Ceram.

1606

Quiros, initially with Torres but separated

Vanuatu

From the east.

Spain

(continued)

Appendix 1: Brief View of History of Early Contact

383

Table A1.1 (continued) Date

Person/Ship

Country of Origin

1606

Torres

1616

Area of PNG

Comment

Spain

Louisiade Archipelago, Southern coast of Papua.

From east

Shouten and Le Maire

The Netherlands

North coast, eastern islands and east coast of New Ireland (they thought that this was the mainland); north coast of New Guinea (by-Geelvink Bay).

Zuid Compagnie of Australische from the east to rival the Dutch East India Company.

1643

Abel Janszoon Tasman

The Netherlands

Solomon Islands, New Ireland, islands to east of these but then down through the Vitiaz Strait realizing the landmasses to the East were separated from New Guinea and making the north coast of New Guinea.

From Batavia via Van Diemen’s land. Tonga and Fiji. A later trip mapped northern Australia but did not pass through the Torres Strait

1699

William Dampier in Roebuck

England

Travelled from North coast to southern islands of Mussau and Erima in the Bismarck Archipelago, strait between mainland and New Britain.

The English East Indies company had been set up in 1599 but no real entrée was achieved until 1699 Missed the passage between New Ireland and New Britain

1705

Geelvink

The Netherlands

Geevink Bay on the North coast.

1740s

Anson and Campbell

England

1756

Philip Carteret

England

Suggested settlement on New Britain Solomons and the From the east New Guinea Islands including St George’s passage between New Ireland and New Britain, Aua and Wuvulu South of Manus. (continued)

Appendix 1: Brief View of History of Early Contact

384

Table A1.1 (continued) Date

Person/Ship

1768

Louis de Bougainville

1771

Country of Origin

Area of PNG

Comment

France

Vanuatu, Solomons, passage between Solomons and Bougainville Island, Buka.

From the east.

James Cook

England

East-West passage between Australia and New Guinea.

Shoals are a problem

1787

Thomas Read, Alliance

American

Possibly realized West coast of Bougainville.

From Sydney to find new route to China. Others followed

1792

Antoine D’Entrecasteaux

French

Detailed Solomons, Bougainville, Buka, strait between them, St George’s strait.

Looking for La Perouse

1849

Owen Stanley

England

Louisiade Archipelago

From Sydney to chart the Great Barrier Reef and the shoals and south coast of Papua

1850

By this stage, the coast of the Mainland, and most of the islands, were at least roughly chartered. Note. The above summary is based on information in Jack-Hinton (1972), Jinks, Biskup and Nelson (1973), and Langdon (1971).

Figure A1.1. The mapping of Papua New Guinea by European explorers—see Table A1.1.

Appendix 1: Brief View of History of Early Contact 385

Italy

Catholic Mission

Captain Owen Stanley

1848

1849

Rev W.W. Gill & A W. Murray

Captain Moresby

George & Lydia Brown, Britain Methodist Mission

Marquis de Rays

Catholic Mission

Catholic Mission

Lutheran Mission

1872

1873

1875

1880

1882

1885

1886

Germany

France

Britain

Britain

Russia

Mikloucho-Maclay

1871 and later

Italy

L.M. d’Albertis

1875, 1877

Lt. C Yule (with Stanley Britain initially)

Britain, Rattlesnake

Country of Origin France

Date Person 1768- De Bougainville 1769

Summary of Early Contact

Table A1.2

Finschhaffen, Morobe

Yule Island, Central Province Papua

Gazelle Peninsula, East New Britain

New Ireland

Duke of York Island

South East Papua

Central Papuan coast

Madang

Along the fly river

Orangerie Bay of South East Papua

Southern coast & islands

Woodlark Island, Papua

Area of PNG South-East of New Guinea

A settlement that failed.

Together with Fijians and Samoans

Positive meeting but reluctant to stop

LMS London Missionary Society

Lived with people as an early anthropologist

A dispute with locals led to violent quarrels.

Brief fight

Warned to be constantly on guard for treacherous locals

Some Italians attempted a settlement but failed and went to NSW

Comment Met by locals in canoes, persuaded to come on board

386 Appendix 1: Brief View of History of Early Contact

Rev A. Maclarin & Rev. Britain C. King, Anglican Mission

Bishop M. J. Stone-Wigg

Catholic Mission

1891

1898

1896

1880- Queen Emma, Phebe 1914 and Richard Parkinson

Methodist Mission

1891

American-Samoan German

Britain

Britain

Germany

Johann Flierl Bavarian Lutheran Mission

1887

Country of Origin

Person

Date

East New Britain and other islands

Aitape, East Sepik

Dogura Plateau, Oro Province, Papua

Dobu Island, Milne Bay Province, Papua

Astrolobe Bay, Madang

Area of PNG

Successful business woman Scientist, coconut plantations, Eugenics perfection with German and Samoan/ Polynesian women on Mortlock Island. Buka men

First Bishop

Away from coastal diseases, they hoped

Comment

Appendix 1: Brief View of History of Early Contact 387

Figure A1.2. Early Patrols in mainland Papua New Guinea.

388 Appendix 1: Brief View of History of Early Contact

Appendix 1: Brief View of History of Early Contact

389

Table A1.3 Mainland Patrols Route(s) Date 1 1876

Patrol Leaders D’Albertis

Route Fly River (end point marked)

2

1885

Everill

Fly and Strickland River (end point marked)

3

1886

Schleinitz

Sepik River (end point marked)

4, 5

1887, 1910

Schultze

Sepik River (end point marked)

6, 6

1890

MacGregor

Fly River (end point marked) and Port Moresby to Oro Province coast

7

1896

Lauterbach

Madang coast to Ramu River

8

1906

Monckton

Lakekamu River to Port Moresby

9

1907

Dammkohler

Markham River to Ramu River

10

1910– 1911

Staniforth Smith

Kikori River to Erave River

11

1913

Pilhofer and Flierl

Markham River to Bulolo

12

1913

Thürnwald

Sepik River

13

1917

Humphries

Owens Stanley Range

14

1922

Austen and Logan

Fly and Alice Rivers

15

1927-8

Karius and Champion

Sepik and Fly Rivers

16

1929-30 Akmana Expedition

From Sepik towards highlands

17

1930

Leahy and Dwyer

Purari to Markham, Lae

18

1933

Leahys and Taylor

Bena Bena, Eastern Highlands to Wahgi and Jimi Valleys

19

1935

Hides and O’Malley

Erave and Kikori Rivers

20

1938-9

Taylor and Black

Hagen, Wabag, Strickland, Kalawari to Sepik, Mt Kari, other tributaries

390

Appendix 1: Brief View of History of Early Contact

Figure A1.3. Lawyer Cane Bridge, Waria River, August/September, 1930 PIC/3234/1-121 LOC Album 962-Papua New Guinea, Waria, Markham, Tauri, Langemar, Purari and Ramu Rivers [picture]./

Appendix 1: Brief View of History of Early Contact

391

Figure A1.4. Papuans making sago, Frank Hurley (1921), Trove digital repository, National Library of Australia.

Figure A1.5. Frank Hurley, 1921, village of Ambasi, Trove digital repository, National Library of Australia.

Figure A1.6. Frank Hurley, 2021. Village of Deva Deva, not far from the coast despite the mountains. Trove Digital repository, National Library of Australia.

Appendix 1: Brief View of History of Early Contact

393

Figure A1.7. Frank Hurley (~1922). Papuan long house, Kaimari village, Urama island. Trove digital repository, National Library of Australia.

Figure A1.8. Frank Hurley, 1921, woman and child making pots, Hanuabada village. Trove digital repository, National Library of Australia.

Appendix 2: Teachers Colleges

Affiliation of Teachers Colleges, Their Inception and Reduction Lutherans Hopoi 1924 Gelem-Siassi 1950 Heldsbach 1914 Rintebe 1957 Amron 1934 Anglican Dogura (Papua) 1898 Later these became Balob Teachers College, Lae, Morobe Province Methodist Duke of York Islands 1882 Kabakada 1890 Vunarima 1958 Ruatoka (Papua) 1894 Roman Catholic Asitavi (Bougainville Girls) Kieta (Bougainville Men) 1899 Yule Island (Papua) 1892 Vuvu (Men) 1901 Vunakanau (Men) Kabaleo (Women) Later these became Kabaleo, at Kokopa, East New Britain Roman Catholic Kairiu Island (Men) 1937 (Catholic) Kunjingini 1955 Later these became Kaindi in East Sepik Province Roman Catholic Fatima 1934 (Western Highlands Province) Later this became Holy Trinity, Mount Hagen, Western Highlands Province Roman Catholic Alexishafen 1908 (Government) (Madang Province) Sek 1909 Lutheran Amele 1923 Karkar 1924 © Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9

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Government Madang 1964 Manus 1955 Later these all became the Government-controlled Madang Teachers College, Madang Unevangelized Fields Mission (UFM) later Asian Pacific Christian (Evangelical Alliance) Mission (APCM) Awaba (Papua) 1965 Later this became APCM Dauli, Tari, Hela Province Seventh Day Adventist Sonoma Kokopo Kabiufa 1950 (SDA) Later this became SDA Sonomo Kokopo, East New Britain Government Port Moresby (PMTC) Malaguna 1934 Dregerhafen 1949 Goroka 1959 Popondetta (Papua) 1955 Later these were replaced by Government Port Moresby Teachers College then Inservice College (PMIC), then PNG Institute of Education

Source. Quartermaine (2001) Figure A2.1. Mathematics Teachers College lecturers meeting in 1978 to prepare syllabus.

Appendix 2: Teachers Colleges

Figure A2.2. Balob Teachers College Diploma Subjects per semester 1998.

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Figure A2.2. Balob Teachers College Diploma Subjects per semester 1998.

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Figure A2.2. Balob Teachers College Diploma Subjects per semester 1998.

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Appendix 3: University Materials

PNG University of Technology – Preliminary Year but then used later in Year 1 of several degrees

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The IEA Curriculum: Mathematics 1997 The Papua New Guinea Context The purpose of the PNG Context of the Mathematics Curriculum is to sensitise teachers, students and parents to the rich fabric of indigenous mathematics. This statement is not prescriptive, but its importance should not be underestimated. Learners who feel that the knowledge they bring to the classroom is valued are more effective in their learning. In many areas of formal education the traditional knowledge, values and attitudes that children bring from their culture to the classroom deserve recognition. Background Information Traditional societies in Papua New Guinea had various, often sophisticated systems of mathematics. Every culture in Papua New Guinea has developed its own mathematical knowledge which is part of the cultural heritage of each individual. Mathematics represents and interprets environment, culture and society in much the same way as language does. Indigenous mathematics forms the basis for links to formal mathematical language. Therefore it is desirable that the mathematical language and prior knowledge that children brig to the classroom will be explored, translated, expanded and then formalised. A great variety of independent mathematical systems have developed. Counting systems based on a variety of bases, from 2 to 68 have been recorded. It is common for different counting systems to operate within the one culture eg. Base four for eggs, base 6 for coconuts and taros and base five for general items. There are traces of this type of variety in the English language also with eggs counted in dozens and shoes in pairs. In PNG society numbers are often firmly linked to objects counted, for example the word for ten coconuts in the Hula language is niu walona and refers exclusively to this group, having no meaning in relation to any other object. Only a third of counting systems in a representative sample of 225 languages showed a well defined abstract concept of number. Teachers of Mathematics must show an awareness of and respect for the traditional cultural heritage of their students whether from Papua New Guinea or overseas. There needs to be an awareness that some language does not have transferable concepts. Children need to be allowed to compare and discuss mathematical words that are different in their language.

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At the same time, the very diversity of the nature of indigenous mathematics offers both a challenge to teachers, and a rich variety from which they may select materials to enhance and enrich their mathematics program. This could be addressed with a possible focus on: • Involving children and the school community in project based exploration of mathematical concepts and practices. This could involve such things as comparing counting systems, bilum making, mapping, basketry, or mat making, • Using natural materials in counting, space and other mathematical topics, • Exploring cyclic time as well as calendar time, • Investigating measurement based upon natural events and processes in contrast to lineal measurement • Exploring patterns and symbols in PNG arts and crafts, • Playing traditional PNG games, • Examining different PNG mathematical systems and comparing these to HinduArabic and Greco-Roman systems.

Creating a PNG Context for Mathematics A PNG Perspective on mathematics will be apparent when teachers, children and the community: • Think about mathematics as another form of cultural knowledge, • Utilise and value the knowledge of traditional practices of the children and the community of their school, • Actively think about how to go about educating children to consider mathematics as part of their culture. • Use ideas from many cultural backgrounds around PNG to enrich the understanding and teaching of mathematics, • Make connections with the local culture and understand more about the kinds of knowledge that children bring into the classroom, • Are aware of cultural issues that may be sensitive. Children may be reluctant to respond to questions because of cultural restrictions. • Are aware that some traditional activities are gender specific an the mathematical language reflects this, • Are encouraged to discuss and compare maths terminology and concepts in different PNG languages and cultures, • Encourage children to find differences, and to explore and value these differences, • See the reason and importance of using their knowledge so that they are not alienated from their cultural knowledge and values. Ensuring Equity The Mathematics Equity statement is provided to promote awareness of issues pertinent to this curriculum area. It provided background knowledge of equity issues relevant to our schoolsin PNG and may promote discussion of ways to address such issues and concerns. This statement represents a perspective, and as such, reflects a viewpoint for consideration. Language. Language is crucial both in the development of each person’s thinking and in communicating the results of that thinking. Spoken language is aneffective means of communication and promotes sharing and solving.

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In the past, children’s language has been undervalued as a vehicle for learning in mathematics. Mathematics lessons were not generally associated with discussion or with communication between learners through spoken language. There has been a tendency in the past to focus upon language as a formal spoken vocabulary related to the transmission and use of symbols. To confine children to mathematical tasks which involve only writing and reading, without the opportunity for talking and listening is unlikely to maximize mathematical growth. Culture and Language. The relationship between linguistic and cultural factors on the learning of mathematics is extremely important. Some communities have ways of organizing the world which are not based on counting and quantifying. Children from such communities could well enter school with very limited number experience, but with perhaps well-developed spatial skills. Teachers need to ensure that ‘ability’ and ‘intelligence’ are not based on narrow interpretations of knowledge which are culturally based. Perhaps the most pertinent point to be made concerning the mathematical ability of second language speakers is that the difficulties experienced with the language should not be translated into a belief that the student is lacking in mathematical ability. If we are to provide a mathematics program which is inclusive of all groups represented in PNG, we must acknowledge the diversity of backgrounds and languages and the expectations laid down by our schools. Mathematics in schools should build upon the experiences children bring with them from their home and community. The development of mathematical expression is important, as concepts are not developed in the absence of mathematical language. Students are more likely to develop mathematical ideas readily when they have clear ways of talking about their experiences and the ability to label mathematical concepts in a way meaningful to them. In the case of second language learners, language patterns need to be established and particular terms used in appropriate situations to make the meaning clear. Teachers should exploit the many mathematical examples found I everyday life and utilise these concepts as a base on which to build and develop further mathematical experiences. The geometric patterns of bilum making, the weaving of mats and the symmetry evident in the creation of some artefacts are just a few of the many examples to be found. Resource material including locally collected or produced items, should reflect cultural diversity and be used alongside commercially produced items. Gender. The under participation of girls in mathematics is well documented. The past decades have shown an increasing awareness of the disadvantage girls have experienced in this curriculum area. In the past, it has been noted that students failed to reach their potential because they could not see the applicability of mathematics to their lives. These students were not encouraged to connect new mathematical concepts and skills to experiences, knowledge and skills which they already had. This was particularly so for girls, for whom the contexts in which mathematics was presented were irrelevant and inappropriate. As a result, deeply entrenched negative attitudes towards mathematics emerged. Strategies which build upon and utilize the strengths and interests that girls bring to mathematics should be promoted. Active involvement of girls in mathematics could be promoted by setting mathematics in relevant social and cultural contexts, assigning cooperative learning tasks and providing opportunities for extended investigations. Girls should be encouraged to develop greater confidence in their mathematical ability. Early success in routine mathematical operations needs to be accompanied by experiences that will develop confidence in skills essential in other areas of the mathematics curriculum. Girls need to be involved in estimation, construction and problem solving skills to broaden the range of their experiences and to develop a foundation for further extension and development of concepts.

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It is particularly important to acknowledge and value the background experiences for girls which have led to the formation of the skills they bring with them to the classroom. Mathematical interest and opportunity should be promoted, regardless of gender, social class or ethnic origins. Other students who may have intellectual, physical or sensory handicaps will require a variety of approaches and strategies to accommodate their needs. All students have the right to participate and be challenged in this area of the curriculum. The Mathematics Curriculum Curriculum Outcomes The mathematics curriculum encompasses many skills which help students function effectively, both personally and vocationally. Mathematics where possible needs to be developed in contexts which are meaningful to real life situations. The mathematics curriculum strives to develop in students a range of skills and understandings. In particular, teaching and learning in mathematics is directed towards enabling all students to: • Apply understandings of operations, relations and patterns with numbers and use equations and functions; • Explore and communicate spatial relations and geometry dealing with locations, shapes, movements and transformations; • Measure; • Communicate and interpret data; • Solve problems, apply mathematics and use appropriate mathematical language; use estimation and approximation and check the reasonableness of results; • Develop positive attitudes towards exploring, learning and using mathematics. Mathematics and the IEA Key Outcomes The IEA’s key outcomes describe achievements which can be made by all students attending IEA schools. Each of the curriculum areas contributes in an holistic way to the achievement of these exit outcomes. At each developmental stage (see below) students become capable of making further steps towards the fulfilment of these outcomes. Following are the key outcomes together with some comments about how learning in Mathematics relates to each. Schools and teachers should strive to plan Mathematics experiences within the context of these key outcomes. IEA schools will assist all children to: Be self-directing. Mathematics is a tool for life. It improves the capacity to analyse and deal with everyday situations, and in this way empowers learners to pursue their own interests. At the same time, mathematics education involves providing students with a chance to investigate their world using the tools of mathematics. These investigations can be open ended and require students to set their own directions. Communicate effectively. The language of mathematics is part of everyday life and communications. Every time we conduct business, do our shopping or enjoy sports we need diverse knowledge of number, shape and measurement to be able to communicate effectively. At the same time, the huge recent growth in the availability of information through radio, television and the newspaper has led to a need for basic skills in statistics and the ability to consider how reasonable the information we are receiving really is. Behave ethically. A knowledge and sense of mathematics allows us to consider the ethics of situations more accurately. Often ethical arguments are supported with numerical and statisti-

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cal evidence. Interpreted incorrectly these can lead to serious errors of judgement. Through learning in mathematics student become more skilled at testing assertions and hypotheses. They begin to realise the power of mathematics in supporting arguments and have a chance to consider the ethical consequences of the misuse of this power. Work collaboratively. Many of the activities which involve working with others in the real world require some skills in mathematics. For example, in most cases we make personal and professional financial and budgeting decisions in consultation with others. Additionally, mathematical investigations and problem solving often require the combined efforts of a group, giving an opportunity for this skill to develop. Analyse and solve problems. Problem solving is at the very core of mathematics. Virtually all mathematical endeavor is concerned with applying learning to real world situations and hence to solving problems. While mathematics is a wonderful tool for the analysis and solution of problems, the skills of problem solving learnt in mathematics can be applied in a wide range of situations. (IEA, 1997, 7-14).

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Source. IEA, 1997, pp. 22, 26, 29, 54. Figure A4.1. Four pages from the IEA Mathematics Curriculum, 1997.

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Reform Elementary School Cultural Mathematics Table A4.1 Cultural Mathematics Outcomes 2003 (from CRIP wallchart) Elementary Prep Number P3.1 Count objects in vernacular using local number system P3.2 Describe the four operations using simple vernacular words P3.3 Solve simple problems using concrete materials P3.4 Describe traditional money and Papua New Guinean money Space

P1.1 Follow and give simple directions for moving in a space P1.2 Identify locally known shapes by their visual appearance

Measurement

P2.1 Measure the length, weight and capacity of things using their own informal measuring units P2.2 Measure how much space is covered by objects using their own informal units P2.3 Use time markers

Chance

P5.1 Identify events that always happen regularly in the community

Patterns

P4.1 Make simple patterns

Elementary 1 Number

1.3.1 Count groups of objects in vernacular 1.3.2 Use number symbols that mean the same as vernacular number words 1.3.3 Solve problems using two-digit numbers to 20 or closest to 20 in vernacular 1.3.4 Use different amounts of money to make up various sums of money

Space

1.1.1 Follow and give directions to move from place to place 1.1.2 Compare and group shapes in the community

Measurement

1.2.1 Measure, and compare the length, weight and capacity of things using local informal units 1.2.2 Compare and measure an area using local ways of measuring 1.2.3 Tell and use time in traditional ways

Chance

1.5.1 Identify and describe events that sometimes happen in the community and environment

Patterns

1.4.1 Recognise various local patterns

Elementary 2 Number

Space

2.3.1 Count objects in vernacular and English using local and standard number patterns 2.3.2 Use vernacular and English words for number symbols and operational signs 2.3.3 Solve problems using two-digit numbers up to 99 2.3.4 Make and solve money problems 2.1.1 Follow directions from simple maps 2.1.2 Investigate and describe the features of geometric shapes

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Measurement

2.2.1 Compare the accuracy of local measures of length, weight and capacity 2.2.2 Estimate the number of objects needed to cover a surface 2.2.3 Identify and sequence events that occur at different times

Chance

2.5.1 Make guesses about events that will happen, may happen or will never happen

Patterns

2.4.1 Collect and compare various patterns

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Materials for Schools

Source. Price, ~2000, p. 10. Figure A4.2. Support material from SIL, Elementary 1.

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Elementary Teachers Guide 2003

(NDOE, 2003, p. i) There was an emphasis on integrating the three syllabuses Culture and Community, Cultural Mathematics and Language by using events that occurred in the community and through other themes. Vernacular phrases were encouraged for mathematical operations and topics and that there be some integration with the strands of language in speaking and listening, writing, and reading. Throughout the syllabus, there was evidence of links to the community and vernacular language. The recommended skills and suggested activities were closely linked.

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Source. NDOEb, 2003, pp. 63-64. Figure A4.3. Junior Primary Content

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Ministerial Policy Statement No. 01/2013 Dated 28/1/2013

(Minister of Education Office, 2013, p. 1)

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Elementary Syllabus 2015 Curriculum Principles Our Way of Life Our customs, traditions and values. The syllabus provides for the growth of our cultural identity through vernacular language skills and activities.it is through language that important aspects of our country’s many cultures and transferred from one generation to the next and between people who live and work together but who originate from different cultures. Our cultures, languages and communities are at the very heart of the vernacular Elementary curriculum Ethics, morals and values. Papua New Guinea National Curriculum Statement emphasises the process of socialization and interaction. Students will communicate their knowledge, skills attitudes and spiritual and moral values in their communities. They will learn how to communicate for different audiences, purposes and situations. In elementary, students will learn to use language confidently in other subject’s areas Multiculturalism. As a multicultural society, we must promote and respect our cultures and languages. The diversity of our cultures is the source of our knowledge, skill, attitudes and Melanesian values. These values will be promoted and knowledge in language and literacy will enable students to share understanding of these with the rest of the world. In the same way, students will learn to exchange understanding from stories and knowledge from the past relating to their own communities and environments. In this way, multiculturism will be maintained and enjoyed while learning experiences will be enriched. Integral human development. Papua New Guinea is a rapidly changing society and faces many challenges. To face these effectively, an individual must strive to reach their full potential socially, intellectually, emotionally, mentally and physically and work with other agents of education such as the home, school and community. The Philosophy of Education for Papua New Guinea, known as the Matane Report, acknowledges the National Goals and Directive Principles in the National Constitution and is based in Integral Human Development. • integral in the sense that all aspects of a person are important • human in the sense that social relationships are basic and • development in the sense that every individual has the potential to grow in knowledge, wisdom, understanding , skill and goodness Citizenship. Through working individually, in pairs or in small groups, the students will be guided how to relate responsibly to others and to respect each other’s opinions, talents, traditions and beliefs. Students will know that each citizen of Papua New Guinea has a role in the growth of their country and that Papua New Guinea herself belongs to a much larger global community. Catering for diversity – gender. Gender is what it means to be a women or man. Gender refers to behaviours and attitudes that are accepted culturally as ways of being a woman, femineity and of being a man, masculinity. Gender is culturally determined. In Papua New Guinea there is a need for the local cultural practices and values with respect to traditional roles of females and males. Catering for diversity – students with special needs. All children have the right to good teaching. Boys and girls must be treated the same during lessons. Teachers must help all children to reach the standards in the syllabus. All children must be given the opportunity to achieve success. Guiding Principles The syllabus is based on three learning principles: • children learn best when we build new learning on what they already know in their culture and home

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• children learn best when they play, use real objects and solve real life problems • children learn best when mathematics is fun, challenging and structured. The syllabus continually refers to pre-existing knowledge and skills and teachers need to set the mathematics into contexts that are familiar and of interest to the students. The students need to use concepts and skills from many areas of mathematics to come up with solutions to problems in real life situations Teaching and Learning. The teaching of mathematics in elementary school will use the textbooks and will be guided by the instructions and teachers notes given in the teacher guides. Teachers must use child-centered teaching strategies to make the lessons enjoyable and challenging. Lessons should be practical and playful and use real objects. Children should be asked open and closed questions and set real-life problems. Children should be given opportunities to talk about mathematics confidently. Children need to learn from what they already know to what they should know in a formal learning situation in the classrooms. The standards are easier to teach and assess. National benchmarks are also set in this syllabus for the end of elementary 2. Student-centered learning. The teaching approach for this syllabus is student-centered learning. The students should be given time to think and do things. Their activities should include investigations, problem-solving and outdoor activities. These provide opportunities for students to discuss, make decisions, plan, organize and carry out activities, record results and report findings. Teaching activities should allow students to listen to each other’s opinions, demonstrate their strategies and critically analyze results. The standards-based approach to learning also emphasizes levels of competence. National benchmarks are set at the end of Elementary 2. These should help teachers to encourage students to become competent, reach benchmarks and be ready for the next level of learning in primary school. Flexibility and relevance. In elementary the school hours will end at 2.30pm each day. The curriculum and learning materials in standard based are also based on activities, stories, culture, beliefs and environment of the community. Teachers need to be flexible in allowing time for spontaneous or unplanned learning experiences to take place any time during the school day. Teachers should encourage students to take part in local activities to make the curriculum more interesting and relevant. Content Overview The syllabus is organised into five units of study: Strand 1 Number and Operation. The strand “Number and Operation” consists of the content that describes the meaning and representation of numbers such as integers, decimal numbers and fractions and methods of calculations In this strand, the objectives are to understand the meaning and representation of 1-digit up to 4-digit numbers, introduction of unit as 10,000, simple fractions. In Operation, students add 1-digit up to 3-digit numbers and do multiplication up to 9 x 9, multiply between 2-digit and 1-digit numbers Strand 2 Quantities and Measurement. In this strand the main objective is to provide students with rich experiences that will become the basis for learning quantities and measurements such as to compare length, area, and volume, and to use objects around them as units of measure and to compare sizes by considering how many such units there are, read clocks in context to their daily life and about the meaning of standard units that are shared by everyone and, and measurements using the standard units. Understand units of length, millimetre (mm), Centimetre (cm) and meter (m) as well as units of volume, millilitre (ml), decilitre (dL), and litre (L), with units of time, day, hour, minute and the relationship among them. Strand 3 Geometrical Figures. The strand “Geometrical Figures” is about the meaning and properties of plane figures and solid figures and the structures of geometrical figures. This strand is very important as it links teaching by closely connecting understanding of the meaning of numbers, quantities, and geometrical figures with activities such as calculations, measurement

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and composition In this strand, the main objective is to provide students with enriched experiences that will form the basis for understanding geometrical figures, recognise shapes, and grasp the characteristics of shapes, make students need to focus on components of geometrical figures such as triangles and quadrilaterals Strand 4 Mathematical Relations. The strand “Mathematical Relations” involves ways of thinking and methods that can be commonly used in handling quantities and geometrical figures. In this strand, the main contents include ideas of functions such as change and correspondences, representation by algebraic expressions of multiplication and their interpretation, organising of numbers and quantities from their everyday lives, represent them by using simple tables and graphs and interpret such representations. Assessment The standards are written so they are easy to assess. There are three types of assessment tasks built into the lesson plans: 1. Assessment for learning Assessment for learning is on-going assessment (also known as “formative assessment”). It is the assessment that teachers do every day during their teaching and at the end of the lesson. Formative assessment helps a teacher to plan their next lesson. 2. Assessment as learning Assessment as learning means that children are involved in assessing their own progress and the work of other children in the class. 3. Assessment of learning Assessment of learning is also called summative assessment. This form of assessment is done at the end of a topic or term. National benchmarking or end of term tests are examples of assessment of learning. Recording It is important for teachers to keep a record of children’s progress and any problems they are having in mathematics. The teachers must use the progress chart in the teacher guides to record children’s learning at the end of each year. Reporting It is compulsory for teachers to; • report the child’s progress to parents at the end of each term; • pass the child’s records to the next teacher before the next school year begins; • pass the child’s records to the primary school when they graduate from elementary. Evaluation Evaluation is when the teacher reflects on their own teaching to improve the children’s learning. For example, • Was the lesson effective? • Did the children reach the expected standard? • How can I improve my teaching? (NDOE PNG, 2015, pp.3, 7, 8, 22)

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Table A4.2 Example of the Detail of the Syllabus Content Strand P.1.2.1 Count the 1.1.2.1 Show objects by 10 as a simple fraction unit for place such as half and value quarter which usually used in their life through dividing into equal parts. Performance Student will Student will Indicators demonstrate demonstrate the achievement the achievement of the above of the above statement when statement when they they; a. Using a. Dividing one composite and object de-composite into two or four such as 10 for parts making or equally and use breaking the relation 10. such as the four b. Counting of quarter part objects by ten. means c. 20 is two sets of the whole of the ten. 23 is two sets original of ten and three one in their life. sets of one. b. Use daily life d. 100 is ten sets experiences of ten or hundred to show sets of one. the idea of dividing equally. Assessment tasks • Memorize the two numbers become 10. • Solve the task for tens place and once place • Counting object by ten or five and communicate which ways of counting is good. (NDOE PNG, 2017, pp. 17, 18)

• Fold paper to show fraction •

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2.1.2.1 Understand the meaning of multiplication, produce multiplication table for easier calculation and use it in their life.

Student will demonstrate the achievement of the above statement when they; a. Knowing situation where multiplication is used and its various ways of representation. b. Finding the simple properties of multiplication, for considering how to calculate, and use of them for making the multiplication table up to 9 times 9 and for checking the results. c. Memorize multiplication table up to 9 times 9 and to multiply one-digit numbers accurately. d. Extend multiplication to the row 0

• Let’s write multiplication sentence using sets of pictures. Memorize multiplication table by using flash cards. • extend the multiplication table using the known multiplication table enjoyably • explain the multiplication table is repeated addition by using blocks and tape diagram • Explain the necessity of multiplication table of row 0

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Junior Primary Syllabus Grades 3-5 Rationale The impact and influence of current developmental and technological trends is significant as Mathematics, Information, Technological Engineering and Science have become driving forces for growing job markets and sustainable development agendas for nations. This Syllabus was revised in response to the public outcry for the declining standards of education including numeracy rates. Papua New Guinea Government introduced SBC aimed at improving the standards of students’ performance through the revision of Papua New Guinea curriculum including Mathematics. Mathematics is a key learning area that underpins many aspects of everyday life such as making sense of natural patterns, information in various forms to make informed decisions. It requires observation, representation, investigation and comparison of patterns in social and physical happenings. The plans of the clearly guided curriculum provides the basis of systematic development of mathematical proficiencies, and values in a quest for a deeper and better understanding of the world around us. Hence, the Mathematics curriculum caters for all 21st century learners, who will think and reason logically, analytically or critically; and become mathematically competent locally and globally overtime. The Primary Syllabus is organised in such a way that teachers are provided clear purpose and focus on disseminating, assessing and monitoring Mathematics proficiencies. Teaching and learning will be effective if the students on the other hand are well informed of their role and purpose in performing the required Mathematics competencies. Aims

The overall aim for Mathematics is to enable all students to: • provide quality mathematics education for all • think and reason mathematically • communicate mathematically by collecting, representing, analysing and evaluating information • apply mathematics processes, knowledge and skills in everyday life • analyse and solve problems using mathematics • make connections within mathematics and with other fields • appreciate mathematics as an essential and relevant part of life • become numerically literate in their daily lives.

It is vital to support the development of student’s proficiencies and competencies as well as the love, appreciation and interest in Mathematics. This will enable them to fully participate effectively and competitively locally and globally. Overarching National Benchmark The overarching goal of the mathematics curriculum is to ensure that all students will achieve a level of mastery of mathematical proficiencies and knowledge that will serve them well in life, and nurture the passion for living that emphasise scholastic ability, a rich heart and mind and the harmony of healthy body as envisioned in vision 2050.

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Level Benchmark By the end of grade five, students should be able to communicate mathematical ideas and thinking and demonstrate systematic approaches in organising and solving mathematical problems. They should gradually develop and apply proficiency from Grade 3 to Grade 5. Grade Benchmarks Grade 3 Standards By the end of grade 3 all students should achieve the following standards: • Recognise, read, write and compare large numbers used in based 10 number value system. • Add and subtract numbers in vertical form with and without carrying two place. • Multiply and divide numbers in vertical form. • Add and subtract simple fraction. • Measure distance using the standard units of measurement metre (m) and kilometre (km). • Use units of weight; grams (g), kilograms (kg) and tonne (t) in given situations. • Use seconds, minutes, and hour to measure duration of time. • Convert hour to minutes and to seconds. • Solve everyday problems on time duration. • Understand value of notes and coins and solve various money problems. • Use the properties of triangles to make patterns. • Identify and name the properties of circles and spheres. • Read and write mathematical sentences in addition, subtraction and multiplication. • Use rules of division in simple calculation. • Collect and represent data on tables and graphs. … Curriculum Principles Curriculum principles identify, describe and focus attention on the important concerns that must be addressed when developing the curriculum at all levels of schooling. They are based on significant cultural, social and educational values and beliefs. The principles of the Standards Based Curriculum (SBC) include the following: • A clear focus on the exit of learning attainments after each grade level • Clear, understandable, consistent and progressive of learning development • Aligned with the National Education Standards which are also aligned to the college, career pathways or other lifelong living after school • Built upon the strengths and lessons learnt from Outcome Based Curriculum. Papua New Guinea National Curriculum Standards are based on the following underpinning principles: 1. Integral Human Development. 2. Our Way of Life. 3. Teaching and Learning. 4. Mathematics Guiding Principles. 1. Integral human development. The Philosophy of Education for Papua New Guinea as described in the Matane Report acknowledges the National Goals and Directive Principles in the National Constitution and is consistent with Vision 2050 and Education for Sustainable

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Development. Papua New Guinea is a rapidly changing society and faces many challenges. To face these effectively, an individual must strive to become an integrated person and to work with others to create a better community. The process of Integral Human Development calls for a National Curriculum, which helps individuals to: • • • •

identify their basic human needs analyse situations in terms of these needs see these needs in the context of spiritual and social values of the community and take responsible action in co-operation with others.

The success of a National Curriculum requires the integrated involvement of all the agents of education such as the home, church, school, and community. Within the National Curriculum, the teachers must integrate knowledge, skills, values and attitudes to allow students to achieve the desired standard expectations of Integral Human Development. 2. Our way of life. Cultural relevance focuses on the richness and diversity of Papua New Guinean cultures and languages. These cultures and languages are examined within their own unique contexts and within historical, contemporary, and future realities. Our traditional life is based on a holistic perspective that integrates the past, present and future. Papua New Guineans are the original inhabitants of Papua New Guinea and live in sophisticated, organised, and selfsufficient societies. Our customs and traditions constitute a cultural mosaic, rich and diverse, including different cultural groups. Our customs and traditions are unique and are featured in the National Curriculum. Therefore, the National Curriculum should enable students to: • demonstrate, understand and practice the values, beliefs, customs, and traditions of Papua New Guinea • demonstrate, understand and apply the unique Papua New Guinean communication systems • demonstrate and recognise the relationship between Papua New Guineans and the global communities • recognise, accept and practice Papua New Guinean arts as forms of cultural expression • give examples of the diversity and functioning of the social economic, and political systems of Papua New Guineans in traditional and contemporary societies and • describe the evolution of human rights and freedoms as they relate to the people of Papua New Guinea. 3. Teaching and learning. The expectations for all students set forth through the National Curriculum Standard Framework strongly emphasise intellectual discipline and high standard attainments through relevant curriculum content, The Standards Based Curriculum intends for a different approach to teaching and learning for all students. This approach emphasises the connections between subject areas and the skills to be acquired and used overtime. The students should develop the ability to reason, solve problems, apply knowledge, and communicate effectively. It also requires that instructional practices encourage students to learn from active, independent inquiry into life situations; and, assumes that they become catalysts for students to pursue lifelong quests for learning and continuous growth. In short, this approach to teaching and learning demands teachers to understand and apply the Standards for Teaching and Learning to the educational environment they create in schools and classrooms. The Standards for Teaching and Learning are higher-order thinking, deep knowledge, substantive conversation, and connections to the world. 4. Mathematics Guiding Principles. The Mathematics curriculum principles identify, describe and focus attention on the important concerns that must be addressed when developing

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and implementing the Mathematics subject. They are based on significant cultural, social and education values, beliefs and norms. The Curriculum Principles also assist in identifying the knowledge, skills and processes and values explicitly stated in the Content Standards. Teaching Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. An effective mathematics teaching and learning program is based on careful thought and design of content. Teachers must prepare clear, specific and focused student centred lessons. The sequence of topics and performances should be based on what is known about how students’ Mathematical knowledge, skills, and understanding is developed over time. This requires teachers who have a deep knowledge of Mathematics and are able to draw on that knowledge with flexibility in their teaching task. Teachers must be supported with ample opportunities and resources to enhance and refresh their knowledge. Learning Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Mathematical ideas should be explored in ways that stimulate curiosity, create enjoyment of Mathematics, and develop depth of understanding. Research has solidly established the important role of conceptual understanding in the learning of Mathematics. By aligning factual knowledge and procedural proficiency with conceptual knowledge, students can become effective learners. They have to recognise the importance of reflecting on their thinking and learning from their mistakes. According to scholars, students become competent and confident in their ability to tackle difficult problems and willing to preserver when tasks are challenging (NCTM, 2000). Therefore, students should be actively engaged in doing meaningful Mathematics, discussing Mathematical ideas, and applying Mathematics in interesting, thought-provoking situations such as asking close and open questions and set real life problems. Equity An excellence in mathematics education requires equity – high expectations and strong support for all students. All students come to school with expectations to learn mathematics that meets their individual interest and need. All students must have the opportunity to learn and meet the same high – quality mathematical instruction. The standards provide for a wide range of students, from those requiring special remedial support to those with talents in mathematics. Every student regardless of race, colour, gender and ability should have the benefit of quality instructional materials, good libraries, and adequate technology. Curriculum A curriculum is more than a collection of activities; it must be coherent, focused on important mathematics, well-articulated across the grades. In a coherent curriculum, mathematical ideas are linked to and build on one another so that students’ understanding and knowledge deepen and their ability to apply mathematics expands. An effective mathematics curriculum focuses on important mathematics that will prepare students for continued study and for solving problems in a variety of school, home, and working settings. A well-articulated curriculum challenges students to learn increasingly more sophisticated mathematical ideas as they continue their studies.

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Assessment Assessment should support the learning of important mathematic concept and furnish useful information to both teachers and students When assessment is an integral part of mathematics instruction, it contributes significantly to students’ mathematics learning. Assessment should inform and guide teachers as they make instructional decisions. The tasks teachers select for assessment convey a message to the students about what kind of mathematical knowledge and performance are valued. Feedback from assessment task helps students’ in setting goals, assuming responsibility for their own learning, and becoming more independent learners. Technology The use of technology is an essential tool to facilitate teaching and learning. It influences the way mathematics is taught and enhances students’ learning. However, technology should not be used as a replacement for basic understanding of arithmetic knowledge and skills. It should be used to foster knowledge, skills and processes in mathematics. (NDOE, 2017, pp. 2-9)

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Figure A4.4. Junior Primary Content (NDOE PNG, 2017, p. 16).

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Similar material may be found in the Syllabuses for Grades 3-5 and Grades 6-8 available on line at https://www.pnginsight.com/MathsExamResources/maths-teachers-resources/ The teachers guide in 2003 provided much of what the syllabus contained in 2015. The terminology and emphasis on assessment is increased and evident. Ideas for teaching are given. The principles advocated in each syllabus include reference to culture but this is not reflected in the syllabus details or teachers guides. Figure A4.5 illustrates the draft guided lessons prepared for elementary schools, randomly selected indicating the kind of guidance given to teachers and the teachers to the class. To extend to inquiry but keeping the same format would mean a second activity of going further (Owens, Edmonds-Wathen, Bino, 2015). The teachers’ guides for higher grades are illustrated by Figures A4.6 and A4.7 which were also randomly selected. Further teacher professional development to ensure that the teachers take ownership of key ideas and to clarify any unclear areas would assist if all teachers had electronic access or a series of text messages to clarify in a timely fashion. There is more emphasis on practice perhaps than inquiry but there is significant depth to each learning plan. The use of colour and layout is attractive and would encourage teacher use. It appears that a student work book may also be available supplying further instructions for example to a set of vertical addition algorithms which appear to be for discussion regarding the errors.

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Draft Teachers Guide for Elementary Teachers 2015

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Figure A4.5. Two pages from Draft Elementary 1 Teachers Guide. (NDOE, 2015, pp. 12, 36)

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Standards Based Teachers Guide

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Figure A4.6. Pages for a topic on fractions from the Grade 6 Teachers Guide. (NDOE, 2019, pp. 53-54).

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Figure A4.7. Page from teachers’ guide for Grade 4. (NDOE PNG, 2019, pp. 38–39, 41–42, 47–48)

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Source. Gough, Lilburn, Rawson, & Sullivan, 1991, p. 144 Figure A4.8. Page from Community School Mathematics Teacher’s Resource Book.

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Lower Secondary Syllabus 2009

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Source. NDOE, 2009, p. 12, 14, 16-19 Figure A4.9. Pages from Lower Secondary Syllabus, 2009 The supporting teachers guide has some great examples for assessment among other things. The important aspect to note is that some excellent materials prepared by the Curriculum Unit have continued to be used since the revision of structure and direction in 2013. An earlier upper secondary syllabus tended to list topics. This is one area where the curriculum was reviewed in 2006 with a high standard and detail but with the greater number of provincial schools teaching years 11 and 12 and the increasing access to information and communication technology (computers) but possibly not equally available, there is likely to be considerable work for the Department in providing the curriculum equitably.

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Index

A Authors Charly Muke, 5, 11–14, 22, 40, 86, 92, 99, 254–255, 257, 263–264, 354 Kay Owens, 12, 14, 45, 64, 75, 88, 231–232, 240, 262, 360–362 Patricia Paraide, 5, 11–13, 90–91, 233, 253–254, 264, 354 Philip Clarkson, 12, 13, 255–256, 274–275 C Cameos Dan and Carrie Luke, 15, 125, 264 David Shield, 181 Deane Arganbright, 15, 165, 186 Miwasa Galligali, 78 Nagong Gejammec, 109 Paulius Matane, 136 Sibona Kopi, 89 Trevor Freestone, 135 Yombu Selden, 12, 15, 75 Churches and missions, 190 Catholic, 100 London Missionary Society, 101 Methodist, Wesleyan, United Church, 99 Colonisation colonialism, 110, 111, 130, 139, 143–145, 171, 313 countering colonisation, 138, 140, 145 impact, 144, 145, 191, 211, 235, 252, 333 impact on mathematics education, 143, 144, 318, 323, 324, 334, 346 racism, 333 Colonised countries West Papua, 350 Colonised nations, 313 Torres Strait Islands, 332 Colonisers Countries, 318 Indonesia, 323 The Netherlands, 350, 352 Computers and computing, 14, 17, 152, 156, 164, 165,167–169, 174, 176, 177, 179, 180, 182, 183, 185–188, 196, 209, 214, 215, 217, 233,289–305, 331, 333, 335, 336 See also Mathematics

Counting systems, 2, 8, 9, 11, 12, 31, 68, 81–84, 125, 132, 153, 174, 179, 184, 193, 215, 216, 221, 246, 256, 266, 268, 275, 278, 319, 346, 368, 370 D Decentralisation, 315 Designing and making, 46 bows and arrows, 37 bridges, 59 canoes, 74, 76, 78, 79 drums, 52 food capture, 35 food preservation, carrying, serving and eating, 58 gardening and animal management, 23, 26, 29, 42–44 houses, 55, 74 hunting and warfare equipment, 39 musical instruments, 53 pottery, 47 shields, 39 stone axes etc., 45 string art, 85 string, binding, bilums, tapa, 32 traps, 36 Discontinuity of mathematics education, 26, 62, 132, 314 E Educational development, 3, 101, 114, 124, 141, 167 Ethnomathematics, 62, 135, 194, 218, 261, 313, 316, 320, 324, 337, 346–354, 357, 370, 372 Examinations, 102, 106, 144, 316 Explorers, Patrols, First Contacts, Early Linguists, Anthropologists and Missionaries Abel, 102 Black, 44, 104 Brown, 15, 16, 31, 55, 99 Chalmers, 101 Chatterton, 107, 112 D’Entrecasteaux, 55 Finsch, 74 Flierl, 8, 31 Heider, 87 Lawes, 31, 82, 101 Leahy, 14, 40, 53, 55, 68, 73

© Springer Nature Switzerland AG 2022 P. Paraide et al., Mathematics Education in a Neocolonial Country: The Case of Papua New Guinea, History of Mathematics Education, https://doi.org/10.1007/978-3-030-90994-9

491

Index

492 Explorers, Patrols, First Contacts, Early Linguists, Anthropologists and Missionaries (cont.) Maclay, 14, 32 Malaccas, 70, 98 Mikloucho-Maclay, 14, 53, 70 Moresby, 8, 13, 45, 53, 74 Queen Emma, 98, 387 Standens, 82, 115 Stanley, 43, 55, 74 Taylor, 44 Thurnwald, 72 F Financial issues, 139, 244–245, 315, 357, 362 Australian Aid, 191, 211, 323, 324, 356, 357, 373 Chinese projects, 314, 335 fees, 209, 230 funding organisations, 211, 319, 336 government funding, 105, 106, 108 head tax, 102 Japanese Aid, 314 lack of funding, 103, 110, 126, 143, 190, 212, 230, 316, 317, 327 Foundational Mathematics, 28, 74, 90, 91, 98, 216, 217, 317, 372 body-part and digit tally systems, 98, 323 classifications, 40, 42 classifiers, 84 communal noticing, 86, 88, 89 counting systems, 81 other systems, 71, 81, 82 cycle and base systems, 82, 84 equivalence and reciprocity, 71, 72 large numbers, 83, 84 mental records and mapping, 22, 45, 46, 71, 86, 88, 89 proportional reasoning–rates, ratio, 35, 51, 80, 90 technology, 22, 52, 55, 58, 59, 68, 70, 89 valuing foundational knowledge and activities, 90, 211 wayfinding, navigation, position, sailing, 72, 74–76, 92, 290 See also Designing and making; Geometry and measuring; Mathematical activities; Mathematics; Reasoning; Trade and reciprocity G Gender, 129, 172, 179, 238, 312, 328, 329, 362, 363, 373 Geography and ecology disasters and weather changes, 85 Geometry and measurement, 35, 80, 185 angles and trigonometry, 29, 87 areas and area measures, 29, 80 curve, circle, circumference, radius, 87, 88 direction three dimensional shapes and volumes, 52

distances and lengths, 25, 78 length measures–fathom, metre, step, cubit, 72 parallel lines and faces, 24, 29, 44, 87 position, 41 shapes, 41 speed, 78, 80 symmetry and transformations, 47 three dimensional shapes and volumes, 50, 58, 88, 131 time, 43, 81 Global bodies, 314, 315, 318 United Nations, 44, 122, 179, 318, 347, 349, 351, 355, 362 World Bank, 122, 140, 206, 323, 357 Globalisation, 314, 316, 362 Glocalisation, 312, 315, 367, 369 H History 1916–1945, 103 1946 to 1970, 119 administrators Groves, 104, 110, 121 Hahl, 7, 101 Macgregor, 8, 14, 101 McKinnon, 130, 141 McNicol, 104 Murray, 7, 8, 105, 108 Australian government Barnes, 126, 139 Hasluck, 121, 124, 139 McKenna, 8, 104 Secretary Hay, 140 distance to government decision-making, 103 PNG government Somare, 108 I Independence, 124, 127 Indigenous education, 27, 123 Indigenous knowledge, 368 Inequalities in education, 126, 193, 327, 329 Intergenerational transfer of knowledge, 22, 23, 26, 29, 70, 84–87, 91, 319, 365, 366, 370 gender and age related, 86, 88 identity, 91, 356–357 loss of culture, 27, 85, 91 observing and discussing, 90 participation, 87, 90 L Languages and cultures, 126, 171, 249 Austronesian Oceanic Malalamai, 70 Tolai, 317 Yabim, 100

Index Motu and Hiri Motu (LF), 76, 98, 323 non-Austronesian or Papuan Gogodala, 78 Trans New Guinea Phylum, 100 valuing languages and language skills, 70, 320, 330 Languages of instruction, 250, 350 bilingual, 125, 192, 218, 231, 257–262, 269–272, 282, 315 English, 104, 106, 124, 125, 137, 146, 230, 317, 318, 322, 359, 370, 372 German, 100, 103 literacy, 100, 101, 319 Tok Pisin, 103, 322 vernacular, 317 vernacular languages, 101, 106, 109, 125, 185, 206, 232, 260, 261, 267–269 M Mathematical activities, 368 decision making, often communal, 46, 79 extraction of minerals and technology, 34 games, play, gambling, 23, 24 making colours, 35 medicinal knowledge and technology, 89, 90 problem solving and inquiry, 79 string designs and modifications, 290 Mathematics communicating, describing and recording, 22, 87, 88, 106, 125 counting and arithmetic - addition, multiplication, division, subtraction, counting, 108, 125, 131 estimating and measuring, 78, 88 higher mathematics, 80, 92, 168, 176, 182 logical reasoning, 22, 44, 132 operations research and large data sets, 168 problem solving and inquiry, 44 spatial designs, 32, 35, 47 statistics, 169 visuospatial reasoning, 32, 47, 86–88, 185, 290 See also Computers and computing; Foundational Mathematics; Geometry and measuring; Trade and reciprocity N Neocolonialism, 145, 211, 311–314, 318, 319, 323–327, 330, 332, 333, 335, 346, 353 Asian turn, 335 educated elite, 142, 318, 330, 353 other countries, 319, 327, 332 P Postcolonialism, 220, 349 Provinces, 5, 12, 22, 27, 31, 36, 39, 44–47, 49–55, 61, 70, 71, 74–76, 81–85, 87, 89, 99–103, 108, 109, 126, 128, 151, 157, 160, 164, 171, 178, 183–185, 188–190, 206, 208, 211, 213, 217, 230, 231, 234, 235, 237, 239, 243, 254, 257,

493 261, 262, 268, 269, 276, 278, 279, 282, 291, 312, 315, 327, 354, 358, 360, 363, 372, 386, 387, 389, 395, 396 R Reports Currie, 122, 124, 170 educational development, 126 Dannevig for League of Nations, 105 education ordinances, 112 Griffiths, 104 Higher Education Committee/Commission of Higher Education, 164, 190, 191 Matane, 139, 202, 269, 317, 356, 365, 367 policy and plans, 102–104, 141, 144, 145 purpose of education, 104, 106, 107, 122, 142 Royal Commission, 102 Tololo, 141, 145, 146, 171, 175, 190, 202, 231, 274, 356 Wedgwood, 109 Weeden, 123, 190 Williams, 107 Research, 29, 207 archaeology and linguistic studies, 31, 43–47, 51, 68, 71, 73, 84 Glen Lean Ethnomathematics Centre, 369 oral histories and personal experiences, 29, 43, 48, 53, 57, 58, 76, 86, 89, 317 Piaget, 130 See also Universities Reasoning, see Mathematics, visuospatial reasoning S School attendance, 99–101, 105, 108, 112, 119, 122, 126, 128, 139, 180 Schools, 240–244 administration/public schools, 102 administration schools, 119 boarding schools, 166 Chinese schools, 108 distance education, 194 elementary schools, 185 mission schools, 99, 104, 105, 112, 121, 122 National High Schools, 127 school boards, 207 secondary schools, 127 Skulanka, 128 technical, 126 technical/industrial training, 100, 102, 105 T schools, 111, 121, 122, 127 School structure reform, 267 Self-government, 4, 104, 115, 124, 150, 172, 188, 235, 333, 356, 360 Student–tertiary, 171 enrolments, 171, 172, 191 quality, 167, 191, 194

Index

494 Syllabus and curriculum, 107, 130, 202, 364–365 assessment, evaluation, and reporting, 239 Australian curriculum, 111 Japanese, 230 manual training, 194 mathematics curriculum, 105, 130, 194 objectives, 230 other curriculum, 100, 111, 364, 365 reform, outcomes-based education, 212, 230, 234, 236 revised reform/standards-based education period, 230, 235–238 T Teacher education, 27, 102, 119, 125, 131–133, 191, 213, 239–240, 273–278, 338, 370 Association of Teacher Education, 190 elementary schools teacher training, 213, 230, 316 National Institute of Teacher Education, 208 other teacher training, 104, 110, 125, 135, 138 Primary and Secondary Teacher Education Project, 215, 277 professional development, 207, 274–277 Teachers, 134 Australian, 109, 133 initiatives, 135 national, 102, 104, 108 National Teaching Service, 133 Teachers colleges, 133, 137, 277–278 syllabus committees, 208 syllabuses and curriculum, 208 Teaching, 131, 334 deficit and duality, 334 Dienes, 120, 130, 133, 273, 274 critique of curriculum, 132 integrating local and western knowledges, 334

Technology in education, 167 computer science, 174 regulation of providers, 164 Territory Australian New Guinea Administration Unit (ANGAU), 109 Australian Territories, 103 Textbooks external production, 166, 168, 348 locally written, 166, 168, 207, 350 Third Culture Indigenous Kids (TCIKs), 331 Trade and reciprocity, 44, 68, 71, 72, 323 engagement, 319, 320 exchange, 332 exploitation and blackbirding, 73 shell money, 72 trade rings, 70, 71 trading partners, 68, 69 Translocal, 367 U Universities community service, 167, 172 degrees and faculties, 171, 180, 181 Divine Word University, 185–186 Mathematics Education Centre, 173, 174 mathematics strands and subjects, 165, 166 overseas students, 164 PNG University of Technology, 170 Preliminary Year, 166 staff, 167, 168, 170–173, 175 staff development, 167, 173 University of Goroka, 125, 208, 355 University of Papua New Guinea (UPNG), 125, 165, 326