Researching and using progressions (trajectories) in mathematics education 9789004396425, 900439642X, 9789004396432, 9004396438

Table of contents :
Acknowledgements --
List of figures and tables --
Notes on contributors --
Introduction / Dianne Siemon, Tasos Barkatsas and Rebecca Seah --
Knowing and building on what students know: the case of multiplicative --
Thinking / Dianne Siemon --
Learning trajectories in early mathematics education / Julie Sarama and Douglas H. Clements --
H lt : a lens on conceptual transition between mathematical "markers" / Ron Tzur --
Using digital diagnostic classroom assessments based on learning trajectories to drive instruction / Jere Confrey, William Mcgowan, Meetal Shah, Michael Belcher, Margaret Hennessey and Alan Maloney --
Researching mathematical reasoning: building evidence-based resources to support targeted teaching in the middle years / Dianne Siemon and Rosemary Callingham --
Reframing mathematical futures ii: developing students' algebraic --
Reasoning in the middle years / Marj Horne, Max Stephens and Lorraine Day --
A learning progression for geometric reasoning / Rebecca Seah and Marj Horne --
Statistics and probability: from research to the classroom / Rosemary Callingham, Jane Watson and Greg Oates --
Investigating mathematics students' motivations and perceptions / Tasos Barkatsas and Claudia Orellana --
Secondary students' mathematics education goal orientations / Tasos Barkatsas and Claudia Orellana --
Epilogue / Mike Askew.

Citation preview

Researching and Using Progressions (Trajectories) in Mathematics Education

Global Education in the 21st Century Series Series Editor Tasos Barkatsas (RMIT University, Australia) Editorial Board Amanda Berry (Monash University, Australia) Anthony Clarke (University of British Columbia, Canada) Yuksel Dede (Gazi University, Turkey) Heather Fehring (RMIT University, Australia) Kathy Jordan (RMIT University, Australia) Peter Kelly (RMIT University, Australia) John Malone (Curtin University of Technology, Australia) Huk Yuen Law (The Chinese University of Hong Kong) Patricia McLaughlin (RMIT University, Australia) Juanjo Mena (University of Salamanca, Spain) Wee Tiong Seah (The University of Melbourne, Australia) Geoff Shacklock (RMIT University, Australia) Dianne Siemon (RMIT University, Australia) Robert Strathdee (Victoria University, Australia) Ngai Ying Wong (Education University of Hong Kong) Qiaoping Zhang (The Chinese University of Hong Kong)

Volume 3

The titles published in this series are listed at brill.com/gecs

Researching and Using Progressions (Trajectories) in Mathematics Education Edited by

Dianne Siemon, Tasos Barkatsas and Rebecca Seah

leiden | boston

All chapters in this book have undergone peer review. Library of Congress Cataloging-in-Publication Data Names: Siemon, Dianne, editor. | Barkatsas, Anastasios, editor. | Seah, Rebecca, editor. Title: Researching and using progressions (trajectories) in mathematics education / edited by Dianne Siemon, Tasos Barkatsas, and Rebecca Seah. Description: Boston : Brill Sense, [2019] | Series: Global education in the 21st century, ISSN 2542-9728 ; volume 3 | Includes bibliographical references. Identifiers: LCCN 2018060016 (print) | LCCN 2019001381 (ebook) | ISBN 9789004396449 (ebook) | ISBN 9789004396432 (pbk. : alk. paper) | ISBN 9789004396425 (hardback : alk. paper) Subjects: LCSH: Mathematics--Study and teaching (Middle school) | Mathematics--Study and teaching (Secondary) Classification: LCC QA13 (ebook) | LCC QA13 .R4637 2019 (print) | DDC 510.71--dc23 LC record available at https://lccn.loc.gov/2018060016

Typeface for the Latin, Greek, and Cyrillic scripts: “Brill”. See and download: brill.com/brill-typeface.

issn 2542-9728 isbn 978-90-04-39643-2 (paperback) isbn 978-90-04-39642-5 (hardback) isbn 978-90-04-39644-9 (e-book) Copyright 2019 by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill, Brill Hes & De Graaf, Brill Nijhoff, Brill Rodopi, Brill Sense, Hotei Publishing, mentis Verlag, Verlag Ferdinand Schöningh and Wilhelm Fink Verlag. All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission from the publisher. Authorization to photocopy items for internal or personal use is granted by Koninklijke Brill NV provided that the appropriate fees are paid directly to The Copyright Clearance Center, 222 Rosewood Drive, Suite 910, Danvers, MA 01923, USA. Fees are subject to change. This book is printed on acid-free paper and produced in a sustainable manner.

Contents Acknowledgements vii List of Figures and Tables viii Notes on Contributors xi Introduction 1 Dianne Siemon, Tasos Barkatsas and Rebecca Seah 1 Knowing and Building on What Students Know: The Case of Multiplicative Thinking 6 Dianne Siemon 2 Learning Trajectories in Early Mathematics Education 32 Julie Sarama and Douglas H. Clements 3 Hypothetical Learning Trajectory (HLT) : A Lens on Conceptual Transition between Mathematical “Markers” 56 Ron Tzur 4 Using Digital Diagnostic Classroom Assessments Based on Learning Trajectories to Drive Instruction 75 Jere Confrey, William McGowan, Meetal Shah, Michael Belcher, Margaret Hennessey and Alan Maloney 5 Researching Mathematical Reasoning: Building Evidence-Based Resources to Support Targeted Teaching in the Middle Years 101 Dianne Siemon and Rosemary Callingham 6 Reframing Mathematical Futures II: Developing Students’ Algebraic Reasoning in the Middle Years 126 Lorraine Day, Marj Horne and Max Stephens 7 A Learning Progression for Geometric Reasoning 157 Rebecca Seah and Marj Horne 8 Statistics and Probability: From Research to the Classroom 181 Rosemary Callingham, Jane Watson and Greg Oates

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9 Investigating Mathematics Students’ Motivations and Perceptions 205 Tasos Barkatsas and Claudia Orellana 10 Secondary Students’ Mathematics Education Goal Orientations 221 Tasos Barkatsas and Claudia Orellana Epilogue 232 Mike Askew

Acknowledgements A warm and heartfelt thanks to all the authors and reviewers who have so kindly donated their time. We have a fantastic network of academics with extensive experience in their respective fields; it was a joy to work with all of you, and your contribution to this book has been valuable. Acknowledgements to all authors and to Professor John Malone, Curtin University, Australia and to Professor Vasilis Gialamas, National and Kapodistrian University of Athens for their participation in the peer review process. A special thank you to Professor Mike Askew for his thoughtful epilogue.

Figures and Tables Figures 1.1

Proportion of students at each level of the emergent numeracy profile in 1999 for years 5 to 9 (n = 1315, 1318, 1467, 1484, & 1276 respectively). 13 1.2 A short task and scoring rubric used in the snmy project (2003–2006). 16 1.3 Proportion of students by laf zone and year level, 2004 (n = 3169). 17 1.4 Proportion of students by laf zone, rmf-p project (n = 1732). 22 3.1 A bar marked vertically into 3 equal parts, with two of the thirds (pink) marked into 1/5th. 68 4.1 “Benchmark percents” LT from the Math-Mapper 6-8 learning map. 82 4.2 The top half of a student report. 84 4.3 The heat maps (teacher reports) for the three constructs in the cluster finding key percent relationships. The levels are listed vertically, with increasing difficulty, and the students, based on performance, are ordered from left to right. Student identifiers can be toggled on or off for confidentiality. 84 4.4 Compound-bar display of proportion correct/incorrect for tested LT levels (L) of construct A, distinguished by district (D1, D2), class section type (regular, accelerated), and grade (6, 7). Each of the five sets of bars corresponds to the cumulative results from all class sections of indicated type. Error bars indicate ±1 S.E. 90 4.5 Compound-bar display of proportion correct/incorrect for tested LT levels (L) of construct B. Information is displayed as in Figure 4.4. 91 4.6 Compound-bar display of proportion correct/incorrect for tested LT levels (L) of construct C. Information is displayed as in Figure 4.4. 92 4.7 Item 695, level 1, construct “Percents as Amount per Hundred.” Correct responses are indicated next to the [1] on right side. 93 4.8 Item 1294, level 4, construct A (“Percent as Amounts per 100”). 93 4.9 Item 698, level 5, construct B (“Benchmark percents”). 94 4.10 Item 1292, level 2, construct C (“Percents as combinations of other percents”). 95 5.1 Three items from the Algebra Tile Task (atilp). 107 5.2 Variable map for statistical reasoning items from MR1 and MR2 data (n = 1570). 112 5.3 Items addressing aspects of angles, gang1 and gang2. 114 5.4 Geometric reasoning items by difficulty with scoring Rubrics, based on MR1 data, February 2017 (n = 769). 117 5.5 Geometric reasoning items by difficulty with scoring Rubrics, based on MR1 & MR2 data, June 2017 (n = 1570). 119

Figures and Tables 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 8.1 8.2 8.3 9.1

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Sample assessment task on balancing equations and associated rubrics. 139 Sample assessment task for function and associated rubrics. 140 Sample responses to ahab3. 141 Map of the responses to the algebra items following Rasch analysis. 142 Framework for developing geometric reasoning. 161 Reasoning items on visualising three-dimensional objects. 170 Examples of student’s reasoning on gnet4. 172 An activity contributing to student experiences across zones 2–5. 175 The lung disease problem (Batanero, Estepa, Godino, & Green, 1996). 197 Initial suggestions for the lung disease problem. 198 The importance of an argument. 199 Boxplot of the four derived factors by year level. 215

Tables 1.1 1.2 1.3 2.1 2.2 3.1 3.2 4.1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 6.1 6.2 6.3 6.4

Emergent numeracy profile (Siemon, Virgona, & Corneille, 2001). 12 The elaborated laf for zone 3. 19 Alignment between snmy materials and master’s (2013) design principles for a learning assessment system. 24 Goals of curriculum research (adapted from Clements, 2007). 42 Categories and phases of the curriculum research framework (crf) (adapted from Clements, 2007). 43 The three-part scheme inferred to underlie counting-all. 64 The 7-step, cyclic process that constitutes student-adaptive pedagogy. 66 Demographics for collaborating school districts. 88 Scoring rubrics for three items from the Algebra Tile Task (atilp). 109 Scoring Rubrics for gang1 and gang2 used in MR1. 115 Fit values for gang1 and gang2 at MR1. 115 Amended scoring rubrics for gang1 and gang2. 116 Fit values for gang1 and gang2 at MR2. 116 Scoring rubrics and rich description for Zone 1 of the learning progression for geometric reasoning. 120 Scoring rubrics and rich description for Zone 1 of the learning progression for geometric reasoning. 121 Initial hypothetical learning progression. 135 Zones in the learning progression for algebraic reasoning. 143 Example of teaching advice for a particular zone of the learning progression for algebraic reasoning. 146 Example of how mountain range challenge can be utilised across zones to support mixed ability teaching. 148

x 7.1 7.2 7.3

Figures and Tables

Hypothetical geometric learning progression. 167 Results for gnet3 and gnet4. 171 Geometric learning progression and broad descriptions of behaviours in each zone. 173 8.1 Question addressing variation in distribution. 186 8.2 Questions addressing variation in expectation and randomness, and variation in inference. 187 8.3 Overview of the statistical reasoning learning progression. 189 8.4 Distribution of students across zones by year level. 190 8.5 Descriptor and teaching advice for Zone 5. 195 9.1 Rotated factor matrix (varimax rotation). 211 10.1 Rotated factor matrix (varimax rotation). 226

Notes on Contributors Mike Askew is Distinguished Professor of Mathematics Education in the School of Education at the University of Witwatersrand, Johannesburg, having previously held Professorships at King’s College, University of London and Monash University, Melbourne. Originally a primary school teacher, Mike moved into teacher education and there developed his interest in research. He has directed many research projects including the influential ‘Effective Teachers of Numeracy in Primary Schools’ and was deputy director of the five-year Leverhulme Numeracy Research Programme. His books include: Transforming Primary Mathematics, A Practical Guide to Transforming Primary Mathematics and the popular Maths for Mums and Dads (with Rob Eastaway). Mike believes that mathematical activity can be, and should be, engaging and enjoyable for all learners and that the majority of learners can come to see themselves as mathematicians, in the sense of having confidence in their ability to do maths. From April 2018 Mike is pleased to be the President of the UK’s Mathematical Association. Anastasios (Tasos) Barkatsas is a Senior Academic in Mathematics and Statistics Education and a Quantitative Data Analyst at the School of Education, rmit University, Australia and has published more than 100 refereed journal and conference research papers, chapters and books. Tasos is also Series Editor of the Brill Sense series: ‘Global Education in the 21st Century,’ an Editorial Board member in a number of international research journals and a reviewer in numerous international research journals and conferences and is currently co-editing two books, which will be published in 2018 as part of his book series. Michael Belcher is a doctoral student in mathematics education at NC State. Prior to joining the sudds team, Mike taught high school mathematics in North Carolina for seven years and spent three years developing a digital middle grades mathematics curriculum. He earned his Bachelor’s degree in mathematics from Wake Forest University and his Masters degree in mathematics education from Teachers College, Columbia University. Rosemary Callingham is Adjunct Associate Professor and a mathematics educator at the University of Tasmania. She has an extensive background in mathematics education

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in Australia, at school, system and tertiary levels, including mathematics curriculum development and implementation, large-scale testing, and preservice teacher education. Her specific research interests include teachers’ pedagogical content knowledge, statistical literacy, mental computation, and assessment of mathematics and numeracy. Douglas Clements is Kennedy Endowed Chair in Early Childhood Learning and Distinguished University Professor at the University of Denver. He is a major scholar in the field of early childhood mathematics education, whose work has relevance to the academy, to the classroom, and to the educational policy arena – he has published over 135 refereed research studies, 22 books, 86 chapters, and 300 additional publications. At the national, level, his contributions have led to the development of new mathematics curricula, teaching approaches, teacher training initiatives, and models of “scaling up” interventions. He has served on the U.S. President’s National Mathematics Advisory Panel, the Common Core State Standards committee of the National Governor’s Association and the Council of Chief State School Officers, the National Research Council’s Committee on Early Mathematics, the National Council of Teachers of Mathematics national curriculum and Principles and Standards committees and is and co-author each of their reports. He has directed more than 35 funded projects. Additional information can be found at http://du.academia.edu/DouglasClements and http://www.researchgate.net/profile/Douglas_Clements/ Jere Confrey is the Joseph D. Moore Distinguished Professor of Mathematics Education at North Carolina State University. She directs the sudds team (“Scaling Up Digital Design Studies”) in building new learning maps and related diagnostic assessments to support personalized learning. She served on the National Validation Committee on the Common Core Standards and built www.turnonccmath.com, a website unpacking the Common Core. She was Vice Chairman of the Mathematics Sciences Education Board, National Academy of Sciences (1998–2004), chaired the nrc Committee, which produced On Evaluating Curricular Effectiveness, and was a coauthor of nrc’s Scientific Research in Education. She authored Math Projects, Function Probe, Precalculus Interactive Diagrams, Graphs N Glyphs and lpp-Sync software. Dr. Confrey received a Ph.D. in mathematics education from Cornell University. Lorraine Day is an experienced academic. Her teaching experience spans many decades in schools and university. She is Editor of Australian Primary Mathematics

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Classroom, a former President and Life Member of The Mathematical Association of Western Australia, member of the Mathematics Education Research Group of Australasia and past member of the National Council of the Australian Association of Mathematics Teachers. Lorraine has been a member of the Reframing Mathematical Futures II research team, working on algebraic reasoning, and as part of this team was a recipient of the 2018 Beth Southwell Practical Implications Award. Currently she is a member of the Australian research team of the Principals as Stem Leaders Project. Lorraine’s passions are engaging students in mathematics and supporting the important work of teachers. She is a regular contributor to professional learning facilitation and has been involved in the development of mathematics education at both a state and national level in Australia. Margaret Hennessey received her BA in Liberal Arts from St. John’s College in 2008. After she received her mat degree from Duke University in 2012, she taught high school mathematics at Durham School of the Arts. She is a recipient of the Knowles Science Teaching Fellowship, which grants financial and professional support to early-career math and science teachers. Since leaving the classroom, she has worked both at Duke and at NC State on grant-funded projects to improve secondary teaching and learning of math and science. Marj Horne is an Adjunct Professor of Mathematics Education at The Australian Catholic University, is an experienced teacher of mathematics at all levels from Early Childhood through to University. She was an active researcher on the Early Numeracy Research Project, Contemporary Teaching and Learning of Mathematics Project, and the Family School Partnerships project. Marj is currently engaged in the Reframing Mathematical Futures II project where her particular interests and expertise are in the development of evidencebased learning frameworks for Algebraic and Geometrical Reasoning and the corresponding teaching advice and activities to support targeted teaching in those areas. Alan Maloney is a research scientist with roots in both biological sciences (PhD, Stanford) and, for the past 15 years, mathematics education. He co-founded Quest Math & Science Multimedia, Inc., and the design of Interactive Diagrams for High School Mathematics. His research focus has included design of mathematics educational software and, more recently, the development of learning trajectories and diagnostic assessment for K-8 mathematics. He is

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the lead editor of Learning over Time: Learning Trajectories in Mathematics Education, and a co-developer of TurnOnCCMath.com. He received his PhD in biological sciences from Stanford University. William McGowan has spent five years as a middle school mathematics teacher, and three years developing a digital mathematics curriculum. He has worked with teachers on implementing new curricula and technology in the classroom. Will earned his Ed.D. in mathematics education at Rutgers University where he studied the ways in which teachers attend and respond to student reasoning. Greg Oates began his career teaching secondary mathematics and statistics for 9 years, before returning to Auckland University in 1997 where he taught undergraduate mathematics (Calculus and Linear Algebra) and post-graduate mathematics education. In 2016, he moved to the University of Tasmania, Launceston Australia, where he currently teaches mathematics education for pre-service teachers in primary and secondary school. His research interests include the integration of technology into mathematics curricula, collaborative learning in mathematics; mathematical reasoning and proof; beliefs and productive disposition; and professional development for teachers with a specific focus on pedagogical content knowledge (pck). Claudia Orellana has been working as Project Manager for the Reframing Mathematical Futures II (rmfii) Project. Her research interests revolve around the use of digital technologies in mathematics education with her PhD thesis having focused on the use of Computer Algebra System (cas) devices in senior secondary mathematics. Having specialised in Mathematics and Chemistry as part of her Science and Education (Secondary) double degree, Claudia also teaches in undergraduate and post-graduate courses within these disciplines. Julie Sarama is Kennedy Endowed Chair in Innovative Learning Technologies and Distinguished University Professor at the University of Denver, Colorado, usa. She has taught high school mathematics, computer science, middle school gifted mathematics and early childhood mathematics. She has directed over 10 projects funded by the National Science Foundation and the Institute of Education Sciences and has authored over 77 refereed articles, 6 books, 55 chapters, and over 80 additional publications. She has also developed and

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programmed over 50 award-winning educational software products. Her research interests include children’s development of mathematical concepts and competencies, implementation and scale-up of educational interventions, professional development models’ influence on student learning, and implementation and effects of software environments. Rebecca Seah is a Mathematics Education lecturer in the School of Education, rmit University, Melbourne, Australia. She is part of the research team in the Reframing Mathematical Futures II project. Her research interests include: spatial and geometric reasoning, assessments and instructional design, numeracy and students with special needs. Meetal Shah is a postdoctoral researcher with the Dr. Confrey’s research team and is interested in validating classroom-based diagnostic assessments, learning sciences, and geometry. Before joining sudds and the doctoral program at NC State, Meetal taught high school mathematics (grades 7 to 12) in Sydney, Australia for 10 years. During that time, she taught the Mathematics Methods course at the University of New South Wales. Meetal has earned her Bachelor’s degree in mathematics from the University of Sydney, Postgraduate Diploma in Secondary Education from the University of New South Wales, and a Master of Education from Macquarie University. Dianne Siemon is a Professor of Mathematics Education in the School of Education at rmit University (Bundoora) where she is involved with the preparation of preservice teachers and the supervision of higher degree students. Di is currently the Director of the Reframing Mathematical Futures project, which is working with 32 secondary schools nationally to develop an evidenced based teaching and learning framework for mathematical reasoning in the middle years. She is also actively involved in the professional development of practicing teachers, particularly in relation to the development of the ‘big ideas’ in number, the teaching and learning of mathematics in the middle years, and the use of rich assessment tasks to inform teaching. Di has directed a number of other large-scale research projects including the Scaffolding Numeracy in the Middle Years Project (2003–2006), the Researching Numeracy Teaching Approaches in Primary Schools Project (2001–2003), and the Middle Years Numeracy Research Project (1999–2001). Di is a past President of the Australian Association of Mathematics Teachers and a life member of the Mathematical Association of Victoria.

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Max Stephens is an Adjunct Professor in the School of Education at rmit University. His research areas focus on: developing and using a construct of Teacher Capacity to improve the teaching and learning of mathematics and investigating the development of students’ algebraic thinking in the primary and early secondary years of school. He has interests internationally in curriculum and assessment, notably in Japan and in China. He is concurrently a senior research fellow in the Melbourne Graduate School of Education at the University of Melbourne. Prior to that, he occupied senior roles with the Victorian Department of Education and at the Victorian Curriculum and Assessment Authority. For the Australian Government he has been a coordinator of numeracy research projects in the Australian States and Territories and has been a reviewer of the Australian Curriculum: Mathematics and has provided interpretations of international assessments in Mathematics for Australian schools. Ron Tzur is a professor of mathematics education at the University of Colorado Denver’s School of Education and Human Development. He earned a PhD degree from the University of Georgia at Athens (1995). Having served as a Principal Investigator on several research projects funded by the US National Science Foundation (nsf), his research program interweaves five interrelated foci: (a) children’s construction of whole number and fractional knowledge, (b) a cognitive mechanism for learning a new concept, (c) a pedagogical approach to promote such learning, (d) mathematics teachers’ professional development of such pedagogy, and (e) linking mathematical thinking/learning with brain processes. Jane Watson is Professor Emerita and has had a long and distinguished career in mathematics education at the University of Tasmania. In addition to gaining many awards for teaching, she has contributed extensively to the field of statistics education research, leading many large research teams. Her work has led to several prestigious awards, including the Clunies Ross National Science and Technology Award, the inaugural Mathematics Education Research Group of Australasia merga Career Research Medal; and the University of Tasmania Vice-Chancellor’s Research Excellence Medal. She is an elected Fellow of the Academy of the Social Sciences in Australia.

Introduction Dianne Siemon, Tasos Barkatsas and Rebecca Seah

The relationship between research and practice has long been an area of interest for researchers, policy makers, and practitioners alike. One obvious arena where mathematics education research can contribute to practice is the design and implementation of school mathematics curricula. This observation holds whether we are talking about curriculum as a set of broad, measurable competencies (i.e., standards) or as a comprehensive set of resources for teaching and learning mathematics. Impacting practice in this way requires finegrained research that is focused on individual student learning trajectories and intimate analyses of classroom pedagogical practices as well as large-scale research that explores how student populations typically engage with the big ideas of mathematics over time. Both types of research provide an empirical basis for identifying what aspects of mathematics are important and how they develop over time. But both types of research also have their limitations. For example, while fine-grained teaching experiments show what students can do when they are taught important mathematics well, their focus and scale mean that they can only impact curriculum design and implementation at the edges. On the other hand, large-scale assessments of what students know and can do over time can only reflect what students have had the opportunity to learn. They do not reflect what might be possible if students were taught different mathematics or if what they were taught was taught well (Watson, 2017). However, where this research also provides valid tools for identifying student’s mathematical thinking, where they are at in their learning journey, and evidenced-based advice as to where to go to next, it can contribute to a more coherent, productive alignment between curriculum, instruction, and assessment (Wilson, 2018) that transcends current practice. This book has its origins in independent but parallel work in Australia and the United States over the last 10 to 15 years. It was prompted by a research seminar at the 2017 pme Conference in Singapore that brought the contributors to this volume together to consider the development and use of evidencebased learning progressions/trajectories in mathematics education, their basis in theory, their focus and scale, and the methods used to identify and validate them. In this volume they elaborate on their work to consider what is meant by learning progressions/trajectories and explore a range of issues associated with their development, implementation, evaluation, and on-going review. © koninklijke brill nv, leideN, 2019 | DOI:10.1163/9789004396449_001

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Implications for curriculum design and future research in this field are also considered. The title of this volume suggests that the two terms, learning progressions and learning trajectories, are synonymous but that is not our intent. Rather, as the chapters in this volume will illustrate, our intent is to problematise both the meaning and use of these terms as they apply in different contexts. A common element in the different interpretations and use of the terms is the notion that learning takes place over time and that teaching involves recognising where learners are in their learning journey and providing challenging but achievable learning experiences that support learners progress to the next step in their particular journey. Another common characteristic is that, to varying extents and in different ways, learning progressions/trajectories are based on hypothesised pathways derived from experience and a synthesis of relevant literature, the design and trial of learning activities aimed at progressing learning within the hypothesised framework, and evaluation methods to assess where learners are in their journey and the efficacy of both the framework and the instructional materials and approaches used. All approaches recognise that this work provides road maps that identify probable routes to progress student learning towards a particular goal – they are not fixed and immutable but can evolve over time to reflect changes in what is valued, taught and learnt. They point to big ideas and the connections between them as well as misconceptions that might impede student learning. As such learning progressions/trajectories provide a valuable, research-based resource for teachers to better identify and target their teaching to point of need in relation to important mathematics, both to consolidate what is already known and to introduce and develop new ideas and strategies within students’ reach. Chapters 1 to 4 consider four different approaches to researching and using learning progressions/trajectories in mathematics education. Chapter 1 describes three, large-scale Australian projects that used rich assessment tasks and Rasch modelling to identify and explore multiplicative thinking in grades 4 through 9. The second of these resulted in an evidenced-based learning progression, validated assessment options, and targeted teaching advice for multiplicative thinking that can be used to identify and address student learning needs. Chapter 2 explores the evolving use of the learning trajectory notion in the context of early childhood mathematics education in the United States. In particular, it points to the value of learning trajectories in bringing coherence to the curriculum-instruction-assessment relationship as a consequence of a common theoretical base and shared goals. Sarama and Clements also remind us that learning trajectories are hypothetical, that is, they serve as a guide to the development of children’s mathematical thinking

Introduction

3

that needs to be reconceptualised by teachers as they interact with their students in their specific context. In Chapter 3, Tzur distinguishes between two types of studies – marker studies and transition studies – that researchers might use to develop hypothetical learning trajectories and guide research on conceptual learning and teaching. A fine-grained analysis of the means by which a child might transition from one marker to another, for example, from counting-all to counting-on, is offered to explain the nature of the processes involved. A contrast in focus and scope is offered in Chapter 4, which introduces a digital learning system based on multiple learning trajectories as a means of providing teachers systematic access to information on student learning that can be used to inform teaching and progress student learning in terms of the underlying cognitive framework. Confrey and her colleagues also make the point that the learning system serves to translate a whole body of research that is largely inaccessible to teachers into a form that is accessible and actionable through the provision of real-time data on student learning for both teachers and students in relation to an evidenced-based cognitive framework. This contribution illustrates the cohesive power afforded by the systematic use of evidenced-based learning trajectories to create a more productive alignment between curriculum, instruction, and assessment that transcends current practice. Because it is digital, it is also responsive to developments in what is known about what students need to know and how that might be accomplished. But, the authors also acknowledge that such systems are not panaceas independent of teacher knowledge, skills and attitudes. Some teachers were more successful than others in using the data to elicit and build on student thinking suggesting that professional development aligned to learning trajectories and the data they generate is also needed. Chapter 5 introduces the work of the Reframing Mathematical Futures II (rmfii) project, the details of which will be considered in the remaining chapters of this book. The project was conducted by a research team based at rmit University in Australia from 2014 to 2018. This large-scale study was designed to build a sustainable, evidence-based learning and teaching resource to support mathematical reasoning initially in grades 7 to 10 that could be used formatively in the way described by Wiliam (2011). It describes the rationale for and methodology used to develop the evidenced-based learning progressions and teaching advice for algebraic, geometrical and statistical reasoning. In particular, it describes how the hypothetical learning progressions (hlps) derived from the relevant literature were tested using rich assessment tasks and partial credit scoring rubrics, and how the resulting empirical evidence was used to construct the learning progressions and teaching advice.

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Day, Horne and Stevens report on the algebraic reasoning component of the rmfii project in Chapter 6. They describe the three big ideas for algebraic reasoning identified by the extensive literature review, Equivalence, Pattern and Function, and Generalisation, and provide an example of the type of tasks used to test the hlp for algebraic reasoning. The analysis of the data show how student thinking in these three areas develops over time and how it can be supported using rich tasks. However, the results also show that many students experience considerable difficulty with some aspects of algebraic reasoning suggesting that current curriculum expectations at this level may need to be reconsidered. In Chapter 7, Seah and Horne consider the nature of geometrical reasoning and its relationship to measurement and spatial reasoning. Based on their review of the literature they argue that how well a concept is learned and reasoned about is dependent to a large extent on the degree of connectedness among the representations used to express or represent the concepts, and an individuals’ ability to visualise and communicate these relationships. As a result, they frame the hlp for geometrical reasoning in terms of three overarching capacities: Multiple Representations, Visualisation and Discourse. In this case, the analysis of the data derived from the student responses to the rich assessment tasks indicate levels of performance at odds with what might reasonably be expected at this level, suggesting that much more time and attention to geometrical reasoning in classrooms is warranted. Watson, Callingham, and Oates consider the development of statistical reasoning in Chapter 8. They identify the key statistical ideas of Variation, Expectation, Distribution, and Inference as a basis for the hlp for statistical reasoning. They provide examples of the tasks used to test the hlp and illustrate the subsequent learning progression and targeted teaching advice. A strong case is made for the use of rich tasks with multiple entry points both as a means of scaffolding student’s statistical reasoning but also as a way of developing teacher knowledge and confidence for teaching statistics. Chapters 9 and 10 report on the student surveys conducted as a part of the rmfii project. In Chapter 9, Barkatsas and Orellana consider the factorial structure of the motivation and perception items from the surveys. Their analysis confirmed four factors: Intrinsic and Cognitive Value of Mathematics, Instrumental Value of Mathematics, Mathematics Effort, and Social Impact of School Mathematics, and identified significant Year (Grade) level differences. In Chapter 10, the authors consider the factorial structure of the goal orientation items from the student surveys. Again, the factors were consistent with those identified in the source studies used to construct the surveys. That is, Performance Approach Goal Orientation, Mastery goal Orientation,

Introduction

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and Performance Avoidance Goal Orientation. Significant differences were found between Year (Grade) level and the second of these factors. Given the paucity of research in this area, Barkatsas and Orellana conclude that further research is needed to explore the role of goal expectations in school mathematics learning. In the final chapter, Mike Askew provides an overview of the work reported and a commentary on the role of learning progressions/trajectories in mathematics education.

References Watson, A. (2017). Researching and using learning progressions (trajectories) in mathematics education Research Symposium: Discussant’s comments. In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st conference of the International Group for the Psychology of Mathematics Education. Singapore: PME. Wilson, M. (2018). Making measurement important for education: The crucial role of classroom assessment. Educational Measurement: Issues and Practice, 37(1), 5–20.

CHAPTER 1

Knowing and Building on What Students Know: The Case of Multiplicative Thinking Dianne Siemon

Abstract Identifying and building on what students know in relation to important mathematics is widely regarded as essential to success in school mathematics. However, determining what is important and identifying what students actually understand in relation to what is deemed to be important are by no means uncontested or straightforward endeavours. In recent years attention has turned to the development of evidenced-based learning trajectories (or progressions) as a means of identifying what mathematics is important and how it is understood over time. But for this information to be useful to practitioners, it needs to be accompanied by accurate forms of assessment that locate where learners are in their learning journey and evidenced-based advice about where to go to next. This chapter traces the origins of the Scaffolding Numeracy in the Middle Years (snmy) research project that used rich assessment tasks and Rasch analysis techniques to develop an evidence-based framework to support the teaching and learning of multiplicative thinking in Years (Grades) 4 to 9. Keywords learning progressions – multiplicative thinking – formative assessment – targeted teaching – middle school

1

Introduction

Teaching informed by quality assessment data has long been recognised as an effective means of improving mathematics learning outcomes (e.g., Black & Wiliam, 1998; Goss, Hunter, Romanes, & Parsonage, 2015; Masters, 2013; National Council of Teachers of Mathematics, 2001; Timperley, 2009; Wiliam, 2011). It is also evident that where teachers are supported to identify and interpret student learning needs, they are more informed about where to start teaching, © koninklijke brill nv, leiden, 2019 | doi:10.1163/9789004396449_002

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and better able to scaffold their students’ mathematical learning (Callingham, 2010; Clarke, 2001; Siemon, 2016). Furthermore, What we do know is that when you invest in teachers using formative assessment … you get between two and three times the effect of class size reduction at about one-tenth the cost. So, if you’re serious about raising student achievement … you have to invest in teachers and classrooms, and the way to do that is in teacher professional development focused on assessment for learning. (Wiliam, 2006, p. 6) Originally, the terms ‘assessment of learning,’ ‘assessment for learning’ and ‘assessment as learning’ were used to draw attention to the different purposes of assessment (e.g., Earl & Katz, 2006). Since then, Wiliam (2011) and others (e.g., Callingham, 2010; Masters, 2013) have blurred this distinction to recognise that any “assessment functions formatively to the extent that evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers to make decisions about the next steps in instruction” (Wiliam, 2011, p. 43). This of course begs the question of what is elicited, how it is interpreted, and to what extent this information is used to support and involve teachers in planning where to go to next.

2

Learning Trajectories, Progressions, and Frameworks

Identifying and building on what students in relation to important mathematics has been the driving force behind most, if not all, of the research on learning trajectories/progressions in school mathematics over the last two decades (e.g., Battista, 2004; Clements & Sarama, 2004; Maloney, Confrey, & Nguyen, 2014; Siemon, Breed, Dole, Izard, & Virgona, 2006; Simon, 1995). However, there are some important differences in the focus, scale and methods of this research that help explain the range of terms used. A brief account of the derivation and use of these terms is included below to help frame the work to be reported in this chapter. More extensive accounts can be found in the special edition of Mathematics Thinking and Learning (2004, Volume 6, Number 2), Daro, Mosher, and Corcoran (2011), Maloney, Confrey, and Nguyen (2014), and Siemon, Horne, Clements, Confrey, Maloney, Samara, Tzur, and Watson (2017). Writing in the special edition of Mathematical Thinking and Learning on learning trajectories in mathematics education, Clements and Sarama (2004) note that “the construct of learning trajectories is less than a decade old, but palpably has many roots in previous theories of learning, teaching, and

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curriculum” (p. 81). This possibly explains why the construct has been interpreted and used in so many different ways since Simon’s (1995) original conception of a hypothetical learning trajectory (hlt) as part of the ‘mathematics teaching cycle.’ That is, as a “consideration of the learning goal, the learning activities, and the thinking and learning in which students might engage” (p. 133). Developed in the context of a teaching experiment underpinned by constructivist perspectives on learning, the hlt was viewed as a “tool for individual teachers to make sense of their own students’ day-to-day progress and to frame their moment-to-moment and day-to-day instructional planning” (Confrey, Maloney, & Nguyen, 2014, p. xiii). Subsequent revisions of the HLT are based on classroom interactions, informal feedback and student responses to particular tasks. Clements and Sarama (2004, 2009, 2014) build on Simon’s (1995) definition but emphasise “a cognitive science perspective and a base of empirical research” (p. 2). This takes the hlt notion beyond individual teachers in dayto-day contexts to a more public space where learning trajectories are viewed as a “device whose purpose is to support the development of a curriculum, or a curriculum component” (2014, p. 1). Here, ‘curriculum’ is taken to mean “a specific set of instructional materials that order content used to support preK-grade 12 classroom instruction” (Clements, 2007, p. 36). As a consequence, Clements and Sarama (2014) describe learning trajectories in terms of “three parts: (1) a goal, (2) a developmental progression, and (3) instructional activities” (p. 2). In a similar fashion to earlier work on Cognitively Guided Instruction (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Fuson, 1997), Clements and Samara (2004) draw on empirical evidence of children’s learning in particular domains to construct hierarchical models of student learning in that domain. They then devise an “instructional sequence … composed of key tasks designed to promote learning at a particular conceptual level or benchmark in the developmental progression” (p. 84). The hypothetical element of the learning trajectory lies in the task sequence, which is hypothesised to be the most likely to support learning. They make no claims about a particular sequence being the only one possible, or that it represents the ‘best’ route for learning and teaching. In this case, learning trajectories are modified as a result of an empirical evaluation of the efficacy of the task sequence as it is implemented in practice over time (e.g., Clements & Samara, 2008). 2.1

A Focus on ‘Big Ideas’ and Coherence in the Teaching and Learning of Mathematics Alongside the growth in research on learning trajectories/progressions and largely in response to the narrowing focus of the curriculum there have been

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calls for an increased focus on ‘big ideas’ in mathematics teaching and learning (e.g., Baroody, Cibulskis, Lai, & Li, 2004; Charles, 2005; Ma, 1999; Siemon, 2006) and for much greater coherence and alignment between curriculum, instruction, and assessment (Black, Wilson, & Yao, 2011; Pellegrino, 2008; Swan & Burkhardt, 2012). This has extended the notion of learning trajectories beyond models to support individual teacher decision making (Simon, 1995) and devices to support curriculum development (e.g., Clements & Sarama, 2014) to integrated, empirically-based frameworks that focus on the development of big ideas and incorporate assessment not only as a means to evaluate the underlying model of learning and the efficacy of the instructional activities, but also to provide validated means of identifying where students are in the learning journey that teachers can use to inform their teaching (Daro, Mosher, & Corcoran, 2011). The work of Confrey and her colleagues on learning trajectories for rational number reasoning (e.g., Confrey & Maloney, 2010, 2014) exemplifies this extended view. Learning Trajectory: A researcher-conjectured, empirically supported description of the ordered network of constructs a student encounters through instruction (i.e. activities, tasks, tools, forms of interaction and methods of evaluation), in order to move from informal ideas, through successive refinements of representation, articulation, and reflection, towards increasingly complex concepts over time. (Confrey & Maloney, 2010, p. 1, emphasis added) The inclusion of diagnostic assessment tools aligned to evidenced-based learning progressions provides a basis for making formative “decisions about the next steps in instruction that are likely to be better, or better founded than, the decisions [that] would have been made in the absence of that evidence” (Wiliam, 2011, p. 43). Understood in this way, learning trajectories or learning and assessment frameworks “support an unprecedented degree of coherence among standards, assessment, … instruction and curriculum” (Confrey & Maloney, 2014, p. 134). Not only because they focus on and provide material support for the teaching and learning of the big ideas in mathematics but because they also support a more systematic and focussed use of the most cost-effective means of improving student achievement, that is, formative assessment (Black & Wiliam, 1998; Wiliam, 2006). This chapter will describe how an empirically-based learning and assessment framework for multiplicative thinking was developed and subsequently used by practitioners to improve student learning outcomes. The framework incorporates diagnostic tools aligned to an evidenced-based learning

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progression and targeted teaching advice designed to progress student learning from one level/Zone of the framework to the next. While aspects of this research have been reported elsewhere (e.g., Siemon, Banks, & Prasad, in press; Siemon, 2016; Siemon, Izard, Breed, & Virgona, 2006), it will serve as a basis for Section 2 of this book, which considers the development of an evidence-based framework for mathematical reasoning.

3

Researching Numeracy in the Middle Years

The journey that led to researching learning progressions for mathematical reasoning (Part 2 of this book) did not start out with formative assessment in mind. It started with the Middle Years Numeracy Research Project (mynrp), a multi-sector, large-scale, ascertaining study1 to explore what was working in numeracy education in the middle years of schooling. For this purpose, numeracy was seen to involve: – core mathematical knowledge, in this case, number sense, measurement and data sense and spatial sense as elaborated in the National Numeracy Benchmarks for Years 5 and 7 (National Numeracy Benchmarks Taskforce, 1997); – the capacity to critically apply what is known in a particular context to achieve a desired purpose; and – the actual processes and strategies needed to communicate what was done and why (Siemon & Virgona, 2001). A quasi-experimental design involving a structured sample of 27 primary schools and 20 secondary schools was used to evaluate student numeracy on two occasions, 14 months apart. In the first phase, data were collected from just under 7000 students in Years 5 to 9 (i.e., Grades 5 to 9) using rich assessment tasks and scoring rubrics based on the dimensions of numeracy described above (Siemon & Stephens, 2001). These data were analysed using spss and Quest, a Rasch modelling tool developed by Adams and Khoo (1993). This confirmed that the tasks were appropriate for the cohort tested and that it was possible to measure a complex construct such as numeracy using assessment tasks that incorporate performance measures of content knowledge and process (general thinking skills and strategies) across a range of topic areas using teachers-as-assessors (Siemon & Virgona, 2001). Key findings included: – there was as much difference in student numeracy performance within each year level as there was between year levels and this difference was equivalent to 7 years of schooling; – there was considerable within school variation at the same year level suggesting individual teachers had a major influence on student performance;

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– the learning needs of many students, particularly those ‘at risk,’ were not being met; and – that irrespective of context, differences in performance were almost entirely due to an inadequate understanding of larger whole numbers, multiplication and division, fractions, decimals, and proportion, and a reluctance/ inability to explain/justify solutions (Siemon & Virgona, 2001).

3.1

Identifying What Is Important

Although most students were able to solve multiplication problems involving relatively small whole numbers, they relied on additive strategies to solve more complex multiplicative problems involving larger whole numbers, rational numbers, and/or situations not easily modelled in terms of repeated addition such as percent and proportional reasoning. Typically addressed as isolated ‘topics’ with little/no reference to one another (Siemon, Bleckly, & Neal, 2012), these aspects of mathematics content were recognised by Vergnaud (1994) as constituting a Multiplicative Conceptual Field. That is, a framework of complex, inter-related ideas and strategies, which for the purposes of the project was described in terms of multiplicative thinking. At the time, there was a significant body of literature pointing to the difficulties students experience with particular aspects of multiplicative thinking (e.g., Anghileri, 1999; Baturo, 1997; Harel & Confrey, 1994; Gray & Tall, 1994; Lamon, 1996; Misailidou & Williams, 2003; Mulligan & Mitchelmore, 1997) and the relatively long period of time needed to develop these ideas (Clark & Kamii, 1996; Sullivan, Clarke, Cheeseman, & Mulligan, 2001). While this work contributed to a better understanding of the ‘big ideas’ involved, very little was specifically concerned with how these ideas relate to one another and which aspects might be needed when, to support new learning both within and between these different domains of multiplicative thinking. Moreover, very little of this work was represented in a form and language that was accessible to teachers or directly translated to practice in the middle years of schooling (Siemon & Virgona, 2001). 3.2 Knowing and Building on What Students Know One of the most promising results of the Middle Years Numeracy Research Project (mynrp) was that the item scaling generated by the Rasch analyses (Rasch, 1960) supported the identification of an Emergent Numeracy Profile comprised of eight distinct developmental levels of numeracy performance from relatively naïve beginnings (Level A) to more sophisticated understandings and capacities (Level H). Although further research was needed to tease

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table 1.1  Emergent numeracy profijile (Siemon, Virgona, & Corneille, 2001)

A B

C

D

E

F

G

H

Uses make-all, count-all strategies to solve a simple number pattern problem Recognises a number pattern and represents it in one way. Makes judgements about data more on the basis of perception than analysis. Little evidence of cognitive monitoring (e.g., estimates or calculates without regard for meaning or applicability). Able to use a number pattern to solve a problem. Monitors cognitive actions and/or goals some of the time (e.g., recognises relevant information but unable to use it efffectively). Beginning to understand and represent simple fraction situations. Generally solves one-step problems involving 3-digit whole numbers, ones and tenths. Describes simple patterns. Consolidating fraction and percent knowledge. Monitors cognitive actions (for 1–2 step problems). Little/no monitoring of cognitive goals (i.e., checks procedures but not their meaningfulness and/or appropriateness to problem context and/or conditions). Consolidating use of data and information appropriate to context. Established in recognising 2D representations of simple 3D space. Beginning to monitor cognitive goals as well as actions (i.e., evaluates what they are doing for sense and relevance). Established in using and interpreting data and/or information appropriate to context, fraction representations, and in describing patterns and relationships. Able to explain solutions to problems. Well established in the use of fractions/ratio. Able to generalise and apply number relationships to solve problems. Monitors cognitive actions and goals (i.e., almost always evaluates what they are doing for meaning and relevance to problem solution).

out and enrich the levels within the Profile (see Table 1.1), it was recognised that this type of empirical data had the potential to inform the design of teaching and learning materials that could identify and build on what was known to develop each student’s numeracy-relevant content knowledge and thinking skills (Siemon, Virgona, & Corneille, 2001). Figure 1.1 shows the proportion of students in each Level of the Emergent Numeracy Profile by year level after the initial round of assessment in 1999. Given that Level A corresponds in curriculum terms to about Year 2 and Level H corresponds to about Year 9 the spread within each year level represents a range in student’s mathematics achievement equivalent to seven years of schooling.

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figure 1.1 Proportion of students at each level of the emergent numeracy profile in 1999 for years 5 to 9 (n = 1315, 1318, 1467, 1484, & 1276 respectively)

In Phase 2 of the project, 20 schools were selected to continue in the project as trial schools on the basis of their results from the first round of assessment (high or low performing) and evidence of their use of the design elements (Hill & Crévola, 1997) in whole school planning (rich or poor). At the end of this phase, a second round of assessment was conducted approximately 14-months after the initial round. As the Emergent Profile was not fine-grained enough to provide trial schools with any specific advice on how to target their teaching, teachers were invited to devise a plan framed in terms of the design elements that was aimed at improving numeracy outcomes given where their students were in terms of the Emergent Numeracy Profile. While the plans varied considerably, and some schools improved more than others for a variety of reasons (see Siemon, Virgona, & Corneille, 2001), the numeracy performance of all schools improved. Overall, the average effect size was over 0.6 indicating that the increase was more than what otherwise would have been expected and suggesting that teaching informed by assessment data and supported by collaborative planning was a valuable tool in addressing the learning needs of all students.

4

Addressing the 7-Year Gap: Exploring the Potential

Having established that multiplicative thinking was responsible for a seven-year range in student numeracy in each year level, and realising the potential of Rasch analysis (e.g., Bond & Fox, 2001) to identify distinct levels of understanding that could be used to inform teaching (e.g., Misailidou & Williams, 2003), an approach was made to the Victorian and Tasmanian Departments of Education to partner in a follow up research project, the Scaf-

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folding Numeracy in the Middle Years project,2 which was aimed at developing an evidenced-based learning and assessment framework to support the teaching and learning of multiplicative thinking in Years 4 to 8. At the time, using assessment data to inform teaching and learning was recognised as a key goal of mathematics education (e.g., Black & Wiliam, 1998; National Council of Teachers of Mathematics, 2001) as was the notion that assessment needed to move beyond “discrete bits and pieces of knowledge to encompass the more complex aspects of student achievement, including how their knowledge is organised and whether they can explain what they know” (Pellegrino, 2002, p. 50). However, this approach was not reflected in practice where the predominant focus was on relatively narrow forms of summative assessment (e.g., Clements & Ellerton, 1995; Schoenfeld, 1999; Swan, 1993). In the United States, the widespread use of external, standardized assessments at almost every grade level, for a wide range of purposes, has had seriously deleterious effects. Since “teaching to the test” these days typically means teaching to a set of skills that have little to do with deep competence, the current incarnations of most assessments serve disruptive rather than productive functions. (Schoenfeld, 1999, p. 21) In an important report, Knowing What Students Know (Pellegrino, Chudowsky, & Glaser, 2001), it was noted that every assessment, regardless of purpose, needs to be based on models of student learning and methods of observation and interpretation consistent with those models that can be used to inform teaching. To this end, the snmy project drew on a sociocultural perspective of mathematical activity that viewed learning “as both a process of active individual construction and a process of enculturation into the mathematical practices of wider society” (Cobb, 1994, p. 13). Simon’s (1995) notion of constructing hypothetical learning trajectories as mini-theories of student learning in particular domains that could be used to analyse and support student growth over time is consistent with this perspective and the primary goal of the project, which was to construct a plausible, probabilistic model for identifying where learners are in relation to multiplicative thinking and a framework to support teachers progress student learning. For the purposes of the snmy project, multiplicative thinking was defined in terms of:

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– a capacity to work flexibly and efficiently with an extended range of numbers (e.g., larger whole numbers, decimal and common fractions, and/or per cent); – an ability to recognise and solve a range of problems involving multiplication or division including direct and indirect proportion; and – the means to communicate this effectively in a variety of ways (e.g., words, diagrams, symbolic expressions, and written algorithms) (Siemon, Breed, & Virgona, 2005). The project was designed in terms of three overlapping phases. In Phase 1 a broad hypothetical learning trajectory (hlt) for multiplicative thinking was derived from an extensive literature review of the key ideas and known areas of difficulty related to multiplicative thinking, that is, multiplication, division, fractions, decimals, rate, ratio, percentage, and proportional reasoning (Siemon, Izard, Breed, & Virgona, 2006). Phase 2 involved the design, trial, and subsequent use of a range of rich assessment tasks to evaluate the various aspects of multiplicative thinking (Siemon & Breed, 2006). An example of one of these tasks is shown in Figure 1.2. The tasks, and their associated scoring rubrics were used at the beginning of the project to test the hypothesised trajectory, and at the end of the project to evaluate student growth. Master’s (1982) partial credit model was used to analyse the data and inform the development of teaching advice, which was presented as a Learning and Assessment Framework (laf) for multiplicative thinking (Siemon, Breed, Dole, Izard, & Virgona, 2006). Phase 3 involved research school teachers and members of the research team in an eighteen-month action research study that progressively explored a range of targeted teaching interventions aimed at scaffolding student learning on the basis of the laf. Just over 1500 Year 4 to 8 students and their teachers from three research school clusters, each comprising three to six primary (K-6) schools and one secondary (7–12) school, were involved in Phases 2 and 3 of the project. A similar group of Year 4 to 8 students from three reference school clusters was involved in Phase 2 only. Data from the first round of assessment were analysed using the Rasch modelling tool Quest (Adams & Khoo, 1993) and the subsequent variable map that listed partial credit items from easiest to hardest on a vertical scale alongside student results was used to link different aspects of multiplicative thinking and identify qualitatively different levels of understanding and strategy usage indicated by student responses (see Siemon, Izard, Breed, & Virgona, 2006). While these levels were largely consistent with the initial

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TILES, TILES, TILES … Task a.

b.

c.

Response

Score

No response or incorrect with no working and/or explanation

0

Incorrect (2 tiles), reasoning based on perceived relationship between dimensions e.g., “2 goes into 4, 2 times and 3 goes into 6, 2 times” or incorrect drawing, or correct but little/no working or reasoning

1

Correct (4 tiles), with appropriate diagram and/or explanation

2

No response or incorrect with little/no working and/or explanation

0

Incorrect (9 or 18 tiles), reasoning based on factors as above, or correct (81 tiles) but little/no working/explanation

1

Correct (81 tiles), with appropriate diagram and/or evidence of additive strategy, e.g., count all or skip count

2

Correct (81 tiles), with appropriate diagram and/or explanation indicating multiplicative reasoning, e.g., factors used appropriately

3

No response or incorrect with little/no working and/or explanation

0

Some attempt, e.g., dimension of larger tile (4cm by 5cm) indicated and/ or incomplete solution attempt, e.g., attempt to draw all

1

Incorrect, calculation based on incorrect dimension of larger tile, e.g., 4cm by 6cm, but supported by correct reasoning of the area required; or correct (500 tiles), with little/no explanation

2

Correct (500 tiles), supported by appropriate diagram and/or explanation based on appropriate diagram or computation strategies

3

figure 1.2 A short task and scoring rubric used in the snmy project (2003–2006)

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figure 1.3 Proportion of students by laf zone and year level, 2004 (n = 3169)

hlt, two levels were collapsed into one and others were qualified and/or elaborated. Rich text descriptions for each level were derived from the performances on each item at each level to form the basis of the laf. As the levels were approximations based on responses identified at similar locations on the scale and to avoid confusion with the use of ‘Levels’ in curriculum documents at the time, the LAF levels were referred to subsequently as Zones. This was felt to be appropriate as it evoked Vygotsky’s (1978) notion of the Zone of Proximal Development, which was consistent with the underpinning sociocultural view of student learning and suggested multiple pathways through Zones. It is also consistent with Battista’s (2004) use of zones of construction. The final laf3 is comprised of eight hierarchical zones ranging from additive, count all strategies (Zone 1) to the sophisticated use of proportional reasoning (Zone 8) with multiplicative thinking not evident on a consistent basis until Zone 4. Figure 1.3 shows the relative proportion of students at each year level in each Zone of the laf after the first round of assessment in 2004.

5

Results

The results confirmed the findings of the mynrp project that there is a seven-year range in student mathematics achievement in in each year level that is almost entirely explained by the extent to which students have access to multiplicative thinking. Importantly, the number and diversity of items used meant that the laf could be elaborated to include teaching advice that teachers could use to scaffold student learning from one Zone to the next. Given that students were located on the item difficulty scale at

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the point where they have a 50% chance of satisfying the scoring criterion (see Siemon, Breed, Izard, & Virgona, 2006), the advice was formulated in terms of specific ideas and/or strategies that needed to be consolidated and established at each Zone as well as specific suggestions as to what to introduce and develop to progress student learning to the next Zone. This second form of advice (i.e., what needed to be introduced and developed) was also used to consolidate and establish what was needed at the next Zone. This provided teachers with some flexibility and promoted fluid movement between Zones. Table 1.2 shows an excerpt from the elaborated laf for Zone 3. In phase 3 of the snmy project, research school teachers worked with research team members to design and trial Zone-specific activities based on the advice in the elaborated laf. Referred to originally as assessmentguided instruction, the use of the Zone-specific activities came to be referred to as targeted teaching in the latter part of the snmy project (Siemon et al., 2006; Siemon, 2016) to distinguish the long-term, multi-faceted nature of the interventions needed to scaffold student’s multiplicative thinking (e.g., Breed, 2011) from the equally valid but short-term or spontaneous teaching decisions that might be informed by a pre-test on subtraction or an informal classroom quiz. The final snmy assessment round was conducted in November 2005. Data were collected from 3350 Year 4 to 8 students and analysed using the Rasch partial credit model (Masters, 1982). The results confirmed the sequence of ideas and strategies in the laf and established the validity of the assessment tasks (Siemon et al., 2006). As the initial and final assessments were conducted in different school years, cohort comparisons rather than matched pairs were used to explore the impact of targeted teaching on student learning (e.g., growth from Year 4 to year 5 in research schools was compared to growth from Year 4 to Year 5 in reference schools). Overall, medium to large effect sizes (in the range 0.45 to 0.75 or more) as described by Cohen (1969) were found in research schools compared to small to medium effect sizes (in the range of 0.2 to 0.5) in reference schools. While the results varied between research schools, overall, they show that teaching targeted to identified student learning needs was effective in improving student’s multiplicative thinking (Siemon et al., 2006). Since 2006, the evidenced-based, formative assessment materials produced by the snmy project (i.e., two assessment options and the elaborated laf) have been used in Victoria, South Australia, Tasmania and Queensland primarily to support interventions in upper primary schools. Although their use in

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Knowing and Building on What Students Know table 1.2  The elaborated laf for zone 3

Zone 3: Sensing (italics refer to task name and item) Demonstrates intuitive sense of proportion (e.g., partial solution to Butterfly House f) and partitioning (e.g., Missing Numbers b)

Consolidate/establish: Ideas and strategies introduced/developed in the previous Zone

Introduce/develop: Place-value based strategies for informally solving problems involving single-digit by two-digit multiplication (e.g., for 3 twenty-eights, THINK, 3 by 2 tens, 60 and 24 more, 84) mentally or in writing Initial recording to support place-value for multiplication facts (see Siemon et al., 2015 and There’s More to Counting Than Meets the Eye) More efffijicient strategies for solving number problems involving simple proportion (e.g., recognise as two-step problems, ask, What do I do fijirst? Find value for common amount. What do I do next? Determine multiplier/factor and apply. Why?) How to rename number of groups (e.g., think of 6 fours as 5 fours and 1 more four), Practice (e.g., by using ‘Multiplication Toss Game’). Re-name composite numbers in terms of equal groups (e.g., 18 is 2 nines, 9 twos, 3 sixes, 6 threes) Cartesian product or for each idea using concrete materials and relatively simple problems such as 3 tops and 2 bottoms, how many outfijits, or how many diffferent types of pizzas given choice of small, large, medium and 4 varieties? Discuss how to recognise problems of this type and how to keep track of the count such as draw all options, make a list or a table (tree diagrams appear to be too difffijicult at this level, these are included in Zone 5) How to interpret problem situations and solutions relevant to context (e.g., Ask: What operation is needed? Why? What does it mean in terms of original question?) Practical sharing situations that introduce names for simple fractional parts beyond the halving family (e.g., thirds for 3 equal parts/shares, sixths for 6 equal parts etc.) and help build a sense of fractional parts, e.g. 3 sixths is the same as a half or 50%, 7 eighths is nearly 1, “2 and 1 tenth” is close to 2. Use a range of continuous and discrete fraction models including mixed fraction models (cont.)

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table 1.2  The elaborated laf for zone 3 (cont.)

Works with ‘useful’ numbers such as 2 and 5, and strategies such as doubling and halving (e.g., Packing Pots b, and Pizza Party c) May list all options in a simple Cartesian product situation (e.g., Canteen Capers b), but cannot explain or justify solutions Uses abbreviated methods for counting groups, e.g., doubling and doubling again to fijind 4 groups of, or repeated halving to compare simple fractions (e.g., Pizza Party c) Beginning to work with larger whole numbers and patterns but tends to rely on count all methods or additive thinking to solve problems (e.g., Stained Glass Windows a and b, Tiles, Tiles, Tiles b) Simple, practical division problems that require the interpretation of remainders relevant to context Thirding and fijifthing partitioning strategies through paper folding (kinder squares and streamers), cutting plasticine ‘cakes’ and ‘pizzas,’ sharing collections equally (counters, cards etc.), apply thinking involved to help children create their own fraction diagrams (regions) and number line representations (see Siemon, 2004) Partitioning – The Missing Link in building Fraction Knowledge and Confijidence. Focus on making and naming parts in the thirding and fijifthing families (e.g., 5 parts, fijifths) including mixed fractions (e.g., “2 and 5 ninths”) and informal recording (e.g., 4 fijifths), no symbols. Revisit key fraction generalisations (see Level 2), include whole to part models (e.g., partition to show 3 quarters) and part to whole (e.g., if this is 1 third, show me the whole) and use diagrams and representations to rename related fractions Extend partitioning strategies to construct number line representations. Use multiple fraction representations Key fraction generalisations – equal parts, as the number of parts increase the size of the part gets smaller; the number of parts names the part (e.g., 8 parts, eighths) and the size of the part depends upon the size of the whole

secondary schools has not been as widespread, where they have been used, there is evidence to suggest that a targeted teaching approach based on the LAF is effective in improving mathematics outcomes for students in Years 7 to 9 (e.g., Reilly & Parsons, 2011).

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21

Fine-Tuning the Targeted Teaching Approach

In 2013, an opportunity arose to investigate the efficacy of and issues involved in implementing a targeted teaching approach in secondary schools using the snmy materials through the Australian Mathematics and Science Partnership Programme (amspp) Priority Project scheme.4 The year-long scheme called for ‘road-ready’ projects aimed at improving student outcomes in school mathematics and building teacher knowledge and confidence for teaching mathematics in primary and junior secondary schools. A condition of funding was that the project had a national focus and involved teachers and students from lower socioeconomic regions. Because time was of the essence, the Reframing Mathematical Futures Priority project (rmf-p) approached those education systems that were aware of the snmy materials to be partners in the project. Their key role was to identify schools that satisfied the funding condition (up to 6 in each jurisdiction) where there was a willingness and capacity to provide some time release (preferably a day/week) for an existing teacher to participate in the project as an snmy ‘Specialist.’ Project funding was provided to support the Specialists attend professional learning days in Melbourne and provide time release (up to 30 days/school) to support at least two other teachers engage with the Specialist in school time to implement a targeted teaching approach to multiplicative thinking. Schools were visited on at least two occasions by members of the research team who acted as mentors to support the targeted teaching approach and meet with school leadership. A total of 28 secondary or middle schools (Years 7 to 9) agreed to participate in the project on this basis and use the formative assessment resources materials with at least 4 classes. The initial two-day residential professional learning program was designed to train the Specialists in the use of the snmy resources, deepen their understanding of multiplicative thinking and its relationship to other areas of the mathematics curriculum, and explore activities and strategies to support a targeted teaching approach based on the laf. In recognition of the need to engage junior secondary students in the assessment and targeted teaching activities, the five essential feelings identified by Sagor and Cox (2004) as crucial to young people’s well-being and success at school (i.e., competence, belonging, usefulness, potency and optimism) were incorporated into the professional learning program. A two-day residential conference was held at the end of the project to document and disseminate project experiences and findings and identify areas for further research. Framed as a design study (e.g., Design-Based Research Collaborative, 2003), one of the objectives of rmf-p project was to identify what targeted teaching

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might look like in secondary schools. As a result, the Specialists and the respective school teams were able to decide exactly how they would use the assessment data, laf and Zone-based activities to support a targeted teaching approach appropriate to their particular circumstances and mindful of the five essential feelings identified by Sagor and Cox (2004). Online, interactive sessions were held every 3 to 4 weeks to address issues raised by Specialists and to further support the implementation of a targeted teaching approach to multiplicative thinking based on the laf. Interestingly, the Specialists reported that one of the most valuable professional learning opportunities they and their colleagues experienced was marking and moderating the student assessments. 6.1 Results The assessments using snmy tools were conducted in August and November of 2013 to evaluate the impact of the targeted teaching approach to multiplicative thinking. Matched data sets were obtained from 1732 students from Years 7 to 10 with the majority (59%) from Year 8. The assessments were marked by the teachers and the de-identified results were forwarded to the research team for analysis. Student raw scores were translated to laf levels using the snmy Raw Score Translator provided by the original research. Matched pairs were used to calculate effect size using the means and standard deviations of the preand post-test laf Level data for each school and effect sizes were extrapolated to one year and adjusted for regression to benchmark the results. Although results varied considerably by school, the overall achievement of students across the 28 schools grew by an average adjusted effect size of 0.65 indicating a medium influence beyond what might be expected (Hattie, 2012). This can be seen in the shift in the relative proportions in each zone of the laf from August to November shown in Figure 1.4.

figure 1.4 Proportion of students by laf zone, rmf-p project (n = 1732)

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While it was always going to be difficult to expect schools to change their practice in such a short time, the fact that so many schools achieved effect sizes well in excess of what would otherwise be expected (i.e., 0.4) suggests that adopting a targeted teaching approach to multiplicative thinking in secondary schools can be effective. However, the considerable differences in school outcomes suggests this is not without its challenges particularly in such a short time frame. A range of factors were nominated by the Specialists in the endof-project surveys as reasons for the differential performance. These included the extent to which the targeted teaching approach was endorsed and practically supported by school leadership, availability of planning and professional learning time, access to appropriate spaces and resources, and varying levels of staff ‘buy in.’ Importantly, while the Specialists reported that the experience of working with the snmy materials had significantly impacted their knowledge and confidence for teaching mathematics, the participating teachers were more ambivalent. This is not surprising as the Specialists had considerably more opportunities for professional learning than the teachers and it takes time to build productive, professional learning communities to support change initiatives (e.g., Erb & Stevenson, 1999). To be fair, the difference in views may also be a function of timing. The surveys were held at the end of the year when teachers were writing reports, so it is possible that their minds were on more immediate concerns. For instance, in their open responses to the survey, the teachers commented on the tension between adopting a targeted teaching approach and ‘covering the curriculum,’ an issue that was particularly noticeable in Term 4 where, for example, one teacher reported that he only offered five targeted teaching sessions for the whole term. Faced with the perceived pressure of addressing content descriptors framed in terms of the three strands, Number and Algebra, Statistics and Probability, and Geometry and Measurement, multiplicative thinking was not necessarily seen as ‘core business.’

7

Conclusion

This chapter has presented an overview of three large scale numeracy research projects. The first, the Middle Years Numeracy Research Project (mynrp) identified that multiplicative thinking was responsible for a seven-year range in student mathematics achievement at this level of schooling. The second, the Scaffolding Numeracy in the Middle Years (snmy) project, explored the development of multiplicative thinking in more depth to produce an evidenced-based learning progression for multiplicative thinking, validated, formative assessment options,

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and targeted teaching advice designed to progress student learning from one level/Zone of the framework to the next. The third, the Reframing Mathematical Futures Priority (rmf-p) project, demonstrated the efficacy of using the snmy resources to target the specific learning needs of students in relation to multitable 1.3  Alignment between snmy materials and master’s (2013) design principles for a learning assessment system Learning Assessment System Design

SNMY Materials

Principles (Masters, 2013, pp. 16–17)

1. Assessment methods should be guided by, and address, an empirically-based understanding of the relevant learning domain

2. Assessment methods should be selected for their ability to provide useful information about where students are in their learning within the domain 3. Responses to, or performance on, assessment tasks should be recorded using one or more ‘rubrics’ 4. Available assessment evidence should be used to draw a conclusion about where learners are in their progress within the learning domain

5. Feedback and reports of assessment should show where learners are in their learning at the time of assessment and, ideally, what progress they have made over time

The hypothetical learning trajectory (hlt) used to inform task design was fijirmly grounded in research pertinent to the development of multiplicative thinking over time. The Rasch partial credit model (Masters, 1982) was used to analyse the tasks and revise the hlt to derive the laf The snmy options use rich tasks (Siemon & Breed, 2006) to assess multiplicative thinking in terms of, core knowledge, the ability to apply that knowledge in unfamiliar situations, and the capacity to explain and justify solution strategies. Rasch analysis was used to test that the tasks were suited to purpose. The snmy options provide scoring rubrics that assess all three aspects of multiplicative thinking in terms of qualitatively described hierarchical levels. Raw Score Translators for each option allow the total score to be translated to the Zones within the laf The analysis of the ordered list of items produced by the Rasch analysis identifijied specifijic levels of achievement in relation to each of the three aspects of multiplicative thinking at each of eight developmental Zones of multiplicative thinking enabling conclusions to be drawn about where a learner was in relation to the laf The two snmy assessment options can be used to report growth against the laf over time. Evaluated in terms of Zones, each Zone provides rich text descriptions of behaviours evident at that level within the progression

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plicative thinking when implemented on a regular basis. However, it also highlighted the need to provide similar evidenced-based frameworks for the three core strands of the mathematics curriculum, that is, Algebra, Measurement and Geometry, and Statistics and Probability, and the connections between those and the development of multiplicative thinking. This prompted the design and implementation of the Reframing Mathematical Futures II (rmfii) project which will be described in Chapters 5 to 9 of this volume. In retrospect, the extensive literature review that underpinned the snmy project might be more aptly described as a hypothetical or theoretical learning progression rather than a trajectory with its connotations of curriculum development and teaching materials (e.g., Clements & Samara, 2014). Taken together the evidenced-based teaching and learning materials produced by the snmy project could be viewed as a learning trajectory but they might also be viewed as a system of assessment (Pellegrino, 2008) or a learning assessment system (Masters, 2013) as is illustrated in Table 1.3. But for the purposes for which they were intended, which was to make the materials available to teachers in a form that they can use in ways that suit them and their particular circumstances, the materials simply came to be referred to as the Learning and Assessment Framework for Multiplicative Thinking (laf).

Acknowledgements This work would not have been possible without the support of the many teachers and schools involved and the collective expertise and efforts of the research team. The author would particularly like to acknowledge the contributions of Sandra Vander Pal, Claudia Johnstone and Claudia Orellana to the work reported here.

Notes 1 The Middle Years Numeracy Research Project (mynrp) 1999–2001 was commissioned by the Victorian Department of Education and Training, the Catholic Education Commission of Victoria and the Association of Independent Schools of Victoria to inform the development of a coordinated and strategic plan for improving the teaching and learning of numeracy in Years 5 to 9 2 The Scaffolding Numeracy in the Middle Years (snmy) Project – An investigation of a new assessment-guided approach to teaching mathematics using authentic tasks was funded by the Australian Research Council from 2003 to 2006.

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3 A description of the elaborated Learning Assessment Framework can be found at http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/ maths/assessment/Pages/learnassess.aspx 4 The Reframing Mathematical Futures Priority Project 2013–2014 was funded by the amspp Priority Project scheme sponsored by the Australian Government, Canberra. See https://www.education.gov.au/australian-maths-and-science-partnerships-programme-amspp

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Siemon, D., Breed, M., Dole, S., Izard, J., & Virgona, J. (2006). Scaffolding numeracy in the middle years – Project findings, materials and resources (Final report). Retrieved from http://www.education.vic.gov.au/school/teachers/teachingresources/ discipline/maths/assessment/Pages/scaffoldnum.aspx Siemon, D., Breed, M., & Virgona, J. (2005). From additive to multiplicative thinking. In J. Mousley, L. Bragg, & C. Campbell (Eds.), Mathematics – Celebrating achievement, proceedings of the 42nd conference of the mathematical association of Victoria. Melbourne: MAV. Siemon, D., Bleckly, J., & Neal, D. (2012). Working with the big ideas in number and the Australian Curriculum Mathematics. In W. Atweh, M. Goos, R. Jorgensen, & D. Siemon (Eds.), Engaging the Australian curriculum mathematics – Perspectives from the field (pp. 19–46). Adelaide: Mathematical Education Research Group of Australasia. (Online Book) Retrieved from https://www.merga.net.au/node/223 Siemon, D., Horne, M., Clements, D., Confrey, J., Maloney, A., Sarama, J., Tzur, R., & Watson, A. (2017, July 17–22). Researching and using learning progressions/ trajectories in mathematics education. In Proceedings of 41st annual conference of the International Group for the Psychology of Mathematics Education. Singapore: NIE. Siemon, D., Izard, J., Breed, M., & Virgona, J. (2006). The derivation of a learning assessment framework for multiplicative thinking. In J. Novotna, H. Moraova, M. Kratka & N. Stehlikova (Eds.), Mathematics in the centre. Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 113–120). Prague: PME. Siemon, D., & Stephens, M. (2001). Assessing numeracy in the middle years – The shape of things to come. In AAMT (Eds.), Mathematics shaping Australia – Proceedings of the eighteenth biennial conference of the Australian Association of Mathematics Teachers (pp. 188–200). Adelaide: AAMT. Siemon, D., & Virgona, J. (2001, December). Roadmaps to numeracy – Reflections on the middle years numeracy research project. Proceedings of the Annual Conference of the Australian Association for Research in Education, Fremantle. Retrieved from https://www.aare.edu.au/data/publications/2001/sie01654.pdf Siemon, D., Virgona, J., & Corneille, K. (2001). The final report of the middle years numeracy research project. Retrieved from http://www.education.vic.gov.au/school/ teachers/teachingresources/discipline/maths/Pages/mynr.aspx Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. Sullivan, P., Clarke, D., Cheeseman, C., & Mulligan, J. (2001). Moving beyond physical models in multiplicative reasoning. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 233–241). Utrecht: IGPME.

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Swan, M. (1993). Improving the design and balance of mathematics assessment: An ICMI study. In M. Niss (Ed.), Investigations into assessment in mathematics education. New York, NY: Springer. Swan, M., & Burkhardt, H. (2012). A designer speaks – Designing assessment of performance in mathematics. Educational Designer, 2(5), 4–41. Retrieved February 14, 2018, from http://www.educationaldesigner.org/ed/volume2/issue5/article19/ Timperley, H. (2009). Using assessment data for improving teacher practice. In ACER (Ed.), Assessment and student learning: Collecting, interpreting and using data to inform teaching. Proceedings of 2009 ACER research conference. Perth: ACER. Vergnaud, G. (1994). Multiplicative conceptual field: What and why? In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 41–60). Albany, NY: State University of New York. Vygotsky, L. (1978). Mind in society. Cambridge, MA: Harvard University Press. Wiliam, D. (2006, September). Assessment for learning – what, why and how. Edited transcript of a presentation to the Cambridge Assessment Network Conference, University of Cambridge, UK. Retrieved from http://www.dylanwiliam.org/Dylan_ Wiliams_website/Presentations.html Wiliam, D. (2011). Embedded formative assessment. Bloomington, IN: Solution Tree Press.

CHAPTER 2

Learning Trajectories in Early Mathematics Education Julie Sarama and Douglas H. Clements

Abstract Approaches to standards, curriculum development, and teaching are diverse. However, increasingly learning trajectories are being used as a basis for each of these. In this chapter, we present our own definition and use of the construct, positing that learning trajectories can serve as one effective foundation for scientifically-validated mathematics standards, curricula, and pedagogy. We discuss the current state of learning trajectories in early childhood mathematics and present the development and expansion of the concept in our work, which began with studies of individuals and progressed to large scale-up implementations and evaluations. Throughout this work, learning trajectories served as the structure that maintained coherence. Keywords mathematics – early childhood – learning trajectories – technology – curriculum

1

Introduction

The term “curriculum” stems from the Latin word for a race course and refers to the course of experiences through which children grow to become mature adults. Thus, the notion of a path, or trajectory, has always been central to curriculum development and study. But on what path do we take our students? How do we decide on a scientific basis? In this chapter, we advance the idea that learning trajectories can serve as an effective foundation for scientificallyvalidated mathematics curricula and instruction.

© koninklijke brill nv, leiden, 2019 | doi:10.1163/9789004396449_003

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1.1 Curriculum in Mathematics Education “Curriculum” has different meanings in different contexts. Some who study curriculum define it as more than just a text, but instead as everything that students encounter in an educational setting. Here, we speak of instructional materials – what is often called the “available curriculum” – in contrast to the ideal, adopted, implemented, achieved, or tested curriculum (Burkhardt, Fraser, & Ridgway, 1989, pp. 5–6). In this meaning, curriculum is an instructional blueprint and set of materials that guides students’ acquisition of certain culturally-valued concepts, procedures, intellectual dispositions, and ways of reasoning (Battista & Clements, 2000). However, we readily agree that the creation, refinement, implementation, scale-up, and evaluation of such materials must be done in a context of the comprehensive definition of curriculum – everything that affects students’ experiences. As stated, the term curriculum has roots in the Latin word for a chariot race course. There are many ways to direct that course, but which path should we choose (Clements & Sarama, 2014b)? Simply a logical mathematical progression? Or should psychology also play a role? If so, should the course be based on sequences of accumulation of connections (Thorndike, 1922), or sequences consisting of a subordinate skill whose acquisition is hypothesized to facilitate the learning of a higher-level skill determined by logical and empirical task analysis (Gagné, 1965/1970)? Should it include psychological research to perform “cognitive” or “rational” analyses (Resnick & Ford, 1981), or use more developmental and cognitive science theories (Case & Okamoto, 1996) emphasizing students as makers of meaning (Baroody, 1987; Carpenter, Fennema, Franke, Levi, & Empson, 2014; Steffe & Cobb, 1988)? Learning trajectories owe much to these previous approaches, which have progressed to increasingly sophisticated and complex views of cognition, learning, and teaching. However, the earliest applications of cognitive theory to educational sequences tended to feature simple linear sequences based on accretion of numerous facts and skills. This was reflected in their hierarchies of educational goals and the resultant scope and sequences. Learning trajectories include such hierarchies but are not as limited as these early constructs to sequences of skills or “logically” determined prerequisite pieces of knowledge. Learning trajectories are not lists of everything children need to learn as are some scope and sequence documents; that is, they do not cover every single “fact” or skill. Most importantly, they describe children’s levels of thinking, not just their ability to correctly respond to a question, and they cannot be summarized by stating the mathematical concept or rule (cf. Gagné, 1965/1970). So, for example, a single mathematical problem may be solved differently by

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students at different (separable) levels of thinking in a learning trajectory. Levels of thinking describe how students think about a topic and why, including the cognitive actions-on-objects that constitute that thinking. In the following section, we provide our definition of learning trajectories. 1.2 Learning Trajectories in Mathematics Education In his seminal work, Simon stated that a “hypothetical learning trajectory” included “the learning goal, the learning activities, and the thinking and learning in which the students might engage” (1995, p. 133). Building on Simon’s definition, and emphasizing a cognitive science perspective and a base of empirical research, “we conceptualize learning trajectories as descriptions of children’s thinking and learning in a specific mathematical domain, and a related, conjectured route through a set of instructional tasks designed to engender those mental processes or actions hypothesized to move children through a developmental progression of levels of thinking, created with the intent of supporting children’s achievement of specific goals in that mathematical domain” (Clements & Sarama, 2004, p. 83). The name “learning trajectory” reflects its roots in a constructivist perspective. Although the name emphasizes learning over teaching, both of these definitions clearly involve instructional tasks and teaching strategies. Some interpretations and appropriations of the learning trajectory construct emphasize only the “developmental (or learning) progressions.” Some terms, such as “learning progressions” are used ambiguously, sometimes indicating developmental progressions, and suggesting a sequence of instructional activities at others. We believe the power and uniqueness of the learning trajectories construct stems from the inextricable interconnection among their components (Clements & Sarama, 2014b).

2

Theoretical Framework

Hierarchic Interactionalism provides the theoretical framework upon which our learning trajectories are based (Sarama & Clements, 2009). The term Hierarchic Interactionalism indicates the influence and interaction of global and local (domain specific) cognitive levels and the interactions of innate competencies, internal resources, and experience (e.g., cultural tools and teaching). Of the 10 tenets of the theory, we briefly summarize those most relevant to learning trajectories (for a full explication, see Sarama & Clements, 2009). Developmental progression and Domain specific progression. Most content knowledge is acquired along developmental progressions of levels of thinking,

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which is most propitiously characterized within a specific mathematical domain or topic. Consistency of developmental progressions and instruction. Instruction based on learning consistent with developmental progressions is more effective, efficient, and generative for the child than learning that does not follow these paths. Learning trajectories. A particularly fecund approach is based on hypothetical learning trajectories complete with instructional tasks that include external objects and actions that mirror the hypothesized mathematical activity of children as closely as possible. These tasks are sequenced, with each corresponding to a level of the developmental progressions. Specific learning trajectories are the main bridge that connects the “grand theory” of hierarchic interactionalism to particular theories and educational practice.

3

Research and Development Approaches

Our work with learning trajectories began with the assumption that the isolation of curriculum development and educational research is deleterious (Clements & Battista, 2000; Clements, Battista, Sarama, & Swaminathan, 1997; Lagemann, 1997) and is, indeed, a main reason curriculum development in the U.S. does not improve (Battista & Clements, 2000; Clements, 2002). For example, although much is usually learned during curriculum development, it is usually not explicated or published (Gravemeijer, 1994b). The educational community needs such knowledge. To address this need, and to build scientific knowledge generally, our 30-year work with learning trajectories (LTs) began with the creation and testing of LTs but has come to span the full range of research and development (R&D) in education. That is, our study and use of LTs have broadened our perspective to now contending that LTs following our multifaceted definition have ramifications for all aspects of curriculum (e.g., ideal, expected, available, adopted, implemented, achieved, or tested, Burkhardt, Fraser, & Ridgway, 1989; Clements, 2007). This requires a wider range of methods (that we will address briefly in subsequent sections). In this chapter, we focus mainly on the R&D methods we use for the creation, refinement, and validation of LTs, with examples of two LTs for geometric measurement and shape composition. Our initial viewpoint considered a learning trajectory as a device whose purpose is to support the development of a curriculum or a curriculum component. As described in our definition, each learning trajectory has three parts: (a) a goal, (b) a developmental progression, and (c) instructional activities. To

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attain a certain mathematical competence in a topic or domain (the goal), students learn each successive level (the developmental progression), aided by tasks (instructional activities) and pedagogical moves designed to help students build the mental actions-on-objects that enable thinking at each higher level (Clements & Sarama, 2004).

4

LTs’ Goals

When we began, we accepted that the first component of an LT, the goal, should be determined by standards (ideal or expected curriculum) created by dialectical process among many legitimate stakeholders (e.g., NCTM, 2000, 2006; NGA/CCSSO, 2010). When more detail was needed, we used reviews of the research literature to identify objectives that contribute to the mathematical development of students, build from the students’ past and present experiences (Dewey, 1902/1976), and are generative in students’ development of future understanding. We now also believe that LTs should play a more active role in determining, as well as incorporating, goals, an issue to which we return in the final section (see “Standards – The ideal and expected curriculum”). Often little addressed in discussions of LTs, the goal of a learning trajectory is more than simply an educational objective – it includes consideration of the mathematical concepts, skills, processes and connections that constitute foundational knowledge of early mathematics (National Research Council, 2009). Our learning trajectories base goals on both the expertise of mathematicians and research on students’ thinking about and learning of mathematics (Clements, Sarama, & DiBiase, 2004; Fuson, 2004; National Mathematics Advisory Panel, 2008; Sarama & Clements, 2009). This results in goals that are organized into the “big” ideas of mathematics: overarching clusters, concepts and skills that are mathematically central and coherent, consistent with students’ (often intuitive) thinking, and generative of future learning (Clements, Sarama, & DiBiase, 2004; NCTM, 2006).

5

LTs’ Developmental Progression

We initially examined others’ attempts to build sequences similar to our LTs. Some have based such work on historical development of mathematics and observations of children’s informal solution strategies (Gravemeijer, 1994a, 1999), anticipatory thought experiments (that often focus on instructional

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sequences, Gravemeijer, 1999; Simon, 1995), or emergent mathematical practices of student groups (Cobb & McClain, 2002). Although we appreciate the guidance of such approaches, we ground our LTs in cognitive science. Creating the developmental progression stands at the heart of our process of building LTs (methods will be summarized in a subsequent section). We begin by learning from others (e.g., Cobb & McClain, 2002), reviewing research to determine if there is a near-universal developmental progression (at least for a given age range of students in a particular culture) identified in theoretically – and empirically – grounded models of children’s thinking, learning, and development (Carpenter & Moser, 1984; Griffin & Case, 1997). That is, we build a cognitive model of students’ learning that is sufficiently explicit to describe the processes involved in the construction of the mathematical goal across several qualitatively distinct structural levels of increasing sophistication, complexity, abstraction, power, and generality. This may be sufficient to establish a cognitive model of children’s thinking, but if details are lacking, we use grounded theory methods and clinical interviews (Ginsburg, 1997) to examine students’ knowledge and ways of thinking in the content domain, including conceptions, strategies, intuitive ideas, and informal strategies used to solve problems (Clements, 2007). We set up a situation or task to elicit pertinent concepts and processes. Once a (static) model has been partially developed, it is tested and extended with constructivist teaching experiments, which present limited tasks and adult interaction to individual children with the goal of building models of children’s thinking and learning (Steffe, Thompson, & von Glasersfeld, 2000). Once several iterations of such work reveal no substantive variations, it is accepted as a working model. This model is subjected to validation and/or refinement through hypodeductive applications of qualitative methods such as teaching experiments, and quantitative methods such as correlational analyses between level scores (Clements, Wilson, & Sarama, 2004) and Rasch modelling (Barrett, Clements, & Sarama, 2017; Szilagyi, Sarama, & Clements, 2013).

6

LTs’ Instruction

Next, the third component is built. The cognitive model includes details of the mental constructions (actions-on-objects) and patterns of thinking that constitute children’s thinking at each level of a developmental progression. Then, it captures how mental constructions are incorporated in each subsequent level, and these details are used to align instructional tasks to a specific level with the intention to promote movement to the succeeding level.

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The learning trajectories construct differs from instructional design based on task analysis as it is based not on a reduction of the skills of experts but on models of children’s learning. LTs also expect unique constructions and input from children, involve self-reflexive constructivism, and continuous, detailed, and simultaneous analyses of goals, children’s thinking and learning, and instructional tasks and strategies. Such explication allows the researcher to test the theory by testing the curriculum (Clements & Battista, 2000), usually with design experiments (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003). To begin, sets of activities are taken from successful interventions in the literature and in expert practice, or created (or tasks adapted from previous work) by the developers. In both cases, the key is ensuring that the activities are theoretically valid in engendering or activating the actions-on-objects that mirror the hypothesized mathematical activity of students in the target level (that is, level n + 1 for students at level n). Design experiments and microgenetic studies (Siegler & Crowley, 1991) are employed, using a mix of model (or hypothesis) testing and model generation such as a microethnographic approach (Spradley, 1979) to understand the meaning that students give to the objects and actions embodied in these activities, and to document signs of learning. As in all phases, equity must be considered, such as in selecting populations with whom to work (Confrey, 2000).

7

Two Examples of LTs: Measurement and Shape Composition

Regarding our examples, the importance and goals for the first, geometric measurement, were well established. However, there was less extant justification for the domain of composing geometric forms. We determined this domain to be significant in that the concepts and actions of creating and then iterating units and higher-order units in the context of constructing patterns, measuring, and computing are established bases for mathematical understanding and analysis (Clements, Battista, Sarama, Swaminathan, & McMillen, 1997; Park, Chae, & Boyd, 2008; Reynolds & Wheatley, 1996; Steffe & Cobb, 1988). Specific goals were dependent not just on the mathematics content, but on research on the developmental progression. The genesis of the shape composition’s developmental progression was in observations made of children using Shapes software (Sarama & Clements, 1998) to compose shapes. Sarama observed that several children followed a similar progression in choosing and combining shapes to make another shape (Sarama, Clements, & Vukelic, 1996). Initially, children learned to fill a hexagonal frame with other pattern blocks, but only by trial-and-error. Later, they created

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combinations systematically. Sarama re-viewed the behaviors all kindergarten children exhibited and found that children moved from placing shapes separately to considering shapes in combination; from manipulation- and perception-bound strategies to the formation of mental images (e.g., decomposing shapes imagistically); from trial and error to intentional and deliberate action and eventually to the prediction of succeeding placements of shapes; and from consideration of visual “wholes” to consideration of side length, and, eventually, angles. We combined these observations with related observations from other researchers (Mansfield & Scott, 1990; Sales, 1994) and some elements of psychological research (Vurpillot, 1976) to refine this developmental progression (Clements, Sarama, & Wilson, 2001; Clements, Wilson, & Sarama, 2004). We subjected this model to validation and refinement through hypo-deductive applications of quantitative methods such as correlational analyses between level scores (Clements, Wilson, & Sarama, 2004). At that point, we confirmed a developmental progression in which children move levels of thinking – from lack of competence in composing geometric shapes, they gain abilities to combine shapes – initially through trial and error and gradually by attributes – into pictures, and finally synthesize combinations of shapes into new shapes, that is, composite shapes. Turning to the third component, we designed instructional tasks in which children worked with shapes and composite shapes as objects. We wanted them to act on these objects – to create, duplicate, position (with geometric motions), combine, and break apart both individual shapes (units) and composite shapes (units). We designed physical puzzles and software environments that required and supported use of those actions-on-objects, consistent with the Vygotskian theory that mediation by tools and signs is critical in the development of human cognition (Steffe & Tzur, 1994). Simultaneously, we documented what elements of the teaching and learning environment, such as specific scaffolding, contributed to student learning (Walker, 1992) – planned a priori or occurring spontaneously. Thus, designs are not determined fully by reasoning. Intuition and the art of teaching (Confrey, 1996; Hiebert, 1999; James, 1892/1958) play critical roles. We refined the tasks and connected scaffolding to elicit each of these hypothesized levels. For example, children may need a “hint” by drawing one or two internal lines into a large ambiguous space in a puzzle. We conducted teaching experiments using these tasks, validating that the actions-on-objects posited to underlie solutions could be observed and were elicited by the tasks at each level. Work with the measurement LT differed in several ways. First, the larger literature in that domain allowed us to use a research synthesis to form the

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initial LT (Sarama & Clements, 2002). We used both these comprehensive research reviews and original research in measurement (Barrett, Clements, & Sarama, 2017; Clements & Barrett, 1996) to describe a detailed developmental progression. Second, the presence of assessment tasks, empirical results and theory allowed us to validate the first LTs with Item Response Theory, creating an equal-interval scale of scores representing both the difficulty of items and the ability of the persons assessed. To measure measurement competence, we sequenced the items, strictly maintaining the order within each measurement domain (length, area, volume), but intermingling items across domains according to the available developmental evidence, including age specifications from the literature and difficulty indices from our pilot testing. Thus, we posited that items were organized according to increasing order of difficulty across domains, but our theoretical claims that this sequencing represented increasingly sophisticated levels of mathematical thinking were made only for items within a given domain. We submitted the results of administering this revised instrument to the Rasch model, validating the developmental progressions for length, area, and volume in multiple studies (Barrett, Clements, & Sarama, 2017; Szilagyi, Sarama, & Clements, 2013). We used confidence intervals to detect segmentation and developmental discontinuity, with non-overlapping intervals interpreted to suggest the possible distinctness of contiguous levels of development in geometric measurement. We similarly used and validated instructional sequences, many again from the extant literature.

8

Different Uses of LTs

In our decades-long work with learning trajectories, we have expanded their roles from serving mainly as the core of the curriculum development project to having implications for all aspects of curriculum. That is, LTs have implications for the ideal, expected, available, adopted, implemented, achieved, and tested curriculum (Clements, 2007) and, at the risk of overstating, the entire educational enterprise. In this section, we describe how, for us, LTs have grown in importance and influence. 8.1 Curriculum Development and Evaluation Our examples in previous sections have briefly described how we created, tested, and refined LTs. The examples are just two of the many we developed for the nsf-funded Building Blocks project and curriculum (Clements & Sarama, 2007/2013), designed to comprehensively address standards for early

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mathematics education for all children. Evaluations have shown that Building Blocks can be effective, with moderate to large effect sizes (Clements & Sarama, 2007), even when compared to another research-based curriculum not built upon LTs (Clements & Sarama, 2008). One issue we have not yet discussed is that LTs are connected. We carefully integrated the LTs across a year of instruction to optimize support among them (as a simple example, developing subitizing to the right level supporting various levels of counting, such as cardinality and vice versa). Indeed, our first version of the LTs were represented on a large matrix, so one could easily view the interrelationships among them. Unfortunately, both the publisher of our curriculum and the publisher of our books said this was too complex and they separated them. We believe, especially years later, that this was a mistake. Arguably the tasks for teachers in mastering LTs is difficult, and one at a time is probably the best way to start (cf. Association of Mathematics Teacher Educators, 2017); however, later on, they should be related. We do what we can to connect them in curricula, as mentioned, and also in presentations and other venues; yet, this remains an important issue for future research and development. Also important are the roles LTs may play in a comprehensive and valid scientific curriculum development program. We believe any such research and development efforts must address two basic issues – effect and conditions – in three domains, practice, policy, and theory, as described in Table 2.1. To achieve these goals satisfactorily and scientifically, developers must draw from existing research so that what is already known can be applied to the anticipated curriculum; structure and revise curricular components in accordance with models of children’s learning such as research-based learning trajectories – the focus of this chapter; and conduct formative and summative evaluations in a series of progressively expanding social contexts (see Table 2.2). With learning trajectories at the core of the curriculum development processes – affecting standards, the path of learning, instruction, and assessments – it is clear that they both play a major role throughout the entire research and development process and bring a coherence to all aspects of that process that was often heretofore lacking. 8.2 Scale-Up LT-based approaches to curriculum or professional development can be valuable, but their contribution to scale-up must also be evaluated – the final phase in Table 2.2. Space constraints prevent us from describing our work in this area, but we note that LTs were again at the core of our scale-up model. Called triad, for Technology-enhanced, Research-based, Instruction, Assessment, and professional Development, the model is based on research and

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table 2.1  Goals of curriculum research (adapted from Clements, 2007)

Efffects

Conditions

Practice

Policy

Theory

a. Is the curriculum efffective in helping children achieve specifijic learning goals? Are the intended and unintended consequences positive for children? (6–10)a b. Is there credible evidence of efffijicacy compared to alternative approaches? (all) i. When and where? – Under what conditions is the curriculum efffective? Do fijinding’s generalize? (8, 10)

c. Are the curriculum goals important? (1, 5, 10) d. What is the efffect size for students? (9, 10) e. What efffects does it have on teachers? (10)

f. Why is the curriculum efffective? (all) g. What were the theoretical bases? (1, 2, 3) h. What cognitive changes occurred and what processes were responsible? (4, 6, 7)

j. What are the support requirements (7) for various contexts? (8–10)

k. Why do certain conditions alter efffectiveness? (6–10) l. Do specifijic strategies produce new results? (6–10) a Numbers in parentheses refer to the specifijic research and development phases in Table 2.2.

enhanced by the use of trajectories and technology. triad places learning trajectories at the core of the teacher/child/curriculum triad to ensure that curriculum, materials, instructional strategies, and assessments are aligned. LTs are the focus of all professional development. Moreover, LTs serve as the connecting tissue that helps one teacher work with others (especially at different grade levels), helps administrators communicate with teachers, and helps educators collaborate with parents and other stakeholders. LTs help promote equity by revealing the hidden potential of underserved children (Clements, Sarama, Spitler, Lange, & Wolfe, 2011; Clements, Sarama, Wolfe, & Spitler, 2013; Schenke, Watts, Nguyen, Sarama, & Clements, 2017). When implemented with fidelity, triad has shown moderate to strong effects including transfer to other domains (Clements, Sarama, Layzer, Unlu, & Fesler, 2018; Clements, Sarama,

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Learning Trajectories in Early Mathematics Education table 2.2  Categories and phases of the curriculum research framework (crf) (adapted from Clements, 2007)

Categories

Questions Asked

Phases and Methods

A Priori Foundations. Literature is reviewed, and wisdom of expert practice sought.

What is already known that can be applied to the anticipated curriculum? Goalsa Phase bcfg 1 bfg 2 bfg 3

Learning Trajectories. Activities are structured in accordance with empiricallybased learning trajectories. Evaluation. Empirical evidence is collected to evaluate the curriculum, realized in some form. The goal is to evaluate the appeal, usability, and efffectiveness of an instantiation of the curriculum and to expand it to include aspects of support for all teachers serving diverse students.

How might the curriculum be constructed to be consistent with models of students’ thinking and learning? Goals Phase bfh 4

Phase 1: Review and content analysis of subject matter. Phase 2: Review of general issues in psychology and education. Phase 3: pedagogy, including the efffectiveness of certain types of activities (phase 3). Phase 4: Grounded theory methods, clinical interviews, teaching experiments, and design experiments tested and revised dynamically the nature and content of the learning trajectories.

How can the curriculum be disseminated? Goals Phase bcf 5 Is the curriculum usable by, and efffective with, various student groups and teachers? How can it be improved in these areas or adapted to serve diverse situations and needs? Goals Phase abfhkl 6 abfhjkl 7 abfijkl 8

Phase 5: Market research, gathering information about mandated educational objectives and surveys of consumers. Phase 6: Interpretive work using a mix of model testing and model generation strategies, including design experiments, microgenetic, microethnographic, and phenomenological approaches to study the meaning that students and teachers give to individuals or small groups. Phase 7: Classroom-based teaching experiments and ethnographic participant observation similarly study whole classrooms, asking the same questions. (cont.)

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table 2.2  Categories and phases of the curriculum research framework (crf) (adapted from Clements, 2007) (cont.)

Categories

Questions Asked

Phases and Methods

Phase 8: Same methods plus content analyses to answer the same questions of a diverse group of new teachers. What is the efffectiveness Phase 9: Summative (e.g., in afffecting teaching research using randomized practices and ultimately fijield trials (or carefully student learning) of the planned quasi-experimental curriculum, now in its designs), incorporating complete form, as it is observational measures and implemented in realistic surveys to study efffijicacy and contexts? to generate political and Goals    Phase public support. In addition, abdfjkl  9 qualitative approaches abcde  10 continue to be useful for fijkl dealing with the complexity and indeterminateness of educational activity. Phase 10: Similar cluster randomized trials but examining the fijidelity or enactment, and sustainability, of the curriculum when implemented on a large scale, and the critical contextual and implementation variables that influence its efffectiveness. a Goals refer to the specifijic questions in Table 2.1, answering these is the second main purpose of the crf, the fijirst being to produce an efffective curriculum, of course.

Layzer, Unlu, Wolfe, et al., 2018; Clements et al., 2013; Sarama, Clements, Starkey, Klein, & Wakeley, 2008). Further, we must look at longitudinal effects, which are particularly important for large-scale interventions, because a full concept of scale requires not

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only consequential implementation, but also endurance over long periods of time and a transfer of responsibility from any external organization to the internal resources of a school district (Coburn, 2003; Dearing & Meyer, 2006). Such lasting effectiveness can be categorized as persistence (continuation of the effects of an intervention on individual children’s trajectories) or sustainability (continued use of program components in the achievement of desirable student outcomes and professional practice over time), with a focus on the maintenance of core beliefs and values, and the use of these core beliefs to guide adaptations (Century, Rudnick, & Freeman, 2012; Scheirer & Dearing, 2011; Timperley, 2011). This is an especially important issue, given the relationship between such fidelity and higher student outcomes (O’Donnell, 2008), the need for evaluations of sustainability past the implementation of the intervention, especially by developers or researchers (Baker, 2007; Glennan Jr., Bodilly, Galegher, & Kerr, 2004; McDonald, Keesler, Kauffman, & Schneider, 2006), and the “shallow roots” of many reforms (Cuban & Usdan, 2003). triad’s effects did not persist at the same level – they decreased over time. However, effects were better maintained in triad’s “follow through” condition that gave some support to Kindergarten and first grade teachers (Clements et al., 2013; Sarama, Clements, Wolfe, & Spitler, 2012). Some groups, especially those historically underrepresented in mathematics, maintained benefits as far out as fifth grade (Clements, Sarama, Layzer, Unlu, Wolfe et al., 2018). Results on sustainability were even more positive. We expected teachers to decrease in the fidelity in which they taught with learning trajectories after project support was discontinued. However, after two years, they increased the quality of their teaching (Clements, Sarama, Wolfe, & Spitler, 2015). Recently, we found even more positive effects six years after support from the project had ceased (Sarama, Clements, Wolfe, & Spitler, 2016). The largest predictor of higher fidelity years out was child gain – teachers sustain and increase the quality of teaching when they observe their children learning. Over 90% of the teachers surveyed explicitly indicated that learning about learning trajectories is what allowed them to see and to understand what activities and teaching strategies supported that learning. 8.3 LT2 – The Learning and Teaching with Learning Trajectories Tool One of the main tools for professional development in triad was the Building Blocks Learning Trajectories (bblt) web application. This was sufficiently successful that we have been funded by the Heising-Simons and Gates Foundations to build a version that is on modern platforms and addresses far more ages – the Learning and Teaching with Learning Trajectories tool, also known as LT2 (see LearningTrajectories.org).

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8.4 Standards – The Ideal and Expected Curriculum Underappreciated is the role that LTs, especially their developmental progressions, might play in determining goals. For example, research on LTs influenced U.S. standards from NCTM (2000, 2006) and more so the Common Core State Standards – Mathematics (nga/ccsso); see the progressions at http://commoncoretools.me. Few understand that when we developed the ccss-m, we first developed progressions, and from those progressions came the standards. This altered our perception of LTs as we realized that the goal – the first component of an LT – should not be determined solely via a top-down determination of a standard (followed by creating the remainder of the LT to achieve that goal). Rather, these processes should be mutually interactive. That is, LTs should influence not just assessments, curricula, and pedagogy, but educational goals as well. Mathematical activity must be understood from the perspective of the students, which may be distinctly different from the thinking of adults. Thus, drafting the initial goals requires analyzing adults’ conceptions of the domain and a model of the concepts and strategies of students as they engage in domain tasks. We should state that we believe that psychologicallybased developmental progressions become increasingly less important as one moves up to the grades where the mathematical content becomes increasingly influential; however, mathematical progressions are important from the beginning and the influence of the psycho-social developmental progressions never disappears. 8.5 Assessment LTs have contributed to assessments of mathematics (Clarke et al., 2001; Horne & Rowley, 2001; Wilson, 2009). Our Research-based Early Mathematics Assessment (rema) is a suite of diagnostic assessments measuring children’s mathematical knowledge and skills (from 3 to 8 years of age) along researchbased developmental progressions for all important topics in early mathematics. These developmental paths are closely related to national and state standards. The measures use an individual interview format with explicit protocol, coding, and scoring procedures. The assessment documents students’ solution strategies and error types. Assessors use an app to enter data and generate reports. These reports include LT levels, making them particularly useful for teaching, as well as research. One limitation of the rema is that it is individually administered, which takes a considerable amount of time. It also does not provide very fine-grained diagnoses of what particular concepts and skills children do or do not possess. We are working on the Comprehensive Research-based Early Math Ability Test

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(cremat) that will use innovative theoretical (Q-Matrix theory and Rule Space procedures), statistical (posets), and computer technology to give teachers more useful and detailed information about children’s knowledge of mathematics in less time than existing assessments. cremat will use Computer Adaptive Testing (cat) to guide the dynamic selection of items and score and analyze the results, providing a comprehensive report including the overall score, level of achievement within each developmental progression (and, if requested, detailed report on achievement on each item in that topic), and a cognitive profile of attributes (Tatsuoka et al., 2016).

9

LT Studies

What is the precise contribution of LTs? Their use in early mathematics instruction has received increasing attention from policy makers, educators, curriculum developers, and researchers (Baroody, Clements, & Sarama, in press; Clements & Sarama, 2014a, 2011; Maloney, Confrey, & Nguyen, 2014; Sarama & Clements, 2009) and are generally deemed as a useful tool for guiding standards, instructional planning, and assessment (Frye et al., 2013; National Research Council, 2009). Despite these recommendations, little research has directly tested the specific contributions of LTs to teaching compared to similar instruction provided without LTs (Frye et al., 2013). The primary goal in the present study was to compare the learning of preschool children who received instruction on shape composition based on an empirically-validated LT to those who received an equal amount of instruction not following the developmental order of an LT. Our ies-funded project “Evaluating the Efficacy of Learning Trajectories in Early Mathematics” includes a series of randomized clinical trials testing different aspects of LTs. These experiments will determine whether LTs are better than other approaches in supporting young children’s learning. For example, in the first of these, we randomly assigned preschoolers who fell at least two levels below the target level of shape composition to receive instruction based on an empirically-validated LT and another, skip-levels, group to an equal amount of instruction focused only on the target level. At a posttest, children in the LT group exhibited significantly greater learning than children in the skip-levels condition. The skip-levels group showed more frustration. Of course, we do not know if this result will generalize to other defining attributes of LTs (such as the sequence of activities) or to other topics or ages of children. We will continue to conduct our series of studies to address these issues.

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Final Words

Research-based learning trajectories can contribute to the development, refinement, and evaluation of standards, curriculum, and teaching practices. In doing so, they can help maintain the coherence and connections of each of these. In curriculum development, for example, using learning trajectories ensures a common theoretical and empirical base for the goals, assessments, instructional activities, and teaching practices. The integration of the research corpi makes the LT construct particular useful. Similarly, specificity of goals helps those wishing to write standards, as we have with the Common Core State Standards – Mathematics and the developmental progressions are invaluable for those creating assessments. Coherence is even better supported, because all these efforts–as well as the domains of research and policy – can use and contribute to the same learning trajectories. This leads to a final issue regarding research. We discussed the benefits of synergism between the three components of LTs for education. Less palpable is that this integration can produce novel research results, even within the local theoretical fields of psychology and pedagogy. The enactment of an effective, complete, learning trajectory may actually alter developmental progressions or expectations previously established by psychological studies, because they open up new paths for learning and development. Such an enactment based on the fine-grain cognitive analysis of the developmental progression and the similarly detailed analysis of the instructional tasks provides a more elaborated theoretical base for curriculum and instruction than is often available and may also open instructional approaches or avenues not previously considered. Finally, we must remember that LTs are hypothetical and should be reconceptualized and re-created by teachers, so that their actual instantiation is based on more intimate knowledge of the particular students involved – their culture, language, knowledge, learning preferences, and engagement in certain task types or contexts. The teacher must construct models of children’s mathematics as they interact with children around the instructional tasks and thus alter their own knowledge of children and future instructional strategies and paths. Thus, the learning trajectories as used are always emergent to some degree. This is less a caveat regarding use of LTs and more an argument for their use in professional development. LTs empower teachers with a well-formed and specific set of expectations about children’s ways of learning and a likely pace along a path that includes major mathematical ideas and objectives. Much work has been done on learning trajectories, but much more remains to be done. Not all topics nor all ages of studies have been equally addressed

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(or addressed at all). Issues of grain size of the developmental progressions and specificity of the instructional tasks need to be creatively modified and evaluated. Our scale up studies need replication and extension. Much needs be done. And much will be learned.

Acknowledgements This research was supported by the Institute of Education Sciences, U.S. Department of Education through Grants R305A120813, R305K05157, and R305A110188. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education. Although the research is concerned with theoretical issues, not particular curricula, a small component of the intervention used in this research have been published by the authors and their collaborators on the project, who thus could have a vested interest in the results. Researchers from an independent institution oversaw the research design, data collection, and analysis and confirmed findings and procedures. The authors wish to express appreciation to the school districts, teachers, and students who participated in this research.

References Association of Mathematics Teacher Educators. (2017). AMTE standards for mathematics teacher preparation. Raleigh, NC: AMTE. Baker, E. L. (2007). Principles for scaling up: Choosing, measuring effects, and promoting the widespread use of educational innovation. In B. Schneider & S.-K. McDonald (Eds.), Scale up in education: Ideas in principle (Vol. 1, pp. 37–54). Lanham, MD: Rowman & Littlefield. Baroody, A. J. (1987). Children’s mathematical thinking. New York, NY: Teachers College. Baroody, A. J., Clements, D. H., & Sarama, J. (in press). Teaching and learning mathematics in early childhood programs. In C. Brown, M. McMullen, & F. N. K. (Eds.), Handbook of early childhood care and education. Hoboken, NJ: Wiley Blackwell Publishing. Barrett, J. E., Clements, D. H., & Sarama, J. (2017). Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume. Journal for Research in Mathematics Education (Monograph series, Vol. 16). Reston, VA: National Council of Teachers of Mathematics. Battista, M. T., & Clements, D. H. (2000). Mathematics curriculum development as a scientific endeavor. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 737–760). Mahwah, NJ: Erlbaum.

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Clements, D. H., Battista, M. T., Sarama, J., Swaminathan, S., & McMillen, S. (1997). Students’ development of length measurement concepts in a logo-based unit on geometric paths. Journal for Research in Mathematics Education, 28(1), 70–95. Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6, 81–89. doi:10.1207/s15327833mtl0602_1 Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the building blocks project. Journal for Research in Mathematics Education, 38(2), 136–163. Clements, D. H., & Sarama, J. (2007/2013). Building blocks, Volumes 1 and 2. Columbus: McGraw-Hill Education. Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45(2), 443–494. doi:10.3102/0002831207312908 Clements, D. H., & Sarama, J. (2014a). Learning and teaching early math: The learning trajectories approach (2nd ed.). New York, NY: Routledge. Clements, D. H., & Sarama, J. (2014b). Learning trajectories: Foundations for effective, research-based education. In A. P. Maloney, J. Confrey, & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 1–30). New York, NY: Information Age Publishing. Clements, D. H., Sarama, J., & DiBiase, A.-M. (2004). Engaging young children in mathematics: Standards for early childhood mathematics education. Mahwah, NJ: Erlbaum. Clements, D. H., Sarama, J., Layzer, C., Unlu, F., & Fesler, L. (2018). Effects on executive function and mathematics learning of an early mathematics curriculum synthesized with scaffolded play designed to promote self-regulation versus the mathematics curriculum alone. Submitted for publication. Clements, D. H., Sarama, J., Layzer, C., Unlu, F., Wolfe, C. B., Fesler, L., … Spitler, M. E. (2018). Effects of TRIAD on mathematics achievement: Long-term impacts. Submitted for publication. Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127–166. doi:10.5951/jresematheduc.42.2.0127 Clements, D. H., Sarama, J., & Wilson, D. C. (2001). Composition of geometric figures. In M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 273–280). Utrecht: Freudenthal Institute. Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2013). Longitudinal evaluation of a scale-up model for teaching mathematics with trajectories and technologies: Persistence of effects in the third year. American Educational Research Journal, 50(4), 812–850. doi:10.3102/0002831212469270

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Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2015). Sustainability of a scaleup intervention in early mathematics: Longitudinal evaluation of implementation fidelity. Early Education and Development, 26(3), 427–449. doi:10.1080/10409289.20 15.968242 Clements, D. H., Wilson, D. C., & Sarama, J. (2004). Young children’s composition of geometric figures: A learning trajectory. Mathematical Thinking and Learning, 6, 163–184. doi:10.1207/s15327833mtl0602_1 Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13. Cobb, P., & McClain, K. (2002). Supporting students’ learning of significant mathematical ideas. In G. Wells & G. Claxton (Eds.), Learning for life in the 21st century: Sociocultural perspectives on the future of education (pp. 154–166). Oxford: Blackwell. Coburn, C. E. (2003). Rethinking scale: Moving beyond numbers to deep and lasting change. Educational Researcher, 32(6), 3–12. Confrey, J. (1996). The role of new technologies in designing mathematics education. In C. Fisher, D. C. Dwyer, & K. Yocam (Eds.), Education and technology, reflections on computing in the classroom (pp. 129–149). San Francisco, CA: Apple Press. Confrey, J. (2000). Improving research and systemic reform toward equity and quality. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 87–106). Mahwah, NJ: Erlbaum. Cuban, L., & Usdan, M. (Eds.). (2003). Powerful reforms with shallow roots: Improving America’s urban schools. New York, NY: Teachers College. Daro, P., Mosher, F. A., Corcoran, T. B., Barrett, J., Battista, M. T., Clements, D. H., … Sarama, J. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. New York, NY: Consortium for Policy Research in Education (CPRE). Dearing, J. W., & Meyer, G. (2006). Revisiting diffusion theory. In A. Singhal & J. W. Dearing (Eds.), Communication of innovations: A journey with Ev Rogers (pp. 29–60). New Delhi: Sage Publications. Dewey, J. (1902/1976). The child and the curriculum. In J. A. Boydston (Ed.), John Dewey: The middle works, 1899–1924. Volume 2: 1902–1903 (pp. 273–291). Carbondale, IL: Southern Illinois University Press. Frye, D., Baroody, A. J., Burchinal, M. R., Carver, S., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. Fuson, K. C. (2004). Pre-K to grade 2 goals and standards: Achieving 21st century mastery for all. In D. H. Clements, J. Sarama, & A.-M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 105–148). Mahwah, NJ: Erlbaum.

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NCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. NCTM. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: National Council of Teachers of Mathematics. NGA/CCSSO. (2010). Common core state standards. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers. O’Donnell, C. L. (2008). Defining, conceptualizing, and measuring fidelity of implementation and its relationship to outcomes in K-12 curriculum intervention research. Review of Educational Research, 78, 33–84. Park, B., Chae, J.-L., & Boyd, B. F. (2008). Young children’s block play and mathematical learning. Journal of Research in Childhood Education, 23, 157–162. Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale, NJ: Erlbaum. Reynolds, A., & Wheatley, G. H. (1996). Elementary students’ construction and coordination of units in an area setting. Journal for Research in Mathematics Education, 27(5), 564–581. Sales, C. (1994). A constructivist instructional project on developing geometric problem solving abilities using pattern blocks and tangrams with young children (Unpublished master’s thesis). University of Northern Iowa, Cedar Falls, IA. Sarama, J., & Clements, D. H. (1998). Shapes [Computer software]. Palo Alto, CA: Dale Seymour. Sarama, J., & Clements, D. H. (2002). Building blocks for young children’s mathematical development. Journal of Educational Computing Research, 27(1–2), 93–110. doi:10.2190/F85E-QQXB-UAX4-BMBJ Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge. Sarama, J., Clements, D. H., Starkey, P., Klein, A., & Wakeley, A. (2008). Scaling up the implementation of a pre-kindergarten mathematics curriculum: Teaching for understanding with trajectories and technologies. Journal of Research on Educational Effectiveness, 1(1), 89–119. doi:10.1080/19345740801941332 Sarama, J., Clements, D. H., & Vukelic, E. B. (1996). The role of a computer manipulative in fostering specific psychological/mathematical processes. In E. Jakubowski, D. Watkins, & H. Biske (Eds.), Proceedings of the 18th annual meeting of the North America chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 567–572). Columbus: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Sarama, J., Clements, D. H., Wolfe, C. B., & Spitler, M. E. (2012). Longitudinal evaluation of a scale-up model for teaching mathematics with trajectories and technologies. Journal of Research on Educational Effectiveness, 5(2), 105–135. doi:10.108011934574 7.2011,627980

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Sarama, J., Clements, D. H., Wolfe, C. B., & Spitler, M. E. (2016). Professional development in early mathematics: Effects of an intervention based on learning trajectories on teachers’ practices. Nordic Studies in Mathematics Education, 21(4), 29–55. Scheirer, M. A., & Dearing, J. W. (2011). An agenda for research on the sustainability of public health programs. American Journal of Public Health, 101(11), 2059–2067. Schenke, K., Watts, T. W., Nguyen, T., Sarama, J., & Clements, D. H. (2017). Differential effects of the classroom on African American and non-African American’s mathematics achievement. Journal of Educational Psychology, 109(6), 794–811. Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for studying cognitive development. American Psychologist, 46, 606–620. doi:10.1037//0003066X.46.6.606 Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205 Spradley, J. P. (1979). The ethnographic interview. New York, NY: Holt, Rhinehart & Winston. Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New York, NY: Springer-Verlag. Steffe, L. P., Thompson, P. W., & von Glasersfeld, E. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Erlbaum. Steffe, L. P., & Tzur, R. (1994). Interaction and children’s mathematics. Journal of Research in Childhood Education, 8(2), 99–116. Szilagyi, J., Sarama, J., & Clements, D. H. (2013). Young children’s understandings of length measurement: Evaluating a learning trajectory. Journal for Research in Mathematics Education, 44, 581–620. Tatsuoka, C., Clements, D. H., Sarama, J., Izsák, A., Orrill, C. H., de la Torre, J., … Tatsuoka, K. K. (2016). Developing workable attributes for psychometric models based on the Q-Matrix. In A. Izsák, J. T. Remillard, & J. Templin (Eds.), Psychometric methods in mathematics education: Opportunities, challenges, and interdisciplinary collaborations (Monograph #15) (pp. 73–96). Reston, VA: National Council of Teachers of Mathematics. Thorndike, E. L. (1922). The psychology of arithmetic. New York, NY: Macmillan. Timperley, H. (2011). Realizing the power of professional learning. London: Open University Press. Vurpillot, E. (1976). The visual world of the child. New York, NY: International Universities Press. Walker, D. F. (1992). Methodological issues in curriculum research. In P. W. Jackson (Ed.), Handbook of research on curriculum (pp. 98–118). New York, NY: Macmillan. Wilson, M. (2009). Measuring progressions: Assessment structures underlying a learning progression. Journal of Research in Science Teaching, 46, 716–730.

CHAPTER 3

Hypothetical Learning Trajectory (HLT): A Lens on Conceptual Transition between Mathematical “Markers” Ron Tzur

Abstract In this chapter I introduce a distinction between two research types, marker studies and transition studies, that I find useful in developing and categorizing hypothetical learning trajectories (hlts). I stress the latter type as one closely related with the third element in Simon’s (1995) definition of hlt. To further depict the linkage between hlt and transition studies, I discuss the starting points and development of hlts, including a model of learning as cognitive change – reflection on activity-effect relationship. I follow those sections with a description of four ways in which I find hlts that specify conceptual transitions to be useful. Keywords hypothetical learning trajectory (hlt) – conceptual transition – constructivism

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In his seminal paper, Simon (1995) introduced and articulated the construct of hypothetical learning trajectory (hlt; hereafter, I use hlt for the singular form and hlts for the plural form of this construct). The various chapters in this volume reflect a growing body of research that this construct has spawned, including ways in which the construct has been grounded, adapted, and used. In this chapter, however, I purposely zoom in on a particular distinction – between marker studies and transition studies – that I find useful for developing and categorizing hlts. To this end, I organize the chapter by first presenting the definition and theoretical grounding that underlie my hlt research. Next, I define and contrast those two study types. Then, narrowing down on transition studies, I describe the development of hlts and how they may be used. This © koninklijke brill nv, leiden, 2019 | doi:10.1163/9789004396449_004

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chapter organization serves the twofold purpose of further articulating how hlt research can contribute to (a) explaining learners’ progress from not knowing to knowing specific mathematical ideas and (b) designing instructional activities/ tasks to promote such progress (Simon, Kara, Placa, & Avitzur, 2018). To illustrate the pertinent and rather abstract constructs in this chapter, I present here an example of two strategies children bring forth to solve addition problems, namely, counting-all and counting-on (Baroody, 1995; Fuson, 1992). Later in this chapter I use these conceptually distinct strategies, which were also identified in research on children’s pre-numerical and numerical reasoning (Steffe & Cobb, 1988), to expound on the distinction and relationship between marker and transition studies. To distinguish the two strategies, let us consider a child who was asked to place two collections of tangible items on a table (say, 8 marbles and 5 marbles). The child is first asked to count and place 8 marbles in one location and then, separately, count and place 5 marbles in another location. Then, when asked how many marbles are in the first collection the child answers “Eight” without recounting – that is, she reestablishes the numerosity of that collection. Once doing the same for the other collection (“Five”), the child is asked: “How many marbles are there in all?” The child may solve this problem by starting at 1, and then counting (pointing to) each and every marble in both collections (e.g., 1-2-3-4-5-6-7-8; 9-10-11-12-13). This strategy, termed countingall, involves operating on units of 1 in spite of having recognized (shortly prior) the numerosity of each collection. Later on in her development, the child may construct a more advanced strategy called counting-on. Here, starting at one of the numerosities (e.g., 8), the child would count only units of 1 that constituted the other collection (e.g., 8; 9-10-11-12-13). Most importantly, if one or both collections are then hidden, the child would supply an activity on other (“figural”) items that she supplies to the situation to keep track of the number of units of 1 and add them in the absence of the tangible marbles (e.g., raising one finger per each count from 9 to 13 and stopping after all five fingers were raised). Research has shown that, often, children who have only developed the counting-all strategy may be unable to solve such a problem with hidden items (Steffe & von Glasersfeld, 1985). The differences in mental operations inferred to underlie each strategy, and in the nature of problems a child can solve spontaneously and independently, underlie the boundary Steffe (1992) had postulated between a pre-numerical marker (counting-all) and a numerical marker (counting-on) in children’s construction of the concept of number. Counting-on is considered conceptually more advanced because it indicates the child can think of the result of her previous counting activity (“8” composed of eight 1s) as a unit in and of itself –

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a composite unit – and thus use it as input for further operations. Moreover, spontaneously supplying some activity for keeping track of the accrual of 1s while counting items in the second (hidden) collection indicates the child has possibly constructed “5” as a unit composed of five 1s. This can be inferred particularly if the child provides some indication of intentionally keeping track to determine when to stop counting. In contrast, a child who can only use counting-all, particularly if unable to take the actions and results of her counting for the hidden items, is inferred to lack such a concept of number as composite unit. A learning trajectory (LT) may include those two conceptual markers, operating only on units of 1 as manifested in counting-all, or operating on both 1s and composite units as manifested in counting-on. A more detailed LT of the progression from counting-all to counting-on may include more intermediate markers (discussed later). Regardless of how minutely closed such markers are, an explanation of the transition between every two markers is still needed (Simon, Kara, Placa et al., 2018). Next, I turn to the definition and grounding of HLT I use, and to how this construct underscores the transition between markers. 1.1 Hypothetical Learning Trajectory (hlt) In my research, I draw on Simon’s (1995) definition of hlt as a constructivist-rooted construct “… made up of three components: the learning goal that defines the direction, the learning activities, and the hypothetical learning process – a prediction of how the students’ thinking and understanding will evolve in the context of the learning activities” (p. 136). As he emphasized further, all three components of an hlt are afforded and/or constrained by the teacher’s knowledge. Specifically, three facets in a teacher’s knowledge influences the selection, articulation, and (if needed) adaptation of a all three hlt components. The first facet is the teacher’s hypotheses (inferences) of students’ available knowledge. The second facet is the teacher’s theories – whether explicit or tacit – on how learning takes place and thus how teaching can promote it. The third facet is the teacher’s knowledge of how students may learn particular content (e.g., counting-on and the concept of number as a composite unit). Furthermore, an hlt the teacher generated prior to instruction is bound to be adapted while she works with students, because this real-time experience “by the nature of its social constitution is different from the one anticipated by the teacher” (p. 137). It should be noted that, within hlt, the notion of “hypothetical” conveys two interrelated meanings. First, the learning trajectory is hypothetical in that it draws on a teacher’s inferences about the learners’ mathematical ways of

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operating (Steffe, 1990, 2002a). Such inferences arise through the interplay between the learners’ observable behaviors, as assimilated by the teacher, and explanations the teacher formulates about what could possibly underlie those behaviors – and why they might make sense to the learners. In the counting-all and counting-on example, the inferences are those pertaining to the units on which the child is operating. Second, hypothetical also implies that the learning trajectory is non-deterministic. Whereas a trajectory may be depicted as solid and straightforward, teaching can occasion and foster but not govern its course and/or outcomes (Kieren, 1995). A hypothesis of how a child may transition from counting-all to counting-on is discussed later. Based on these two meanings of hypothetical, I deem central the third component of an hlt, that is, the hypothetical learning process. The first two components (learning goal, activities) seem common to most LTs and/or teaching plans. The third component, however, requires explicating a theoretical stance on conceptual learning, so it provides a rationale for (a) linking the learning goal(s) with students’ available knowledge (e.g., why would counting-on be a reasonable goal for students who use counting-all) and (b) for articulating how the learning activities are likely to promote the intended progress (see more on this in the How Far From the Start example later). Thus defined, I consider the construct of hlt as a vital, twofold “tool of the trade” for making theoretical and practical progress in mathematics education. Theoretically, hlts help explain how a person may transition from not having to having a particular way of reasoning mathematically, that is, articulate the process of promoting particular conceptual advances (Simon, Kara, Placa et al., 2018). Practically, hlts can help deepen and systematize the design, enactment, and adaptation of teaching specific mathematics to particular students (Clements & Sarama, 2004; Simon, 2018; Sztajn, Confrey, Holt Wilson, & Edgington, 2012; Wilson, Mojica, & Confrey, 2013). As noted above, in this chapter I focus on the role that hlts, as content-specific products of theory-based conceptual analyses, can serve in explaining the transition from previously constructed markers to more advanced ones (e.g., from counting-all to counting-on). 1.2 Marker and Transition Studies The last sentence in the previous paragraph stresses a distinction I have been using, and am introducing in this chapter for the first time, between two types of research on LTs, marker studies and transition studies. This distinction draws on the premise that mathematics teaching is a goal-directed activity, aimed at promoting students’ conceptual progress towards intended mathematics (Skemp, 1979; von Glasersfeld, 1995a). The first type, marker studies,

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foregrounds conceptual landmarks that comprise a learning trajectory (e.g., counting-all, counting-on, and four markers between them – see more later). The second type, transition studies, foregrounds the specification of conceptual transformations involved in progressing from less to more advanced markers (e.g., how starting at 1 may change to starting at the numerosity of one collection). Because a primary goal of my work on hlts is to understand how learning of particular mathematics may progress and how teaching may foster such progression, in the main my research program conists of transition studies. As the construct of hlt is rooted in a constructivist theory of knowing and learning (Piaget, 1970, 1985; von Glasersfeld, 1995b), I consider it as a vital tool for and product of transition studies. My reason for this consideration is the aforesaid definition of hlt, which merges three elements (Simon, 1995): the teacher’s learning goal(s) for students, her plan for learning activities (tasks), and her hypothesis of how the learning process may unfold from what the teacher inferred to be the students’ extant mathematics. The first element must go beyond stating/naming a concept to be mastered. It requires articulating the nature of abstract linkage – anticipation – one would construct between her goal-directed activity and its results (Simon, Kara, Placa et al., 2018; Simon, Tzur, Heinz, & Kinzel, 2004; Skemp, 1979). The third element (unfolding of learning process) draws on the constructivist premise that new mathematical concepts emerge through reorganization of previously available concepts (Olive, 1999; Piaget, 1985; Steffe, 2002b). It thus requires using a stance on cognitive change (see more later) to formulate meticulous hypotheses of how learners may bring forth available concepts and transform those into the intended ones (Simon et al., 2004). Combined, these two elements (first and third) constitute the hypothesized conceptual transition, while the second element of an hlt (learning activities) provides guidance for teacher-learner interactions to foster that transition. Simply put, an hlt can guide research and/or teaching by specifying (a) the conceptual starting point(s) and the learning target (i.e., markers), (b) the process of change between them (i.e., transition), and (c) tasks a teacher can use to engage students in mental activities hypothesized to foster the intended change (Simon, Kara, Placa et al., 2018). Accordingly, a prime method for studying learners’ transition between particular markers that I have been using is the constructivist teaching experiment (Steffe, 2002a; Steffe, Thompson, & von Glasersfeld, 2000). In a teaching experiment, a team of researchers sets out to test hypotheses about how learners’ available ways of reasoning mathematically (i.e., markers) may (a) enable assimilation of and engagement in tasks that could (b) bring forth reorganization of these markers into new, conceptually more advanced ones

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(i.e., transition). To this end, team members serve in the dual role of researcherteacher, which entails they reflexively promote, and study, transitions between markers. Indeed, a research team might intend to promote a particular conceptual advance, such as the transition from counting-all to counting-on (Tzur & Lambert, 2011). Yet, teaching experiments are conducted with an a-priori, built-in expectation that during a teaching experiment unintended conceptual advances, and/or barriers to such advances, may arise. In both cases of teaching experiments, with expected or unexpected findings, my articulation of transitions from available to more advanced markers draws on a two-stage model of learning as a reflective process of cognitive change, called Reflection on Activity-Effect Relationship (Simon et al., 2004; Tzur & Simon, 2004). Next, I explicate that model, while further depicting the starting point and development of HLTs.

2

Starting Points and Development of hlts

2.1 Starting Point The core constructivist principle of assimilation (Piaget, 1985) provides the starting point for any hlt study I conduct. Assimilation entails that any new learning can only be as good as the goal-directed activities afforded and/or constrained by learners’ available ways of reasoning, that is, their extant schemes (von Glasersfeld, 1995b). For example, learners whose current schemes include counting-all as the strategy for finding the total of 1s in two, previously counted collections, are afforded with a way to correctly interpret a task (e.g., adding 8+5 marbles described earlier) and figure this total out – while being constrained to starting at 1 and operating on 1s. To teach and study how learners transform (reorganize) available schemes into new ones, with my teams I thus first engage in articulating fine details of the three parts that might constitute the learners’ available schemes (von Glasersfeld, 1995b). Below, I describe the parts and illustrate them with the counting-all example. The first part of a scheme is the mental template (‘situation’) by which learners may make sense of a given ‘input’ (e.g., mathematical task), which triggers the goal(s) they would set out to accomplish. In the counting-all example, the child’s situation is constituted by having established two collections of items, each with its own numerosity; the goal is to find the numerosity of 1s in both collections. This goal would call up the second part of the scheme – a mental activity sequence that the learners have been using to reliably accomplish their goal(s). In our example, this activity involves starting at 1 and counting each and every item (unit of 1) until exhausting all items in both collections.

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While the scheme’s activity is being carried out, the learners’ goal(s) regulate their noticing of effects that either match or do not match the third part of the scheme – a result they expected to ensue from the activity. In our example, stating “13” at the end of counting constitutes the effect of this activity. If, for example, the child expected to have a total larger than 10, then this would have been her predicted result; if she thought the answer would be 10 – then she may notice the actual effect differs from what she anticipated. Detailing all three parts of learners’ available schemes is crucial for a transition study, because conceptual change is postulated to commence, and thus possibly be fostered, through their noticing of actual effects of their activity that differ from the effects they expected before the activity (Simon et al., 2004). To articulate learners’ available schemes that would serve as a starting point for studying hlts, as well as the hypothetical process of change (reorganization) those schemes may undergo, we combine two main sources: taskbased interviews with participating learners and scrutiny of previous, relevant research. Using these two sources reflexively, our goal is to detail the precise boundary between schemes we infer students have already constructed and schemes into which the available schemes could possibly be transformed (yet to be constructed). In our example, we may engage the child in the task of adding 8 and 5 marbles. Once she counted and re-established the numerosity of each collection, we would hide both collections. A child whose most advanced scheme for finding the total of 1s in two collections is counting-all would typically get stuck, because the activity she has available cannot be executed. In contrast, a child whose most advanced scheme is counting-on would typically call up that activity – stating the numerosity of the first collection and using some figural items (e.g., fingers) to substitute for the hidden items of the second collection. She would stop the count based on the track-keeping method she used to indicate “having reached 5 counts” – and state the last number word (“13”) as the total. The difference between the two children is further explicated once we unhide the collections: the first child would then begin the counting activity while the second child would use it to confirm her result (or, if the two worked together, state: “This is what I thought/said”). An intermediate stage (marker) in the transition from counting-all to counting-on could be inferred, for example, if a child started counting at 8, but has not kept track of the number of 1s she has been adding (e.g., 8; 9-10-11-12-13-14-15). The notion of precise boundary between markers entails further attention to one of two stages at which Simon and Tzur (2004) postulated learners’ schemes might be established. An anticipatory stage of a scheme is inferred if the learners can bring the scheme forth and use it spontaneously and independently when solving relevant tasks. In our example of an intermediate child

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above, the child knew to start at 8 – but not to keep track of the 1s in the second collection. A participatory stage is inferred if the learners can bring forth and use a newly constructed scheme albeit not yet spontaneously and independently (e.g., by somehow being incited for a novel use of an activity). In our example of a child who counted to 15, the child may notice (“Oops”), during or after she stopped counting, that she did not keep track to make sure only 5 additional items were counted. In the other example, of a child who uses counting-all, as she counts the 8th item (last in the first collection) she may notice independently, or after being asked, it coincided with the numerosity of the first collection – and thus, instead, restart the count from “8.” Recent studies by Simon and his team (Kara, Simon, & Placa, in press; Simon, Kara, Norton, & Placa, 2018; Simon, Kara, & Placa, 2018; Simon, Placa, Avitzur, & Kara, 2018; Simon, Placa, Kara, & Avitzur, 2018) provided further empirical distinctions of and support for the participatory stage in detailing hlts of children’s construction of multiplicative and fractional schemes. The hypotheses of how an intended conceptual change may be fostered differ based on the stage of learners’ available schemes. If we infer an anticipatory stage, we identify a relevant participatory stage of a new scheme to serve as the goal for their next learning. Accordingly, we detail ways to proactively promote Reflection Type-I, which consists of a learner’s comparison between effects they expected and actual effects they noticed to ensue from their activity. An example of this type of reflection was given with the child who, after executing counting-all to “8,” noticed on her own (or with others’ input) that this number word and the move to “9” were linked to the initial establishment of the numerosity of the first collection. Such comparisons provide the mental mechanism for creating a novel, provisional relationship between the learners’ goal-directed activity and its actual effects. Importantly, such comparisons do not require attributing to the learners knowledge more advanced than what has been previously available to them (Simon et al., 2004). Tzur (2011) postulated that this type of reflection is needed to promote a transition to the participatory stage of the next scheme (Tzur, 2011). If, however, we infer learners’ schemes to be at the participatory stage of a new scheme, we set the goal for their next learning to be the anticipatory stage of that scheme. In our example, this would be a child who initially did not keep track and erroneously counted to 15, but then recounted while keeping track to reach at “13.” In fact, the child may be aware of, and explain, that her mistake was in not keeping track. Accordingly, we detail ways to proactively promote Reflection Type-II, which consists of comparisons across mental records of instances in which an activity did or did not yield particular effects. For example, we may ask the child to compare between the experiences in

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which they did and did not keep track, and specify how were they similar (e.g., starting at “8” in both) or different (e.g., “using my fingers to stop at 5”). Such a comparison provides the mental mechanism for abstracting the intended activity-effect anticipation and reasoning why the effects necessarily come out of the activity. Tzur (2011) postulated that this second type of reflection is needed to promote the transition from a participatory to an anticipatory stage of a new scheme. To further illustrate how the above constructs are being used as a starting point, I provide an example from Tzur and Lambert’s (2011) study. In that study, we identified an HLT consisting of 4 sub-stages (markers) in the shift from counting-all to counting-on, that is, from having no concept of number as a composite unit to the early onset of that concept. For that study, we sampled all students who spontaneously and independently used the counting-all strategy for adding two previously counted collections (e.g., 7 cubes and 4 cubes). Table 3.1 specifies our inferences of the scheme that underlies such a strategy. For a child at the anticipatory stage of this scheme (underlying counting-all), we set out the learning goal of constructing a participatory stage of a scheme that would give rise to the concept of number as composite unit, which can be indicated by the development of a counting-on strategy (Steffe & von Glasersfeld, 1985). To this end, our HLT would include engaging students in a playful activity I have created, called How Far From the Start (hffs). In this activity, Player A would role a die, stand on a floor tile marked Start, walk from it while counting the number of steps implied by the dots on the die, write that numeral on a sticky note, and place it on the tile where she just stopped. Player B would then role the die, walk the corresponding number of tiles while starting from the tile with Player A’s sticky note, write the numeral on another sticky note, and place it on the tile at which she stopped. Then, both learners work together to figure out how far is the sticky note of Player B from the start (e.g., 11). table 3.1  The three-part scheme inferred to underlie counting-all

Situation + goal

Activity sequence

Result

Having separately counted all 1s in each of two given collections of tangible items to fijind their numerosities, set out to fijind the numerosity of a combined collection of those items.

Starting over from 1, count every tangible item in the combined collection by creating 1-to-1 correspondence between those items and number words in the conventional sequence.

Reaching the fijinal item to be counted and stating the number word that corresponded to this item to indicate the numerosity of the entire combined collection.

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As part of this hlt, we expect learners who begin playing the hffs game to find the total number of steps by assimilating the task into their available scheme, that is, by initially using counting-all. While they play, the researcherteacher will begin probing for their reflection on the effect they can notice, namely, always calling out the number on Player A’s sticky note (e.g., 7), and next the number 8, when counting to find the combined total. For example, we may ask the players to stop their count while stepping on that tile and tell us if they are surprised to have said “Seven.” We may also ask what would be the next number they would state, and/or if they could consider starting at a spot and a number word other than 1. We will likely then shift from real tiles to a drawn out board game marked with Start and End tiles. This allows us, later, to cover some of the tiles on Player A’s path to further orient the learners’ reflection onto the possibility to use Player A’s location/numeral as a start. Letting players switch roles between Player A and B, and repeating these experiences, fosters their creation of a provisional link between their counting-all activity up to the stopping point of Player A (e.g., 7) and the effect it ensued – starting with the number-after (Baroody, 1995; Fuson, 1992) when resuming their count (e.g., 8). This new, provisional linkage opens the way not only to starting the count from Player A’s stopping point (7) but also to keeping track of the count of 1s in Player B’s walk. That is, a new stage of anticipating where to start (Tzur & Lambert, 2011) is formed at a participatory stage, while the learners replace 1 as the start for finding the combined total by their noticed effect of starting from Player A’s stoppage (e.g., 7; 8-9-10-11). 2.2 Development of hlt The core constructivist stance on learning as a conceptual reorganization (Piaget, 1985), coupled with a corresponding, student-adaptive pedagogy (Tzur, 2013), underlies my development of hlts. Above, I provided a brief description of the two types of reflection and two stages (participatory, anticipatory) that constitute reorganization of available schemes into new ones. By studentadaptive pedagogy (AdPed), I refer to the cyclic, 7-step process Tzur (2008) postulated as an elaboration of Simon’s (1995) notion of hlt (Table 3.2). When conducting teaching experiments for transition studies, we use this 7-step cyclic process to develop hlts while engaging in two types of analysis – ongoing and retrospective (Steffe et al., 2000). Ongoing analysis focuses on inferring individual learners’ conceptual progress during the most recent teaching episode(s). Here, the team makes inferences about changes in the learners’ anticipation and explanation of effects they notice to ensue from their activity, and the extent to which the learners can use the newly abstracted anticipation spontaneously. Those tentative inferences constitute Step 1 of the

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table 3.2  The 7-step, cyclic process that constitutes student-adaptive pedagogy

AdPed step

Illustration: counting-all to counting-on

1. Specifying students’ available schemes.

Counting-all with the 1-to-1 count, starting at 1, to fijind numerosity of a combined collection. “Anticipating where to start the combined count” or “anticipating the need to keep track so one knows where to stop.”

2. Specifying the intended mathematics in terms of the child’s situation/goal, activity, and efffects to be noticed. 3. Identifying a mental activity sequence through which the conceptual change may evolve. 4. Selecting and/or adapting tasks to promote the intended learning.

5. Engaging learners in the task while letting them bring forth and use previously constructed schemes fijirst. 6. Monitoring learners’ progress

7. Introducing follow-up questions and probes to foster reflection type-I and/or reflection type-II (return to Step 1 based on learners’ responses).

Stoppage of counting-all with reflection on the noticed stoppage place and how/ where counting is resumed. hffs – a game that is fijirst being played on floor tiles, then as a board game allowing to introduce constraints, such as covering some of the tiles to foster noticing of counting start other than 1. Using counting-all in hffs.

Developing facility with the game played over tiles opens the way to shifting to the board game. Asking “Were you surprised to have said 7 when, to fijind how far was Player B from the start, you recounted from 1?”; or, covering a few tiles on Player A’s path when using the board game; or asking, “How were the ways you and your partner solved these last three problems similar/ diffferent?”

7-step cycle, which informs Steps 2, 3, and 4 in the design of teaching for the next episode. After completing all teaching episodes, further development of hlts occurs though retrospective analysis, which focuses on distinguishing and explaining plausible ways in which learners’ mental systems may give rise to their observable behaviors (actions and language). Drawing on the principles of grounded theory

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methodology (Glaser & Strauss, 1967), retrospective analysis identifies commonalities across different learners’ solutions while striving to specify schemes that, we infer, could serve as conceptual underpinnings of those solutions. Such inferences were illustrated in the examples of counting-all and counting-on, focusing on the units (1s only, or 1s and composit units) on which the child seems to operate. Those schemes, for which we detail both the participatory and anticipatory stages, become the markers of hlts. Then, going back to the data, we search for ways in which transition from one scheme (marker) to the participatory and then anticipatory stage of the next one might have taken place, along with instructional moves that seemed essential in fostering that learning. Here, an example would be to ask a child who did not keep track when starting at the numerosity of one collection, but could notice this lacking, to compare across those experiences and explain “to her younger sibling” the advantage of keeping track.

3

Different Uses of hlts

One obvious way I use previously developed hlts is to inform new hlt studies. In particular, previous hlts are used (a) to identify participants for a study based on their available schemes and (b) as a suggestive, developmental framework for determining what to teach next. In this final section, however, I focus on four other uses: studying adults’ mathematical reasoning, promoting teachers’ shift toward a student-adaptive pedagogy (AdPed), which draws on empirically grounded hlts, guiding new hlt studies with special-needs populations, and examining the applicability of hlts for learners in different cultures (e.g., China). 3.1 Promoting Adults’ Mathematical Reasoning Most hlts have been developed and refined while working with K-12 student populations, whereas hlts arising out of studies with adult learners are largely lacking. Thus, in my teaching of mathematics to adult learners – teachers and non-teachers alike – I have been using hlts from research on K-12 students’ construction of schemes for multiplicative (Hackenberg & Tillema, 2009; Tzur et al., 2013; Ulrich, 2015, 2016) and fractional reasoning (Steffe & Olive, 2010). Quite often, I found that university students and practicing teachers, who attained high school and perhaps college-level mathematics, seemed to lack foundational schemes for multiplicative and/or fractional reasoning. By-andlarge, experiences I designed for teaching them based on hlts gleaned from research with children have repeatedly proven viable for those adult learners. For example, substantial efforts were needed to promote K-12 practicing teachers’ shift away from the limited and limiting understanding of fractions as

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figure 3.1 A bar marked vertically into 3 equal parts, with two of the thirds (pink) marked into 1/5th

“parts-of-wholes.” All of them, including those teaching high school algebra, geometry, and calculus, struggled to solve tasks that eventually led them to construct a scheme by which to make sense of diagrams such as the one seen in Figure 3.1 (here, teachers were asked why 1/5 of 2/3 is 2/15 of the whole, while the whole appears to consist of only 11, unequal parts). Moreover, I have led two studies in which hlts from research with children informed fostering mathematical progress in adults. One study (Tzur, Hodkowski, & Uribe, 2016) used an hlt about children’s construction of a scheme for partitioning fractional parts to inform an elementary teacher’s understanding of decimals. For example, she would partition 1/10 into ten parts, and (looking at 19 unequal parts) explain that the resulting unit fraction is 1/100 of the original whole because the whole is 100 times as much of it (10 times as much of 1/10, which is 10 times as much of the resulting unit fraction). This allowed her, for the first time, to make sense of why, for example, 0.9 is equivalent to 90/100, and why 0.9 is larger than 0.87. Another study (Tzur & Depue, 2014) used an hlt about children’s construction of a scheme of unit fractions (1/n) as multiplicative relations to inform an fmri study on how the brain of adults process comparisons of whole numbers and of unit fractions. This study showed that a teaching plan rooted in the hlt developed for children promoted not only participants’ conceptual understanding of unit fractions but also the time it took their brains to process numerical comparisons. In both studies, results indicated that supporting adult learners’ reorganization of their available schemes could follow reorganization found in hlts from research with children. 3.2

Promoting Teachers’ Shift toward Student-Adaptive Pedagogy (AdPed) In the past three years, I have been directing a 4-year project1 in which my team attempted to promote and study the impact of a professional development

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(PD) program on upper-elementary teachers’ transition toward AdPed and on their students’ learning and outcomes. A substantial part of this PD program engages teachers in learning to notice, infer, and use hlts about students’ multiplicative and fractional schemes. Our project has been conducted in 5 elementary schools in urban school districts (western usa) where most students are identified as non-white (~65% Latino/a and ~20% African-American), and over 50% are identified as English Language Leaners. Already during the first year of the job-embedded PD, participating teachers seemed to begin grasping problem-solving strategies linked with those schemes, and construct initial meanings for the conceptual structures and operations that underlie those strategies. Personal testimonies from teachers, and their school principals, indicated that learning about students’ reasoning as captured in those HLTs is non-trivial and challenging – yet viable and rewarding. After a second, 5-day Summer Institute in which we oriented teachers’ focus onto incorporating the student-adaptive classroom activities into their district curriculum, the hlts proved invaluable in providing conceptually differentiated instruction, and hence in increasing students’ learning and outcomes. For example, our research pointed to a particular link between the spontaneous strategy a child uses to add two, single digit numbers (e.g., 8+7), such as counting-all, counting-on, doubling, or break-apart-make-ten (8+2=10, 10+5=15) – and children’s extant ability to reason multiplicatively. Introducing this linkage to the teachers, including how to use an assessment of the child’s concept of number based on those strategies, led teachers to adapt goals/activities to students’ available schemes and, subsequently, to ststistically significant growth in students’ multiplicative reasoning (forthcoming). 3.3 Guiding hlt Studies with Special Needs Populations In the past 10 years, I have been collaborating with special education colleagues in using hlts, developed through studies with ordinarily achieving students, also for research with struggling students (see, for example, Hunt, Tzur, & Westenskow, 2016; Tzur et al., 2016). Overall, our studies with children identified by their school systems as “students with learning disabilities (sld) in mathematics” provided evidence that both markers and transitions found in previous hlts were highly applicable to sld. That is, promoting sld progress along multiplicative and fractional hlts seemed to proceed similarly to the ones found with other learners (K-12 students and adults). Accordingly, I have become more and more convinced that, quite often, struggling students are not “learning disabled,” but rather “teaching disabled” (Tzur, 2013). Simply put, I contend that application of conceptually focused HLTs can provide a judicious way forward for the most vulnerable student populations.

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3.4 Examining hlts in Different Cultures An important issue arises about hlts developed through studies in western cultures, namely, can they be extended to learners in other cultures. A recent study (Norton, Wilkins, & Xu, 2018) examined and demonstrated such crosscultural applicability of hlts for fractions, between US and Chinese students. Specifically, they showed that Chinese students began their progress through the sequence of fractional schemes (markers) with part-whole, and then moved through unit fractions as multiplicative relations to the partitive scheme (proper fractions) and reversible scheme (e.g., re-produce a whole when given an unpartitioned piece that is 5/8 of it). In another, ongoing, collaborative teaching experiment I have been conducting in China, we set out to further address this issue. Specifically, we asked an elementary teacher in a school affiliated with a college of education in northern China to identify among his 44 students all those who (a) struggle in mathematics and (b) he does not know how to support their learning. The teacher identified five (~12%) such students, a percentage similar to reports about SLDs in the west (see Fuchs, Fuchs, & Hollenbeck, 2007). Our preliminary analysis of data collected through individual, task-based interviews with four of those 4th graders indicated available schemes consistent with distinct markers of known, western-born hlts (Tzur et al., 2013). Student A could (only) use counting-all to add two collections of tangible items. Student B could spontaneously use counting-on and, with prompting (participatory), make sense of the Break-Apart-Make-Ten (bamt) strategy (Murata & Fuson, 2006). For example, Student B could follow and make sense of an explanation for adding 8+7, by decomposing 7 into two sub-units (5+2), then adding 8+2 to compose 10, and finally adding the remaining 5 to find the total (15). Student C could spontaneously use bamt and indicated the independent (anticipatory) use of the first scheme in multiplicative reasoning – but struggled with operating on problems that require the second multiplicative scheme. Student D’s spontaneous solutions indicated construction of the two first multiplicative schemes at the anticipatory stage, but only a participatory stage of the fourth multiplicative scheme. Indeed, all four students seemed to struggle because their available schemes, as informed by western-born hlts, fell short of supporting the mathematics taught in their 4th grade classroom.

4

Concluding Remarks

In this chapter, I have introduced a distinction between two types of studies – marker and transition – researchers may use to develop hlts. I emphasized

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that the former type provides conceptually distinct progressions in how people may reason mathematically (schemes), whereas the latter type augments those progressions by specifying how learners may reorganize available schemes into more advanced ones – and how teachers may foster such reorganization. I pointed out that the third element in Simon’s (1995) defintion of hlt, namely, the hypothetical learning process, is the one for which transition studies, informed by a model of learning as a cognitive change, are required. Using the example of counting-all and counting-on as two markers, I presented the model of reflection on activity-effect relationship to explain learning (conceptual transition) as a two-stage process occurring through two types of reflection. Using this model, and the example of foundational transition from counting-all to counting-on (i.e., from not having to having a concept of number as composite unit), I have depicted how this model could be used to design a transition study leading to the development, and possible uses, of empirically grounded hlts. In short, in this chapter I have centered on hlt as a construct to guide research on conceptual learning and teaching.

Note 1 This project is funded under the US National Science Foundation Award No. 1503206. Any opinions expressed herein are those of the author and do not necessarily represent the views of the Foundation.

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Tzur, R. (2013). Too often, these children are teaching-disabled, not learning-disabled. In Proceedings of the 11th annual Hawaii international conference on education. Honolulu, HI: Author (DVD). Tzur, R., & Depue, B. E. (2014). Conceptual and brain processing of unit fraction comparisons: A CogNeuro-MathEd study. In S. Oesterle, C. Nicol, P. Liljedahl, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 5, pp. 297–304). Vancouver: PME. Tzur, R., Hodkowski, N. M., & Uribe, M. (2016). A grade-4 teacher’s mathematics: The case of Annie’s understanding of decimal fractions. In Proceedings of the 14th annual Hawaii international conference on education. Honolulu, HI: Author. Tzur, R., Johnson, H. L., McClintock, E., Kenney, R. H., Xin, Y. P., Si, L., … Jin, X. (2013). Distinguishing schemes and tasks in children’s development of multiplicative reasoning. PNA, 7(3), 85–101. Tzur, R., & Lambert, M. A. (2011). Intermediate participatory stages as zone of proximal development correlate in constructing counting-on: A plausible conceptual source for children’s transitory ‘regress’ to counting-all. Journal for Research in Mathematics Education, 42(5), 418–450. Tzur, R., & Simon, M. A. (2004). Distinguishing two stages of mathematics conceptual learning. International Journal of Science and Mathematics Education, 2, 287–304. doi:10.1007/s10763-004-7479-4 Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (part 1). For the Learning of Mathematics, 35(3), 2–7. Ulrich, C. (2016). Stages in constructing and coordinating units additively and multiplicatively (part 2). For the Learning of Mathematics, 36(1), 34–39. von Glasersfeld, E. (1995a). A constructivist approach to teaching. In L. P. Steffe & J. Gale (Eds.), Constructivism in education (pp. 3–15). Hillsdale, NJ: Lawrence Erlbaum. von Glasersfeld, E. (1995b). Radical constructivism: A way of knowing and learning. Washington, DC: Falmer. Wilson, P. H., Mojica, G. F., & Confrey, J. (2013). Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking. The Journal of Mathematical Behavior, 32(2), 103–121. doi:http://dx.doi.org/10.1016/ j.jmathb.2012.12.003

CHAPTER 4

Using Digital Diagnostic Classroom Assessments Based on Learning Trajectories to Drive Instruction Jere Confrey, William McGowan, Meetal Shah, Michael Belcher, Margaret Hennessey and Alan Maloney

Abstract Learning trajectories (LTs) hold promise to effectively convey to teachers significant patterns in student thinking. The research on learning that undergirds LTs has accumulated over several years, and the research base continues to develop. However, most of this research is not easily accessible– much less actionable–by most teachers. We introduce a digital learning system (dls), based on LTs, as a means of providing teachers systematic access to information on student learning. In particular, we illustrate the use of this system’s LT-aligned diagnostic assessments, on percents, to provide valid and timely data to teachers and students alike. The DLS itself tracks student progress over time, and data from field testing supports research on the refinement of the trajectories and their measures.

Keywords learning trajectories – classroom-based assessment – feedback – middlegrades mathematics – learner-centered instruction

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Introduction

The field of mathematics education, despite numerous advances in the learning sciences and cognition, remains unable to systematically prepare a highly qualified teaching force and support the ongoing improvement of practicing teachers. The research on learning has accumulated over a number of years. Much of it is summarized periodically in handbooks in mathematics and statistics education (Ben-Zvi, Makar, & Garfield, 2017; Cai, 2017; English, 2002, 2008; Grouws, 1992; Lester, 2007). The field continues to develop new insights into © koninklijke brill nv, leideN, 2019 | DOI:10.1163/9789004396449_005

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students’ ideas, especially as a result of new technologies and tools (Blume & Heid, 2008; Hoyles & Lagrange, 2010) and emerging novel contexts. However, the bulk of this research is neither readily accessible, nor actionable, by most classroom teachers. Teachers’ knowledge of student learning patterns still relies on a traditional set of teacher preparation courses (Kieran, Krainer, & Shaughnessy, 2013). While the courses provide examples of student learning, they lack systematic and sufficient access to the research base. The challenge of providing access to contemporary research in education, to both pre- and in-service teachers, is exacerbated by the growing trend of lateral entry to teaching (Darling-Hammond, 2010). We contend that professional practice needs ongoing professional development grounded in the learning sciences: professional development focused around learning trajectories could provide teachers with a solid foundation regarding how students reason about particular topics, and would support instruction to become more effectively student-centered. Lacking a comprehensive understanding of research on student learning, far too many teachers in the US approach instructional planning using a standard-bystandard approach (Confrey, 2015). This is inadvisable because individual standards within the Common Core State Standards in Mathematics (ccss-m) vary widely in grain-size. Some standards are highly specific (e.g. 7.SP.C.8.a “Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.”) whereas others incorporate a broad range of concepts within a particular domain and require unpacking (e.g. 7.RP.A.3 “Use proportional relationships to solve multistep ratio and percent problems.”). Furthermore, the authors of the ccss-m state that the standards were not designed as a “curriculum” (Council of Chief State School Officers, 2010). Although standards identify the knowledge students should achieve and indicate when they should be able to demonstrate that knowledge, they do not sufficiently address how those ideas are learned across time and grades. This implies that the implementation of the standards varies based on the resources available and the pedagogical content knowledge of teachers. Learning trajectories (LTs) hold promise as means to convey to teachers what has been learned about significant patterns in student thinking in an efficient and effective manner. LTs are composed of a target concept and the delineation of a number of “progress levels” that describe students’ thinking as they progress from naïve to sophisticated understandings. LTs assume that students confront a sequence of curricular tasks that help them to move through these levels. Moreover, LTs can serve as a means to organize and search for student-centered curricular materials. Successful instruction requires that teachers leverage discourse around student ideas and

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help students gradually mature in their thinking. Thus, LTs help teachers to plan lessons and to evaluate the success of instruction. In this chapter we introduce Math-Mapper 6-8 (MM6-8; www.sudds.co), a digital learning system (dls) designed as a synthesis of a large segment of the research on student learning in mathematics into a LT-based framework and a diagnostic assessment system. MM6-8 is organized around nine big ideas from middle school to demonstrate how LTs can form a foundational access to research on learning in middle grades mathematics and provide actionable feedback to teachers and students on mathematics learning. MM6-8 is comprised of a learning map, a set of curated resources, and a diagnostic assessment and reporting system. The learning map is organized hierarchically: the nine big ideas are composed of 24 relational learning clusters (rlcs),1 which, in turn, comprise a total of 62 constructs. Each rlc is a collection of constructs/LTs that complement each other and are best taught in relation to each other within a unit of instruction. A learning trajectory is specified for each construct. The Common Core State Standards are indexed to the LTs. Thus, the map is designed to keep teachers’ attention on students’ cognitive progression while simultaneously assuring them that expected grade level standards are being addressed. The digitally administered diagnostic assessments measure student understanding of the progress levels of the learning trajectories; each comprises 8–12 items selected from a cluster. Testing occurs within the course of instruction, often approximately two-thirds of the way through the time assigned to an instructional unit. Students receive immediate feedback on how they did, and can revise their responses and/or choose to practice items on individual constructs. Simultaneous reports to teachers, in the form of “heat maps,” display correct and incorrect answers both by student and by LT progress level, making it simple for a teacher to quickly make fine-grained instructional decisions for the entire class, small groups within the class, and/or individual students. To fully illustrate the resources provided in MM6-8, we describe its design and implementation in relation to one cluster, “Finding Key Percent Relationships,” in the big idea “Compare quantities as ratio, rate or percent and operate with them.” This cluster is comprised of three constructs, “percent as amount per 100,” “benchmark percents,” and “percents as combinations of other percents.” We report on the data collected at three partner schools and describe the diverse populations of students they serve. We share and analyze the patterns of student results, citing particular items to illustrate the trajectories and student achievement. Our data suggest that teachers often overestimate the success of students after instruction and that timely data can provide real opportunities for them to adapt instruction to address the specific needs of students. The data from students also suggest that students find the immediate

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access to data helpful and that they begin to pick up and use the language of the LTs in describing where they need to study or practice more. The chapter is organized as follows: we discuss the concept of LTs and describe how ours were constructed, illustrating the ideas using the cluster “Finding Key Percent Relationships.” Then, we describe our diagnostic assessment system and show how students at our partner schools are empowered to take more ownership of their learning as they learn to use data from MM6-8’s assessments. In addition, we describe how our partner teachers use MM6-8 in their classes and professional learning community (plc) meetings to strengthen their understanding of student reasoning and the related mathematics. Finally, we report on the results of students’ performance on the assessment for the same cluster to illustrate how the research team uses data from MM6-8 to refine the tool.

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The Concept of Learning Trajectories in Math-Mapper 6-8

Our definition of a LT builds on Simon’s (1995) “hypothetical learning trajectories (hlts),” which described teacher-conjectured paths of the development of students’ thinking as they engage with instructional tasks. hlts are hypothetical because the teacher cannot know in advance how the trajectory will manifest during instruction. In practice, students’ thinking may develop in unexpected ways, resulting in revisions to the hypothesized trajectory. Teachers engage in a continual process of conjecturing paths of students’ development and revising those paths based on experiences and observations during instruction (Simon, 1995). Like Simon, we acknowledge the hypothetical nature of LTs and recognize the essential role of instruction in students’ development. We emphasize that these conjectures, while hypothetical, are based on a synthesis of research on student thinking and are empirically-supported. We define an LT as: A researcher-conjectured, empirically supported description of the ordered network of constructs a student encounters through instruction (i.e., activities, tasks, tools, forms of interaction, and methods of evaluation), in order to move from informal ideas, through successive refinements of representation, articulation, and reflection, toward increasingly complex concepts over time. (Confrey, Maloney, Nguyen, Mojica, & Myers, 2009, p. 3) LTs are probabilistic conjectures, expressed as progress levels, that describe the landmarks and obstacles students are likely to encounter through engaging

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with instruction, as their thinking develops from less sophisticated to more sophisticated over time (Confrey, Maloney, Nguyen, & Rupp, 2014). They are probabilistic in that they describe likely paths of development based on a synthesis of the research on student thinking related to a target concept, and this development is influenced and constrained by instruction (Nguyen & Confrey, 2014). Students are expected to require instruction in order to progress along LTs (Clements & Sarama, 2004; Confrey, Maloney, Nguyen, & Rupp, 2014; Nguyen & Confrey, 2014; Simon & Tzur, 2004). LTs are not developmental stage theories (Confrey, Maloney, & Nguyen, 2014; Lehrer & Schauble, 2015). Although they are presented as a ladder in our DLS for ease and space considerations, LTs do not imply a strictly linear progression toward a sophisticated understanding of the target concept. Instead, we conceptualize our LTs using the metaphor of a climbing wall for which a variety of starting points are likely and expected, and multiple paths, containing predictable obstacles and footholds, are possible for successfully ascending the wall. Some paths may be more challenging and lead to more obstacles than others and some footholds may be more difficult to reach than others. However, these paths are predictable, and when used to formatively assess students’ understanding, can inform teachers’ instruction and guide students’ learning. 2.1 Research on Learning Percents In this section, we present a brief overview of some of the research, on students’ learning of percents, which contributed to the development of our LTs. Prior to receiving formal instruction on the concept, students possess an informal and intuitive understanding of percent, stemming in part from widespread societal use of percent terminology. From describing effort (“give it 100%”) to describing how much battery life remains on a phone (“50% charged”), the use of such terminology is ubiquitous in most students’ outof-school experiences. As a result, students come to instruction with an informal understanding of many commonly used percents. For example, Moss and Case (1999) reported elementary students understanding “that 100% meant ‘everything,’ 99% meant ‘almost everything,’ 50% meant ‘exactly half,’ and 1% meant ‘almost nothing’” (p. 129). Although such familiarity with the topic provides a foundation upon which to build more sophisticated conceptions of percent, it can also contribute to students developing misconceptions. For example, students’ understanding of 100% as “everything” can be useful but it may also lead to difficulties understanding percents greater than 100% (Parker & Leinhardt, 1995). Understanding 50% as “half,” offers an opportunity to leverage fractional reasoning to accurately generate benchmark percents

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(Middleton, van den Heuvel-Panhuizen, & Shew, 1998), but such fractional reasoning may hide the comparative base of 100, or could lead students to confuse fraction, decimal, and percent procedures, interpreting 0.75%, 75%, and ¾ as equivalent (Parker & Leinhardt, 1995). The potential for fractional reasoning to both contribute to and interfere with students’ understanding of percents highlights a broader challenge inherent to learning this concept. Depending on when and how percents are introduced, students may come to instruction with knowledge about fractions, decimals, ratios, proportions, multiplication, and division, and the similarities among these concepts may interfere with their understanding of percent if the distinctions among them are never examined or exposed (Parker & Leinhardt, 1995). One approach to teaching percents that leverages students’ previous knowledge is to treat percents as a ratio with a comparative base of 100 (Parker & Leinhardt, 1995; van den Heuvel-Panhuizen, 1994). Parker and Leinhardt (1995) argued that teachers can leverage students’ understanding of comparing ratios by letting students first compare ratios with a common value of 100 before introducing percents. After students have been introduced to the concept, they should be supported to recognize the importance of the referent in understanding and comparing percents. For example, problems that ask students to compare the value of a small percent of a large quantity (e.g. 5% of 200) with a larger percent of a smaller quantity (e.g. 60% of 10) can highlight the relative nature of percents (Parker & Leinhardt, 1995; van den Heuvel-Panhuizen, 1994), a particularly challenging concept for students. The selection and use of visual representations is also essential to students’ understanding of rational number concepts generally, and, specifically, of percents (Behr, Lesh, Post, & Silver, 1983; Parker & Leinhardt, 1995). Moss and Case (1999) introduced percents by asking students to estimate the water level in a beaker and assign it a number from 1 to 100. This representation leveraged students’ visual estimation strategies and connected to their whole-number strategies, while also introducing the comparative base of 100 (Moss & Case, 1999). In a similar approach, Middleton and colleagues (1998) used the visual representation of a bar model to build on students’ prior knowledge and to support students to connect their fractional reasoning with ratios and percents. Using a bar model, students could split the bar into benchmark fractions and represent both the percent and the quantity. Moreover, the bar model avoided obscuring the comparative base of 100, a common pitfall of such problems, by keeping the 100 central to the representation. Another common representation used is the hundreds board: a square 100’s grid. The hundreds board can been used to represent decimals (the whole square represents 1 and each

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smaller square represents 0.01) and percents (the whole square represents 100% and each smaller square 1%). This representation allows students to convert among decimals, fractions, and percents; however, it makes representing and understanding percents greater than 100% more difficult (Parker & Leinhardt, 1995). Whether representing percents using a bar model (Middleton et al., 1998), beakers with different water levels (Moss & Case, 1999), or hundreds boards, students benefit from opportunities to experience and connect a variety of visual representations (Behr, Lesh, Post, & Silver, 1983). Thus, an effective approach to the teaching of the topic could be one that presents it as emerging from ratio (van den Heuvel-Panhuizen, 1994), and leverages students’ visual estimation (Moss & Case, 1999) and fractional reasoning (Middleton et al., 1998), while exposing the comparative base of 100 (Parker & Leinhardt, 1995), and using and connecting several visual representations (Behr et al., 1983; Parker & Leinhardt, 1995). The research reviewed in this section informed our development of the percents LTs. 2.2 Key Percent Relationships LTs The “Finding Key Percent Relationships” cluster is located within the big idea “Compare quantities as ratio, rate, or percent and operate with them.” The cluster contains three constructs, and we briefly review the structure of each to illustrate how they were developed. All three constructs are mapped to 6thgrade standards in the ccss-m. Construct A2, “Percent as Amount per 100,” focuses on recognizing how percents simplify the comparison among ratios. Conceptually, we view percent as most closely connected to ratio (as relative size) and emphasize the importance of a referent unit in the form of a proportion [of] or a ratio [of]. Construct A links percents to a ratio a:b in which b=100, and a, the percentage, is the missing value. The first two levels draw on what students learned previously about comparing ratios3 by finding a common value, which is, in this case, 100. This introductory construct also relies heavily on the use of percent bars, as well as a limited use of 100’s grids. The next two levels emphasize percentages of the same quantity as adding to 100, and, in anticipation of ratios greater than 100%, stress percent as relative size. Construct B, “Benchmark Percents” (Figure 4.1) builds out from many students’ cultural experiences with percents prior to formal instruction, drawing on students’ familiarity with percents in everyday life such as (retail) sales or in sports. Using the percents associated with common fractional parts, students are encouraged to use mental strategies for increasingly complex fractional parts. At level 1, students work only with 0% and 100%. By level 2, they add

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figure 4.1 “Benchmark percents” LT from the Math-Mapper 6-8 learning map

in 50%, to begin the association with one half. Level 3 adds in the common benchmarks associated with one-fourth (25%) and three-fourths (75%), as well as 1/10th (10%), 1/10th of 1/10th (1%), and half of 1/10th (5%). Level 3 also specifies the range of representations including percent bars, 100’s grids, or sector graphs. Level 4 specifies that the student can not only associate the benchmark percents with fractional amounts, but can compute the x% (for those values) of a given quantity, using numerical expressions. By level 5, students can find the original whole, given the percentage as a quantity, for benchmark percents. Construct C, “Percents as Combination of Other Percents,” continues to build on those familiar benchmarks to provide strategies for more complex approaches. Construct C is the “top” LT in this cluster. In this LT, students learn to extend beyond the benchmark percents, initially by combining percents through multiplication (30% as 3x10%) and addition (15% as 10%+5% of a quantity). Using combinations to find unknown percents is first worked out for increments of 5% or 10% and then for increments of 1%. Understanding these combinations, students then extend percents beyond 100, by multiplication (e.g. 200%, 300%), and then below 1%, by division by 10. The limitations of the part-whole misconception of fraction can likewise show up in percents, leading students to expect that a percent of a number must be less than that number. In order to successfully extend percents beyond 100 and to reason flexibly with any percent, students must generalize percents as a description of relative size. Seeing percent as relative size is strongly supported by viewing it in relation to ratio as an operator.

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Diagnostic Assessments and Reports

Elaboration documents. To coordinate MM6-8’s learning trajectories and diagnostic assessments, the project’s learning sciences team developed an “elaboration document” for each of the 62 constructs. Each elaboration document provides a high-level summary of the skills and understandings students develop as they move through a construct’s LT levels, identifies prerequisite constructs, and identifies associated standards (ccss-m in the current version). Then, each LT level is described, providing information about relevant misconceptions (if any) and cases or problem types that should be addressed by various items at that level. Elaboration documents continue to serve as foundations for assessment item development. Item development. Items are designed to elicit student thinking reflected in each LT level. They are targeted conceptually to examine how well students can relate ideas together, identify relevant concepts, interpret representations, or reconcile misconceptions. Each item is written to align to (and thus, to measure) a single LT level. Items undergo extensive development before being field-tested by students in our partner schools. MM6-8 currently includes multiple choice (33%), multiple select (20%), numeric (28%), one-letter (18%) and open-ended (1%) item types. Test administration. Test forms are written for each of the 24 rlcs, with multiple forms for each grade that has a LT level in the rlc. Each test form contains 8–12 items, sampled from grade-level LT levels across the cluster, and takes approximately 30 minutes to complete. Teachers can assign each student one of up to four equivalent test forms. Assigning multiple forms to a class allows a greater array of items to be explored by a class, setting the class up for richer discussions of the underlying meaning of the tested levels (instead of only the details of a particular problem), and a more complete class report. Student reports. Upon completing a test, a student receives immediate feedback in the form of a report (Figure 4.2) containing scores by construct, level, and individual items. The report presents the percent correct score for each construct (as a dial), as well as a color-coded “learning ladder” that shows performance on the tested items, by level. Below the report (not pictured), an item matrix displays the detailed results on each item. It allows the student to review items from the test, and then to revise and resubmit incorrectlyanswered items. After revisions are submitted, the student’s construct scores are automatically updated. The improved scores are displayed in a different color (aqua) on relevant construct-score dials. Class reports. Teachers also receive immediate feedback on student performance on the test. The primary mode of display is a “heat map” for each construct,

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figure 4.2 The top half of a student report showing a student’s percent correct by construct and performance by LT level

figure 4.3 The heat maps (teacher reports) for the three constructs in the cluster Finding Key Percent Relationships. The levels are listed vertically, with increasing difficulty, and the students, based on performance, are ordered from left to right. Student identifiers can be toggled on or off for confidentiality

which shows, through color-coding, each student’s performance on each item. Items are ordered by their associated levels, from bottom to top (Figure 4.3). Each heat map corresponds to a single construct. Each column in the heat map corresponds to a color-coded learning ladder for one student in one construct (see Figure 4.2). Columns (students) are ordered from lowest (left) to highest (right) overall performance in the construct. As in the student report, orange cells represent 0% correct on the item, shades of blue represent varying levels of correctness, dark blue represents 100% correct on the item, and white cells represent untested levels. As one “goes up” the LT levels, item difficulty tends to increase (increasing proportions of orange in the upper levels). Taken as a whole, the heat maps give a teacher a high-level snapshot of students’ learning in a cluster, providing visual cues to help decide which concepts would be more effectively revisited with the whole class or, alternatively, a particular subgroup of students.

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A teacher can tap on a level and reveal the students’ item. The class can then rework the problem. This display also allows the teacher to reveal the correct answer, to display the item analysis, which reports the common responses for the item, and, if applicable, can show the percentage of students who revealed any particular misconception associated with the item.

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Student and Teacher Uses of Assessment Data

Through design studies, classroom observations, teacher reports, student surveys, and summer workshops, we have identified a number of ways the diagnostic assessment data are used formatively in the classroom. First, we share how students make use of the data in their score reports. Then we describe ways teachers analyze the heat maps, and their typical instructional moves resulting from the assessments. We conclude the section with a description of teachers’ analyses of class reports in plc meetings. Students’ uses. We collected 6052 responses4 to a brief student survey about all of MM6-8’s assessments in the 2016–2017 school year. Students reported that the assessments reflected the material taught in class (95%), and that the tests were challenging but not too hard (85%). In the same year, we collected 874 responses to a survey about the student reports in which 76% of the responses indicated that the information in the reports was useful. Students indicated that they used the test report to make specific claims about the constructs that they performed well on, as well as those on which they required additional work. For example, when asked for one interpretation of their reports about percents, responses included the following: I need to work on percents and benchmark percents. I noticed that I need to keep working on percents as an amount per 100, when they are not in a 1x10 or 10x10 grid. I know most of the ratio topic but I also need to work on the multi-step percentage problems. I will retry the incorrect problems and work through them. During design studies and user-testing interviews, students expressed appreciation of the revise and resubmit features. These features engaged them as partners in their learning, allowing them to respond to the assessment results,

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and demonstrated learning gains as the result of reviewing and reflecting on feedback from assessments. MM6-8 also provides students with opportunities to practice items within single constructs. Students have been observed examining the color-coded ladder in their data reports to identify the levels of the items on which they scored poorly, and then targeting those particular levels of the LT in the practice mode: the LT levels become meaningful to students through the items associated with each level. Overall, the students showed benefits of being engaged in “assessment for learning” (Black, Harrison, Lee, Marshall, & Wiliam, 2003) as they recognized the system as a means of self-awareness and of providing opportunities for them to improve. As they began to use the language of the map and trajectories to describe their strengths and weaknesses, they made plans for revising or reviewing content, demonstrating empowerment to act on their data. However, the degree to which students come to use the data formatively depends significantly on the degree to which the teacher shifts the class’s orientation to assessment from one of teacher evaluation of the student to one of student and teacher sharing the responsibility for supporting growth (Confrey, Toutkoushian, & Shah, 2018). Teachers’ uses. The data consisted of videos of teachers reviewing results with their students in class. The observation methods, coding of qualitative data, and the results from our qualitative study are the focus of another publication (Confrey, Maloney, Belcher, McGowan, Hennessey, & Shah, 2018). In this section of the chapter, we summarize the findings from our study of classroom observations. We found that teachers deploy a variety of methods to review the data. They select items for class review, decide on what roles students have in this review process, use various feedback mechanisms including item analysis data, and leverage the structure of the map and the LTs in classroom discussion. Frequently, they begin by noticing patterns of correctness and incorrectness as color-coded in the heat maps, identifying transitions from non-understanding to understanding. For example, when reviewing “Finding Key Percent Relationships,” one teacher noted: “We have done well understanding benchmark percentages and their estimated values, but are having trouble equating percents in other situations such as percent of a whole [Construct B Level 5] or more than 100% [Construct C Level 3].” Having noticed the patterns, a common instructional move is to review selected levels, and specific assessment items with the whole class. A teacher might select particular items on which student performance was weaker, or items taken by all her/his students. Teachers use the features described previously concerning item analysis and misconceptions to generate spirited class discussions. A major source of variability among teachers in their use of the heat map data is the degree to which the teachers are student-oriented. When teachers first use MM6-8, a typical tendency is to narrate to students how to solve a

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difficult problem (despite the designer’s intent for the problems to encourage discussions). Many teachers, with professional development, eventually learn to use the system’s features to prompt themselves to ask a student to describe the goal of the problem, or to suggest a first step towards the solution, and then let the class (individually or as a group) work on solving that item. At other times, students are provided complete opportunities to share their approaches. With experience, teachers use other instructional moves, such as engaging in small group instruction, peer tutoring, targeted practice, and re-testing. Teachers at one of our partner districts regularly use stations once a week in their mathematics classes, one of which involves meeting with the teacher to discuss targeted needs based on their MM6-8 assessment results. Another station involves students working through practice items in MM6-8, where students can choose the constructs and LT levels on which to practice, as informed by their score report. We have observed that teachers’ own review of the data seems to be necessary for them to develop a deeper understanding of the learning trajectories. Even when provided with substantial professional development around an LT prior to teaching a unit, teachers often need to see the value of the LTs based on the data from their own class’s reports. At first, they treat the results for any single item as evidence about that item in isolation, but over time and with more experience, they learn to see an item as representative of a LT level. For instance, an item requiring students to informally calculate 23% might be treated as adding 20% and 3x1%, but more experienced teachers encourage students to work with a variety of combinations of benchmark percentages to respond to such items. Professional Learning Communities (plc). To support teachers in interpreting and acting on the data from MM6-8’s assessments, we collaborated with teachers to develop a data review protocol to use in their grade-level plc meetings. After sharing best practices for reviewing data, acting on formative assessments, and norms for plcs (Heritage, 2007; Kruse, Louis, & Bryk, 1994; Love, 2004), we established a process for reviewing common assessment data and discussing the reviews’ results. The process involved individual teachers reviewing the data in preparation for the PLC meeting, sharing perceived general trends and noticeable deviations, and discussing common difficulties across classes. These common difficulties are used to identify potential weaknesses in the curriculum or instruction on a topic. To support continuous improvement, the discussion includes time for sharing instructional strategies and reviewing the structure of the curriculum to address the identified weaknesses, possibly through re-teaching, inclusion of topics in future units, or revising curricular content for the next year. During a summer workshop, when teachers discussed their schools’ data for the “Finding Key Percent Relationships” cluster, they observed that “... [students] know Benchmark Percents; about half of the students are able

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to do Percents as Amount per 100.” They added, “students seem to know when percentages should and should not add to 100 [level 3 of “Percents as Amounts per 100”].” After reviewing the results for “Percents as Amounts per 100,” level 4 (“Describes percent (using percent bars, ratio boxes, or 100ths grids) as amount relative to the size of the total and distinguishes absolute (measure or count) from relative amounts (percent)”), teachers developed a plan to revise instruction, agreeing to put more emphasis on the conceptual idea of a percent, and planning to “reteach the concept of what a percent is.” Teachers also decided to incorporate ratio boxes into their percent lessons, to support students in understanding percentages as ratios. At the plc meetings during the school year, teachers also discuss how they help students interpret the MM6-8 scores, which are often lower than teacherdeveloped unit test scores. They also express a need to help students to understand that the value of classroom assessments is different than that of the typical high-stakes tests and district-led benchmark assessments. One teacher, who refers to the tests as “check-ins,” values the conceptual nature of the problems, and maintains high expectations for her students while supporting them to see mathematical learning as a process of growth. When discussing ways to present the assessments with her plc, she shared with teachers the way she described the tests for her students: “[Math-Mapper] is above and beyond [the end of year assessment]. If you can do [Math Mapper], you can do anything.”

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Student Results on Key Percent Relationships by Class Type

This section reports on results from field testing by district and class type. We note that our partnership with collaborating districts is ongoing. We design table 4.1  Demographics for collaborating school districts

District 1 District 2 Population served 977  African-American (%) 27  Asian (%) 1  Hispanic and Mixed (%) 10  White (%) 53 Percent free and reduced lunch 56.9 Number of years implementing 1-1 computing 3 Number of teachers participating 19 Total number of tests taken in 2015-16 and 2016-17 school year 9,197

1163 4 9 8 79 9.9 5 33 21,696

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new features, study their implementation, carry out iterative phases of validation, and refine the tool based on data analysis using a model of continuous improvement. We support teachers’ professional growth as we observe and learn from their interactions with their students and peers. MM6-8 continues to be field tested at three partner schools in two districts. District 1, listed as low-performing at the state level, has only recently transitioned to digital resources. District 2, serving upper middle-class children, has been recognized as high-performing, has used digital resources extensively (Table 4.1). As described previously, students and teachers receive assessment feedback during regular instruction. The research team subsequently examines the results for two other general purposes: to identify overall patterns in student outcomes, and to validate and refine the tool. In this section, we report the overall results on the cluster on “Key Percent Relationships” to discuss what can be said about student learning in this cluster. Results for District 1 (D1) are reported for grade 6 because the cluster is targeted at sixth grade. All sections are reported together (there is no tracking). District 2 (D2) differentiates regular and accelerated classes at each grade (6, 7); their grade 7 data come from an administration of the cluster test as a pretest in preparation for the next cluster. Figures 4.4, 4.5, and 4.6 show the cumulative student results for each construct, by proportion correct and incorrect, for different types of sections in the two districts. The number of item responses per level ranged from n=1 to 259. Some rows of the compound bar diagrams are empty (e.g. Level 1, D1, grade 6) because the levels were not sample on forms. While different grades and sections exhibit different percent correct at corresponding levels, when percent correct for levels is ranked (for instance, 1 to 5 for a 5-level construct) the ranking of levels is the same for every grade and section. The results from students in the regular sixth grade class in D1 showed slightly higher overall performance than those of the regular classes in D2, but slightly lower than those of the accelerated classes in D2. Patterns of results across the different constructs also reveal important characteristics of the rlc itself. On the learning map, constructs A and B are positioned as equally difficult; however construct A is clearly more difficult than B, but easier than construct C (Figures 4.4–4.6). Our interpretation of this result is that students are effectively leveraging informal knowledge of benchmark percents as they enter into the study of percents. And we believe that the increased difficulty of “Combinations” is consistent with the importance of the earlier constructs in learning it, particularly, construct A, which sets up percent as a ratio. A detailed validation study of this rlc is beyond the scope of this chapter and the focus of a separate article (Confrey et al., 2018).

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figure 4.4 Compound-bar display of proportion correct/incorrect for tested LT levels (L) of construct A, distinguished by district (D1, D2), class section type (regular, accelerated), and grade (6, 7). Each of the five sets of bars corresponds to the cumulative results from all class sections of indicated type. Error bars indicate ±1 S.E

Discussion of Construct A Results. The two lower levels of the LT in construct A showed some surprising student weaknesses. Items at the first level were designed to determine whether, when comparing ratios a:b and c:d, students understood that having a common value (for instance, if a=c), they could then simply compare the other values (b,d) to determine which ratio was greater. The second level builds from there to see if they could use 100 as that common value within the context of using a ratio box or table. To illustrate the lower level construct A items, we present an example. In it, of the 67 responses on item 695, a level-1 item of construct A, 30 (45%) found the least common multiple (LCM) (Figure 4.7); ten answered with 2, the least common factor (other than 1) among the serving sizes; 12 gave other fac-

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figure 4.5 Compound-bar display of proportion correct/incorrect for tested LT levels (L) of construct B. Information is displayed as in Figure 4.4

tors such as 5 or 10. On the last three responses, which depended on the first response, the correct responses dropped to 22, 21, and 25 percent, suggesting that, overall, about half of the students who successfully found the LCM could solve the entire problem. However, the incorrect answers ranged from less than 1 gram to over 3000 grams. A weak performance on this introductory level of the LT suggests that students may have an inadequate grounding in ratios and/or fractions, specifically, with regard to recognizing how having a common value facilitates comparison (concepts that were covered in the previous big idea, which has to do with one-dimensional numbers). Another pattern observed in the data concerned behavior of certain levels within constructs, especially when the compound bar diagram deviates from

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figure 4.6 Compound-bar display of proportion correct/incorrect for tested LT levels (L) of construct C. Information is displayed as in Figure 4.4

the Guttman5 shape. Both items for construct A, Level 3 “Quantifies subsets of a total as percents and recognizes when percentages should or should not add to 100%” were easier than expected (Figure 4.4). One item presented a 10x10 board with three colored regions, two of which were gridded, allowing counts. Students’ task was to find the number of squares in the remaining grid and interpret those as a percent of the area (its item difficulty6 b was –1.1; SE=.56.). A second item at this level, which referred to three kinds of jawbreaker candies, reported the percentage of two and asked for the percentage of the third, performed nearly identically (b=-1.1, SE=.20). In a validation study of this cluster, this level was flagged as too easy (across all items in the cluster, the b parameter

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figure 4.7 Item 695, level 1, construct “Percents as Amount per Hundred.” Correct responses are indicated next to the [1] on right side

figure 4.8 Item 1294, level 4, construct A (“Percent as Amounts per 100”)

ranged from –2.5 to 3.6) (Confrey et al., 2018). This analysis suggests that the higher-than-expected performance on this level may imply a need to designate the level itself to a lower position in the LT. At construct A’s top level (level 4), “Describes percent (using percent bars, ratio boxes, or 100ths grids) as amount relative to the size of the total and distinguishes absolute (measure or count) from relative amounts (percent),” student performance was weak. The data from the item aligned to this level

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in Figure 4.8 suggest the students were not distinguishing between relative and absolute measures. Only 9 of 66 students (13.6%) correctly chose both the second and third response. Nearly half (n=34) chose the second option as the only correct answer, perhaps indicating that they know how to build to 100 lbs to figure out amount of corn. Two students chose the third option as the only correct answer. Overall, 20 students included the first option in their choices, nearly all (59) included the second option, 21 students included the third option, and 17 included the fourth. It appears that with the exception of option 2, the other options were equally appealing. We plan to conduct some think-aloud interviews with this item and, at the same time, to advise teachers to do more curricular work on the differences between relative and absolute measures. Discussion of Construct B Results. Students performed well on the first two levels of benchmark percents, suggesting a level of familiarity with 100% and 50%. At level 3, “Associates benchmark percents of 25%, 75%, 10%, 5%, and 1% with corresponding ‘fractional amounts of’ a quantity and describes interrelationships using figures, percent bars or circles, or 100ths grids,” students performed better than expected (Figure 4.5). Responses ranged from 91–94 % correct on both level-3 items; closer scrutiny revealed that the items could be solved simply by matching and eliminating options. These items illustrate a kind of construct-irrelevance, which undermines their validity, elimination of which is a high priority; these items are being revised. Between 10% (accelerated) and 60% (regular) of students had difficulty with applying benchmark percents to find an amount of a quantity (level 4), and between 40% (accelerated) and 90% (regular) of students had difficulty reversing that process to find the original amount (level 5). A number of challenges were evident in responses to a level-5 item (“Given the benchmark percentage and the numeric value of the subset, finds the size of the collection”). Of 231 students, 99 (43%) gave the correct answer to Item 698 (Figure 4.9). But many students showed signs of the misconception “Recognizes no clear distinctions among 10%, 1%, .1%, and .01%”: 73 students (32%) gave answers that were off by powers of ten. Of those, nearly half gave an answer of the distance to Mars being 3400 thus both reversing the

figure 4.9 Item 698, level 5, construct B (“Benchmark percents”)

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figure 4.10 Item 1292, level 2, construct C (“Percents as combinations of other percents”)

direction of the calculation and providing an absurd answer given the context. Also 12 students showed evidence of subtracting either 1% or 10% of 340,000 to produce their answer. These results suggest teachers need to work more with early forms of equations using benchmark percents, and they also need to strengthen students’ estimations working with 1% and below especially for items that had high numeric values. Discussion of Construct C Results. For this construct one sees overall weaker student performance. At level 1, students work only with combinations of 5 and 10%. By level 2, when 1% is included, performance drops precipitously. Among sixth graders, 40% (accelerated) to 75% (regular) show difficulties with the items. On item 1292 (Figure 4.10), of 78 responses, only 25 (slightly less than one third) correctly recognized the equivalence of the two methods shown in the item and gave the correct answer. Nearly half the students were able to calculate the percent correctly, but not identify that both strategies of reasoning were correct. This may suggest that students are being taught the procedures for such problems, without being able to reason more informally with benchmarks. In a problem from level 3 (“Extends benchmark of 100% to scale to, for example, 200% or 300% as 2 times or 3 times as large”) of construct C, students were asked to identify pairs of values where the second value was 300% of the first. 70% selected (1 and 300) as one of their answers, showing evidence of a common misconception that teachers should address with the class. Overall, these results demonstrate the value of the structure of the individual learning trajectories and of the clusters in helping teachers to see trends in their classes and summarize the results of their instruction. In the context of this “Finding Key Percent Relationships” cluster, a primary takeaway across all the classroom sites involved is that students are still strug-

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gling to distinguish relative from absolute amounts with respect to percents, and need to work with this idea more. Many students show facility with benchmark percents, but weaknesses begin to appear in applying benchmarks over 100% or under 1%. Difficulties that are in evidence for the first two constructs accrue in construct C, where there is some evidence of relying too heavily on procedural approaches and a lack of skills to evaluate the reasonableness of responses. These patterns of responses apply to more than half the students in the regular classes, and to between 20% and 40% of the students in accelerated classes.

6

Conclusions

This chapter reports on the use of a new digital learning system (dls), MathMapper 6-8, as a systematic way to use learning trajectories to improve instruction and strengthen teachers’ knowledge base regarding empirically established patterns in student learning. Learning trajectories are embedded in the dls as a means to leverage students’ initial knowledge and gradually transform that knowledge to reach an understanding of a target construct, via descriptions of likely progress levels. MM6-8 represents a major step forward in providing both a cognitive framework for mathematics learning and coherent, detailed, real-time data on student learning to both teachers and students. Our approach to learning trajectories is not to assume one single most efficient path to the target construct, despite the fact we represent growth as if it involved climbing a ladder. Instead, we conceptualize learning trajectories as more akin to a rock-climbing wall, to acknowledge that students may exhibit diverse paths of reasoning rather than simply traversing linear sequences of equally-spaced steps as they struggle to master a target construct. Organizing our map around big ideas, clusters, and LTs allows us to avoid the pitfalls of using standards to guide and sequence instruction. By using LTs, our approach is (1) more resistant than standards to shifting political and proprietary sensitivities, (2) more oriented to student learning, and (3) grounded in cognitive research and more consistently organized around fine-grained LT progress levels. We develop diagnostic assessments associated with LTs to provide precise information on how students are progressing in relation to a specified cognitive framework. Thus, MM6-8’s LTs are not tied to any single curricular or instructional sequence, and the dls can support implementation of diverse curricula and comparison of outcomes.

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A critical component of our dls is its support for students to become active participants in their own learning process. We showed how their participation and reflection are supported by the data provided to them via reports, opportunities to revise and resubmit answers, to practice and review the items, and, when facilitated by teachers, to retake tests. Another important realization from our work was based on our observations of teachers. Some teachers were more successful than others at making in-class data reviews into opportunities to elicit student thinking and thus to engage in “assessment for and as learning.” We demonstrated how teachers began by treating the rows of the heat maps as a means to simply review an item, using the tool as if they were reviewing a regular test, and telling the student what they did wrong and how they could correct their responses. However, they gradually learned to extend to a more generalized perspective of a LT level. Then, once they were provided with the item analysis and indicators of misconceptions, they found these to be important tools that enabled them to create a dialogue about the item responses and to invite students to share their own thinking. Increasingly, we observed how teachers began to view items as indicators of behavior in relation to the levels, rather than simply as examples, and how they then could in turn invite students into a learning trajectory perspective, one that supports and is supported by a growth mindset (Dweck, 2006; Boaler, 2015). Despite all these advances in our understanding of our dls and the potential contributions of focusing on LTs, our student results for the “Key Percent Relationships” cluster, presented by school and class, showed that many students are still mastering only lower levels of the trajectories. One important contribution of MM6-8 is to provide teachers with data concerning all the students in the room. Our partner practitioners have been showing us how this leads them to develop new routines that are responsive, via the assessment data and displays, to all the students. Our original premise for this chapter was that the concept of learning trajectories offers a feasible means of providing systematic professional development to all teachers in a school. Our current collaborations suggest that this approach is especially viable when a school’s leadership legitimates and supports the data practices. A final reflection addresses a claim made early in the chapter concerning the state and evolution of the learning trajectories. Because MM6-8 is a digital application and its design and implementation anticipates change, it is possible to make ongoing updates to the application. This is undertaken regularly as items are added and revised and as learning trajectories are revised in light of data, professional collaborations, and new research findings.

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Notes 1 From hereon we will refer to the relational learning cluster (rlc) simply as “cluster.” 2 Since this chapter was published, the levels of Construct A have been revised based on the analysis documented later section in the chapter. Revised levels can be found on sudds.co 3 In MM 6-8, when working on a given big idea, we recommend that teachers begin with the bottom left rlc and progress towards the rlc on the top right. 4 Individual students may have completed a survey after two different cluster tests. Those results are reported independently of each other. 5 Decreasing proportion of blue from bottom level to top level. 6 Item difficulty was established through the conduct of an Item Response Theory (irt) analysis of the data.

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Confrey, J., Maloney, A., Belcher, M., McGowan, W., Hennessey, M., & Shah, M. (2018). The concept of an agile curriculum as applied to a middle school mathematics Digital Learning System (DLS). International Journal of Educational Research, 92, 158–172. Confrey, J., Maloney, A., Nguyen, K., Mojica, G., & Myers, M. (2009). Equipartitioning/ Splitting as a foundation of rational number reasoning using learning trajectories. Paper presented at the in 33rd Conference of the International Group for the Psychology of Mathematics Education, Thessaloniki, Greece. Confrey, J., Maloney, A. P., Nguyen, K. H., & Rupp, A. (2014). Equipartitioning, A foundation for rational number reasoning. In A. Maloney, J. Confrey, & K. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (1st ed., p. 61). Charlotte, NC: Information Age Publishing. Confrey, J., Toutkoushian, E., & Shah, M. (2018). A validation argument from soup to nuts: Assessing progress on learning trajectories for middle school mathematics. Assessment in Education: Principles, Policy and Practice, 1–16. Council of Chief State School Officers. (2010). Myths vs. facts. Retrieved from http://www.corestandards.org/about-the-standards/myths-vs-facts/ Darling-Hammond, L. (2015). The flat world and education: How America’s commitment to equity will determine our future. New York, NY: Teachers College Press. Dweck, C. S. (2006). Mindset: The new psychology of success. New York, NY: Random House Publishing Group. English, L. D. (Ed.). (2002). Handbook of international research in mathematics education. New York, NY: Routledge. English, L. D. (Ed.). (2008). Handbook of international research in mathematics education (2nd ed.). New York, NY: Routledge. Grouws, D. A. (Ed.). (1992). Handbook of research on mathematics teaching and learning (1st ed.). New York, NY: Macmillan. Heritage, M. (2007). Formative assessment: What do teachers need to know and do? Phi Delta Kappan, 89(2), 140. Hoyles, C., & Lagrange, J. (Eds.). (2010). Mathematics education and technologyrethinking the terrain: The 17th ICMI study. New York, NY: Springer. Kieran, C., Krainer, K., & Shaughnessy, J. M. (2013). Linking research to practice: Teachers as key stakeholders in mathematics education research. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Third international handbook of mathematics education (pp. 361–392). Dordrecht, The Netherlands: Springer. doi:10.1007/978-1-4614-4684-2_12 Kruse, S., Louis, K. S., & Bryk, A. (1994). Building professional community in schools. Issues in Restructuring Schools, 6(3), 67–71. Lehrer, R., & Schauble, L. (2015). Learning progressions: The whole world is NOT a stage. Science Education, 99(3), 432–437. doi:10.1002/sce.21168

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Lester, F. K. (Ed.). (2007). Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (2nd ed.). Charlotte, NC: Information Age Publishing. Love, N. (2004). Taking data to new depths. Journal of Staff Development, 25(4), 22–26. Middleton, J., van den Heuvel-Panhuizen, M., & Shew, J. A. (1998). Using bar representations as a model for connecting concepts of rational number. Mathematics Teaching in the Middle School, 3(4), 302–312. Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122–147. doi:10.2307/749607 Nguyen, K. H., & Confrey, J. (2014). Exploring the relationship between learning trajectories and curriculum. In A. P. Maloney, J. Confrey, & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 161–185). Charlotte, NC: Information Age Publishing. Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421–481. doi:10.3102/00346543065004421 Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205 Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91–104. doi:10.1207/s15327833mtl0602_2 van den Heuvel-Panhuizen, M. (1994). Improvement of (didactical) assessment by improvement of problems: An attempt with respect to percentage. Educational Studies in Mathematics, 27(4), 341–372. doi:10.1007/BF01273377

CHAPTER 5

Researching Mathematical Reasoning: Building Evidence-Based Resources to Support Targeted Teaching in the Middle Years Dianne Siemon and Rosemary Callingham

Abstract This chapter traces the origins of the Reframing Mathematical Futures II (rmfii) research project before describing how rich assessment tasks and Rasch analysis techniques were used to develop an evidence-based resource to support the teaching and learning of algebraic, statistical and geometric reasoning in Years (i.e., Grades) 7 to 10. Hypothesised learning progressions were developed for each of the three target domains based on prior research. Potential assessment tasks, together with detailed scoring rubrics, that addressed the learning progressions were developed and trialed. A detailed examination of the Rasch analysis output indicated modifications that were needed and the modified questions were retrialed. Finally, the resulting variable was interpreted and segmented into Zones that were used as the basis for developing teaching advice. In describing this process, we comment on the application of learning trajectories more generally and suggest further avenues for research in this field.

Keywords mathematical reasoning – Rasch measurement – formative assessment – middle school – professional development

1

Developing an Evidence-Based Framework for Mathematical Reasoning

Despite the endorsement of important mathematical practices such as problem-solving and mathematical reasoning in the acm since its inception in 2010, the mathematics to be taught is typically represented as a set of disconnected topics and skills to be demonstrated and practiced rather than explored, © koninklijke brill nv, leideN, 2019 | DOI:10.1163/9789004396449_006

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discussed and connected (Shield & Dole, 2013; Siemon, Bleckly, & Neal, 2012; Sullivan, 2011; Vincent & Stacey, 2008) and prompting relatively narrow forms of assessment that value performance over mastery (Dweck & Leggett, 1998; Sullivan, 2011). This particular relationship between curriculum, instruction and assessment has been characterised by Black, Wilson, and Yeo (2011) as a “vicious triangle” and by Swan and Burkhardt (2012) in terms of What You Test Is What You Get (wytisyg). Where curriculum drives both instruction and assessment, assessment has a “damaging backwash effect” (p. 4) on what is taught and how, with the result that, along with important mathematical practices, ‘big ideas’ such as multiplicative thinking and mathematical reasoning are rarely given the attention they deserve (Siemon, 2017). One of the major reasons that secondary school teachers in the initial rmf-p project were reluctant to adopt a targeted teaching approach to multiplicative thinking was their perception that this was not related to the curriculum they were expected to teach. An analysis of the Australian Curriculum: Mathematics (acm) at the time (Australian Curriculum, Assessment & Reporting Authority [ACARA], 2018) found that approximately 75% of the Year 8 curriculum required or assumed student access to multiplicative thinking (Siemon, 2013) but this fact was not easily apparent to the teachers in the project. This situation points to the critical importance of alignment between curriculum, instruction (i.e., teaching) and assessment (Swan & Burkhardt, 2012), and has important implications for the development of Zone-based activities to support a targeted teaching approach to all areas of the mathematics curriculum. A focus on mathematical reasoning is needed to equip teachers with the knowledge, confidence and disposition to go beyond narrow skill-based approaches to teaching mathematics in the middle years. Defined broadly in the acm as a “capacity for logical thought and actions,” mathematical reasoning has a lot in common with mathematical problem solving, but it also relates to students’ capacity to see beyond the particular to generalize and represent structural relationships, which is a key aspect further study in science technology, engineering and mathematics (i.e., stem) related fields (Wai, Lubinski, & Benbow, 2009). This situation is not unique to Australia. Compared to the curricula of countries that do well on international assessments of mathematics achievement, the elementary and middle school mathematics curriculum in the United States has been “characterised as shallow, undemanding, and diffuse in content coverage” (Kilpatrick, Swafford, & Findell, 2001, p. 4) leading to calls by these authors and others (e.g., Stacey, 2010; Sullivan, 2011; Swan & Burkhardt, 2014) for a much greater focus on mathematical problem solving and reasoning.

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1.1 The Reframing Mathematical Futures II Project (RMFII, 2014–2017) The second tranche of amspp funding provided an opportunity to explore the extent to which it was possible to work with teachers to develop evidencedbased resources for mathematical reasoning that brought about a better alignment between curriculum, instruction and assessment (Pellegrino, 2002; Swan & Burkhardt, 2012). The aim of the rmfii project was to build a sustainable, evidence-based, learning and teaching resource to support the development of mathematical reasoning in Years 7 to 10 that could be used formatively in the way described by Wiliam (2011). That is, to inform a deeper, more connected approach to teaching mathematics that recognises and builds on what learners already know and takes them beyond low-level skills and routines. For this purpose, mathematical reasoning was seen to encompass: i. core knowledge needed to recognise, interpret, represent and analyse algebraic, geometric, statistical and probabilistic situations and the relationships/connections between them; ii. an ability to apply that knowledge in unfamiliar situations to solve problems, generate and test conjectures, make and defend generalisations; and iii. a capacity to communicate reasoning and solution strategies in multiple ways (i.e., diagrammatically, symbolically, orally and in writing) (Siemon, 2013). This focus was both pragmatic – it had the potential to address the concerns of the rmf-p teachers who felt constrained from adopting a targeted teaching approach by the perceived demands of the curriculum; and theoretical – it offered an opportunity to build probabilistic models of student learning in relation to mathematical reasoning that could be used to improve the quality of educational practice. As a result, and because the aim of the rmfii project aligned with the goals of the Australian Association of Mathematics Teachers (aamt), aamt was invited to partner in the project with a view to disseminating and scaling up project outcomes to a wider professional audience (Cobb & Jackson, 2011). Given this focus and the commitment to work with teachers on the design and trial of assessment tasks to support the development and subsequent use of the learning progressions, design-based research methods were seen to be most appropriate (e.g., Barab & Squire, 2004; Design-Based Research Collective, 2003). An important goal of design-based research is to “directly impact practice while advancing theory that will be of use to others” (Barab & Squire, p. 8). Design studies are typically interventionist and conducted in naturalistic settings to better understand the “messiness of real-world practice” (p. 3). They

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generally involve a multi-disciplinary team working with practitioners in successive iterations of “design, enactment, analysis, and redesign” (Design-Based Research Collective, 2003, p. 5) to develop “theories about both the process of learning and the means … to support that learning” (Cobb, Confrey, di Sessa, Lehrer, & Schauble, 2003, p. 10). The following research questions framed the work of the rmfii project. i. To what extent can we develop rich tasks to accurately identify key points in the development of mathematical reasoning in the junior secondary years? ii. To what extent can we gather evidence about each student’s achievements with respect to these key points to inform the development of a coherent learning and assessment framework for mathematical reasoning? iii. To what extent does working with the tasks and the knowledge they provide about student understanding assist teachers to improve student’s mathematical performance at this level? iv. What strategies and/or teaching approaches are effective in scaffolding mathematical reasoning in the middle years? v. What are the key features of classroom organisation, culture and discourse needed to support/scaffold students’ mathematical reasoning at this level? The first step in this process was to secure the services of a research team with recognised expertise in each area, that is, algebraic reasoning, geometric reasoning, statistical reasoning, assessment task design, the use of Rasch analysis, and the design of online professional learning. The research team, which also included two aamt representatives, met for the first time in August 2014 to refine the methodology, develop detailed time frames, and discuss individual responsibilities. Their contributions to the rmfii project are described in later chapters. 1.2 Participants Project partners in all Australian States and Territories identified between four to six secondary schools in their jurisdiction that met the amspp funding requirements to participate in the rmfii project (i.e., located in lower socioeconomic regions with diverse student populations). This resulted in a total of 32 secondary schools nationally, 20 of which had participated in the rmf-p project. A Specialist teacher, one from each school was identified and supported to work with up to 4–6 other teachers in their school to trial assessment tasks and reasoning activities and support a targeted teaching approach to mathematical reasoning. From 2015 to 2017, approximately 80 teachers, and 3500 students in Years 7 to 10 were involved in the project. Project schools were visited at least twice a year by a member of the research team and residential

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professional learning opportunities were provided on an annual basis. Online professional learning sessions were held at least once per school term to address learning needs identified by Specialists. An additional 1500 or so Year 5 to 10 students from other schools participated in the trialling of the assessment tasks at various stages of the research. 1.3 The Approach A similar approach to the one used to develop the evidence-based learning and teaching resources for multiplicative thinking was adopted for the rmfii project (Siemon & Breed, 2006; Siemon, Izard, Breed, & Virgona, 2006). That is, hypothetical learning progressions were derived from the research literature for each area of mathematical reasoning. The progressions were then used to inform the design of rich assessment tasks that in turn could be used to test the learning progressions through successive iterations using Rasch analyses (Bond & Fox, 2015) to inform the resulting teaching advice. Professional learning opportunities, mentor visits, and project funding was provided to the school-based Specialists to enable them to work with other teachers at their school to conduct the assessments, trial mathematical reasoning activities and provide feedback to the research team. Phase 1 focussed on the identification of the ‘big ideas’ in mathematical reasoning and the derivation of hypothetical learning progressions in each area. The members of the team with specific expertise in each area were charged with identifying assessment tasks that, where possible, could assess reasoning vertically (i.e., at different levels of complexity within the same hypothetical learning progression) and horizontally (i.e., at similar levels of difficulty across different hypothetical learning progressions). An example of one such task is given in Figure 5.1. The ‘big ideas,’ hypothesised learning progressions and assessment tasks were interrogated by other members of the research team at an extended faceto-face meeting prior to the residential professional learning workshop for Specialists and project partners in November 2014. The workshop introduced the rmfii project and explored a range of exemplary mathematical reasoning activities. New schools were introduced to the snmy materials and targeted teaching by the rmf-p Specialists. During this phase, the Specialists were asked to provide feedback on the trial assessment tasks and scoring rubrics. While this process provided valuable information about the readability and suitability of the items for the students in the research schools, it did not yield information about the types of reasoning hypothesised to be at the upper end of the progressions. To this end, and to test the suitability of the assessment tasks more broadly, multiple task booklets, referred to as Forms, were prepared

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and trialed in a range of non-project schools. The forms comprised five to six tasks – either from the one area (Standard Forms) or from two areas (Mixed Forms). Common tasks were included across forms to support analysis as well as anchor items from the snmy assessments to investigate relationships between mathematical reasoning and multiplicative reasoning. The extended trialing generated interest from a diverse range of schools from different states and territories that resulted in over 1000 responses to the Forms from students across Years 5 to 10, which were coded by a team at rmit University and analysed using Master’s (1982) Rasch partial credit model and Winsteps 4.0.0 Rasch Measurement (Linacre, 2017). The resulting ordered lists of item rubrics were used to review and refine the hypothetical learning progressions in each area, the refined versions of which became known as Draft Learning Progressions (dlps). This process identified some gaps in the progressions that prompted the redesign of some items and/or rubrics and/or the design of additional items to further test and elaborate the progressions. Phase 2 focussed on the design and trial of additional assessment items to test under-evidenced aspects of the dlps. This led to the preparation of a revised set of mathematical reasoning forms (MR1) that were used by research schools between September 2016 and March 2017. The forms were marked by the teachers at the school using the scoring rubrics provided and the de-identified results were forwarded to the research team for analysis Valid responses were obtained from over 3360 students and analysed using the Rasch partial credit model (Masters, 1982) and Winsteps 4.0.0 Rasch Measurement (Linacre, 2017). While this confirmed many aspects of the dlps, the findings prompted a further round of assessment (MR2) in the Australian autumn of 2017 to ‘fleshout’ certain aspects of the dlps. The data from MR1 and MR2 were then considered together to refine the dlps and inform the development of the teaching advice. This phase also included the analysis of student and teacher on-line surveys, and the development of teaching advice and professional learning modules to support a targeted teaching approach to mathematical reasoning. The final phase of the project will evaluate data from the MR3 and MR4 rounds of assessment conducted in late 2017 or early 2018 to determine the efficacy of a targeted teaching approach to mathematical reasoning. It will also focus on the development and publication of project outcomes and reports for publication on a web-based platform hosted by aamt. The remaining sections of this chapter will focus on key aspects of Phase 2: task design; the use of Rasch modelling to test the dlps for algebraic, statistical and geometric reasoning; and the derivation of domain-specific teaching advice from the analysis of student responses to the final versions of the assessment forms (i.e., MR1 and MR2).

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1.4 Task Design As indicated above and as will be illustrated in Chapters 6, 7 and 8 (this volume), a range of assessment tasks were developed to asses an inferred sequence of ideas/strategies within the same learning progression and, where possible, connections to other learning progressions for mathematical reasoning or multiplicative thinking. In general, tasks were developed around a meaningful context (e.g., packaging a gift to be sent overseas, finding the best route in an emergency, or making sense of statistical claims made in relation to a school-wide survey) but in other circumstances tasks were set in decontextualized settings (e.g., explaining why a given relationship is true or false). Figure 5.1 shows three items from the Algebra Tile Task (atilp) together with their rubrics. These items were designed to assess aspects of algebraic reasoning but they also support inferences in relation to key aspects of the geometry learning progression (reasoning about perimeter and area).

figure 5.1 Three items from the Algebra Tile Task (atilp)

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Scoring rubrics were written for each item and revised on the basis of feedback from research school teachers and the Rasch item fit analysis (see below). The rubrics valued evidence of: i. core knowledge needed to recognise, interpret, represent and analyse algebraic, geometric, statistical and probabilistic situations and the relationships/connections between them; ii. ability to apply that knowledge in unfamiliar situations to solve problems, generate and test conjectures, make and defend generalisations; and iii. a capacity to communicate reasoning and solution strategies in multiple ways (i.e., diagrammatically, symbolically, orally and in writing) (Siemon, 2013). Because this introduces a subjective element into the assessment process, project funds were provided to enable research school teachers to meet together in school time to mark and moderate student responses. Although this was aimed at ensuring the rubrics were applied consistently, the debates it generated about how the rubrics and student responses were being interpreted was felt to be a very valuable professional learning experience. For instance, one of the issues that arose in relation to the Algebra Tile (atilp) problem was the higher score(s) given for simplest form (see Table 5.1) with many teachers feeling that it was ‘unfair’ to penalise students who had answered the question correctly but had not expressed this in simplest form. The team’s response was to remind the teachers that the students had been advised in the sample question at the start of the exercise to “use as much mathematics” as they could in presenting, explaining or justifying their responses. Our view was that a disposition to look for and express relationships in simplest form was a fundamental mathematical practice that supported reasoning with mathematical objects and that responses such as this were more likely to indicate a capacity to reason mathematically than responses that did not attend to these relationships. The construction of the rubrics in this way allows Masters (1982) partial credit model to be used to order responses according to their difficulty. For instance, atilp3.5 which indicates a score of 5 on the third item above, is ranked very much higher on the variable map produced as a result of the Rasch analysis than a response scored as atilp3.2 (Algebra map shown in Chapter 6, this volume). The following sections provide a detailed account of how Rasch analysis was used to inform the development of the learning progressions, assessment tools, and teaching advice. 1.5 Using Rasch Analysis to Develop Learning Progressions Rasch (Rasch, 1960) analysis is a technique that is widely used in education to produce measurement scales of a construct of interest (Bond & Fox, 2015). Unlike other modelling approaches, data are fitted to the appropriate Rasch model rather than the model being tested against the data. Although this may

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table 5.1  Scoring rubrics for three items from the Algebra Tile Task (atilp)

ATILP 1 Score

Description

0

No response or irrelevant response

1

Calculation based on numbers (e.g., assumes a = 5 or 6 cm) that shows an understanding of perimeter (e.g., 5 + 1 + 5 + 1 = 11 or 12 + 2 = 14)

2

Correct response but not in its simplest form (e.g., a + a + 1 + 1)

3

Correct, simplifijied response (2a + 2 or 2(a + 1))

ATILP 2 Score

Description

0

No response or irrelevant response

1

Incorrect, but attempt made to solve for the perimeter using symbols (e.g. a + a + 3 + 3; 4 a + 3) or assumes a = 5 or 6 cm (e.g., 10 + 3 + 10 + 3 or 12 + 12 + 3 + 3)

2

Correct response but not in its simplest form (e.g., 2a + 2a + 3 + 3)

3

Correct symbolic response in simplifijied form (e.g., 4a + 6 or 2(2a + 3))

ATILP 3 Score

Description

0

No response or irrelevant response

1

Incorrect , but partially identifijiable (e.g., 6a + 6), with little/no working or explanation to support response or incorrect calculated response based on a = 5 or 6 cm

2

Incorrect response due to minor errors but with working that shows understanding of perimeter or correct based on a = 5 or 6

3

Correct symbolic response but not simplifijied (e.g., a + 1 + 1 + a + 1 + a + 1 + a – 3 + a – 1 + a + 1 + 2 + 1) and without clear explanation

4

Correct symbolic response (6a + 4) without clear explanation (e.g., just added all the sides together) OR correct response but not simplifijied with a reasonable explanation/working to support solution

5

Correct symbolic response (6a + 4) with clear explanation for sides that are less than a (e.g., a – 1 or a – 3) or explanation based on visualisation.

appear to be a trivial difference, in reality it provides several advantages for a project such as rmfii. These advantages are: 1. A demonstration of the validity of the items used to create the scale through fit to the model;

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2.

Provision of an invariant measurement scale that can be used to identify longitudinal changes in students’ performance; 3. Removal of the need for every student to attempt every item providing the opportunity for a larger item pool to be used. These ideas will be explored in more detail in later sections of this chapter. Rasch models are underpinned by three assumptions: 1. The target construct is unidimensional. This assumption is sometimes regarded, as controversial but is the same assumption underpinning any examination or test. The construct may be ‘thick,’ such as mathematics, or more focussed, such as mental arithmetic, which is a subset of the larger construct of mathematics. In the rmfii project, the focus was on reasoning with specific domains of algebra, geometry, and statistics. 2. The items act additively so that a higher total indicates a greater quantity of the target construct. This assumption intuitively makes sense in any assessment process. 3. Items work independently from each other. This assumption is sometime violated in tests of mathematics, where a correct response to an early question is sometimes required for success on subsequent questions. It is not, however, an ideal situation and steps were taken to avoid this situation in the design of the reasoning questions. Students can get correct answers using incorrect reasoning, and identifying this situation was important for the teachers involved in the project. Hence, the questions developed often required an elaborated response. These responses were coded using a scoring rubric that needed to be sufficiently robust that any teacher could use it reasonably accurately with minimal training. The scoring rubrics for most questions allowed for an increasing quality of response and were coded using a score code of 0 to n, where n was the maximum score for any particular question. Each score code was assigned according to a rubric that identified a distinct behaviour identified in the response. This partial credit approach is familiar to teachers, and they provided feedback at meetings about the score codes. Because of this approach, the specific Rasch model applied to the data was the Partial Credit Model (pcm) (Masters, 1982). This model can be expressed mathematically as

where πix is the probability of a person responding in category x (x = 1,2, … m) of item i;

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β is the person’s ability in the domain being measured by this set of items; and δix is the difficulty of the step threshold that governs the probability of the response occurring in category x rather than category x – 1. The analyses were undertaken using Winsteps 4.0.0 Rasch Measurement computer program (Linacre, 2017). The software provides an estimate of each person’s ‘ability,’ the measured performance against the specific set of items, and of item difficulty, in effect the measured performance against the specific group of test takers. The unit used is the logit, the logarithm of the odds of success. Item difficulties are constrained to be distributed around a mean of 0.00 logits. The powerful aspect of this approach is that all measures, of items and persons, are on the same scale, so that the actual position of test takers against the items can be identified. Output from the analysis can be in the form of a variable map. This map shows the position of both test-takers and items arranged on a vertical logit scale. Figure 5.2 shows the variable map for the Statistical Reasoning variable (see Chapter 9 this volume for details of the development). The logit scale is shown on the left-hand side and the # symbols indicate the distribution of persons along that scale. The items are shown on the right-hand side, identified by the item identifier used in the project. The vertical line separating the persons and the items is marked M for the mean value, S for one standard deviation from the mean and T for two standard deviations from the mean. Maps for Algebraic Reasoning and for Geometric Reasoning are shown in Chapters 6 and 7 respectively (in this volume). Persons whose measured ability lie at the bottom of the map are those with the lowest capacity for statistical reasoning. In general, they have provided the weakest responses to the questions asked. The opposite is true for persons appearing at the top of the map. These are the most capable students in terms of statistical reasoning. In parallel, the items are also arranged along the vertical scale from easiest at the bottom to hardest at the top. The mean item difficulty value is 0 logit, as constrained by the analysis, and the mean person ability value is shown around -0.5 on the map (the measured value was -0.8 but the pictorial output is an approximation). The proximity of the two values for persons and items, indicates that in general the items are well targeted to this sample of students, that is the items provide opportunities for all students to demonstrate their capacity. Students lying at the very bottom of the map generally scored 0 or 1 on the items that they attempted, and often these students did not attempt many items presented. Because the Rasch model is probabilistic, where students lie at the same logit value as an item, they have a 50% chance of success on that item. As an example,

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figure 5.2 Variable map for statistical reasoning items from MR1 and MR2 data (n = 1570)

a student having a measured ability of 0.00 logits would have a 50% chance of gaining a code pf 2 on the items srash and scon2 that both lie around this value on the scale. These students would have more that 50% chance of obtaining the

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relevant score codes for all items having difficulties below 0.00 logits and less than 50% chance on items above 0.00 logits of difficulty. These students would not be expected to score 2 on item spsyc for example. This comparison between students and items provides the basis for using score bands as a foundation for teaching advice. It also provides a ‘diagnostic’ element because teachers mark their own students’ work, and can identify unexpected responses, both better or worse than expected (Misailidou & Williams, 2003). 1.6 Using Rasch Analysis to Refine Questions and Items The Rasch model provided one way of testing the validity of the underlying construct through the idea of fit to the model. Because it is a requirement that data must fit the model, it follows that any data that do not fit the model must be examined. Winsteps gives four fit indices: Outfit and Infit mean squares, and a standardised t statistic for each of these. Outfit is an unweighted index and is sensitive to the presence of outliers. Infit is a weighted index and is more sensitive to unexpected patterns of response from persons where the items are roughly targeted on them, and vice versa for items. In general, for the Infit and Outfit indices, values lying between 0.5 and 1.5 are useful for measurement purposes. The standardised statistics take the usual values for statistical significance of ±1.98 (Linacre, 2017). A multi-stage process was used to establish that all the items were working together consistently to provide a usable measurement scale, in effect establishing the construct validity. The steps used were 1. Look at the overall fit for the test. 2. Look at the fit for each item. Those showing underfit (Infit/Outfit values large and positive; t values outside expected range) indicated some randomness in the data, suggesting that responses and/or coding were inconsistent. 3. Each underfitting question was examined in detail by the specialist teams to diagnose the problems. This examination included a consideration of the item map to see whether the question was covering a reasonable spread of the variable, and a focus on the underlying Big Idea to establish whether the question was useful and needed in the data set. 4. Changes were made to the question wording, to the rubric wording, or rubric categories were collapsed, or the question was deleted. Using this process ensured that a large number of questions was retained for further trialing, ensuring a wide pool of items that could be used to develop pre- and post-assessments that teachers could use to evaluate their teaching. Two questions from the geometry strand provide an example of this process. The questions, gang1 and gang2, address aspects of properties of shapes. They are shown in Figure 5.3.

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figure 5.3 Items addressing aspects of angles, gang1 and gang2

The initial scoring rubrics for these two items are shown in Table 5.2. These were based on experts trying to anticipate student responses, and order these to show increasing quality of response. The MR1 analysis for these two items showed considerable misfit. This is summarised in Table 5.3. It is clear that gang1 shows considerable misfit to the point that the contribution of the question to the overall scale would be compromised. gang2 shows less misfit but the relatively large Outfit values indicate that there are outliers that may be influencing the interpretation of the rubric. An examination of the location of each score code on the item map showed that for GANG1 the codes 1, 2 and 3 were located very close together on the map with difficulties of -0.79, -0.60, and -0.52 logits respectively. The two questions together addressed aspects of reasoning about properties of shapes and knowledge of angles, and the team wanted to retain both.

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GANG1 Score 0 1 2 3 4

Description No response or irrelevant response Disagree it is a rhombus but specify some of its properties correctly Disagree it is a rhombus but claim it is a parallelogram with some properties Agree it is rhombus but insufffijicient or incorrect properties to defijine it or claims it is a parallelogram and includes all properties Agree it is rhombus. Explanation needs to include necessary and sufffijicient properties, that is, it is a parallelogram with one of the following properties • 4 equal straight sides • Opposite angles equal, sides equal • Two lines of symmetry

GANG2 Score 0 1 2 3 4

Description No response or irrelevant response Incorrect angles At least 2 angles correct but no reason Two angles found correctly with sensible reasons or all angles correct with no reasoning All angles correct with clear reasons given relating to the folding and properties. F = 45o; h = 45o; s = 135o (e.g., Folding corner to centre creates half right angle; All angles around centre of side equal so any 2 make 45o or Four angles of quadrilateral add to 360o)

table 5.3  Fit values for gang1 and gang2 at MR1

Question

InfijitMSQ

Infijit t

OutfijitMSQ

Outfijit t

GANG1 GANG2

1.39 1.15

4.2 1.5

2.27 1.52

6.1 3.5

After discussion, a new set of rubrics was developed. For GANG1 the changes were considerable, collapsing the rubric to show score codes of 0 to 2 only. GANG2 was better exemplified in an attempt to eliminate any misunderstanding of the rubric that might have led to inconsistency in marking. The new rubrics are shown in Table 5.4. The new rubrics were used in the Trial schools, and in the second round of testing, MR2. The fit statistics were greatly improved as shown in Table 5.5.

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table 5.4  Amended scoring rubrics for gang1 and gang2

GANG1 Score 0 1

2

Description No response or irrelevant response Disagree it is a rhombus but specify some properties correctly (e.g., it is a parallelogram) OR agree that it is a rhombus but insufffijicient or incorrect properties given. Agree it is rhombus. Explanation needs to include necessary and sufffijicient properties, that is, it is a parallelogram with one of the following properties • 4 equal straight sides • Opposite angles equal, sides equal • Two lines of symmetry

GANG2 Score 0 1

2 3 4

Description No response or irrelevant response Incorrect angles but some evidence that angle measure is understood, angle measures are roughly accurate (e.g., angle f is larger than angle h but smaller than a right angle). At least 2 angles correct but no reason given Two angles found correctly with sensible reasons or all angles correct with no reasoning All angles correct with clear reasons given relating to the folding and properties. F = 45o; h = 45o; s = 135o (e.g., Folding corner to centre creates half right angle; All angles around centre of side equal so any 2 make 45o or Four angles of quadrilateral add to 360o)

table 5.5  Fit values for gang1 and gang2 at MR2

Question

InfijitMSQ

Infijit t

OutfijitMSQ

Outfijit t

GANG1 GANG2

1.07 0.83

0.3 -0.7

1.13 0.69

0.4 -0.6

1.6.1 Valid Formative Assessment Tools This process was used across all strands so that after MR2 a final pool of items could be identified for development into pre- and post-assessment tasks. This has enabled the team to produce four, overlapping and anchored assessment Forms for each area that teachers can use together with the

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figure 5.4 Geometric reasoning items by difficulty with scoring Rubrics, based on MR1 data, February 2017 (n = 769)

respective scoring rubrics and raw score translators to identify where each student is in their respective learning journey in relation to the particular learning progression. 1.7

Using the Item Difficulties to Identify Zones and Inform Teaching Advice Once a stable construct was established, the variable was ‘segmented’ (Kennedy & Wilson, 2007) into a set of Zones. The purpose of providing teachers with more generalised information was based on feedback from the snmy project that suggested that teachers liked having sufficient information to plan for teaching without being overwhelmed by detail. The Zones became the framework for providing teaching advice. In the first instance, judgements about approximate Zone boundaries within the learning progressions were made on the basis of the ordered lists of item difficulties. For instance, in the variable map (Figure 5.2) the boundary between Zone 1 and Zone 2 was initially set at -2.39 logits on the basis that there was a 0.36 logit difference between the logit measures for SHGT1.1 and SHAT8.2. However, to consider what these differences actually represented in terms of what could be inferred about students’ mathematical reasoning, the corresponding scoring rubrics were added to the ordered list of items as shown in Figure 5.4. At this stage, relevant specialist members of the research team met for a full day to interrogate what could be inferred from the scoring rubrics located at similar points on the scale to decide whether or not they represented

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qualitative differences in the nature of adjacent responses with respect to the sophistication of the mathematics or mathematical reasoning involved and/ or the extent of cognitive demand required. For example, in the ordered list of geometric reasoning items for the lower end of the scale shown in Figure 5.4, the initial boundary between Zone 1 and 2 was set at -2.05 logits (marked by the heavier line) and the boundary between Zones 2 and 3 was set at -1.44 logits. This process, which came to be referred to as ‘this goes with that,’ prompted rich discussions about the nature of reasoning indicated by responses at similar levels of difficulty and led to a reconsideration of some of the Zone boundaries on the basis of perceived commonalities in the nature of reasoning inferred. For instance, it was agreed that item measures gspsq1.1, grect3.1, and grect2.1 should be included in Zone 1 as the behaviour they represented suggested a similar type of reasoning to that suggested by the item measures already in Zone 1 (i.e., reasoning based on appearance). This is reflected in the following excerpt from the meeting notes taken at the time. … can recognise some rectangles and simple 2D shapes but limited knowledge of geometric language (e.g., vertices, edges etc), recognises and names shapes and objects but reasoning based on appearance (i.e., what it looks like). (Siemon, 14 February, 2017) As indicated above, the MR2 assessment round in the latter part of 2017 included a number of previously trialed but additional or revised items to tease out particular aspects of the learning progressions. Once these data were added to the analysis and it became clear that the scales were stable, the ‘this goes with that’ process was repeated. That is, the same members of the research team met again to interrogate what was meant by behaviours evidenced at similar levels of difficulty and to consider similarities and differences in the nature of reasoning that could be inferred. This prompted some further but minor changes to the Zone boundaries to accommodate the new items as can be seen in Figure 5.5. This process facilitated the development of rich descriptions of the characteristic behaviours evidenced at each Zone that indicate what students are able to do and what they might find difficult at each Zone within the learning progressions. While these illustrate the qualitative differences in thinking over successive Zones, their main advantage is that they are in a form and at an appropriate level of generality and specificity to inform teacher’s in-themoment noticing and attending to student learning. Table 5.6 illustrates the rich description derived from the scoring rubrics for Zone 1 of the learning progression for geometric reasoning.

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figure 5.5 Geometric reasoning items by difficulty with scoring Rubrics, based on MR1 & MR2 data, June 2017 (n = 1570)

At the full day meeting that interrogated the scoring rubrics and items, consideration was also given to the question, “If the students at this Zone are able to do …, or find … difficult, what is it that they need to explore/experience next that will support their learning?” Initially, this consideration led to some general statements about likely ways forward based on the progression. Following this process, a considerable amount of time was spent by specialist research team members in devising teaching advice that teachers could use in ways that suited them and their students. Because the Rasch analysis assigns students to the scale on the basis of where they have a 50% chance of success at an item at the same point on the scale, this means that there are some aspects of the behaviours identified within the relevant Zone that need to be consolidated and established to deepen students’ understanding and others that need to be introduced and developed to progress their learning to the next Zone. This relationship is illustrated in Table 5.7.

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table 5.6  Scoring rubrics and rich description for Zone 1 of the learning progression for geometric reasoning

Zone 1 Scoring Rubrics (from Figure 5.5)

Draw shapes (2D) or objects (3D) from the student’s perspective At least 2 shapes identifijied correctly but 2 or more shapes incorrectly identifijied as rectangles At least one of the response correct Incorrect grid reference provided (e.g., C6 or other) or inappropriately referenced (e.g., 6B) Names at least two objects (3D) in terms of their faces (e.g., square, rectangle, hexagon or triangle), may name one object correctly (e.g., cuboid or rectangle prism) or names faces or objects from dog’s perspective

Rich Text Description of Zone Zone 1: Recognise simple shapes by appearance and common orientation; show emerging recognise of objects from diffferent perspectives; naming and describing 3D objects base on common 2D shape names; identifying some standard nets; identifying location using simple referencing system. In measurement situations, recognising comparisons in 1 dimension without using units. Hierarchy and properties Recognise shapes by appearance and common orientation, Transformation of Relationships • Show emerging recognition of objects from diffferent perspective • Show emerging recognition of reflectional symmetry of objects and shapes • Show emerging recognition of a coordinate system. Geometric Measurement Understand the attribute of length, area and mass in terms of comparison

Given the strong research base for using low-threshold high-ceiling tasks in mixed ability groups (e.g., Sullivan, 2011), and feedback from project school teachers that they wanted to explore more effective and engaging ways of teaching, the research team also focussed on identifying indicative rich tasks that would address a range of learning needs across a number of Zones. These aspects of the rmfii project will be examined further and illustrated in more detail in later chapters in this volume. 1.7.1 Project Outcomes to Date As a result of the processes described above, stable, evidenced-based learning progressions have been identified for algebra, geometry and statistical

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table 5.7  Scoring rubrics and rich description for Zone 1 of the learning progression for geometric reasoning

Rich Text Description of Zone

Teaching Advice

Zone 1: Recognise simple shapes by appearance and common orientation; show emerging recognise of objects from diffferent perspectives; naming and describing 3D objects base on common 2D shape names; identifying some standard nets; identifying location using simple referencing system. In measurement situations, recognising comparisons in 1 dimension without using units.

What do they Need?

Hierarchy and properties Recognise shapes by appearance and common orientation, Transformation of Relationships – Show emerging recognition of objects from different perspective – Show emerging recognition of reflectional symmetry of objects and shapes – Show emerging recognition of a coordinate system. Geometric Measurement Understand the attribute of length, area and mass in terms of comparison

Consolidate and Establish 2D – Experience with a large of diffferent shapes, particularly non-prototypical ones. 3D – Name common 3D objects, identify some of the features in terms of faces, vertices and edges Transformation and Location – – Exploring different perspectives on objects and collections of objects (e.g., bird’s or spider’s eye view of classroom from own perspective) – Experience identifying symmetry of shapes and patterns with mirror lines in different positions and in creating symmetric patterns – Directional language of left, right, top bottom – Use of a coordinate systems (street map) to identify locations and give directions Measurement Ordering physical object base on attributes of length, area and mass Introduce and Develop – Language associated with describing simple shapes/objects and their properties – Rotational symmetry as well as reflectional symmetry – Exploring the faces of solids and deconstructing solids to nets – Using informal measures to compare (length, angles) – Estimating length measure, introduce angle as a measure of turn

reasoning, together with valid assessment forms that can be used to identify where students are in relation to the ‘big ideas’ in each area and monitor their growth over time. At this stage, indicative teaching advice has been derived from the learning progressions and on-line professional learning modules have been developed to support teachers’ use of these materials. It is too early to report on the efficacy of using the evidenced-based learning and teaching resources to support a targeted teaching approach to algebraic,

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geometric and statistical reasoning as the MR3 and MR4 data are yet to be analysed but the signs are hopeful that attending to the big ideas and being aware of where students are and how to progress their learning will make a difference to student learning.

2

Where to from Here

The relationship between curriculum, instruction (teaching), and assessment has not always been conducive to encouraging teaching based on a deeper understanding of the big idea and the connections between them (Sullivan, 2011; Swan & Burkhardt, 2012). While there are excellent pockets of research on effective practices at the level of individual classrooms and schools, very little of this is at sufficient scale to warrant attention from policy makers and curriculum developers. Recent work suggests that research on learning progressions/trajectories has a powerful role to play in generating sustainable and defensible links between what we know about learning, curriculum, and assessment (e.g., Shepherd, Penuel, & Pelligrino, 2018; Wilson, 2018) that, if harnessed and made accessible to teachers in a form that they can adapt to their own students and circumstances, can provide structured and evidence-based frameworks to support formative assessment. As for the work reported here, there are several opportunities for further development. Given that all three scales developed have a focus on reasoning, it would seem likely that all of the items could form a thick construct of Mathematical Reasoning. Some initial exploration of this idea has indicated that in general the items from algebra, geometry and statistics do work together consistently to form a single underlying construct. At this stage, however, the detailed work indicated in this chapter such as a close consideration of fit to the Rasch model, has not been completed. Items that do not fit the composite model of reasoning will need to be removed from the item pool because the possibility of updating rubrics is not available. In this instance, the items will be examined to try to understand what they contribute to the individual reasoning domains that is different from that of the overall reasoning construct. Once the final item pool is established, then the same process of discussion and decision making around the zone boundaries and the meaning of the zones will be undertaken. A second intriguing possibility is of linking to the multiplicative thinking progression. Again, it seems reasonable, because of the underpinning nature of multiplicative thinking, that there will be an association between the snmy and the rmfii constructs, especially the Algebraic Reasoning Progression. To this end, link tests have been given to several volunteer schools, which include

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some items from snmy alongside items from the rmfii domains. The snmy items will allow a link to be made to archived snmy data to test the notion of an association. The journey continues!

Acknowledgements The work reported here was conducted as part of the Reframing Mathematical Futures II research project based at rmit University with funding provided by the Australian Department of Education and Training through the Australian Mathematics and Science Partnership Program 2014–2017. The opinions expressed are those of the authors and do not represent views of the Department of Education and Training. The authors acknowledge with appreciation the schools, teachers, and students who participated in the research. This work would not have been possible without the support of the many teachers and schools involved and the collective expertise and efforts of the research team. The authors would particularly like to acknowledge the contributions of Sandra Vander Pal, Claudia Johnstone and Claudia Orellana to the work reported here.

References Australian Curriculum, Assessment and Reporting Authority (ACARA). (2018). The Australian Curriculum: Mathematics. Sydney: Author. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/ Barab, S., & Squire, K. (2004). Design-based research: Putting a stake in the ground. The Journal of Learning Sciences, 13(1), 1–14. Black, P., Wilson, M., & Yao, S. (2011). Road maps for learning: A guide to the navigation of learning progressions. Measurement: Interdisciplinary Research and Perspectives, 9, 71–123. Bond, T., & Fox, C. (2015). Applying the Rasch model: Fundamental measurement in the human sciences (3rd ed.). Mahwah, NJ: Lawrence Erlbaum. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13. Cobb, P., & Jackson, K. (2011). Towards an empirically grounded theory of action for improving the quality of mathematics teaching at scale. Mathematics Teacher Educaiton Development, 13(1), 6–33. Design-Based Research Collective. (2003). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5–8.

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Dweck, C. S., & Leggett, E. L. (1988). A social-cognitive approach to motivation and personality. Psychological Review, 95(2), 256–273. Kennedy, C. A., & Wilson, M. (2007). Using progress variables to interpret student achievement and progress (BEAR Technical Report No. 2006-12-01). Berkeley, CA: Berkley Evaluation and Assessment Research Centre. Retrieved from https://bearcenter.berkeley.edu/sites/default/files/Kennedy_Wilson2007.pdf Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Linacre, J. (2017). Winsteps Rasch measurement V4.0.0 [Computer Program]. Chicago, IL: Winsteps.org. Masters, G. (1982). A Rasch model for partial credit modelling. Psychometrika, 47, 149–174. Misailidou, C., & Williams, J. (2003). Diagnostic assessment of children’s proportional reasoning. Journal of Mathematical Behaviour, 22, 335–368. Pellegrino, J. (2002). Knowing what students know. Issues in Science & Technology, 19(2), 48–52. Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research. (Expanded Edition, 1980. Chicago, IL: University of Chicago Press) Shepard, L., Penuel, W., & Pellegrino, J. (2018). Using learning and motivation theories to coherently link formative assessment, grading practices, and large-scale assessment. Educational Measurement: Issues and Practice, 37(1), 21–34. Shield, M., & Dole, S. (2013). Assessing the potential of mathematics textbooks to promote deep learning. Educational Studies in Mathematics, 82(2), 183–199. Siemon, D. (2013). Reframing mathematical futures: Building a learning and teaching resource to support mathematical reasoning. AMSPP Competitive Grant Round funding application. Melbourne: RMIT University. Siemon, D. (2017). Reframing mathematical futures: Using learning progressions to support mathematical thinking in the middle years. In A. Downton, S. Livy, & J. Hall (Eds.), 40 years on: We are still learning! Proceedings of the 40th annual conference of the mathematics education research group of Australasia (pp. 651–654). Melbourne: The Mathematics Education Research Group of Australasia Inc. Siemon, D., Bleckly, J., & Neal, D. (2012). Working with the big ideas in number and the Australian Curriculum Mathematics. In W. Atweh, M. Goos, R. Jorgensen, & D. Siemon (Eds.), Engaging the Australian curriculum mathematics – Perspectives from the field (pp. 19–46). Mathematical Education Research Group of Australasia. (Online Book) Retrieved from https://www.merga.net.au/node/223 Stacey, K. (2010). Mathemaitcs teaching and leanring to reach beyond the basic. In ACER (Ed.), Proceedings of 2010 ACER research conference (pp. 17–20). Melbourne: ACER.

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Sullivan, P. (2011). Teaching mathematics: Using research-informed strategies (Australian Education Review No. 59). Melbourne: ACER. Swan, M., & Burkhardt, H. (2012). A designer speaks – Designing assessment of performance in mathematics. Educational Designer, 2(5), 4–41. Retrieved February 14, 2018, from http://www.educationaldesigner.org/ed/volume2/issue5/article19/ Swan, M., & Burkhardt, H. (2014). Lesson design for formative assessment. Educational Designer, 7(2). Retrieved from http://www.educationaldesigner.org/ed/volume2/ issue7/article24/ Vincent, J., & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS Video Study criteria to Australian Eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82–107. Vygotsky, L. (1978). Mind in society. Cambridge, MA: Harvard University Press. Wai, J., Lubinski, D., & Benbow, C. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101(4), 817–835. Wiliam, D. (2011). Embedded formative assessment. Bloomington, IN: Solution Tree Press. Wilson, M. (2018). Making measurement important for education: The crucial role of classroom assessment. Educational Measurement: Issues and Practice, 37(1), 5–20.

Chapter 6

Reframing Mathematical Futures II: Developing Students’ Algebraic Reasoning in the Middle Years Lorraine Day, Marj Horne and Max Stephens

Abstract Mathematical reasoning is an important component of any mathematics curriculum. This chapter focuses on algebraic reasoning in the middle years of schooling, often termed as a predictor of later success in school mathematics. It describes the design research process used in the Reframing Mathematical Futures (RMFII) project to develop an evidence-based Learning Progression for Algebraic Reasoning framed by three big ideas: Equivalence, Pattern and Function, and Generalisation. The Learning Progression for Algebraic Reasoning may be used to identify where students are in their learning journey and where they need to go next. Once the Learning Progression for Algebraic Reasoning was developed it was used to design Teaching Advice to help teachers to provide appropriate activities and challenges to support student learning. Two implied recommendations of this chapter are that algebraic reasoning based on these three key ideas should precede symbol use; and, that algebraic reasoning as described here needs to be cultivated in the primary school years. Keywords algebra – algebraic reasoning – learning progressions – design research – middle years

1

Introduction

Twenty years ago, it would have been common for teachers and mathematics educators to believe that algebraic reasoning was almost synonymous with the capacity to employ literal algebraic symbols. It is no longer credible to make this assumption with recent curriculum documents in countries as diverse as Australia (ACARA, 2017), usa (Common Core State Standards Initiative, 2018), England (Department of Education, 2013), China (Ministry of Education, 2011), © koninklijke brill nv, leiden, 2019 | doi:10.1163/9789004396449_007

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Japan (Ministry of Education, Culture, Sports, Science and Technology, 2008a, 2008b) and Singapore (Ministry of Education, 2013) consistently linking Number and Algebra, starting from the early primary years. The 2004 12th International Commission on Mathematical Instruction (icmi) Study, The Future of the Teaching and Learning of Algebra, had several chapters devoted to the development of algebraic reasoning which both reflected and anticipated emerging trends in national curriculum documents. Carolyn Kieran (2004) in her chapter, entitled The Core of Algebra: Reflections on its main activities, distinguished four approaches that are crucial to making algebraic learning meaningful to students. These were: – Generalisation of numerical and geometric patterns and of the laws governing numerical relationships – Problem solving – Functional situations – Modelling of physical and mathematical phenomena (p. 21). In a subsequent chapter, Lins and Kaput (2004) drew attention to two key characteristics of algebraic thinking: – First, it involves acts of deliberate generalisation and expressions of generality. – Second, it involves … reasoning based on the forms of syntactically structured generalisations (p. 48). What these authors recognised clearly is that the study of Number necessarily includes consideration of number patterns and relationships as well as becoming proficient at calculation; and that all students should be expected to finish primary school familiar with different types of mathematical thinking that we describe as algebraic, though not necessarily dependent on symbolic representation. Rather than thinking that algebra should follow arithmetic in some temporal sense that implies that algebra should be studied later in school, our view is that the study of arithmetic, especially working with number relations and operations, potentially embodies a degree of generalisation that should provide an important foundation for and entry into algebraic reasoning. Through children’s structured experience of patterns and relationships, several big ideas need to be fostered. These key ideas include Pattern and Function, Equivalence, and Generalisation. These three provide the focus of this chapter. They are not intended to be treated in isolation from each other, and are related to students’ ability to understand and use simple mathematical formulations and representations, and to their developing understanding of the concept of a variable. The assumption being made here is that unless these foundational ideas of algebraic reasoning are fostered in the primary and middle years of

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school, students’ subsequent encounters with and effective use of symbolic expressions are likely to be viewed as novel sets of rules and procedures rather than developing from and supported by the careful study of patterns and relationships. While it is our argument that algebraic reasoning needs to grow out of students’ rich experiences with patterns and relationships, it is also clear that for some students this development is quite difficult, or is hindered by difficulties they experience in moving away from purely arithmetical thinking. When arithmetical thinking tends to focus solely on calculating an answer, it can be difficult for some students to see the possibility of a general pattern, or to move beyond a single instance where a mathematical relationship may be true. Being able to distinguish numbers that may vary in a mathematical expression from those that stay the same is an important step from thinking only about specific cases. Equally important is being able to explain the basis for this thinking verbally and/or symbolically. At a further stage, students realise that equivalent expressions can be represented in a variety of ways; and that later when they use symbolic forms to represent equivalent expressions, they understand that the symbolic expressions themselves can be varied without changing the fundamental mathematical relationships. One of the biggest problems confronting the teaching and learning of algebra is that while the end points or goals of algebraic reasoning may be clear to teachers and textbook writers, not enough is known about how important and sophisticated concepts, such as the three big ideas, develop and how they can be supported throughout the primary and junior secondary years. Within the same classroom, some students may have achieved a relatively deep understanding of key algebraic ideas while other students may be operating at a much more basic level. Expressed another way, teachers need to know how to gather reliable evidence to show the level at which students are operating, and what specific teaching is most likely to move everyone’s thinking forward.

2

The Need for Evidence-Based Learning Progressions

Our focus on evidence-based learning progressions is intended to assist teachers and students to improve the quality of their algebraic reasoning and performance. Early in this chapter, there is a brief review of hypothetical learning trajectories and progressions as reported by researchers in the field such as Clements and Sarama (2014), Confrey, Maloney, and Corley (2014), Fonger, Stephens, Blanton, Isler, Knuth, and Gardiner (2018), Ronda (2004) and Simon

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and Tzur (2004). The work reported here on algebraic reasoning comes from a larger research project on mathematical reasoning, the Reframing Mathematical Futures II (rmfii) project, which was conducted between August 2014 and December 2017 (see Siemon, Day, Stephens, Horne, Seah, Watson, & Callingham, 2017). This part of the chapter is intended to locate the work of rmfii in respect of algebraic reasoning within a wider context of research and scholarly writing. The focus of this chapter then turns to how a sound empirical foundation can be used to support and refine these ideas. To achieve this, the research team designed and developed a set of approximately 25 learning and assessment tasks to be used with students starting from Year 5 to Year 10. Two of these tasks will be discussed in this chapter. Each assessment task addressed one or more of the big ideas which frame this chapter, and almost all of the tasks included sub-tasks amounting to about 75 sub-tasks overall. Each sub-task was accompanied by its own scoring rubric designed by the research team. Most scoring rubrics graded student responses on a three-point (0, 1, 2) scale; but there were a few with a two point (0, 1) scale or a four (0, 1, 2, 3) or five-point (0, 1, 2, 3, 4) scale. The design and use of these scoring rubrics was intended to assist teachers to distinguish between lower and higher order responses to the tasks, to value student reasoning and consequently to utilise students’ different responses through classroom discussion and analysis. The multi-level numerical scoring employed by the scoring rubrics was also an essential element for statistical analysis of the items themselves and the construction of zone descriptors. The scoring rubrics will be illustrated with respect to two of the tasks used, and samples of actual student responses will be used to illustrate how the scoring rubrics have been applied. The chapter will provide a short description on how these assessments in algebra were trialled across Australia. Earlier in this book, there is a general and more extensive discussion of how this was done across the whole rmfii project, including the use of Item-Response-Theory to score the responses to rank the tasks and sub-tasks according to their level of difficulty and in order to establish eight proficiency strands or zone descriptors that describe a progressive development in students’ levels of algebraic thinking. If they are to be effective, evidence-based learning progressions need to be complemented with practical suggestions for teaching that are informed by evidence of students’ performance. This will comprise the second last section of this chapter. The final section of the chapter will attempt to provide answer to the question: Where has this research taken us? In particular, we conclude with a brief discussion of what we now understand more clearly about students’ algebraic reasoning, how it can be measured and its progress

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charted, and how this component of rmfii has provided a more effective and evidence-based framework for improving the teaching and learning of algebra in our schools.

3

The Context

The development of the Learning Progression for Algebraic Reasoning took place within the context of the rmfii Project which is described in Chapter 5. The project involved teachers and students in Years 7–10 from schools classified as low socio-economic across each state of Australia. Each school within the project nominated a specialist teacher and the specialist teachers worked with at least two other teachers in their schools. Altogether 32 schools and 1563 students participated, however the number of students answering questions in any domain depended on the main focus of the teachers in the schools. These schools are henceforth referred to as project schools. As well as these project schools there were 12 schools who were not part of the project but who trialled assessment tasks. These schools are referred to as trial schools.

4

The Development of an Evidence-Based Learning Progression for Algebraic Reasoning

The process began with the development of a hypothetical learning progression, a synthesis of key ideas and their likely development over time, based on evidence from the literature. This hypothetical learning progression provided the basis for a set of rich assessment tasks to be sourced or designed with scoring rubrics designed to enable learners to demonstrate their understanding and reasoning (Day, Horne, & Stephens, 2017; Day, Stephens, & Horne, 2017). An initial trial of these tasks with students in Years 7–10 and subsequent analysis of student work and teacher feedback led to a refinement of the tasks and rubrics. The analysis of the trialling led to suitable tasks being placed in assessment forms, called MR1 and MR2, for algebraic reasoning. These forms were accompanied by detailed scoring rubrics that teachers used to mark student work. These tasks were then used with a large number of students across the project schools. Data from these trials of the forms were analysed using a Rasch model (Bond & Fox, 2015) to produce a map of the difficulty of the tasks with respect to each other. The three big ideas were not analysed separately, as they intertwine too much. Rather, the items were analysed together to provide a ‘thick’ construct of Algebraic Reasoning.

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Members of the research team met to discuss the big ideas of algebra and how the development of these big ideas was mapped by the data. By looking at the full range of behaviours that were exhibited by students in tasks in close proximity to each other on the map, delineations were drawn that distinguished what were to become the eight Zones for algebraic reasoning. There was some concern within the Algebraic Reasoning research team that very few responses were being received on tasks that addressed the higher-order algebraic content and reasoning (Day, Horne, & Stephens, 2017). This concern led to the research team seeking schools from outside the project schools, from a wide variety of socio-economic backgrounds, to participate in the trialling of the next two rounds of MR3 and MR4. In this way it was hoped to collect data that would inform about students in Zones 7 and 8 in particular. It was thought that this extra data may shift some of the Zone boundaries at the upper end, but in fact the data remained quite stable even with the extra trial schools’ data. Once the eight Zones were determined, a rich description of what students in each Zone could do was developed by the research team. From this description the Teaching Advice was designed so that teachers could see what a student in each zone needed to be consolidated and established from the previous Zone and what teachers could introduce and develop to assist students to transition to the next Zone. The Teaching Advice also points to rich tasks that may assist teachers to help students to access the algebra in that Zone. Most of these tasks are low threshold-high ceiling tasks so that teachers can use them with students across a range of zones. All of the materials that have been developed as a result of this project, the assessment forms with scoring rubrics and Raw Score Translators, the tasks, the Teaching Advice and lists of suitable tasks linked to a range of Zones will be made freely available to teachers of mathematics in Australian schools. Additionally, members of the research team have made a series of six professional learning modules to support teachers in developing algebraic reasoning with their students.

5

Development of the Initial Hypothetical Learning Progression

Algebra as an area of mathematics covers a wide range of content in the curriculum at the Years 7–10 level. Research generally focusses on one or two aspects at a time such as the move from arithmetic to algebra through generalisation (Kieran, 2004; Carraher, Martinez, & Schliemann, 2007), the concept of equality and equations (Baroody & Ginsburg, 1983; Knuth, Alibali, Hattikudur,

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McNeil, & Stephens, 2008; Warren, Mollinson, & Oestrich, 2009), the development of symbolic understanding (Caparo & Joffrion, 2006; Kaput, Blanton, & Moreno, 2008), understanding of relations and functions (Ronda, 2004; Smith, 2008), and the fluency of the translations among the different representations of functions including the use and interpretation of graphs (Amit & Fried, 2005; Kaput, 1998). These elements are not only in the domain of research but are also mentioned in the curriculum documents to which teachers refer. Very few references to learning trajectories and progressions are directly related to the big ideas of algebra. The development of a learning framework focussing on the understanding of function (Ronda, 2004) was one of the first to specifically consider the domain of algebra in secondary school mathematics. In her work Ronda divided the understanding of function into four domains: graphs, equations, linking representations and equivalent functions. Early writing on hypothetical learning trajectories include the work of Simon and Tzur (2004) whose focus was fine-grained looking at learning of a specific concept over a short period of time, particularly as it developed in the classroom through planned interactions. Their focus, however, was not algebra. Since then Clements and Sarama (2014) have provided further theoretical considerations and connected the learning trajectories closely to curriculum development, suggesting that curriculum should be based on what students can actually do and how they usually progress (Siemon et al., 2017). Looking at the big ideas in algebra, Fonger, Stephens, and Blanton (2018) have been working on a learning progression which considers the big ideas of equivalence, expression, equations and inequalities with the core concept of the equals sign across Years 3–5, with classroom activities linked to the development of the progression. These authors distinguish between trajectory and progression by the grain-size that is being investigated, with trajectory being fairly fine-grained and progression stepping back and looking at the larger picture. While starting specifically with equi-partitioning in rational number, Confrey, Maloney, and Corley (2014) have included three learning progressions which directly relate to an algebraic curriculum: early equations and expressions, linear equations, and function and variation, distributions and modelling. Confrey et al. describe fine-grained learning trajectory approaches to algebraic content development. These pieces of research informed our work, although our focus on algebraic reasoning in particular has added a new dimension. The breadth of concepts included in algebra during the middle years of schooling is wide, so it was not possible to hypothesise a trajectory, or single path. The overarching big ideas of Pattern and Function, Equivalence, and Generalisation are strongly connected to each other so rather than

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establishing many learning trajectories, a progression indicating some of the signposts in the landscape of algebra through which learners would most likely pass was constructed drawing on aspects from all three big ideas. Since one of the main purposes of this progression was as a tool for teachers it was important that it reflected, in some teacher-friendly form, aspects of the curriculum. Based on our reading of the literature we identified what we saw as the big ideas of algebra. Initially five big ideas of algebra were identified as Pattern and Sequence, Generalisation, Function, Equivalence and Equation Solving (e.g. Blanton & Kaput, 2011; Blanton, Stephens, Knuth, Gardiner, Isler, & Kim, 2015; Carraher, Schliemann, Brizuela, & Earnest, 2006; Fujii & Stephens, 2001; Mason, Stephens, & Watson, 2009; Panorkou, Maloney, & Confrey, 2013; Perso, 2003; Stephens & Armanto, 2010; Watson, 2009). Using evidence from prior research the hypothetical learning progression was developed under these headings or big ideas. As there appeared to be an unnecessary degree of overlap among these important orienting ideas, it was decided to collapse the descriptors into three big ideas, those of Pattern and Function, Equivalence, and Generalisation, recognising that in the final iteration there would also be overlap between these leading to a single progression rather than three different ones. Pattern and Function is strongly based on the idea of structure and included such aspects as the structure of arithmetic, identification of relationships and inverse relationships, identification of variables and constants, flexible movement between multiple representations, rates of change, ideas of continuous and discrete functions, domain and range, and families of functions. Equivalence has at its core idea the notion of balance and includes such aspects as the meanings of the equals sign, relational thinking, equivalent expressions and equations. Generalisation strongly connects arithmetic and algebraic thinking focussing on similarities and differences. Generalisation includes moving from specific to general and general to specific. This includes the employment of a variety of representations such as models, words, pictures, and/or symbols. The structure of arithmetic, number and algebraic laws, patterns, functions and equivalence situations may all offer opportunities to develop generalisation. The meaningful use of mathematical language and symbols may also be developed through the process of generalisation. Although the focus of the hypothetical learning progression was to be on algebraic reasoning, it was considered appropriate to identify the progression in terms of algebraic content, as students, at different levels, need content about which to reason. Another reason was because the progression was being developed to support teachers who generally have a strong commitment

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to algebraic content. Underpinning this content focus was the understanding that to reason algebraically at the highest level involves visualisation, being able to move fluidly between multiple representations and having the language and discourse to reason mathematically. The design of the original hypothetical learning progression then took form in a table with headings based on big ideas in algebra as described above – Pattern and Function, Equivalence, and Generalisation with clear gradation under each heading and an attempt to match across sections. The headings of the three big ideas were used for convenience, even though this was intended to be a single hypothetical learning progression. This initial three column framework is shown in Table 6.1. This initial hypothetical learning progression provided the basis for the writing of assessment tasks which would have a clear focus on enabling the students to demonstrate reasoning in algebra. Many tasks were written covering concepts and reasoning related to the initial hypothetical learning progression and trialled with students enabling refinement and selection of tasks. The tasks were designed to allow students to demonstrate their thinking and problem solving in algebraic contexts across different levels. For example the Balancing Scales task, shown in Figure 6.1, was designed around the lower levels of equivalence while the Hot Air Balloon task, in Figure 6.2, had its focus more on the middle levels of function. Some assessment tasks incorporated more than one big idea. The tasks have had space for student responses restricted in the figures shown here. The rubrics follow each task. The rubrics were designed to provide a gradation which at the upper levels focussed more on the reasoning. ahab1 (Algebra Hot Air Balloon part 1) and ahab2 did not require any written explanation but students had to reason about the graph scale and interpret data with a connection between the table form and the graph form of the relation. Figure 6.3 shows some illustrative student responses to ahab3 which required explanation. The first row responses were from students in Year 9. The one on the left refers to only one of the two possible times and, while there is an explanation, it is not clear what is meant so would score a two (2). The one on the right has both times given in the correct range but again the explanation is lacking. The second row responses are both from students in Year 10. The one on the left has answered with only one of the two correct times. While the explanation is not very clear it does show an understanding of proportion and that the student is correctly interpolating from the graph, so the score would be a three (3). The student on the right of the second row has given both times with a reasonably clear explanation so would score four (4). The higher responses though were not necessarily related to year level.

Recognise, continue and describe simple growing and repeating patterns, identifying the growing and repeating elements and representing the same pattern in diffferent ways. Use functional thinking to identify the rules that relate two varying quantities

Recognise, describe, ‘debug’ and use growing patterns and patterns involving operations on whole numbers.

3

Recognise patterns in daily life copying, continuing and creating both repeating and growing patterns. Describe qualitative and quantitative change.

2

1

Pattern and function

table 6.1  Initial hypothetical learning progression

Explain generalisations by telling stories in words, with materials and using symbols.

Explore and conjecture about patterns in the structure of number, identifying numbers that change and numbers that can vary.

Explain a generalisation of a simple physical situation.

Generalisation

(cont.)

Investigate equivalence using the balance concept and understand that the equals sign indicates equivalent sets rather than calculate. Use number lines to work with unknown quantities. Write and solve equations with single unknowns relating three whole numbers. Use own strategies to maintain equivalence between two amounts, compensating changes made to one with changes to the other using relational thinking. Use comparative language to describe balance situations. Solve simple word problems based on part-partwhole thinking with unknowns in diffferent positions. Use multiple forms to represent unknowns, including informal use of symbols. Reason with rational numbers using equivalence and compensation strategies.

Equivalence

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Extend the use of patterns to include rational number. Identify functions as linear or non-linear and contrast their properties from tables, graphs, or equations. Interpret tables and graphs showing a quantity changing.

Move flexibly between diffferent representations of patterns explaining how they are connected. Explore the relationships of two numbers or quantities as they vary simultaneously.

4

5

Describe rules for how materials can be linked by changes in shape and size. Identify and describe situations with constant or varying rates of change and compare them.

Pattern and function

Follow, compare and explain rules for linking successive terms in a sequence or pair quantities using one or two operations.

Explain generalisations using symbols and explore relationships using technology.

Generalisation

table 6.1  Initial hypothetical learning progression (cont.)

(cont.)

Examine the relationships between diffferent number pairs that when multiplied give the same result and how diffferent number pairs when divided can give the same quotient. Extend the solution of equations and word problems to include two steps and use symbols to represent the equations. Use equivalent strategies with more complex numerical and non-linear expressions. Using understanding of number relationships investigate equations where there may be more than one solution. Extend word problems to two and three steps which involve multiple operations and rational numbers, including those where ‘the whole’ is unknown. Explore equivalent expressions using counterexamples to demonstrate non-equivalence. Find numbers or number pairs that satisfy inequalities expressed within a context.

Equivalence

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6

Understand proportional relationships and explore their features by using algebraic expressions, tables and graphs. Generate and plot data in fijirst-quadrant coordinate graphs, describing patterns and the resulting scatter of plots. Classify number patterns which are linear, square or involve a power of a whole number. Interpret, construct and clarify rules for describing patterns and functions, and apply them to familiar or concrete situations. Recognise and represent at least linear and quadratic relationships in tables, symbols and graphs and describe informally how one quantity varies with another, identifying the dependent and independent variables. Plot, sketch and interpret graphs, interval lengths, increases and decreases over an interval, and gradient expressed as a ratio.

Pattern and function

Use and interpret basic algebraic conventions for representing situations involving a variable quantity.

Generalisation

table 6.1  Initial hypothetical learning progression (cont.)

(cont.)

Explain why two expressions are equivalent. Set up equations to represent constraints in a situation. Solve equations of the form ax + b = cx + d and ax2 + bx = c using balance and graphical methods, and solve linear equations using analytic methods. Use CAS technology to formulate equations and interpret solutions.

Use CAS technology and spreadsheets to solve contextual problems, explaining the processes used.

Equivalence

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8

7

Extend the classifijication of number patterns by considering the behaviour of successive terms in sequences, parameters and the types of general rules that can be used to describe them, and relate these patterns to everyday situations. Extend recognition and representation to include reciprocal, exponential and quadratic functions in tables, symbols and graphs. Describe the assumptions needed to use these functions as models. Plot, sketch and interpret graphs in four quadrants, considering local and global features, including maxima and minima and cyclical changes. Extend patterns to include more complex functions and explain how the rules relate to the context. Use CAS technology to investigate families of functions and transformations of general rules. Understand, analyse and compare properties of families of functions and relations and use them to model quantitative relationships where appropriate.

Pattern and function

Equivalence Extend the use of equivalence to a variety of symbolic expressions including brackets. Set up equations and inequalities that represent one or two constraints in a situation. Extend the solution of equations and the use of constraints in context to include simultaneous equations.

Move flexibly between equivalent forms including factorisation. Write, recognise or choose equivalent forms of equations, inequalities and systems of equations, and solve and analyse solutions, using technology when appropriate.

Generalisation Use and interpret algebraic conventions for representing generality and relationships between variables and establish equivalence using the distributive property and inverses of addition and multiplication.

Combine facility with symbolic representation and understanding of algebraic concepts to represent and explain mathematical situations.

table 6.1  Initial hypothetical learning progression (cont.)

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figure 6.1 Sample assessment task on balancing equations and associated rubrics

The assessment tasks were initially trialled with students in Years 7–10 from a range of schools in different states of Australia who were not part of the project (trial schools). The tasks were then used with the students in the project schools (MR1 data) which overall resulted in three sets of trial data being received. As a result of this trialling and analysis a Draft Learning Progression (dlp) for Algebraic Reasoning was designed (Day, Horne, & Stephens, 2017). After further trialling to test the dlp identified by the initial Rasch analyses two more data sets (trial schools, and MR2 data) were received and analysed. The scale was unidimensional with good fit to the Rasch model by all items, indicating that the items worked together to create a single construct which was interpreted as algebraic reasoning. The Rasch analysis was used with the data obtained to derive a Learning Progression for Algebraic Reasoning which consisted of a set of eight zones based on the evidence of what the students at these levels could actually do (see Chapter 5). While the questions and

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figure 6.2 Sample assessment task for function and associated rubrics

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figure 6.3 Sample responses to ahab3

concepts were based on the initial hypothetical learning progression the dlp which arose was not based on the hypothetical learning progression but rather on the student responses. Each of the assessment items in algebra was given a code so ahab1 refers to the first part, a, of the Algebra Hot Air Balloon task shown above. ahab1.1 refers to an answer scoring a one (1) on the rubric for this task. ahab2.3 thus refers to a score of three (3) on the task ahab2. The map in Figure 6.4 shows the model generated by the Rasch analysis. The items pertaining to the questions aeqb and ahab are highlighted. From the map it can be seen that the question aeqb1 elicited responses which on the modelling were located in Zones 1 and 2 while aeqb2 was more difficult, linking to Zones 4 and 5. This second item required the students to translate their drawing to symbols and this was found to be a higher level response than balancing the scales. The item ahab allowed responses which, while mostly located in Zones 1–3, included Zone 6 and Zone 8 where the students were asked to recognise both the height on ascent and descent and to provide an explanation. It should be noted that a zone is determined by the full range of behaviours within that zone and not just as the result of one question. The theoretical scientific framework and practical steps in partitioning the performance results into zones have been described in greater detail in Chapter 5. To prevent the inappropriate use of raw scores, each assessment form had different maximum score totals assigned. Identification of the eight zones allowed each of the assessment forms to have a Raw Score Translator added to enable teachers to locate students on the Learning Progression for Algebraic Reasoning (Siemon, Callingham, Day, Horne, Seah, Stephens, & Watson, 2018). The next stage was to look at all the behaviours that were demonstrated in each zone and to use them to describe the zone generally. The result of this is shown in the zone descriptors in Table 6.2. The zones, as described here, are a single evidence-based Learning Progression for Algebraic Reasoning (lpar) which is different to the hypothetical learning progression which was initially

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figure 6.4 Map of the responses to the algebra items following Rasch analysis

proposed. Aspects of the three big ideas are clearly discernible but are combined in this single lpar. Generally the zones here are not as mathematically challenging as the levels hypothesised. In the area of Pattern and Function the move from additive thinking to multiplicative thinking came through strongly in the evidence as beginning at Zone 4 and being used in simple situations in Zone 5. It had not been identified in the hypothetical learning progression although rational number, which relies on multiplicative thinking, appeared in levels 3 and 4. While evidence of generalisation in students’ responses was not apparent until Zone 2, Zone 3 is close to the hypothesised level 3, though with very simple use of symbolic language. It is not until Zone 6 that we see students able to generalise simple arithmetic relationships with justification which had been hypothesised at levels 4 and 5 in the hypothetical learning progression. In the area of Equivalence, equivalent relationships are just beginning to be recognised in Zone 3 and even at Zone 5, while able to use a strategy to maintain equivalence, generalisation is not occurring. In the hypothetical learning progression using one’s own strategies to maintain equivalence was in level 2

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table 6.2  Zones in the learning progression for algebraic reasoning

Zone Typical observed behaviours 1

2

3

4

5

Can continue simple patterns, but likely to build them additively. Reasoning is confijined to specifijic incidences and numerical examples of simple physical situations. Arithmetic thinking is used. Abstraction and generalisation not evident at this stage. Beginning to recognise patterns and relationships and conjecture about this. Able to identify numbers that vary and numbers that stay the same. Engage with the context, but arithmetic reasoning typically based on calculations is still being used. Recognise some multiples and some relationships like 6 more/6 less, while not necessarily recognising equivalence. Can work with simple scales and transfer from a table of values to a graph. Beginning to use symbolic expression and elementary reasoning. While still using arithmetic approaches there is evidence of relational reasoning with the numbers and providing some explanation. Beginning to recognise simple multiplicative relationships but without explanation. There is some evidence of co-ordination of two ideas but explanation is limited. Algebraic expressions are used rather than equations. Beginning to recognise equivalent relationships. Can explain simple generalisations by telling stories, manipulating materials and very simple use of symbolic language. Beginning to work multiplicatively and simultaneously co-ordinate variables, although still uses specifijic examples to convince. Able to reason and generalise in simple situations. Can recognise and interpret relevance of range from table and/or graphs and to recognise functional relationships. When faced with more complex algebraic situations are unable to use the full range of explanation or handle all of the information simultaneously. Beginning to transition to abstraction by inserting a number for a pronumeral. Able to use multiplicative reasoning in simple situations. Can reason with more complex additive situations involving larger numbers and subtraction but usually by examples. Has moved from algebraic expressions to using equations. Can derive a strategy that maintains equivalence, but cannot yet generalise. Able to use symbols to express rules. Can follow, compare and explain rules for linking successive terms in a sequence. Beginning to generalise using words or using some symbolic generalisations in simple situations. Recognises and represents simple functional representations. Can justify an argument using mathematical text. Beginning to generalise but connects closely to building on in context. (cont.)

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table 6.2  Zones in the learning progression for algebraic reasoning (cont.)

Zone Typical observed behaviours 6

7

8

Can use and interpret basic algebraic conventions to represent situations involving a variable quantity. Beginning to explain using logical language and to use if … then reasoning. Use symbolic language but the need for simplifijication is still being developed. Able to generalise simple arithmetic relationships with justifijication, including multiplicative relationships, but are often still context bound. Can show why several expressions are equivalent, typically employing numerical (non-symbolic) justifijications. Is able to use and interpret algebraic conventions for representing generality and relationships between variables. Beginning to use sound logical reasoning with appropriate reasoning language (e.g. if … then, must) evident. There is more co-ordination of multiplicative thinking and the associated language to notice algebraic structure. Can recognise and use the relationships between multiple entities and connections between and within diffferent representations. Able to establish and describe equivalence explaining relationships using the distributive property and the inverses of addition and multiplication. Can generalise quite complex situations and in more direct situations beginning to use simplest form. Is able to combine a facility with symbolic representation and an understanding of algebraic concepts to represent and explain mathematical situations. Explanations are sophisticated using logical thought and the language of reasoning. Can use multiple representations in a co-ordinated manner to solve, analyse, convince and conclude. Can visualise the form and structure of a function, at least graphically, from a real context. Is able to work in a context free environment using symbolic language and treat algebraic expressions (e.g. 3x + 2) as single entities. Can generalise more complex situations. Is able to establish and describe equivalence involving the four operations explaining relationships in symbolic terms. Can use abstract symbols to solve problems in context with multiple steps.

and strategies for equivalence with more complex numerical and non-linear expressions were suggested as level 4. The differences between the hypothetical learning progression and the lpar may be partially explained by the project schools being low socio-economic schools without strong records of success in mathematics, however the trial schools were more representative of the population as a whole and the analysis of their data supports the zones as shown here. Probably a more critical aspect is that the assessment used, and the evidence-based zones thus developed, were focussing on algebraic thinking and reasoning rather than

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on the algebraic skills which are often presented to the students as the curriculum. Students who have learned mathematics as a set of unconnected or loosely connected procedures often cannot explain their thinking and the processes they use. As teachers in the project and trial schools were able to assist students by reading the questions and scribing their verbal reasoning, it would not appear that the difficulty students had with explanation was as a direct result of low literacy skills. It is possible, however, that there may be a disconnection for students of using explanation within mathematics classes. The assessment provides teachers with a window on student thinking about algebra and enables them to know more about their students’ understandings which should enable them to better target their teaching. With each of the assessment forms a Raw Score Translator allows the teachers to identify the zone where each student is generally working. The subsequent Teaching Advice recognises that students in every group are different and have slightly different needs but knowledge of the zones can assist the teachers to better inform their teaching.

6

The Development of the Teaching Advice

The next phase of the project was to provide teaching advice targeted at the zones to assist teachers to provide appropriate activities and challenges to support student learning. The intention was not to write curriculum but rather to provide advice and supporting activities and resources that teachers could use in their programs. The zone descriptions were elaborated for teachers by connecting some of them to actual assessment items to illustrate the behaviours. Then for each zone, behaviours were described that would assist teachers to consolidate and establish that zone as well as behaviours which would assist in introducing and developing the following zone. An example of this teaching advice for Zone 6 is provided in Table 6.3. The text in brackets (e.g., altrns2.3) indicates an item from an assessment task – in this case an item that used a toy train of a particular length and asked students to calculate and show their reasoning for how many wheels it would have, with the score code (3) allocated to the response. The italicised text in the Teaching Implications column gives the titles of activities available to teachers via a drop box or from indicated websites. During the course of the project, participating teachers received professional learning around the three big ideas in algebra, and were introduced to selected activities in the Teaching Advice. A range of activities for use with the whole class were provided to teachers via a drop box and through internet links. It is recognised that in any classroom students are operating across many

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table 6.3  Example of teaching advice for a particular zone of the learning progression for algebraic reasoning

Zone 6 behaviours

Teaching implications

Can use and interpret basic algebraic conventions to represent situations involving a variable quantity. Beginning to explain using logical language and to use if … then reasoning. Use symbolic language but the need for simplifijication is still being developed. Able to generalise simple arithmetic relationships with justifijication, including multiplicative relationships, but are often still context bound. Can show why several expressions are equivalent, typically employing numerical (non-symbolic) justifijications.

Consolidate and Establish (from Zone 5): Investigate relationships in real life contexts that involve ratio and other multiplicative relationships. Explore situations where a variable can be used multiple times (e.g. the perimeter of the long tiles) and express the fijindings in symbols. Interpret and create graphs intuitively as they relate to real life situations (e.g. distance from home against time on a trip to the shops, speed of a racing car as it goes around a given track). (Maths300: Speed Graphs) Explore inverses and maintaining equivalence, particularly additively. Generalise from multiple examples in arithmetic situations. Investigate questions that ask students to work backwards. E.g. ABRT5 given the number of tables work out the size number. (Maths300: Backtracking, nRich: Think of Two Numbers 1170) Explore questions for which there is not a sensible answer within the context e.g. given an odd number of tables, what is the size number? Move flexibly between multiple representations. Move from symbolic structure to simplest form by recognising equivalent expressions. (ReSolve: Tens and Units) Explore activities where lots of answers are possible – “now another one, now another one …” followed by generalising, and describing and justifying the rule. Scafffold the generalisation process. (Maths300: Game of 31) Introduce and Develop: Explore meaningful situations where there is more than one relationship using multiple representations. (nRich: How Much Can We Spend? 6650) Explore multiplicative reasoning with direct and inverse variation and proportional reasoning. (nRich: Burning Down 497) (cont.)

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table 6.3  Example of teaching advice for a particular zone of the learning progression for algebraic reasoning (cont.)

Zone 6 behaviours

Teaching implications

Move between diffferent representations of a function in a variety of contexts, including abstract ones. Explore generalisations about number relationships where justifijication (leading to proof) can be developed (e.g. if the sum of the digits is divisible by 9 the number is divisible by 9). (Maths300: Consecutive Sums, nRich: Please Explain 1006) Explore simple examples of operations research involving constraints. Appreciate the relationships between functions and diffferent ways of expressing general functional notation. Examine the impact of gradients and constants on linear graphs and use diffferent letters, rather than just common ones like y = mx + c. (Maths300: Algebra Walk, ReSolve: Think of a Number – Linear Equations) Equivalence Explain what is happening to someone else or in Recognises the additive another real situation using multiple representations. inverse and maintaining Formalise the recognition and description of number equivalence (e.g. ARELS3.2). Can understand patterns and use algebraic text. (Maths300: Four Piles Problem) and strengthen more Solve algebraic problems involving multiplicative complex relational relationships and develop a greater range of strategies. reasoning situations. (nRich: Please Explain 1006) Simplify algebraic expressions, manipulate algebraic Generalisation Able to generalise simple text and move flexibly between equivalent forms. arithmetic relationships (Maths300: Algebra Charts, ReSolve: Working with (can see the general in the Algebra, Addition Chain) particular) (e.g. ABRT3.2). Express generalisations in diffferent ways, using multiple representations. Pattern and Function Is able to describe and justify rules involving multiplicative relationships (e.g. ATRNS2.3). Can use ratios (e.g. ALEM3.1). Is beginning to work with symbols to express rules, but not necessarily in the simplest way. Is able to describe and generate rules when a prompt is present (ARELS3.2 compared with ARELS1.3).

zones. Teachers need to cater for this diversity within the classroom. Most of the activities recommended provide opportunities for students to learn across a number of different zones. This can be illustrated by looking at one such activity, the Mountain Range Challenge (adapted from Unseen Triangles, lesson 20 maths300), which uses

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the context of a mountain range to explore a visual growing pattern based on equilateral triangles. Students are encouraged to notice the structure of the pattern and generalise the pattern on the structure they identify. This leads to exploration of equivalent expressions, solving equations and using multiple representations. Mountain Range Challenge is suitable for students in Zones 3–7. Table 6.4 illustrates how one activity can be used to cater for the diversity within the classroom, given that the activity is also dependent on the students sharing and discussing their findings (Zone 6 is not included as it is shown in Table 6.3). table 6.4  Example of how mountain range challenge can be utilised across zones to support mixed ability teaching

Zone specifijic behaviours

Teaching implication

3

Consolidate and Establish (from Zone 2): Include symbolic representation in the diffferent representations words/pictures/ tables/graphs/symbols Use comparative language answering the question what is the same and what is diffferent, in diffferent contexts. Go from a generalisation to a number of specifijic examples and from some specifijic examples to a generalisation. Describe and represent multiplicative relationships in multiple ways. Notice the structure of concrete, contextualised growing patterns, identifying what changes and what stays the same. Encourage explanations using words, diagrams, graphs, symbols and examples within a variety of contexts.

Beginning to use symbolic expression and elementary reasoning. While still using arithmetic approaches there is evidence of relational reasoning with the numbers and providing some explanation. Beginning to recognise simple multiplicative relationships but without explanation. There is some evidence of co-ordination of two ideas but explanation is limited. Algebraic expressions are used rather than equations. Beginning to recognise equivalent relationships. Can explain simple generalisations by telling stories, manipulating materials and very simple use of symbolic language.

Introduce and Develop: Identify what varies and what is constant in relation/function situations. Move from specifijic examples to generalisations and generalisations to specifijic examples (cont.)

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table 6.4  Example of how mountain range challenge can be utilised across zones to support mixed ability teaching (cont.)

Zone specifijic behaviours

4

Beginning to work multiplicatively and simultaneously co-ordinate variables, although still uses specifijic examples to convince. Able to reason and generalise in simple situations. Can recognise and interpret relevance of range from table and/or graphs and to recognise functional relationships. When faced with more complex algebraic situations are unable to use the full range of explanation or handle all of the information simultaneously. Beginning to transition to abstraction by inserting a number for a pronumeral.

Teaching implication Justify arguments and generalisation by using specifijic examples. Discuss how many examples you need before you are “sure.” Identify variables (what changes), recognise that a letter can stand for multiple numbers. Recognise equivalent expressions by substituting numbers > generalisation. Record and describe situations in multiple ways (multiple representations). Consolidate and Establish: What was Introduced and Developed in previous Zone Introduce and Develop: Move to a diffferent representation with simple functional representations Relate the structure of a situation to the function generated Explore diffferent strategies for maintaining equivalence Explain and justify using both words and mathematical text Investigate relationships containing number using concrete materials to see how patterns are structured. Move from just recognising patterns to identifying relationships. Notice structure to form generalisations Use thinking strings to see diffferent structures. Identify variables and constants by recognising what changes and what stays the same Use multiple embodiments to help move away from context. Explain and justify using mathematical text and diagrams in a variety of contexts. (cont.)

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table 6.4  Example of how mountain range challenge can be utilised across zones to support mixed ability teaching (cont.)

Zone specifijic behaviours

Teaching implication

5

Consolidate and Establish What was Introduced and Developed in previous Zone

7

Able to use multiplicative reasoning in simple situations. Can reason with more complex additive situations involving larger numbers and subtraction but usually by examples. Has moved from algebraic expressions to using equations. Can derive a strategy that maintains equivalence, but cannot yet generalise. Able to use symbols to express rules. Can follow, compare and explain rules for linking successive terms in a sequence. Beginning to generalise using words or using some symbolic generalisations in simple situations. Recognises and represents simple functional representations. Can justify an argument using mathematical text. Beginning to generalise but connects closely to building on in context. Is able to use and interpret algebraic conventions for representing generality and relationships between variables. Beginning to use sound logical reasoning with appropriate reasoning language (e.g. if … then, must) evident.

Introduce and Develop Explore inverses and maintaining equivalence, particularly additively. Generalise from multiple examples in arithmetic situations. Investigate questions that ask students to work backwards. Explore questions for which there is not a sensible answer within the context Move flexibly between multiple representations. Move from symbolic structure to simplest form by recognising equivalent expressions. Explore activities where lots of answers are possible – “now another one, now another one …” followed by generalising, and describing and justifying the rule. Scafffold the generalisation process.

Consolidate and Establish What was Introduced and Developed in previous Zone

(cont.)

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table 6.4  Example of how mountain range challenge can be utilised across zones to support mixed ability teaching (cont.)

Zone specifijic behaviours There is more co-ordination of multiplicative thinking and the associated language to notice algebraic structure. Can recognise and use the relationships between multiple entities and connections between and within diffferent representations. Able to establish and describe equivalence explaining relationships using the distributive property and the inverses of addition and multiplication. Can generalise quite complex situations and in more direct situations beginning to use simplest form.

7

Teaching implication Introduce and Develop Solve problem situations with general solutions, using mathematical argument to justify solutions.

What Have We Learned and Where to Next?

At the start of this chapter we outlined our reasons for focussing our learning progressions on the three big ideas of Equivalence, Pattern and Function, and Generalisation. The data obtained shows clearly how the development of students’ thinking in these three areas can be charted using well-designed tasks supported by scoring rubrics that value evidence of algebraic reasoning. The evidence-based lpar demonstrated that students find algebraic reasoning, especially in terms of writing explanations, more challenging than the hypothetical learning progression first proposed. The students in both the trial and the project schools appear to have had limited experience at explaining their reasoning in algebraic contexts. While they were able to perform some traditional algebraic procedures, their explanations were often sparse and sometimes non-existent. They also had difficulty coordinating multiple representations. The implication is that students in primary schools would benefit from an introduction to algebraic reasoning

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as discussed in this chapter. It is important not only in consolidating and extending their experiences of number patterns and relationships but also in laying a strong basis for a more formalised treatment of algebra in the junior secondary years. What is also surprising from the data reported here is that many secondary age students continue to find algebraic reasoning challenging. One may indeed ask what kinds of algebraic learning experiences they have been receiving since leaving primary school. We believe that many students, and possibly some teachers, perceive the focus to be almost exclusively on symbol transformation and manipulation. That impression is supported by what many textbooks for Year 7 upwards seem to pay most attention to. We argue that symbolic aspects of algebra need to be supported by a sound understanding of Equivalence, as well as of Pattern and Function, and Generalisation. These big ideas need to go hand in hand with the proficient use of symbolic expressions. One of the challenges of our Teaching Advice is to show how this complementary relationship can be fostered in the middle years, especially, if students are to become more confident algebraic thinkers, and if participation in higher level mathematics courses in the senior secondary school years is to improve. This can only occur by addressing the challenge of intelligent use and interpretation of symbolic expressions in the preceding years. Based on the evidence of what students in Years 7–10 can do in terms of algebraic reasoning the curriculum at these levels needs to be reconsidered. Teaching needs to have a focus on visualisation, the language of algebra and discussion and explanation, and the different representations used with fluid movement between them. In this project the focus was on the development of the evidence-based learning progression, assessment that supported teachers to target their teaching and advice and support materials for teachers. Studies are needed on what are the enablers for teachers and schools to move towards the teaching of reasoning and developing classrooms where there is a culture that encourages discussion and explanation, visualisation and multiple representations. One of the purposes of our scoring rubrics is to show teachers what they need to look for, and how to foster the kinds of classroom discourse and feedback that will move students’ reasoning forward. This is complemented by the Teaching Advice that we have prepared. In its current form, our Teaching Advice begins to address these issues, but more time needs to be given to showing how the three big ideas that have formed the basis of this research should continue to be evident in curriculum statements, school mathematics textbooks and associated assessment tasks.

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We are not arguing that the three big ideas that we have chosen are the only way of conceptualising the development of students’ algebraic reasoning. We have demonstrated that these three ideas are well supported by relevant research literature. Any construction of big ideas is to some extent arbitrary, as others may interpret the literature to include different key ideas. What is important is that the big ideas have to make sense to classroom teachers; and, as opposed to longer lists of goals, need to be concise enough to inform teaching, the design of instruction, and assessment to monitor students’ progressive development of algebraic reasoning through both the primary and secondary years.

References Amit, M., & Fried, M. (2005). Multiple representations in 8th grade algebra lessons: Are learners really getting it? In H. Chick & J. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 57–64). Melbourne: PME. Australian Association of Mathematics Teachers (AAMT). (2017). Maths300. Retrieved from http://maths300.com/library.htm Australian Curriculum, Assessment and Reporting Authority (ACARA). (n.d.). The Australian curriculum: Mathematics. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/ Baroody, A., & Ginsburg, H. (1983). The effects of instruction on children’s understanding of the ‘equals’ sign. Elementary School Journal, 84, 199–212. Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: Advances in mathematics education (pp. 5–23). doi:10.1007/978-3-642-17735-4_2 Blanton, M. L., Stephens, A., Knuth, E., Gardiner, A., Isler, I., & Kim, J. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87. Bond, T., & Fox, C. (2015). Applying the Rasch model: Fundamental measurement in the human sciences (3rd ed.). Mahwah, NJ: Lawrence Erlbaum Associates. Caparo, M., & Joffrion, H. (2006). Algebraic equations: Can middle school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27(2–3), 147–164. doi:10.1080/02702710600642467 Carraher, D. W., Matinez, M., & Schliemann, A. (2007). Early algebra and mathematical generalization. ZDM Mathematics, 40, 3–22. doi:10.1007/s11858-007-0067-7

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Chapter 7

A Learning Progression for Geometric Reasoning Rebecca Seah and Marj Horne

Abstract This chapter reports the development of a learning progression for geometric reasoning. Geometric reasoning is the ability to critically analyse axiomatic properties, formulate logical arguments, identify new relationships and prove propositions. All types of geometric concepts develop over time, becoming increasingly integrated and synthesized as individuals learn to visualise beyond the physical images, and participate in ‘taken-as-shared’ mathematical discourse to describe, analyse, infer and deduce geometric relationships, leading to engaging in formal proof. By analysing data collected through a series of assessment items we have designed and verified an eight-zone learning progression. Examples will be provided to show how the assessment items, activities and teaching advice can be used to help develop and nurture geometric reasoning.

Keywords spatial and geometric reasoning – visualisation – mathematical discourse – learning progression

1

Setting the Context

As one of the oldest disciplines, geometry may be considered as a diverse subject, with over 50 methodologies and theories, including algebraic, analytic, and differential geometry to name a few. This presents a challenge when making curriculum decisions. With new topics such as probability, statistics and computer science introduced to mathematics education, the amount of time devoted to the teaching of geometry, at all school levels, has declined in favour of teaching arithmetic (Clements & Sarama, 2011; Mammana & Villani, 1998). Many research agendas, especially those in the early years, also tend to focus on © koninklijke brill nv, leideN, 2019 | DOI:10.1163/9789004396449_008

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number, algebra and measurement rather than geometry (MacDonald, Davies, Dockett, & Perry, 2012). Decades of neglect in some countries has resulted in geometry learning today characterised by memorising the vocabulary and applying formulae in routine arithmetic calculations (Barrantes & Blanco, 2006), thus creating significant gaps in student knowledge. Many teachers share with their students’ similar difficulties: not recognising geometric properties or perceiving class inclusions of shapes; unable to visualise shapes and objects from different perspectives; facing difficulties when reasoning about measurement concepts such as surface area and volume and solving problems (Marchis, 2012; Owens & Outhred, 2006; Sáiz & Figueras, 2009). In this chapter, we present a case for teaching geometric reasoning, and delineate the key ideas underpinning learning to reason in a geometric context. Following discussions on geometric reasoning, we describe the process of drafting and validating a learning progression for geometric reasoning, which resulted in eight incremental thinking zones. Examples of the assessment tasks, student responses, and the item analysis used in the development will be provided. We then explain how activities and advice to target specific zones were developed to support teachers to nurture geometric reasoning. This study is part of the Reframing Mathematical Futures II project, which has developed a number of learning progressions in different mathematical domains (see Chapter 5 for details). See Siemon et al. (2017) for further discussions on various aspects of learning progressions.

2

A Case for Developing Geometric Reasoning

A major part of school geometry is concerned with the concepts of shapes and objects, with all related aspects such as size, position, orientation, and hierarchy. Being able to reason geometrically is more than memorising terminologies and applying theorems to known situations. Shapes and objects are not standalone entities, but rather, a connected network of concepts. Learning to reason about geometric relationships means developing a knowledge and understanding of the similarities and differences among shapes and objects despite their orientations and the mediums in which they are presented. Being able to invent and apply formal conceptual systems to investigate geometric relationships is part and parcel of reasoning geometrically (Battista, 2001, 2007). For example, knowing that shapes can be joined together to create new shapes and objects. Geometric reasoning also involves the development and use of conjecture, deductive reasoning and proof (Brown, Jones, Taylor, & Hirst, 2004). For example, deducing the size of an angle based on known properties of shapes or

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constructing necessary and sufficient definitions of shapes and understanding the differences between the meanings of terms used. Since the word ‘geometry’ literally means ‘earth measure,’ physical measurement is an integral part of learning geometry. Measurement concepts such as length, area and volume require secure knowledge of geometric properties and ‘dimensionality’ – the magnitude of attributes (Fernández & De Bock, 2013). Rules such as ‘length by width,’ or ‘length by width by height’ apply only to selected examples. Computational errors occur if they are applied indiscriminately without understanding of the geometric relationships. Many students and teachers alike have superficial understanding of measurement concepts (Lieberman, 2009; Owens & Outhred, 2006; Tan Sisman & Aksu, 2016). Common difficulties include not understanding partitioning in terms of array and grid structure and confusion of area with perimeter, volume with surface area, and volume with capacity. When learning about measurement, the focus tends to be on formula memorisation and routine application of rules rather than understanding why such rules work. Inability to visualise geometric structures hinders the learning of measurement. In a lesson study, teachers realised the need to get their students to learn to visualise the three dimensional (3D) lateral surface area in two dimensions (2D) and make connections between height in 3D and width in 2D (Lieberman, 2009). Only then, could the students comprehend the concept of surface area and apply the formula correctly. Being able to reason geometrically then, is about drawing on knowledge of axiomatic properties and their relationships to formulate a logical, coherent line of argument in order to justify a proposition and generalise its use in measurement and problem solving situations. Its success is dependent upon one’s geometric knowledge and spatial reasoning ability. The context of geometry is a vehicle for developing spatial reasoning (Clements & Sarama, 2011). The concept of spatial reasoning or spatial ability came to the fore with the work of Wai, Lubinski, and Benbow (2009). After tracking over 50 years of cumulative empirical research on 400,000 youth, they found spatial ability a predictor in educational achievement and attainment in Science, Technology, Engineering and Mathematics (stem) related disciplines. A later report from Pricewaterhouse Coopers (April, 2015) stated that an estimated 75% of the fastest growing occupations require stem. These factors have heightened research interest in this field. So, what is spatial reasoning? It is generally agreed that spatial reasoning is not a unitary construct. As a form of ability, psychological researchers disagree on its exact constituents, as well as the process and steps of spatial development (Yilmaz, 2009). The many terminologies generated from this field, such as spatial abilities/thinking, visuospatial reasoning, visual imagery, visualisation, and

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perception, complicate research efforts and reflect the multifaceted nature of the construct as well as the types of research methods employed. For Lohman (1994), spatial ability is defined as the capacity to generate, retain, retrieve, and transform well-structured visual images. McGee (1979) sees spatial ability as involving two main factors: spatial visualisation and spatial orientation while Carroll (1993) detected five major factors: visualisation, spatial relations, closure speed, flexibility of closure and perceptual speed. Yilmaz (2009) aptly pointed out that factor analyses on spatial ability often do not detect the same underlying factors, nor do they account for the dynamic interaction between spatial abilities and environmental issues. Marrying psychometric research with educational applications, Ramful, Lowrie, and Logan (2017) viewed spatial reasoning in terms of mental rotation, spatial orientation and spatial visualisation to capture much of the middle school mathematics curricula requirements. Mental rotation refers to one’s ability to imagine how 2D and 3D objects would appear after they have been turned around. Spatial orientation is the ability to imagine how an object looks from a different vantage point. Spatial visualisation, for them, refers to any spatial tasks that do not involve mental rotation or orientation. While analysing the spatial ability factors provided useful boundaries for designing multiple choice test items, there may be other skills such as mental translation or mental reflection at play that such tests did not address. There is also a need to consider environmental and cultural factors that influence how individuals reason in non-clinical situations. Framing the ability within ‘well-structured visual images,’ as defined by Lohman (1994) is also problematic as reasoning in mathematics is often about finding patterns in seemingly unstructured situations. From a mathematics education research perspective, spatial reasoning is about the capacity ‘to see, inspect and reflect on spatial objects, images, relationships and transformations’ (Battista, 2007, p. 843). It is about the interaction between what one sees and the meaning the representations seek to portray. This ability is an important component of geometric reasoning. Regardless of the variations in factors, the learning of geometry and geometric reasoning has implication beyond the boundary of geometry. Research found that early exposure to geometric concepts, with a focus on visualising and analysing the orientations of shapes, correlates positively with subsequent performance on arithmetic tasks (Gersmehl & Gersmehl, 2007; Gunderson, Ramirez, Beilock,  &  Levine, 2012; Uttal et al., 2013; Verdine, Golinkoff, Hirsh-Pasek, & Newcombe, 2017) and improves mathematics reasoning (Casey, Lombardi, Pollock, Fineman, & Pezaris, 2017). Significantly, children who grew up in low socioeconomic areas with limited exposure to constructional toys

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tend to have poorer spatial skills and lack the capacity to apply these skills to improve their academic performance (Carr et al., 2017; Jirout & Newcombe, 2015). While there is a gender difference in individuals’ spatial ability, such differences are more pronounced in older children and adults (Sima, Schultheis, & Barkowsky, 2013). Taken together, research findings point to the importance of early exposure of geometric ideas in rich learning environments on students’ mathematics learning outcomes.

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A Theoretical Framework for Developing Geometric Reasoning

The learning of geometry does not necessarily follow a linear path. Children do not learn all about triangles first before learning other shapes. Instead, all types of geometric concepts develop over time, becoming increasingly integrated and synthesised (Jones, 2002). How well a concept is learned and reasoned about is largely dependent on the degree of connectedness among the representations used to express the concepts, and individuals’ ability to visualise and communicate the relationships (Figure 7.1). In the following sections, we will discuss each of these aspects and how they are connected when reasoning. multiple representations reasoning visualisation

discourse

figure 7.1 Framework for developing geometric reasoning

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Representations of Geometric Concepts

The use of representations to learn and explain abstract concepts is central to mathematics learning. A representation is a configuration of signs, characters, icons, or objects that is used to stand for something else (Goldin, 2003; Goldin & Shteingold, 2001). These may include symbols to represent numerals and diagrams depicting 2D shapes, parallelism, or nets of a solid. Individual representational configurations form part of a larger representational system with primitive characters, rules and structure. There are two types of systems: (1) the external representational system, with shared characteristics, bound by specific rules where meanings and conventions of the mathematical

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structure are established, and (2) the internal representational system (or visual images), personal reconstructions of images derive from own interpretations and interactions with the concepts. Visual representations possess both figural and conceptual characters (Fischbein, 1993). Figural characters can be external, embodied on paper or with other materials, or iconical, centred on visual images. Conceptual characters are ‘concept image’ – the collective mental pictures, the corresponding properties and processes that are associated with the concept (Tall & Vinner, 1981). As an external system used in geometry, a representational configuration in the form of a diagram is not simply a depiction of an actual object experienced in the world. Rather, they are used in an attempt to take an abstract concept and make it concrete (Phillips, Norris, & Macnab, 2010). When used for instructional purposes, representations can be classified as external (embodied materially on paper or other support) or iconical (figurative, centred on visual images) (Mesquita, 1998, p. 183). They can also be determined in terms of ‘finiteness’ and ‘ideal objectiveness.’ For example, a figure may be shown as a particular triangle with sides 3, 4, and 5 in any units – an example of ‘finiteness’; the same figure may also be depicted as a geometric figure with three sides, with no reference made to specific concrete material – an example of ‘ideal objectiveness.’ Some representations are also more typical than others. For example, a square is typically shown with its sides horizontal and vertical. To accept that a tilted square is still a representation of a square requires a conceptual knowledge of what a square is and being able to identify it regardless of its orientation. Identifying a tilted square may be at a low level of reasoning just because it looks like a square or at a higher level a student may reason that because it has four equal sides and right angles, it must be a square.

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Visualisation of Geometric Concepts

In everyday situations, our sensory organs are bombarded with different stimuli. Individuals form visual images of what is seen, either by modifying the image or viewing it from a certain perspective (by rotation, translation, reduction in size, etc.). These images, believed to be similar to the visualisation objects, are stored in the visuospatial sketchpad – part of the working memory responsible for handling visual and spatial information (Kosslyn, Thompson, & Ganis, 2006). It holds the information it gathers, allows one to recreate images either based on the presentation of a stimulus or an image retrieved from the long-term memory. Research has shown that visuospatial sketchpad is involved in early arithmetic attainment (Simmons,

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Singleton, & Horne, 2008) and the solution of all kinds of deductive reasoning situations (Knauff, Mulack, Kassubek, Salih, & Greenlee, 2002). Visualisation then, is taken to include both the product and the processes of constructing, interpreting, using, and transforming visual images, in our minds, on paper or with technological tools with the purpose of depicting and communicating information, thinking about and developing previously unknown ideas and advancing understandings (Arcavi, 2003). As a product, the visualisation object includes the visual perception of a present stimulus or visual mental imagery in the mind. A visual mental image is defined as a construction of visual (shape information, colour and depth) or spatial (location, size, and orientation of entities) information one perceived (Presmeg, 1986; Sima et al., 2013). Presmeg (2006) categorised five kinds of imagery: concretepictorial imagery (picture in the mind that resembles real-life objects/situations), kinaesthetic imagery (of physical movement), dynamic imagery (the image itself is moved or transformed), memory images of formulae, and pattern imagery (pure relationships stripped of concrete details). They are introspective (personal concept image, see Tall & Vinner, 1981) of what individuals believe the object looks like. Visualisation as a process involves two pathways. A bottom-up processing takes place during visual perception of an unfamiliar visualisation object where all necessary component parts are viewed to construct the image introspectively. A top-down processing takes place when a familiar object is seen under degraded conditions (partially screen or in poor lighting, etc.). In this situation, the viewer relies on contextual information to retrieve images from long term memory to help recognise and interpret what is presented. Hence, visualisation is a cognitive ability, a dynamic neuronal interaction between perception and visual mental imagery for comprehending the world around us (Ishai, 2010). To successfully visualise an image, the person viewing the image must have some repertoire of experiences, mental skills and volitions to begin the process of interpreting the images within the person’s existing network of beliefs, experiences and understanding (Phillips et al., 2010). Otherwise, the visualisation object is merely a source of optical data. Yet many people ‘don’t know what they see, they see what they know’ (Goethe cited in Arcavi, 2003), as evidence by their tendency to have a stereotypical view of what a shape should look like (Fujita, 2012; Levenson, Tirosh, & Tsamir, 2011; Seah, Horne, & Berenger, 2016). The context within which perception takes place plays a critical role in determining the type of imagery that attracts our attention (Arcavi, 2003; Ishai, 2010). What concepts are taught, how they are being introduced and explained, and the impact of emotions on the interactions influence what knowledge is learned and how a ‘concept image’ is formed.

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Discourse about Geometric Concepts

To reason is to logically reflect, explain and justify a position (Kilpatrick, Swafford, & Findell, 2001). It necessitates three core elements: (1) conceptual knowledge needed to recognise, interpret, represent and analyse algebraic, statistical, and geometric situations, (2) an ability to apply that knowledge in unfamiliar situations to solve problems, generate and test conjectures, make and defend generalisations; and (3) a capacity to communicate reasoning and solution strategies in multiple ways (i.e., through diagrams, symbols, orally and in writing) (Siemon et al., Chapter 4). To reason about geometry means to engage with mental entities constructed through the use of geometrical representations in the form of points, lines, angles, shapes and objects. All geometric concepts encompass both sensory images and the associated formal definition, what Fischbein (1993) termed as a figural concept. Every ‘concept image’ corresponds to a ‘concept definition’ – a form of words used to specify that concept (Vinner, 1991). Individuals develop their own personal concept images and concept definitions through experience. Initially, geometric representations are understood purely from visual recognition. Once a mental image is formed, the definition becomes dispensable or even forgotten (ibid.). Difficulties arise when the properties that define the shapes are not brought to the students’ attention, thus creating a disjuncture between the students’ personal figural concepts (concept image and definitions) derived from experience and formal figural concepts derived from axioms, definitions, theorems and proofs. Concept images or mental imageries are more than a means by which information is learned, stored and retrieved. They play a key role in developing reasoning and comprehension of verbal information (Kosslyn, Behrmann, & Jeannerod, 1995). Research shows that students who can interpret a situation and produce an accurate visual-schematic representation are almost six times more likely to solve a word problem correctly whereas those who produce inaccurate visual images or pictorial images that are close to the visual appearance of objects are more likely to give an incorrect answer to a word problem (Boonen, van Wesel, Jolles, & van Der Schoot, 2014). Importantly, the way students perceive and talk about geometric visual representations reveals their thought processes and in turn shapes their thinking. Terms such as square, triangles and circles are condensations of definitions (Duval, 2014), not necessary when used in a designative or descriptive way but crucial when used to infer or justify a particular geometric argument. If the term ‘rectangle’ is used whenever a quadrilateral with two long sides and two short sides is shown, students are likely to view all shapes with these features as rectangle and define the term according.

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Their personal figural concept of a rectangle may include parallelograms and trapeziums but not squares – the formal figural concept for a rectangle rejects some parallelograms and trapeziums but includes squares (Seah et al., 2016). To accept that a square is a rectangle, an understanding that a rectangle is a closed planar four straight sided shape with right angles is required. In summary, learning mathematics is about changing a discourse (Sfard, 2008). The type of keywords and representations used to describe and define a term, the narratives of the discussion, and the learning routines to which these are enacted are keys in shaping students’ mathematical reasoning to match those that are ‘taken-as-shared’ within the community. If a student sees a rectangle as a four-sided shape, s/he is likely to assume that all quadrilaterals are rectangles. Similarly, if a rectangle is defined as having ‘two long sides and two short sides,’ then the student is unlikely to see a square as a rectangle but may include many non-rectangles. Hence, it is in the interplay between visualisation, representations and mathematical discourse that the design of the geometric learning progression, the assessment tasks and teaching advices are anchored.

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Constructing the Geometric Learning Progression

In constructing the progression, we acknowledge that the dominance of van Hiele levels on geometric thinking research, and their neglect of visualisation, meant that the framework we developed needed to move beyond these factors and recognise that students can reason at multiple levels, at different rates for different concepts (Shaughnessy, 1986). The following sections documents the three research phases we undertook to construct and refine the geometric learning progression. 7.1

Phase 1: Constructing a Hypothetical Geometric Learning Progression Unlike other researchers, who used national curriculum statements and/or standards as a starting point for their work (for examples, see Siemon et al., 2017 or Clements & Sarama in Chapter 2), we found that the Australian Curriculum: Mathematics (Australian Curriculum Assessment and Reporting Authority (ACARA), n.d.) alone did not adequately encapsulate the concepts, procedures, dispositions, and reasoning required to promote geometric reasoning. A search on ‘big ideas’ for teaching geometry also threw out a number of different foci. For Jones (2002), it is about invariance, symmetry and transformation whereas Johnston-Wider and Mason (2005) look at geometric thinking as involving invariance, language and points of view, reasoning and visualising and representing.

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Our synthesis of available research findings showed that the work of Battista (2007) provided a useful platform to develop an initial hypothetical geometric learning progression that included both geometry and measurement concepts (see Table 7.1). Battista proposed an alternative route to the original van Hiele levels to involve four levels of thinking: visual-holistic reasoning, analytic-componential reasoning, relational-inferential property-based reasoning, and formal deductive proof. He expanded the development of property-based thinking to progress from visual-informal, to informal insufficient-formal reasoning and finally sufficient formal property-based reasoning. The hypothetical learning progression we proposed made a distinction between descriptive reasoning – using informal language to explain what one sees and analytic reasoning – using formal language to analyse properties of shapes. This allowed us to better investigate the growth in the connectedness between visualisation and mathematical discourse and contribute to the design of instructions that target specific thinking. The hypothetical geometric learning progression presented here in Table 7.1 also took account of the curriculum requirements and provided a basis for the development of rich assessment tasks that would enable the identification of what students in years 7–10 could actually do. Note that the actions described in the table relate to the learning behaviours, but it is in the discussions and explanations students made that provide windows to understand their reasoning. The term ‘progression,’ as opposed to ‘trajectory’ – implying a single pathway, is used to reflect the nature of learning as moving within and across domains. We wrote assessment tasks, scoring rubrics, and classroom activities that could help us assess and teach each of the geometric areas and the levels of the progression. The design of all tasks was guided by the interconnectedness between representation, visualisation and discourse (see Figure 7.1). We were particularly interested in tasks that would assess inferential and formal deductive reasoning as well as geometric measurement situations. Classroom tasks were trialled at the rmf II project schools and student responses collected through this exercise assisted in deciding which tasks were best suited as classroom activities and those which could be used as assessment items. Feedback on the assessment items was also sought from other members of the team as well as colleagues in the field in order to refine the tasks and marking rubrics prior to further trialling. The first trial assessment consisting of 62 items was administered to Year 4 to 10 students outside the project school cohort to determine its reliability and validity. These students were from across social strata and States to allow for a wider spread of data being collected. Up to 585 students participated. These data were marked by two markers and validated by a team of expert consultants to ensure the accuracy

Descriptive reasoning

Visual reasoning

1. Pre-recognition: Recognise the diffference between 2D shapes and 3D solids according to their appearance. 2. Recognition: Identify the 2D faces on a 3D solid they see. Identify and name spheres, cylinders, cubes and pyramids.

3. Visual informal reasoning: Identify faces, edges and vertices of 3D solids.

1. Pre-recognition: Recognise shapes by their appearance as visual wholes, e.g., a rectangle “looks like” a door. Names simple 2D shapes 2. Recognition: Attend to at least one feature of the shape but not all shapes in the family, either omitting some or including shapes with the general image but not all the required features for polygons of order 3–6 and 8.

3. Visual informal reasoning: Identify and explain shapes by necessary properties using informal language based on visual rather than conceptual knowledge (e.g. straight sides, corners). 4. Informal and insufffijicient formal reasoning: Begin to acquire formal language that can be used to describe what they ‘see’ and spatial relationships between parts of shapes. 4. Informal and insufffijicient formal reasoning: Describe the characteristics of right pyramids and prisms with regular bases. Name and explain the diffference between pyramids and prisms using visual and informal language. Identify the component faces and use those 2D shapes to construct the solid.

3D

Shape 2D

table 7.1  Hypothetical geometric learning progression

3. Visual informal reasoning: Identify isometric transformations with simple shapes. Describe a simple structure or group of objects from other perspectives. 4. Informal and insufffijicient formal reasoning: Identify components of more complex shapes and recognise translation, reflection and rotation of components. Construct simple tessellations.

1. Pre-recognition: Recognise ‘like’ shapes in situ when orientation has not changed. 2. Recognition: Identify reflective symmetry in 2D with single axis of symmetry

Transformations

(cont.)

3. Visual informal reasoning: Measure length with informal units; estimate length measure; recognise properties of units such as equivalence and iteration. Use informal units to measure area/volume. Describe an angle informally; identify angles larger and smaller than a given angle. 4. Informal and insufffijicient formal reasoning: Transfer between length units; measure lengths over 1m accurately. Understand the diffference between perimeter and area. Use an angle measuring implement without scale to measure angles; transfer between degrees and fractions of a full turn; estimate angles; identify angle types.

1. Pre-recognition: Describe the characteristic/attribute being measured for length; correctly identify objects longer than and shorter than a given object. 2. Recognition: Order objects by length and can recognise attribute of area, volume and mass using terms like heavier, holds more, covers more.

Measurement

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Analytic reasoning

3D 5. Sufffijicient formal property based reasoning: Describe all the properties of a given object. Identify the shape of cross sections of simple polyhedral. Construct the net of a cuboid and other simple 3D solids. Draw 2D representations of 3D solids.

Shape 2D

5. Sufffijicient formal property based reasoning: Reason about shapes based on properties. However, their defijinitions are not minimal. Extend classifijication to the diffferent types of triangles and quadrilaterals; recognise regular shapes as special cases; describe all the properties associated with a shape, including diagonal properties, angle and side features. Identify parallel lines and a transversal; equivalent, complementary and supplementary angles.

table 7.1  Hypothetical geometric learning progression (cont.)

5. Sufffijicient formal property based reasoning: Identify similar and congruent 2D shapes. Identify reflectional and rotational symmetries of 2D shapes and patterns.

Transformations

5. Sufffijicient formal property based reasoning: Use formal units to measure area and volume; performing appropriate calculations for rectangles and cuboids. Calculate perimeters of composite shapes. Transfer between area and volume units understanding the reciprocal nature of the relationship. Calculate areas of triangles and other composite shapes; calculate the volume of simple prisms. Use a standard protractor to measure angles.

Measurement

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Formal proof

Relational-inferential property-based reasoning

6. Empirical relations: Recognise right angles and parallel faces in 2D representations of 3D solids. 7. Analysis: Identify diagonals in 3D solids and position them accurately on 2D representations of the cross sections. 8. Logical inference: Recognise, name & understood that there are only fijive Platonic solids. 9. Hierarchical shape classifijication based on logical inference: Construct arguments based on the properties of 2D shapes in 3D solids

6. Empirical relations: Use empirical evidence to conclude that if a shape has one property, it has additional properties. 7. Analysis: Understand that when one property occurs, another property must occur. 8. Logical inference: Students are not only able to identify necessary properties but have also understood the idea of sufffijiciency in relation to identifijication. 9. Hierarchical shape classifijication based on logical inference: Recognise the hierarchy associated with the shapes.

10. Construct arguments based on the properties of shapes and recognise the requirements for congruent and similar shapes. Recognise tangents and can identify right angles and equivalent angles in a circle.

3D

Shape 2D

table 7.1  Hypothetical geometric learning progression (cont.)

6. Empirical relations: Identify dilations. Identify the reflective planes in 3D objects.

Transformations

9. Hierarchical Shape: Find the surface area of simple polyhedral, cylinders and spheres. Extend their volume calculations to pyramids, cylinders and spheres.

6. Empirical relations: Begin to understand the diffference between surface area and volume. 7. Analysis: Identify the components of a circle and identify π as the ratio between the circumference and the diameter. Calculate the circumference and area of a circle. 8. Logical inference: Use Pythagoras’ theorem to calculate the third side of a right-angled triangle when given the other two sides. Apply measurement knowledge to fijinding perimeters and areas of composite shapes involving circles and triangles.

Measurement

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of the marking rubric and data entry. Rasch modelling (Bond & Fox, 2015) was used to analyse the trial data, refine the marking rubric and inform the drafting of the hypothetical geometric learning progression (see Siemon et al., Chapter 5). 7.2 Phase 2: Formalising the Geometric Learning Progression Phase 2 involved formalising, testing and validating of assessment tasks in order to chart the learning progression based on evidence of what students can do. The assessment items were grouped into three domains: (1) properties and hierarchy, (2) transformation of relationships, and (3) geometric measurement, though some items contributed to more than one domain. For example, the dog’s perspective task involves the identification of three dimensional objects (properties and hierarchy), viewing these from different perspective and drawing these views (transformation of relationships) (Seah & Horne, 2018). The items were designed to assess what middle school students (grades 5–10) were able to do and focused on reasoning rather than procedural skills. We generated multiple forms to be used in the project schools. Each question was given a code, for example, gnet3, indicates Geometry Net item 3. There were 36 items collated into two forms in the second trial (MR1) and administered to 755 students in project schools with 742 valid responses.

figure 7.2 Reasoning items on visualising three-dimensional objects

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Figure 7.2 shows an example of an assessment task and the corresponding marking rubric. This item requires students to demonstrate an ability to visualise a cube based on the two-dimensional format of a net. gnet3 requires the students to visualise but gnet4 requires explanation, thus allowing students to demonstrate their reasoning. Data were collected from a total of 166 trial school students and 271 project school students in Year 7 to 10. The corresponding zones for the responses were generated after Rasch modelling (see Phase 3). table 7.2  Results for gnet3 and gnet4

gnet3 Score MR1 0 1 2 3

N = 166 % 14 8.4 10 6.0 49 29.5 93 56.0

gnet4 MR2 N = 271 17 14 64 176

% 6.3 5.2 23.6 64.9

MR1

MR2

N = 166 % 27 16.3 17 10.2 54 32.5 68 41.0

N = 271 37 23 106 105

% 13.7 8.5 39.1 38.7

We analysed student responses by comparing data collected from trial schools and MR1. As shown in Table 7.2, the students were generally able to draw the net of a cube (86%, 88%). However, nearly 30% (24% in MR1) reproduced the net already given, rotating it, rather than creating a different net. As for gnet4, 73% (78% in MR1) were able to draw a net that matched the scenario with 41% (39%) able to give an appropriate reason. Around 28% trial and 31% MR1 students respectively scored maximum on both questions. Further analysis of gnet4 focussed on the nature of the explanations given. Students provided a range of explanations, from no explanation, idiosyncratic responses (claiming that there is only one way), explanation implying a prototypical image of the net of a cube, to explanations that show their ability to reason about their visualisation from different perspectives (Figure 7.3). This type of analysis provided valuable information on students’ visualisation and reasoning abilities. It assisted in shaping the boundaries of the zones and drafting of the teaching advice, done in Phase 3. For example, the explanations given by students in Figure 7.3 show their ability to visualise an incorrect net and explain the mental images in their mind. Further, from the limited range of incorrect nets provided, as well as the lack of explanation, one can infer that many students have little experience with working with 3D objects, seeing things from different perspectives, and making a logical argument to support their claim.

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figure 7.3 Examples of student’s reasoning on gnet4

7.3 Phase 3: Developing Targeted Teaching Advice To refine the learning progression, repeated trials of the assessments were necessary to ensure that the scales produced as a result of the Rasch modelling are stable. We scrutinised the data to determine if there were qualitative differences in the nature of adjacent responses on the scale with respect to the sophistication of the mathematics or mathematical reasoning involved and

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the extent of cognitive demand required. Responses exhibiting similar levels of reasoning were grouped together to form eight relatively discrete, hierarchical zones (see Table 7.3). table 7.3  Geometric learning progression and broad descriptions of behaviours in each zone

Zone 1: Pre-cognition – Recognise simple shapes by appearance and common orientation; show emerging recognition of objects from diffferent perspectives; name and describe 3D objects based on common 2D shape names; identify some standard nets; identify location using simple referencing system. In measurement situations, recognise comparisons in 1dimension without using units. Zone 2: Recognition – Identify simple shapes in situ and on simple solids; recognise some reflective symmetry, some nets of simple solids and some simple shapes. Show emerging representation of 3D objects; use some simple geometric language; show emerging perception of measurement concepts such as length, area, and angle but do not coordinate information or justify thinking. Beginning to represent and move between representations but focuses mostly on one property (isolated features). Zone 3: Emerging informal reasoning – Use one or two properties or attributes (insufffijicient) to explain their reasoning about shapes and measurement but often do not recognise properties in non-standard representations. Demonstrate awareness of measurement attributes. Tend to visualise objects from own perspective. Use simple coordinates. Tend to see objects and groups of objects as a whole but unable to analyse components independently. Zone 4: Informal and insufffijicient reasoning – Use some geometric language in context, name some 3D objects and are able to visualise some objects from a diffferent perspective but show incomplete reasoning in geometric and measurement situations, attending to necessary properties but not recognising redundancy. Use some properties to identify shapes/objects. Perform measurement calculations but attend to only one attribute. Give directions from a map from personal rather than other viewer’s perspective when situations are more complex. Zone 5: Emerging analytical reasoning – Able to visualise and represent 3D objects using 2D platforms (such as Nets); recognise properties in non-standard orientations and are starting to use properties to identify classes; begin to use but not recognise sufffijicient conditions; use either properties or orientations to reason in geometric situations; access relevant geometric language; demonstrate knowledge of dilation and coordinate systems and recognise some rotational symmetry; use landmarks but retain personal orientation when providing directions; provide partial solutions and explanations when calculating measurement situations. Begin to coordinate multiple components. (cont.)

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table 7.3  Geometric learning progression and broad descriptions of behaviours in each zone (cont.)

Zone 6: Property based analytical reasoning – Use properties accurately when reasoning about spatial situations but lack knowledge of geometry hierarchy. Understand properties of 2D shapes but not special cases (e.g. regular). Geometric and measurement arguments rely on examples/counter examples. Provide accurate directions from a map using appropriate language and describe directions from walker’s perspective. Understand impact of doubling dimensions on volume, able to visualise volume and calculate when numbers are small. Omit one step when calculating multi-step measurement problems. Able to make deductions about angle situations with limited explanations. Beginning to reason deductively but not able to coordinate all aspects. Zone 7: Emerging deductive reasoning – Work analytically with properties of rectangles. Beginning to recognise necessary and sufffijicient conditions. Use sound reasoning in argument/explanations, though explanations often are procedurally based or based on an example. Able to recognise the relationship between length, area and volume. Using multiple properties to reason but in measurement situations may rely on procedural explanations. Zone 8: Logical inference-based reasoning – Construct arguments based on multiple properties of 2D shapes and 3D objects, use the necessary and sufffijicient conditions to reason about geometric and measurement situations, conjectures and propositions (theorems), demonstrate analysis of both reflectional and rotational symmetry.

Broad descriptions of behaviours, detailed teaching advice, and the ideas and strategies needed to progress learning to the next zone were written. The advice to teachers focussed on the behaviours that the students should participate in to consolidate and establish the particular zone and to introduce and develop ready for the next zone. There is also a collection of activities for teachers to use in the class which foster some of those behaviours and the use of language, visualisation and representations. These activities usually allow for students across a few zones to gain needed experiences. For example, Figure 7.4 shows how getting students to build cubes or other 3D objects such as triangular prisms from shapes that link, such as Geoshapes, connects to behaviours in the zones teaching advice. Once students have built a 3D object such as a cube they can open it up again to form a net. The challenge is for them to find as many different nets as they can, recording them on paper. During the activity they need to explain how they know all the shapes they have found are different and to explain

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figure 7.4 An activity contributing to student experiences across zones 2–5

how they know they have found all possible nets. An important component of this is group work with the students explaining to each other and sharing their findings. For activities like this to really assist with the development of geometric reasoning the aspect of group work and the continual challenge to explain are critical components.

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Barriers and Journey Ahead

Research conducted with teachers to effect change necessarily faces a number of challenges. First, decisions have to be made on what should be included in the assessment tasks. Our principal aim was to ensure that the tasks had real life applicational value. They also needed to be sufficiently challenging to attract student’s interests in attempting the task. Because we did not have sufficient data on Australian students’ level of understanding, many of the tasks written were based on our judgement of what we expect students to be able to do. However, many students in trial schools did not respond to the more difficult tasks we wrote during the trialling phase. These were students across social strata and States. Their lack of response to more challenging tasks revealed the state of geometry learning in many classrooms. It influenced the type of questions that were included in subsequent assessment tasks. A preference for algebraic reasoning also meant that the amount of data collected for geometric reasoning tended to be less than that collected for algebraic reasoning (about 50%) and statistical reasoning (60%). Consequently, the zones

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(Table 7.3) obtained were much lower than the hypothetical learning progression (Table 7.1) we originally proposed. More data that captured students’ performance at a higher level is clearly needed. The geometric learning progression is not a replacement of the curriculum. Rather, it is an attempt to provide clarity of what geometric reasoning entails and how to promote such thinking. For this to take effect, a systematic change in recognising the importance of visualisation and mathematical discourse in reasoning, and that geometry is the most viable tool to promote both abilities is needed. Already, we have seen that in some tasks, students in the project schools, who came from lower social strata, outperform students in the trial schools. The next phase of the research will be to interrogate the effectiveness of the targeted teaching advice and activities on teaching and learning.

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Chapter 8

Statistics and Probability: From Research to the Classroom Rosemary Callingham, Jane Watson and Greg Oates

Abstract This chapter presents the outcomes associated with the statistical reasoning strand of the Reframing Mathematical Futures (rmfii) project. Background is provided on the emergence of statistical reasoning and statistical literacy as constructs included in the middle school curriculum. A description of the key statistical ideas of Variation, Expectation, Distribution, and Inference follows, with a description of how these were used in the context of the project. Examples of questions and responses are provided that were used in the development of the Statistical Reasoning Learning Progression. Following a description of the eight zones of the learning progression, there is a discussion of the development of teaching advice for assisting students to progress through the zones. Finally, an example of the professional learning activities for statistical reasoning is given. Keywords statistics and probability – statistical reasoning – middle school – rich statistics tasks – professional development

1

Introduction

This chapter presents the outcomes associated with the statistical reasoning strand of the Reframing Mathematical Futures (rmfii) project. Background is provided on the emergence of statistical reasoning and statistical literacy as constructs included in the middle school curriculum. A description of the key statistical ideas of Variation, Expectation, Distribution, and Inference follows, with a description of how these were used in the context of the project. Examples of questions and responses are provided that were used in the development of the Statistical Reasoning Learning Progression. Following a © koninklijke brill nv, leideN, 2019 | DOI:10.1163/9789004396449_009

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description of the eight zones of the learning progression, there is a discussion of the development of teaching advice for assisting students to progress through the zones. Finally, an example of the professional learning activities for statistical reasoning is given.

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Statistical Understanding

Unlike Algebra and Geometry, Statistics is a relative newcomer to the mathematics curriculum. Data and chance became part of the school curriculum in 1989 in the usa (National Council of Teachers of Mathematics [NCTM], 1989), followed shortly after by other countries (e.g., Australia (Australian Education Council, 1991) and New Zealand (Ministry of Education, 1992)). The inclusion of the domains of statistics and probability in the curriculum led to a growing research agenda on teaching and learning statistics and probability. Early research focussed on student learning in particular components of the domain such as average (Mokros & Russell, 1995), sampling (Watson & Moritz, 2000), graphing (Friel, Curcio, & Bright, 2001), and probability (Jones, Langrall, Thornton, & Mogill, 1997). Over time the interest in students’ understanding has expanded to include variation as the underlying concept of all statistics based on its omnipresence (Cobb & Moore, 1997, p. 801). Research has focused on general understanding (e.g., Reading & Shaughnessy, 2004; Watson, Kelly, Callingham, & Shaughnessy, 2003), as well as more specifically in relation to data distributions (e.g., Lehrer & Schauble, 2004; Noll & Shaughnessy, 2012) or chance outcomes (Shaughnessy, Canada, & Ciancetta, 2003; Watson & Kelly, 2004). Further combining the underlying concepts, research began to consider students carrying out complete investigations, for example by designing surveys to administer and analyse topics of interest (Lavigne & Lajoie, 2007; Meletiou-Mavrotheris & Paparistodemou, 2015; Watson & English, 2015). The appreciation of the need to study students as they complete their own investigations arose from the implications of standards and guidelines produced by the leading organisations supporting mathematics and statistics. The outline provided by the NCTM (1989) was expanded in the American Statistical Association’s (asa) Guidelines for Assessment and Instruction in Statistics Education (gaise) Report (Franklin et al., 2007). Acknowledging variation at each stage, gaise set out the process of statistical problem solving at the school level: Formulate questions (including clarification of the problem); Collect data (including designing a plan); Analyse data (including appropriate methods); and Interpret results (including relating to the original question).

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Statistical Literacy

In parallel to this mathematical view of chance and data, the concept of statistical literacy emerged (Wallman, 1993; Gal, 2002). This notion was concerned with the application of statistics in daily life as an essential component of being an informed citizen and clearly drew on the knowledge and understanding of statistical ideas developed in school. Frameworks were developed to assess statistical literacy (Gal, 2000; Watson, 1997), which took a hierarchical view with critical evaluation at the highest level. These frameworks acknowledged the fundamental contextual nature of statistics (Moore, 1990; Rao, 1975). Although statistical literacy requires statistical knowledge, it does not involve actively carrying out statistical investigations. Rather it involves being able to recognise and understand the usage of statistical terminology and tools in unrehearsed contexts. Spontaneous decision-making then takes place based on general contextual knowledge, and critical literacy and thinking skills. This happens, for example, each time a media announcement is read or heard that makes a strong claim on a public issue based on statistical “evidence.” Watson and Callingham (2003) used Rasch analysis approaches with archived data from several studies that had focussed on components of statistical understanding to create a hierarchical scale of statistical literacy. This scale was subsequently validated using surveys to create a new data set with some of the archived items and some new items (Callingham & Watson, 2005). The hierarchical construct of statistical literacy took account of both the mathematical and contextual nature of statistical literacy and respondents at the highest level could integrate statistical skills with contextual interpretation, in particular drawing on proportional reasoning. This construct has remained stable over time (Callingham & Watson, 2017), and provided the basis for the development of a statistical reasoning construct in the rmfii project. The previously validated scale (Watson & Callingham, 2003) clearly drew on statistical reasoning at an increasingly sophisticated level but required reinterpretation to address the focus on reasoning in the rmfii project.

4

Teachers’ Knowledge for Teaching Statistics

One intended outcome from the rmfii project was a package of materials to support teachers’ development of statistical reasoning in their students. The focus on reasoning, which is in line with the expectations of the NCTM (1989) and asa (Franklin et al., 2007) for students, puts pressure on teachers, many of whom may have had limited experience with learning or teaching statistics

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in their own school or pre-service education. Teachers need to go beyond simply knowing the statistical content to understanding how students’ develop statistical concepts. Using Shulman’s (1987) notion of pedagogical content knowledge, Callingham, Carmichael, and Watson (2016) found that students’ statistical learning outcomes were related to the level of teachers’ measured pedagogical content knowledge for statistics. Others, including Groth (2007), have built upon the work of Hill, Rowan, and Ball (2005) and Hill, Schilling, and Ball (2004) on the “mathematical knowledge for teaching.” Based on their distinction between Common Knowledge (e.g., course content) and Specialised Knowledge (e.g., issues for teaching), Groth added an additional distinction between mathematical knowledge (e.g., computation) and non-mathematical knowledge (e.g., related to context) for teaching statistics. He then provided four categories of examples for each of the four stages of statistical problemsolving in gaise (Franklin et al., 2007). The asa has also provided detailed guidelines for teachers in its document, Statistical Education of Teachers (set) (Franklin et al., 2015). The rmfii project was predicated on the belief that being exposed to and working with learning progressions that indicate students’ developing understanding can help teachers gain deeper knowledge of both the content and their students’ learning (Siemon, Breed, Dole, Izard, & Virgona, 2006). In statistics, this belief implies that working with a suitable, validated learning progression can support teachers to develop the Common and Specialised statistical knowledge, as well as the mathematical and non-mathematical knowledge, for teaching. Thus a principle aim was to develop and validate a learning progression for Statistical Reasoning, and to provide advice to teachers about appropriate teaching approaches at different stages in students’ development. The surveys that had been used previously (Callingham & Watson, 2005; Watson & Callingham, 2003) provided a starting point for this work.

5

Developing the Statistical Reasoning Construct

A framework for the development of the learning progression in statistics was provided by Watson (2006) that identified five interlinking key ideas for statistics in school. In the context of the rmfii project, these were reorganised into three Big Ideas that also linked to the curriculum. Variation is the key underpinning idea for all of statistical reasoning. For the purposes of the Statistical Reasoning Learning Progression and associated teaching advice, this very large idea is described in ways that link more easily to the curriculum. This process recognised the Australian Curriculum: Mathematics (Australian Curriculum,

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Assessment and Reporting Authority [ACARA], 2018), which includes a focus on graphing (Distribution), measures of central tendency and chance (based on Expectation and Randomness), and some indication of interpretation of findings (Inference).The three Big Ideas based on variation used in the project are briefly described here. Variation with Expectation and Randomness (e.g., chance, probability, averages) provides links to the curriculum ideas of chance and uncertainty, and measures of central tendency. Ideas about chance and randomness are well understood in the context of variation. An average, whether mean, median, or mode, expresses an expectation about a data set. The average is a description that typifies the data as given but which varies as new data are added or data points are removed. Variation with Distribution (e.g., graphs, tables, representations) is linked to curriculum ideas about tables and graphing. Distribution describes the ways in which the data are spread out or distributed. A graph is a picture of a distribution and the associated variation. There are many ways of representing data distributions and some of these are highly technical and used only by professional statisticians. In school it is useful to develop the idea that tables, graphs or other representations are ways of visualising variation through the distribution of the data. Variation with Informal Inference (e.g., sampling, populations, decisionmaking) is linked to the curriculum ideas of justifying an answer and drawing conclusions from data that vary. Collecting data samples is a purposeful activity that describes a situation and can help people make decisions. Professional statisticians use a variety of tools to draw inferences from data but at the school level the aim is to encourage students to make informal inferences with justifications based on the data, or to ask questions about the nature of the data. In the rmfii project, a pool of 41 questions was developed based on many that had been used in previous studies (e.g., Callingham & Watson, 2017), together with some newer questions that specifically addressed inference, and some rewritten questions to bring them up-to-date in terms of context. Some questions had multiple parts (e.g., shgt, shown in Table 8.1); others were single questions (e.g., shat8 and sjmes, shown in Table 8.2). Table 8.1 provides an example of a question, consisting of three items, shgt1, shgt2, shgt3, which addresses variation in distribution. The three items begin with straightforward graph reading, then ask explicitly about variability, and request an explanation for the response. Students can demonstrate different levels of understanding, which are associated with zones as explained later.

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Table 8.1  Question addressing variation in distribution

Domain

Question

Variation in The following graphs describe some data collected about Grade 7 Distribution students’ heights in two diffferent schools. School A School B

[SHGT1] How many students are 156 cm tall in each school? [SHGT2] Which graph shows more variability in students’ heights? [SHGT3] Explain why you think this. Coding and scoring rubric SGHT1 0 No response or irrelevant response 1 Reads correct values from graph, School A=9 and School B=10 SGHT2 0 No response or irrelevant response 1 Incorrect (School B or they are the same) 2 Correct (School A) SGHT3 0 No response or irrelevant response 1 Explanation misapplies notion of variability and focuses on an average height (e.g., A, a lot of people are around about the same height in that school) or B – it goes up higher, it is more spaced out) 2 Explanation focuses on the size or the number of the individual bars without regard to what they represent, need to state some feature of the graph (e.g., B – because it goes up and down and varies more, A – has more lengths, more numbers) 3 Explanation implicitly refers to the wide range/diffference of heights (e.g., A, because they have at least one person in every height except 147 cm, School A takes up the whole graph and B doesn’t) 4 Explanation explicitly refers to the wide range/spread and/or variety of heights (e.g., School A has more of each height. School B has lots of one, there’s more variety in heights)

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Table 8.2 shows two single item questions addressing the two domains of Variation in Expectation and Randomness, and Variation in Inference. Despite being single item questions, the scoring rubrics allow for a wide range of responses. Table 8.2  Questions addressing variation in expectation and randomness, and variation in inference

Domain Variation in Expectation and Randomness

Question

[SHAT8] A mathematics class has 13 boys and 16 girls in it. Each pupil’s name is written on a piece of paper. All the names are put in a hat. The teacher picks out one name without looking. Tick the box to show which outcome is more likely – the name is a boy or – the name is a girl or – the name could be a boy or a girl Please explain your answer using as much mathematics as you can. Variation in [SJMES] Inference Every morning James gets out of the left side of the bed. He says that this increases his chance of getting good marks. Explain what you think of this claim.

Coding and scoring rubric 0 No response or irrelevant response 1 Incorrect, little/no reasoning (e.g., it’s just luck) 2 Incorrect (e.g., name is a boy or girl) but reasoning that recognises variation in some way (e.g., depends on mix, same chance, could be anything) 3 Correct (name is a girl) with either no explanation or explanation does not reference total (e.g., 16 is bigger than 13) 4 Correct, fraction included in explanation (e.g., 16/29 chance)

0 No response or irrelevant response 1 Agrees with James or reflects a belief in superstition (e.g., I think he is right; I get out the same side too. It’s bad luck getting out of the other side) 2 Rejects superstition or disagrees with little/no explanation (e.g., Not true; It’s just superstition; It doesn’t matter what side of the bed he gets out on; You still could fail or pass the test ) (cont.)

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Table 8.2  Questions addressing variation in expectation and randomness, and variation in inference (cont.)

Domain

Question

Coding and scoring rubric 3 Rejects claim, recognises that James’ belief does not afffect his chance of good marks in some way, may suggest alternative reasons for his marks at school (e.g., It won’t help him get better marks. You only get good marks if you study and try hard; I don’t think he would unless there’s a wall on the right hand side) 4 Response indicates that beliefs or psychological states concerning luck may influence physical outcomes which are under personal control (e.g., I think he is wrong because whether he gets good marks or not depends on him, the test, whether or not he is concentrating, etc.; Getting out the left side might possibly help a bit though, because if he thought he would get good marks, he might)

Each item was scored using a coding or scoring rubric developed from previous use of the question or based on a pilot administration. These rubrics were important because teachers themselves marked their students’ work. This requirement provided teachers with an opportunity to develop both Common and Specialised knowledge for teaching statistics. The questions were organised into test forms that contained overlapping questions so that they could be combined using Rasch analysis. These forms were given to over 1500 students in Years 7 to 10 across two testing periods (MR1 and MR2). The data from both administrations (MR1 and MR2) were combined for the purpose of scale development (Bond & Fox, 2015) and Rasch-analysed using Winsteps 4.0.0 (Linacre, 2016). (See Chapter 5 for details of the analysis approach.) The scale produced was interpreted with a focus on the reasoning required to respond successfully to clusters of items along the variable, resulting in an eight-level learning progression of statistical reasoning. An overview of this progression is provided in Table 8.3. The

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Zone

Characteristics

Example of behaviour

Zone 1

Idiosyncratic approach or single procedural focus Considers aggregated information and single aspects of sampling but without recognising value Emerging statistical appreciation but without explanation

Reads single value from graph or table; uncertainty is 50%. Reads information from simple graphs; recognises uncertainty “anything can happen.” Recognises expectation and variation but cannot integrate these ideas. Associates two variables or compares graphs but unable to explain or justify reasoning.

Zone 2

Zone 3

Zone 4

Zone 5

Zone 6

Zone 7

Zone 8

Recognises influence of variation and expectation in more complex scenarios but interprets inappropriately Straightforward explanation and simple numerical justifijication

Reasons using some relevant aspects of data but may ignore some important features. Informal appreciation of Constructs reasonable argument uncertainty and variation in chance based on chance and probability; uses measures of central tendency. Makes inferences across ideas using Demonstrates statistical thinking, proportional reasoning including data, context, and representation, in integrated mathematical way. Integrates proportional, statistical, Recognises, coordinates and and contextual reasoning to justify integrates all relevant information conclusions to make evidence based decisions using proportional reasoning.

levels within the learning progression were termed zones, in line with the project nomenclature. In some respects the number of zones is arbitrary, but eight zones appears to provide teachers with enough information both to see how students’ reasoning has developed and where it needs to move to, with sufficient detail to be able to meet students’ learning needs. In practice, most teachers need to work across about three zones for the majority of students in their classroom. Zone 1 was characterised by a focus on a single aspect of the data or context, whereas in Zone 8 respondents could construct a reasoned argument applying

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proportional reasoning to make evidence-based decisions related to context. The zones showed increasing sophistication of statistical reasoning. In the lower levels, responses indicated mainly procedural aspects, such as reading information from a table or graph, or quantifying simple chance situations such as dice rolling. Students provided explanations that simply repeated the question or focussed on irrelevant aspects of context, such as the code 1 response for sjmes shown in Table 8.2. Gradually they became more able to make comparisons and by Zone 4 were able to compare two graphs that were similar or relate two different variables together, such as the code 2 responses to shgt3 shown in Table 8.1. Explanations and justifications were generally very limited or missing altogether, or, as in shgt3, focused on the appearance of the data. In Zone 5, however, there was a deeper consideration of the data and the context, although critical elements of either of these aspects might have been missing. The higher zones demanded integration of mathematical understanding of statistics, increasing use of proportional reasoning, and a sophisticated understanding of context in increasingly complex situations. It is notable, however, that students did not apply proportional reasoning unprompted until Zones 7 or 8, for example the code 4 response to shat8 (see Table 8.2) that appeared in Zone 7. An additional analysis considered the proportion of students from each school year level in each zone to provide a basis for developing benchmarks for teachers to use. The distribution of students across zones is shown in Table 8.4. There are a number of points that need to be made about the distribution in Table 8.4. The schools involved in the project were not a representative sample of schools in Australia. In general they came from lower socio-economic areas as targeted by the project funding. The project also focussed on students in the middle years of schooling from Year 7 to Year 10. The progression, however, could potentially be used outside this year range. In many schools, only students in lower ability classes were included, especially in Years 9 and 10, which was reflected in the surprisingly large proportions in Zones 1 and 2 for these Table 8.4 Distribution of students across zones by year level

Yr 7 % Yr 8 % Yr 9 % Yr 10 % Overall %

Zone 1

Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8

7.6% 8.4% 10.4% 11.4% 9.1%

6.7% 9.5% 6.7% 14.4% 8.5%

25.1% 22.2% 20.2% 15.2% 21.6%

29.7% 27.0% 28.0% 34.8% 28.6%

22.4% 15.9% 15.2% 13.6% 16.9%

5.5% 13.8% 13.0% 7.6% 11.2%

1.7% 2.8% 5.0% 3.0% 3.2%

1.2% 0.3% 1.5% 0.0% 0.8%

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year levels. A big proportion of students was located in Zone 4, regardless of year level. Even with the limitations described, this finding suggests that students did not get practice at explaining and justifying their statistical thinking. Zone 5 marks the actual beginning of statistical reasoning, drawing on data to make decisions. Earlier zones expect only lower level skills, such as reading from graphs and recognising uncertainty. A more representative student sample might show a somewhat different distribution, but these results are similar to the findings described in earlier work (Callingham & Watson, 2005; Callingham & Watson, 2017; Watson & Callingham, 2003). The implications for teaching are discussed in the following sections.

6

Developing Advice for Teachers

A key outcome from the project was to develop a package of materials for teachers providing both tools to determine approximately where their students were along the learning progression and advice about teaching approaches that met the needs of students in the different zones. A selection of questions was organised into test forms that included all three Big Ideas of Statistical Reasoning. These were relatively short (12 to 15 items) and designed to be given to students either as a whole or in parts. Some teachers, for example, preferred to give students a couple of questions in each lesson until all questions had been presented. Teachers were also provided with a Raw Score Translator that converted the score obtained by each student as a result of marking the responses against the scoring rubric into an approximate zone location. The zone scores were obtained by identifying where the rubric code scores were located on the statistical reasoning scale and summing the score values within each zone, creating a cumulative total. This approach had the advantage of allowing teachers to use a familiar system of attributing partial credit for questions only partly correct and obtaining a summary score within a quality-ofresponse framework linked to a hierarchical scale. The zone identified for each student was recognised as approximate but did provide a starting point for planning for classroom teaching. The characteristics of each zone provided the foundation for teaching advice and rich tasks explicitly linked to the learning progression. The zone descriptors shown in Table 8.3 were elaborated, based on the data collected, by examining the skills, knowledge, and understanding that students needed to gain the appropriate code score for items located within each zone. Rather than describing specific items, the cluster of items in each zone was considered holistically to provide a rich description of the statistical reasoning

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that was typical within each zone. The elaborated descriptor for Zone 4, for example, was: – Compares data in two graphs but focuses on single elements only. Associates two variables with a single value and provides descriptive explanations. – Recognises variability and expectation in more complex random situations but explanation refers to uncertainty in general terms and is not quantified, or is based on strict probability (expectation). May not recognise the importance of equal likelihood. – Recognises relative order in language of uncertainty but does not appreciate some subtleties. – Reasons quantitatively in familiar situations involving related comparison and in the context of uncertainty. Relies on additive thinking in situations involving measures of central tendency, and is unlikely to question the quality of data. Critiques sampling approaches using single aspects only (i.e., size or method) in an evaluative situation. Falls back on personal beliefs in more complex situations when asked for an explanation. Students located in Zone 4 clearly had some skills and understanding. They could use numerical approaches but these tended to be additive rather than multiplicative. They could describe situations, sometimes quite elaborately, but did not appreciate the more subtle aspects. Advice for teachers considered two aspects for learning (see Table 8.5). Consolidate and Establish recognised skills and knowledge that were borderline, often those that differentiated the particular zone from the previous zone. Introduce and Develop prepared students to move to the next zone. Within each of the categories a nutshell statement gave an indication of the underlying statistical reasoning that needed to be addressed, followed by suggested activities, including links or suggestions where appropriate to relevant resources.

7

Rich Statistical Activities

Much attention in this project was given to the critical nature of the task design and activities. As Clark and Roche (2009) observed, “there is a strong consensus in the research literature that the nature of student learning is determined by the type of task and the way it is used” (p. 722). Task design is a highly complex process as frequently observed in the literature (e.g., O’Shea & Peled, 2009; Ratnayake, Oates, & Thomas, 2016; Sullivan, Clarke, & Clarke, 2012). Liljedahl, Chernoff, and Zaskis (2007) describe good task design as a recursive process, where tasks are trialled and “analysed for the mathematical and pedagogical affordances that the task, as designed and implemented, actually

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accesses (p. 241). In developing the learning progressions, the mathematical affordances of the questions or tasks were key to the identification of the zones. In developing teaching advice, the zone descriptors informed development of the classroom tasks. Thus the processes used were complementary, one informing the other. One chapter of the 22nd icmi study Task Design in Mathematics Education (Watson & Ohtani, 2015) considers frameworks and principles for task design and traces the history of task design along with the psychological and theoretical perspectives that have informed and influenced it. Kiernan, Doorman, and Ohtani (2015) describe task design within three frames that include the overarching theoretical perspective within which the task is embedded (for example, socio-constructivism as one of many possible influences), through to the more local, domain-specific frames that address particular mathematical concepts, procedures, or processes, such as statistical reasoning. They highlight a number of key principles for effective task design identified in the literature, for example the five principles reported by Prusak, Hershkowitz, and Schwarz (2013) (cited in Kiernan, Doorman, & Ohtani, 2015, p. 37), who suggest that rich tasks should: – Encourage the production of multiple solutions (Levav-Waynberg & Leikin, 2009). – Create collaborative situations (Arcavi, Kessel, Meira, & Smith, 1998). – Engage in socio-cognitive conflicts (Limón, 2001). – Provide tools for checking hypotheses (Hadas, Hershkowitz, & Schwarz, 2001). – Invite students to reflect on solutions (Pólya, 1945/1957). Although these principles may not all be essential for the tasks used in the initial assessment, they do highlight the critical and complex nature of the tasks needed as follow-up activities. There are a multitude of internet sites and books containing a wealth of rich tasks for teachers that include statistical reasoning and can be seen to embody the principles as described above. Consider for example the nRich project (https://nrich.maths.org), or in the Australian context the rich tasks developed by Maths300 project (http://www.maths300.com), and the challenging tasks being developed by both the reSolve project (https://www.resolve.edu.au) and by Peter Sullivan and the team in the Task Types in Mathematics Learning (ttml) project (Sullivan, 2017; Sullivan, Clarke, & Clarke, 2012). The Australian Association of Mathematics Teachers (aamt, http://aamt.edu.au/) publishes and sells a wide range of resources on its website (including Maths300), for example Sullivan (2017) and Teaching with Rich Learning Tasks: A Handbook (Flewelling & Higginson, 2005) that provide mathematical tasks spanning the curriculum.

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Given the wide spread of students across zones (see Table 8.4) of the Statistical Reasoning Learning Progression, it is clear that tasks were needed that required multiple access points for students. This characteristic is not always a feature of tasks aimed at specific curriculum levels. The study by Clarke and Roche (2009), for example, identified particular challenges for teachers when using focussed mathematical tasks situated within a specific context, which is particularly relevant for statistical reasoning because of the contextual nature of statistics. Teachers found the tasks difficult to use because they were too challenging for some while others finished early, with less confident students not knowing how to start the tasks without assistance. Thus, a significant feature of the tasks developed for use in the rmfii project is that they are suited to a multitude of levels, and can be readily adapted for further extension. Resources such as Mathematical Assessment for Learning: Rich Tasks and Work Samples (Downton, Knight, Clarke, & Lewis, 2013) provide scaffolding for teachers in adapting the tasks, whereas tasks from other projects (e.g., reSolve) provide detailed advice for teachers in structuring the tasks for their classes, providing very valuable approaches for teachers looking for activities to follow up the learning progression assessments. The evidence from rmfii is that working with the learning progressions also facilitates teachers in adapting tasks themselves to extend students in zones at either end of that which the curriculum level of the original task may be aimed (Moore, 2017). The Moore study, however, focuses on the Algebraic Reasoning progression, and similar evidence for statistics remains to be tested. Such evidence would provide a valuable confirmation of the usage of Groth’s (2007) ideas about Common and Specialised statistical knowledge for teachers.

8

Links between Zones of the Learning Progression and Teaching Advice

Table 8.5 shows the descriptor and teaching advice for Zone 5. This advice was built on Zone 4 and addressed the outcomes described in Zone 5, while also setting up knowledge and understanding needed in Zone 6. For example, in Zone 4, students were beginning to use measures of central tendency but did not appreciate the underlying relationships because they were reasoning additively. By Zone 6, the focus is on the use of these statistics to describe distributions. Hence the advice for Zone 5 focussed on the appropriate choice of a summary measure of central tendency in relation to a particular context, and introduced the ideas about summarising a distribution (e.g., Balancing

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Statistics and Probability table 8.5  Descriptor and teaching advice for Zone 5

ZONE 5

Teaching advice

Consolidate and establish Provides a statistical Consolidate and make explicit ideas about explanation but this may be fairness e.g., Odds Fair and Unfair (MCTP Chance incomplete (e.g., SAMEA.3), and Data Investigations Vol. 1); Two Coins, Fairness and recognises equal outcomes of dice (Available through the RMF2 project). for all numbers (e.g., SCON3.2). Continue to construct a variety of graphs Recognises simple proportion from stories, news articles, e.g., use in chance contexts (e.g., http://graphingstories.com/ to provide videos BOX9.3). that students can then graph. Write a short report Orders language of chance about the process. qualitatively (e.g., WORD.2). Continue to explore large samples with Recognises key aspects of technology to make predictions about central tendency but reverts increasingly complex situations, e.g., data from to non-statistical justifijications the Melbourne Cup or the Bureau of Meteorology. (e.g., SAMED.2, SAOUT.2). Explore all measures of central tendency with Implicitly recognises that all an emphasis on the most appropriate for context combinations of numbers have e.g., Balancing Act (Rubin & Mokros, 1990); House the same chance (e.g., STATS3). prices; MCTP Chance and Data Investigations Vol. Intuitively suggests association 2, Bikes, Monkey Bars and Skeletons. expressed in non-quantitative Introduce and develop way (e.g., SSKIN.2), and Critique diffferent graphical representations can recognise important to interpret, describe, and compare afffordances information in making and constraints of diffferent representations, e.g., comparisons (e.g., STWN3.2). Numeracy in the News: Data Representation Recognises relevant aspects http://www.mercurynie.com.au/mathguys/ of graphical representation introduc.htm and uses these to reason Introduce the relationship between statistical statistically (e.g., spread in data and algebra through function graphs, e.g., STWN2.2 or STWN1.2) but may Guessing the graph (nrich) https://nrich.maths. not include all aspects (e.g., org/6990 SGHT3.3). Represent and quantify relationships between Recognises appropriate sample two variables, e.g., use percentages in two way size (e.g., SCLIM1.2) and tables; Collect data about ‘favourites’ such as ice provides appropriate critiques cream flavours, food, TV shows, etc., by boys, girls of sampling method (e.g., or Year 7/Year 8. Start with only two choices to SMV10.2, SMV12.2) but does not create 2x2 tables. Then expand to more choices. explicitly include randomness. Balancing Act Associations.

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Act activity (Rubin & Mokros, 1990) that collects data about balancing on one leg with eyes closed). Teachers were given access to a range of rich tasks from which they could choose approaches that would suit their particular context, and were also encouraged to seek out rich tasks of their own, using some of the resources suggested earlier.

9

Professional Learning

In addition to the materials, teachers were also able to access professional learning sessions either face-to-face at project meetings, or online using a collaborative software program that allowed them to ask questions in real time. These online sessions were recorded so that teachers who were unable to be at the sessions could access them later. One of the approaches to professional learning for teachers was to present them with hypothetical classroom scenarios where students and teachers interacted across zones. The objective was to build teachers’ confidence in their abilities to scaffold classroom discussion to help students develop deeper understanding, rather than merely provide a “correct” answer. The presentation was designed to be used in several ways, depending on whether a group of teachers was present in a “live” setting with a leader, or whether teachers were alone, wishing to consider their pedagogies. A leader of a group session was expected to stop at various points in the presentation to ask the audience to suggest the next move on the part of the teacher or perhaps the next expected response from a student. Teachers might make suggestions at any point based on their own previous experience, as well as try out new ideas on their colleagues. A single teacher encountering the scenario would need to pause the presentation of the stimulus and consider what should happen (or be said) next before moving on. Three different media were available in the rmfii project: PowerPoint slides that could be progressed manually by a presenter or individual teacher; a video presentation that again could be paused; or written text (e.g., a conference paper). Although the media differed, the purpose was to present a realistic dialogue between a teacher and students, providing a range of responses across zones. This idea is illustrated with extracts from a PowerPoint presentation based on student responses collected in previous student surveys. The problem considered by the class in the PowerPoint is shown in Figure 8.1. It is a two-way table conditional probability problem with data that suggest independence of two variables in a context where this is likely to be a surprising result.

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figure 8.1 The lung disease problem (Batanero, Estepa, Godino, & Green, 1996)

In the introductory part of the lesson the teacher and class discuss students’ beliefs about the context of the problem and sampling issues, which cover the lower zones of the Learning Progression with respect to this problem. The teacher then moves the class to looking at the data in the table (Figure 8.1) and invites a dialogue among the students in the class, with different responses struggling with proportional reasoning in Zones 4 to 6. She allows the students to disagree with each other as various reasons are presented, as shown in Figure 8.2. The response of Mason in Figure 8.2(a) is typical of Zone 3, as he identifies a single value (90) because it is “the biggest” after considering all the other values in the table. At this point, teachers in a professional learning session could discuss how they would respond. In a live presentation to a group of rmfii teachers, many suggestions were made, some jumping straight to the issue of the totals and proportional reasoning. In Figure 8.2(b), however, the teacher asks for other students’ responses. Initially this is a good strategy as it gives the students a chance to interact and reason with each other. Dealing with the second response in Figure 8.2(b) raises a second dilemma for teachers to consider, a “correct” answer but inappropriate reasoning to justify it. The situation is resolved by the teacher in Figure 8.2(c) by presenting a counter example similar to the student’s consideration, focussing only on the highlighted cells, with other cells reduced to zero because the student had ignored them, and then asking for a conclusion. A further interaction occurs with students suggesting other reasoning based on two cells, leading to the issue of the different sample sizes for the two conditions, which had been noted earlier by a student. After reaching the appropriate decision of independence for the data in the table using proportional reasoning, there remains an issue for the teacher about “right” answers

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figure 8.2 Initial suggestions for the lung disease problem

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figure 8.3 The importance of an argument

to the question as shown in Figure 8.3. The teacher makes a very important point about the evidence supporting an answer being as important as the answer itself. The idea of presenting teachers with examples of students’ responses and possible classroom dialogue and discussion was powerful in developing teachers’ own understanding of the statistical ideas, as well as providing them with an authentic classroom context. Examples of the use of rich tasks in classroom scenarios such as the one presented here have been provided by Watson (2016, 2017a, 2017b) and Callingham, Watson, and Siemon (2017) in different contexts. In particular, these suggestions and scenarios reflect Groth’s (2007) categories for non-mathematical Specialised knowledge for teaching statistics.

10

Implications for Classroom Teachers

Identifying an approximate initial zone for students’ understanding provides a useful approach for teachers. In essence this approach targets the point at which teaching is likely to be maximally effective, because it focusses on the zone in which students have some understanding but need consolidation of current ideas and development of new concepts in order to progress. The use of rich tasks with multiple entry points allows for students across zones to participate fully in the class at their current level of understanding. Targeted teaching of this type has been shown to be effective (Siemon, Breed, Dole, Izard, & Virgona, 2006) in the context of multiplicative thinking. Combining the professional learning and teaching advice with the surveys that teachers

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were able to use at the start, throughout, or at the end of units of work, creates a complete package for schools to use. In particular, the use of rich tasks provides opportunities for students to explain and justify their thinking. Unless these opportunities are presented and expected in the classroom setting, the evidence suggests that students do not progress easily to the higher levels of thinking that have been deemed desirable for statistical literacy (e.g., Gal, 2000, 2002; Wallman, 1993; Watson, 1997). The components of this project – research involving the identification and characterisation of learning progressions and the practical aspects of providing teaching advice directly linked to the research outcomes – provide an exemplar for linking research and practice in potentially powerful ways to further the development of Statistical Reasoning in the school context.

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Chapter 9

Investigating Mathematics Students’ Motivations and Perceptions Tasos Barkatsas and Claudia Orellana

Abstract The purpose of this study was to explore the factorial structure of motivation and perception items from a student survey utilised as part of the Reframing Mathematical Futures II (rmfii) Project. Data were collected in 2016 from students in Years 7 to 10 from various different states and territories across Australia. An exploratory factor analysis identified four factors which were consistent with the studies the items were adapted from: Intrinsic and Cognitive Value of Mathematics, Instrumental Value of Mathematics, Mathematics Effort, and Social Impact of School Mathematics. A multivariate analysis of variance (manova) also revealed that there were statistically significant differences between Year Level for some of these factors. The results from the study have confirmed that the survey items continue to be valid and reliable in the mathematics context. The findings also highlight the need for further investigations to examine how students’ motivations and perceptions of mathematics develop and differ across the different states and territories in Australia. Keywords motivation – perceptions – mathematics – factorial structure

1

Theoretical Framework

Over the past decades, mathematics education researchers have recognised the importance of affective factors in the teaching and learning of mathematics (Di Martino & Zan, 2010; Goldin, 2002; Hannula, 2002). Seen as playing a significant role in this field, the affective research area investigates “the interplay between cognitive and emotional aspects in mathematics education” (Di Martino & Zan, 2010, p. 1). Although the term affect has been interpreted in many ways (Zan, Brown, Evans, & Hannula, 2006), McLeod (1992) made a © koninklijke brill nv, leideN, 2019 | DOI:10.1163/9789004396449_010

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key contribution to this area of study, conceptualising the affective domain as composed of three mayor constructs – attitudes, beliefs, and emotions. In addition, he proposed that: We can think of beliefs, attitudes and emotions as representing increased levels of affective involvement, decreased levels of cognitive involvement, increasing levels of intensity of response, and decreasing levels of response stability. (p. 579) Of the three constructs presented by McLeod (1992), attitudes is considered to have the longest history but is also a construct that has been difficult to define (Di Martino & Zan, 2010; Leder, 1992). In the simplest sense, attitudes can be described as, “someone’s basic liking or disliking of a familiar target” (Hannula, 2002, p. 25), however various definitions in the literature contain overlapping aspects with beliefs and emotions. Early definitions of attitudes by Neale (1969) and Hart (1989) included beliefs about mathematics and its usefulness to the learner as important aspects of this construct, with Hart (1989) also considering “one’s emotional reaction to mathematics” (p. 39). Exploring the relationship between attitudes, beliefs, values, and emotions or feelings further, Leder and Grootenboer (2005) conceptualised attitudes as a moderately stable construct which can fluctuate more than beliefs and values but not as much as emotions or feelings. This aligns with McLeod’s (1992) distinction between attitudes and beliefs in that he considered beliefs as more stable and not as easily changed. Despite its ambiguous nature in research, there have been a number of instruments developed to measure students’ attitudes towards mathematics. Early scales developed by Michaels and Forsyth (1977) and Sandman (1980) saw attitude as a multidimensional construct and incorporated items measuring various factors such enjoyment or confidence. In accordance with this view, the Attitudes Toward Mathematics Inventory developed by Tapia and Marsh (2004) explores four underlying dimensions for attitudes including selfconfidence, value, enjoyment, and motivation. While there is a wide selection of instruments available, one of the most popular instruments in this field of research in mathematics education is the Fennema-Sherman Mathematics Attitudes Scales (Fennema & Sherman, 1976). Also viewing attitudes as a multidimensional construct, this instrument measures nine scales each composed of 12 items using a 5-point Likert scale. These include Attitude towards Success in Mathematics, Mathematics as a Male Domain, Mother, Father, Teacher, Confidence in Learning Mathematics, Mathematics Anxiety, Effectance Motivation, and Mathematics Usefulness

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(Fennema & Sherman, 1976). From the aforementioned instruments, it is clear that the “formation of a mathematical attitude is a complex process involving the interaction of many factors. It cannot be explained simply or completely” (Taylor, 1992, p. 12). Another affective variable, which has implications for students’ involvement in learning, is the construct of motivation (Churchill et al., 2013; McLeod, 1992). Although not as “popular” in the context of mathematics education (Hannula, 2006), current studies in this field can provide profound insights into the reasons behind student achievement and/or failure (Middleton & Spanias, 1999). As highlighted by Fielding-Wells, O’Brien, and Makar (2017): Motivation affects learning and behaviour by focussing attention towards a particular goal, in turn, leading to an increased energy and effort, an increased initiation of activities and a greater persistence in carrying out those activities. (p. 238) As with other affective dimensions, motivation has also been defined in various ways and from numerous different theoretical perspectives. Generally, “motivations are reasons individuals have for behaving in a given manner in a given situation” (Middleton & Spanias, 1999, p. 66). From the behaviourist perspective, some of these reasons stem from external incentives, such as a reward or to avoid punishment, referred to in the literature as extrinsic motivation. (Churchill et al., 2013; Middleton & Spanias, 1999). The social cognitivist view, however, links motivation with a sense of self and self-efficacy (Churchill et al., 2013). When learning is considered valuable and done for “its own sake” this drive or desire is known as intrinsic motivation (Middleton & Spanias, 1999). As summarised by Deci and Ryan (1985): When people are intrinsically motivated, they experience interest and enjoyment, they feel competent and self-determining, they perceive the locus of causality for their behavior to be internal, and in some instances they experience flow. (p. 34) Middleton and Spanias (1999) also claimed that students who are intrinsically motivated display pedagogically desirable behaviours, such as persistence and risk taking, which are essential for achievement in the mathematics classroom. Although there are numerous theories for this construct, motivation has, in some studies, been seen as a commonly accepted element of attitudes (Nisbet & Williams, 2009). Many of the instruments, which measure mathematics motivation, stem from the Fennema-Sherman Mathematics Attitudes Scales

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(Fennema & Sherman, 1976) and the Attitudes Toward Mathematics Inventory (Tapia & Marsh, 2004) both of which include selected items on motivation. In an attempt to provide a more comprehensive instrument for measuring this construct, Lim and Chapman (2015) developed the Academic Motivation Towards Mathematics Scale (amtms) based on the self-determination theory of motivation, which encompasses the subdomains of extrinsic motivation, intrinsic motivation, and amotivation (an absence of extrinsic and intrinsic motivation). Although only tested in an Asian context, the instrument showcases the multifaceted and complex nature of motivation. Wigfield and Eccles (2000) noted in their review, that various motivation theorists have posited a variety of constructs to explain the influence of motivation on choice, persistence, and performance. The Expectancy-Value theory of motivation, proposed by Eccles et al. (1983) argues that an individual’s choice and persistence in a task can be predicted by their expectations for success and how much they value the task. Success expectations are defined as an individual’s “beliefs about how well they will do on upcoming tasks, either in the immediate or longer term future” (Wigfield & Eccles, 2000, p. 70). Values, on the other hand is composed of four components: attainment value (the importance of doing well in the task), intrinsic value (enjoyment of the task), utility value (usefulness of the task), and cost (negative aspects of the task such as effort) (Wigfield & Eccles, 2000). Intrinsic value can be seen as similar to the idea of intrinsic motivation as both relate to completing a task out of enjoyment, whereas utility value focuses on more extrinsic motivations for completing a task (future benefits) (Wigfield & Eccles, 2000). The Expectancy-Value theory has been utilised in various studies to explore students’ choices and achievement in mathematics, including in Australian contexts (Lazarides & Watt, 2015; Watt, 2004, 2010). This chapter will incorporate a survey instrument based on this framework to explore students’ mathematics motivations and perceptions in order to gain insights into their learning experiences in mathematics.

2

Aims

The aims of the study were to investigate: – The factorial structure of the motivations and perceptions items in the Reframing Mathematical Futures II (rmfii) Project survey. – The existence of statistically significant differences among the derived factors and independent variable Year Level.

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3

Methods

3.1 Data Source and Sample An online survey was undertaken as part of the rmfii Project, which aims to find ways to improve the teaching and learning of mathematics students in Year 7 to 10. The purpose of the survey was to examine students’ views regarding their learning experiences in mathematics. The participants came from Australian State and Catholic schools involved in the rmfii project across various Australian States and Territories. A total of 606 Year 7 to 10 students from nine schools across New South Wales, Queensland, Northern Territory, and South Australia responded to the survey. 3.2 Instrument The survey consisted of 95 items and was designed by adapting items from instruments developed in prior studies (Dweck, Chiu, & Hong, 1995; Frenzel, Goetz, Pekrun, & Watt, 2010; PISA, 2006; Watt, 2004, 2010; Wyn, Turnbull, & Grimshaw, 2014; You, Ritchey, Furlong, Shochet, & Boman, 2011). The survey examined the following constructs: Mathematics Learning Climate, Friends Perceptions of Mathematics, Perceptions of naplan, Homework, Mathematics Motivations and Perceptions, Gender Perceptions of Mathematics, Personal Goals in Mathematics, Mindset, Perceptions of School, Perceptions of Mathematics Teaching, and Mathematics career. For the purposes of this chapter only the 2016 Mathematics Motivations and Perceptions item responses will be examined. There are a total of 21 items adapted from Frenzel, Goetz, Pekrun, and Watt (2010), PISA (2006), and Watt (2004, 2010) examining factors that influence students’ perceptions of mathematics and their beliefs about themselves as mathematics learners. 3.3 Data Collection A link to the online survey was provided to participating students by their teachers from September 2016 and it was completed either in the students’ own time at home or during class time. The survey was anonymous and students and their respective parents were made aware of the purpose of the survey.

4

Results

An initial data screening was carried out to test for univariate normality, multivariate outliers (Mahalanobis’ distance criterion), homogeneity of

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variance-covariance matrices (using Box’s M tests) and multicollinearity and singularity (tested in the manova analysis). Descriptive statistics normality tests (normal probability plot, detrended normal plot, Kolmogorov-Smirnov statistic with a Lilliefors significance level, Shapiro-Wilks statistic, skewness and kurtosis) showed that assumptions of univariate normality were not violated. Mahalanobis’ distance was calculated and a new variable was added to the data file. There were fewer than twenty outlying cases, which is acceptable in a sample of 606 students. These outliers were therefore retained in the data set. Box’s M Test of homogeneity of the variance-covariance matrices (which tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups) was not significant at the 0.001 alpha level and we therefore concluded that we have homogeneity of variance. The questionnaire items were subjected to an Exploratory Factor Analysis (efa) by using spsswin. Reliability tests were also conducted. A Multivariate (or Multiple) Analysis of Variance (manova) statistical test was used to investigate statistically significant differences by Year Level. 4.1 Exploratory Factor Analysis (efa) Given the exploratory nature of the study and that the structure could vary, three factor analyses one for each of the possible combinations between the three Year Levels (7, 8 and 9) categories with sufficient student numbers were performed in order to investigate possible differences between Year Levels. Since no differences were observed in the four initial analyses, a final factor analysis using data from 606 complete students’ responses to the 21 items forming the questionnaire, indicates that the data satisfy the underlying assumptions of the factor analysis and that together four factors (each with eigenvalue greater than 1) explain 74.6% of the variance, with 42.9% attributed to the first factor – Intrinsic and cognitive value of mathematics (see Table 9.1). Further, according to Coakes and Steed (1999), if the Kaiser–Meyer–Olkin (kmo) measure of sampling adequacy is greater than 0.6 and the Bartlett’s test of sphericity (bts) is significant then factorability of the correlation matrix is assumed. A matrix that is factorable should include several sizable correlations. For this reason (Tabachnick & Fidell, 1996) it is helpful to examine matrices for partial correlations where pairwise correlations are adjusted for effects of all other variables. Tabachnick and Fidell (1996) further stated that: Significance tests of correlations in the correlation matrix provide an indication of the reliability of the relationships between pairs of variable. If the correlation matrix is factorable, numerous pairs are significant (and) there are mostly small values among the variables with effects of the

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other variables removed. Finally, Kaiser’s measure of sampling adequacy is a ratio of the sum of squared correlations to the sum of squared correlations plus sum of squared partial correlations. The value approaches one if partial correlations are small. Values over 0.6 and above are required for good FA (factor analysis). (p. 642) The Kaiser–Meyer–Olkin (kmo) measure of sampling adequacy in this study is greater is 0.923 and the Bartlett’s test of sphericity (bts) is significant at 0.001 level, so factorability of the correlation matrix has been assumed. Reliability analysis yield satisfactory Cronbach’s alpha values for each factor: Factor 1, 0.95; Factor 2, 0.90; Factor 3, 0.87 and Factor, 0.78. This indicates a strong degree of internal consistency in each factor. table 9.1  Rotated factor matrix (varimax rotation)

1 Factor 1: Intrinsic and Cognitive Value of Mathematics Q46 I fijind mathematics enjoyable Q45 I fijind mathematics interesting Q48 I would like to fijind out more about some of the things we deal with in our mathematics class Q51 Being good at mathematics is an important part of who I am Q44 I like mathematics more than other subjects Q49 I want to know all about mathematics Q47 After a mathematics class, I look forward to what we are going to do in the next lesson Q52 It is important for me to be someone who is good at solving mathematics problems Q50 Being someone who is good at mathematics is important to me Factor 2: Instrumental Value of Mathematics Q33 What I learn in mathematics is important for me because I need this for what I want to study later on Q35 Studying mathematics is worthwhile for me because what I learn will improve my career prospects Q32 Making an efffort in mathematics is worth it because this will help me with what I want to do

Factor 2 3

4

.866 .856 .820 .807 .806 .803 .784 .738 .709

.854 .849 .835 (cont.)

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table 9.1  Rotated factor matrix (Varimax rotation) (cont.)

1 Q34 I study mathematics because I know it is useful for me Q36 I will learn many things in mathematics that will help me get a job Factor 3: Mathematics Efffort Q40 It worries me that mathematics courses are harder than other courses Q41 I am concerned that I won’t be able to handle the stress that goes along with studying mathematics Q39 Achieving in mathematics sounds like it really requires more efffort than I’m willing to put in Q37 When I think about the hard work needed to get through in mathematics, I am not sure that it is going to be worth it in the end Q38 Considering what I want to do with my life, studying mathematics is just not worth the efffort Factor 4: Social Impact of School Mathematics Q42 I’m concerned that working hard in mathematics classes might mean I lose some of my close friends Q43 I worry about losing some valuable friendships if I’m studying mathematics and my friends are not

Factor 2 3 .795

4

.787

.856 .827 .735 .696

.535

.903 .899

The naming of the four factors was guided by the relevant literature and the nature of the questionnaire items associated with each factor. This resulted in the following four factors (F1–F4): F1: Intrinsic and Cognitive Value of Mathematics, F2: Instrumental Value of Mathematics, F3: Mathematics Effort and F4: Social Impact of School Mathematics. We will discuss each factor in detail, in what follows. Factor 1: Intrinsic and Cognitive Value of Mathematics The first component consists of nine items, which examine the intrinsic and cognitive value of mathematics. Three of these items were adapted from Watt (2004) and examine students’ intrinsic value of mathematics (i.e., how likeable or enjoyable students find the subject). Three items were adapted from Watt’s (2010) step study which examine the attainment value of mathematics (i.e., how important it is to do well in mathematics). The final three items were

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adapted from Frenzel, Goetz, Pekrun, and Watt (2010) and examine students’ interest in mathematics. All nine items explore mathematical value in terms of personal enjoyment, importance, or interest hence the construct has been labelled to encompass these factors (intrinsic and cognitive value). F2: Instrumental Value of Mathematics The second component consists of five items that examine the instrumental value of mathematics (i.e., that the learning of mathematics is valuable for students’ futures). The five items have all been adapted from the PISA (2006) questionnaire and specifically examined students’ instrumental motivation to learn science subject(s) – the term science subject(s) was replaced with mathematics. These items were the only items taken from the PISA (2006) questionnaire and have loaded to develop a construct consistent with the original study. F3: Mathematics Effort The third component consists of five items and examines students’ perceptions of the effort required in mathematics. The items were adapted from Watt’s (2010) step study and examine the “costs” associated with mathematics. Three items examine the Effort Costs and two items examine the Psychological Costs associated with mathematics. As the latter two items can be related to the greater effort expended in mathematics (harder and more stressful) the five items have been grouped together under the overall construct of Mathematics Effort. F4: Social Impact of School Mathematics The fourth component consists of two items examining the social impact of school mathematics. These items were adapted from the instrument used in Watt’s (2010) step study, which specifically examined the Social Cost perceived by students as a result of studying or working hard in mathematics. Consistent with this study, the two items have loaded to form the construct labelled here. 4.2 Analysis of Variance (anova) The existence of statistically significant differences on each of the four derived factors by Year Level was investigated by conducting a Multivariate Analysis of Variance (manova) statistical test. The dependent variables (DVs) were the four factors derived from the EFA and the independent variables (IVs) were Year Level (Levels 7–9). Year 10 students’ responses have not been included in this analysis because of the small number of students. Pillai’s Trace criterion was used to test whether there are no significant group differences on a linear combination of the dependent variables. Pillai’s

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Trace criterion is considered to have “acceptable power and to be one of the most robust statistics against violation of assumptions” (Coakes & Ong, 2011, p. 158). Since the multivariate effect for Year Level is significant (p < 0.05), we have to interpret the univariate between-subjects effects by adjusting for family-wise or experiment-wise error using a Bonferroni-type adjustment, and we derive the adjusted alpha level 0.01 (0.05/4). According to Coakes and Ong (2011), “adjusting for familywise or experiment-wise error decreases the chance of Type I error” (p. 158). Using this alpha level, we have significant univariate main effects for the following variables: – Factor 2: Instrumental Value of Mathematics [F(2, 561) = 9.06, p < 0.001, η2 = .04] Effect sizes were calculated using eta squared (η2). In our interpretation of effect sizes we have been guided by Cohen, Manion, and Morrison’s (2007) proposal that 0.1 represents a small effect size, 0.3 represents a medium effect size, and 0.5 represents a large effect size. In our case, the effect size was .04 (medium effect). Post hoc tests The purpose of a post hoc tests is to determine which Year Levels are statistical significant different from each other. A Games-Howell post hoc multiple comparisons test was performed in order to explore the differences for each component. The Games-Howell test has been used because the Year Level sizes differ. It was found that Year 8 and Year 9 and the Year 7 and Year 9 students’ scores had significantly different mean values (p < 0.001) for Factor 2: Instrumental Value of Mathematics. The mean scores indicate that Year 8 students had a marginally higher mean than Year 7 students and a higher mean than Year 9 students. Also, Year 7 students had a higher mean than Year 9 students. – Factor 3: Mathematics Effort [F(2, 564) = 11.13, p < 0.001, η2 = .02] Effect sizes were calculated using eta squared (η2). The effect size was .02 (small effect). A Games-Howell post hoc multiple comparisons test was performed in order to explore the differences for each factor. It was found that Year 8 and Year 9 students’ scores had significantly different mean values (p < 0.001) for Factor 3: Mathematics Effort. It was also found that the Year 7 and Year 9 students’ scores had significantly different mean values (p < 0.001) for Factor 3. It was also found that Grade 7 and Grade 9 students’ scores were statistically significantly different (p < 0.001) for Factor 3. The mean scores indicate that

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Year 9 students had a higher mean than Year 7 and Year 8 students. Also, Year 8 students had a marginally higher mean than Year 7 students. – Factor 4: Social Impact of School Mathematics [F(2, 568) = 20.71, p < 0.01, η2 = .02] Effect sizes were calculated using eta squared (η2). The effect size was .02 (small effect). A Games-Howell post hoc multiple comparisons test was performed in order to explore the differences for each factor. It was found that Year 8 and Year 9 students’ scores had significantly different mean values (p < 0.01) for Factor 4. The mean scores indicate that Year 9 students had a higher mean than Year 8 students. We have been using Boxplots to demonstrate the manova analysis outcomes pictorially. Box and whisker plots represent an application of the median. The lowest quartile is the 25th percentile, meaning that 25% of the scores are at or below the lowest quartile and 75% are at or above it. The median (the horizontal line inside the box) is the 50th percentile, meaning that 50% of the scores are at or below the median and 50% are at or above it. The upper quartile is the 75th percentile, meaning that 75% of the scores are at or below the upper quartile and 25% are at or above it. A large spread of scores inside the box indicates greater variability. Figure 9.1 summarises the four factors and the Years 7–9 scores. The horizontal axis (independent variable) represents the Years 7–9 scores. The vertical axis (dependent variable) represents the factor scores. It can be seen that

figure 9.1 Boxplot of the four derived factors by year level

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there were variations between Year Levels for each of the four factors, F1: Intrinsic and Cognitive Value of Mathematics, F2: Instrumental Value of Mathematics, F3: Mathematics Effort and F4: Social Impact of School Mathematics.

5

Discussion

The aims of the study presented in this chapter were to investigate the factorial structure of the motivation and perception items of the rmfii Project survey and to determine if there were statistically significant differences on the identified factors between Year Levels. An exploratory factor analysis revealed that there were four factors, with eigenvalues greater than 1, explaining 74.6% of the variance. Items loaded on factors similar to the studies they were adapted from (Frenzel et al., 2010; PISA, 2006; Watt, 2004, 2010) and were labelled Intrinsic and Cognitive Value of Mathematics, Instrumental Value of Mathematics, Mathematics Effort, and Social Impact of School Mathematics. Analysing the results by Year Level, an Analysis of Variance (anova) revealed that there were statistically significant differences for the factors Instrumental Value of Mathematics, Mathematics Effort, and Social Impact of School Mathematics. Year 9 students had a significantly lower mean score for Instrumental Value of Mathematics compared to Year 7 and Year 8 students. However, Year 9 students had a significantly higher mean score than both Year 7 and Year 8 students for Mathematics Effort and a significantly higher mean score than Year 8 students for Social Impact of School Mathematics. Thus, it appears that the Year 9 students in this study see mathematics as less valuable for their future, requiring more effort, and having more of a social cost than their Year 7 and Year 8 counterparts. The findings with respect to effort can be expected given that mathematical ideas and concepts grow in complexity as students move into higher year levels. Examining the Australian Mathematics Curriculum (Australian Curriculum, Assessment and Reporting Authority, 2010 to present), Year 9 students in particular are introduced to new concepts that they have not encountered before in earlier years (e.g., non-linear relations, Pythagoras theorem, trigonometry). An Australian study by Watt (2004) also found that all participating students perceived mathematics as requiring slightly more effort from the end of Grade 7 through to Grade 10. The increase in the complexity of mathematics across the secondary years of school may also serve to explain the lower instrumental value scores in the participating Year 9 students. Watt (2004) found that mathematics utility value declined through Grade 7 to Grade 10 with the decline becoming greater over time. She suggests that “the sharp declines could be due to math becoming

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progressively more abstract and hence less obviously useful by senior years” (p. 1570). A decline in mathematics importance was also evident in an earlier longitudinal American study conducted by Fredricks and Eccles (2002) from the primary years up until Year 10. However, it is important to note that students’ perceptions of mathematics importance in this study increased beyond Year 10. The authors attributed this change to students’ greater focus on their future careers. As Australian students select subjects based on their future career prospects in their final two years of secondary schooling (Year 11 and Year 12), it may also be the case that students’ instrumental value of mathematics would increase during this time. A longitudinal study including these final two years of secondary schooling would be intriguing to examine any changing trends. Fredricks and Eccles (2002) also noted that, “as children grow older, they are more likely to engage in social comparison, which may result in a more critical evaluation of their abilities” (p. 528). The period of adolescence is a significant developmental period for students as they are developing their social identities and are undergoing various physiological and emotional changes during this time (McInerney & McInerney, 2006). Year 9 in particular has been identified as a key phase during this developmental time and a Victorian document by the Department of Education and Training highlights the importance of understanding the learning needs of students in this year level (Cole, Mahar, & Vindurampulle, 2006). Such a key time in students’ social development may explain the higher scores on the Social Impact of School Mathematics factor by Year 9 students as students in this year level may be more concerned about their peers’ perceptions of them and may fear losing their friendships as a result of studying mathematics. However, contextual factors, such as the classroom environment, may also play an important role and further investigation would be required to explain these findings further. The factor solution that has been proposed in this chapter will be crossvalidated by using a Confirmatory Factor Analysis (cfa) on the combined 2016-8 data, once the 2018 data have been collected.

6

Conclusion

The results of this study have shown that the items used for the rmfii Project survey continue to produce similar factors to the studies they were adapted from confirming their validity and reliability in the mathematical context. Examining differences in the four identified factors by Year Level revealed that participating Year 9 students had lower instrumental value for mathematics,

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perceived mathematics as requiring more effort, and felt that studying mathematics would incur a greater social cost than the participating Year 7 and Year 8 students. According to the Expectancy-Value Theory of achievement motivation, both expectancies and values directly influence achievement related choices and also performance, effort, and persistence (Wigfield & Eccles, 2000). The finding that Year 9 students in this study developed more negative perceptions of mathematics may influence their motivation and achievement within this domain. Therefore, it is important to further explore the reasons behind these negative perceptions and examine contextual factors, which may have influenced the results such as classroom environment, teacher influences, or socioeconomic factors. Examination across a broader range of year levels could also showcase how these perceptions develop and change over time, particularly across different Australian States and Territories as each has their interpretation of the Australian Curriculum.

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Michaels, L., & Forsyth, R. A. (1977). Construction and validation of an instrument measuring certain attitudes toward mathematics. Educational and Psychological Measurement, 37(4), 1043–1049. Middleton, J. A., & Spanias, P. A. (1999). Motivation for achievement in mathematics: Findings, generalizations, and criticisms of the research. Journal for Research in Mathematics Education, 30(1), 65–88. Neale, D. C. (1969). The role of attitudes in learning mathematics. The Arithmetic Teacher, 16(8), 631–640. Nisbet, S., & Williams, A. (2009). Improving students’ attitudes to chance with games and activities. Australian Mathematics Teacher, 65(3), 25–37. Programme for International Student Assessment [PISA]. (2006). Student questionnaire for PISA 2006. Retrieved from http://www.oecd.org/pisa/pisaproducts/database-pisa2006.htm Sandman, R. S. (1980). The mathematics attitude inventory: Instrument and user’s manual. Journal for Research in Mathematics Education, 11(2), 148–149. Tabachnick, B. G., & Fidell, L. S. (1996). Using multivariate statistics (3rd ed.). New York, NY: Harper Collins. Tapia, M., & Marsh, G. E. (2004). An instrument to measure mathematics attitudes. Academic Exchange Quarterly, 8(2), 16–21. Taylor, L. (1992). Mathematical attitude development from a Vygotskian perspective. Mathematics Education Research Journal, 4(3), 8–23. Watt, H. M. G. (2004). Development of adolescents’ self-perceptions, values, and task perceptions according to gender and domain in 7th- through 11th-grade Australian students. Child Development, 75(5), 1556–1574. Watt, H. M. G. (2010). STEPS: Study of transitions and education pathway. Retrieved from http://www.stepsstudy.org/ Wigfield, A., & Eccles, J. S. (2000). Expectancy-value theory of achievement motivation. Contemporary Educational Psychology, 25(1), 68–81. Wyn, J., Turnbull, M., & Grimshaw, L. (2014). The experience of education: The impacts of high stakes testing on school students and their families. Sydney: Whitlam Institute. You, S., Ritchey, K., Furlong, M., Shochet, I. M., & Boman, P. (2011). Examination of the latent structure of the psychological sense of school membership scale. Journal of Psychoeducational Assessment, 29(3), 225–237. Zan, R., Brown, L., Evans, J., & Hannula, M. S. (2006). Affect in mathematics education: An introduction. Educational Studies in Mathematics, 63(2), 113–121.

Chapter 10

Secondary Students’ Mathematics Education Goal Orientations Tasos Barkatsas and Claudia Orellana

Abstract The aims of this study were to explore the factorial structure of the goals items from a student survey used as part of the Australian Reframing Mathematical Futures II (rmfii) project, and to examine whether statistically significant differences were evident between the derived factors and the variable Year Level. Three factors were derived using an Exploratory Factor Analysis (efa). The three factors were consistent with prior studies based on Goal Achievement Theory: Performance Approach Goal Orientation (F1), Mastery Goal Orientation (F2), and Performance Avoidance Goal Orientation (F3). Statistically significant differences were also found between Year Level and one of the factors (F2). The findings have highlighted potential avenues for further research with respect to students’ goal orientations across different Year Levels. Keywords goals – secondary – mathematics – students – motivation

1

Theoretical Framework

Within the field of mathematics education, it has been widely acknowledged that affective variables, such as attitudes, beliefs, and motivations, play a significant role in the teaching and learning process (Di Martino & Zan, 2010; Goldin, 2002). However, despite the large amount of work exploring these variables in educational psychology, the construct of motivation within the mathematics domain has not been as extensively explored (Hannula, 2006; Middleton & Spanias, 1999). Motivation is defined as the “reasons individuals have for behaving in a given manner in a given situation” (Middleton & Spanias, 1999, p. 66). Some individuals are motivated to obtain rewards or avoid punishments (extrinsic © koninklijke brill nv, leideN, 2019 | DOI:10.1163/9789004396449_011

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motivation), and others are motivated by their desire or drive to learn (intrinsic motivation). However, one of the most prominent theories around motivation in educational research has been the Achievement Goal theory (Senko, Hulleman, & Harackiewicz, 2011). Goal theorists focus more on achievement behaviour (i.e., behaviour to demonstrate high ability – Nicholls, 1984) and early research in this field aimed to describe how “different goals elicit qualitatively different motivational patterns” (Ames, 1992, p. 261). As a result, two main types of goal orientations arose in the 1980s: mastery goals, which focus on personal improvement, and performance goals, which focus on demonstrating ability (Ames, 1992; Dweck, 1986; Patrick, Ryan, & Kaplan, 2007). In reviewing these two orientations, Senko et al. (2011) identified two key distinctions that separate mastery goals from performance goals. The first is that these two types of goal orientations are based on differing views of ability. In examining children’s behaviour, Dweck (1986) conceptualised that children who believed that ability (or intelligence) is malleable tended towards a mastery goal orientation. These children sought challenges and were more persistent regardless of whether they had high or low confidence in their present ability. In contrast, children who tended towards a performance goal orientation believed that ability is a fixed attribute and their levels of confidence in their ability affected their behaviour patterns (Dweck, 1986; Senko et al., 2011). High confidence led to challenge seeking and persistent behaviours, whereas low confidence led to challenge avoidance and low persistence. The second main difference between these two goal types is their definition of success and failure. From a performance goal perspective, success is achieved when you outperform your peers, whereas from a mastery goal perspective, success is achieved when you meet task-based or self-defined criteria such as obtaining a particular grade or having a feeling of improvement (Senko et al., 2011). Of the two types of goal orientations discussed above, the mastery goal orientation has tended to provide the most favourable results in educational research (Senko et al., 2011; Tulis & Ainley, 2011; Wolters, 2004). Patrick et al. (2007) noted that the development of a mastery-oriented classroom could encourage students to have a greater desire for self-improvement: Conveying support and promoting respect among students will contribute to an environment in which students can focus on understanding content rather than diverting attention to how they are being perceived by others or contributing to anxiety about ridicule if they experience difficulty or uncertainty. (p. 85)

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On the other hand, research on performance goal orientations has been mixed and inconsistent. Although there have been studies that demonstrate positive outcomes (see review by Midgley, Kaplan, & Middleton, 2001), other studies have linked the performance goal orientation with negative outcomes (Senko et al., 2011; Tulis & Ainley, 2011). These inconsistencies in research led to the development of an additional dimension to mastery and performance goals. Mastery goals were divided into mastery-approach and mastery-avoidance goals (avoid a lack of mastery) and performance goals were divided into performance-approach and performance-avoidance goals (avoid appearing incompetent) (Senko et al., 2011). The addition of an “avoidance” dimension helped to address the inconsistencies evident in early performance goal research as many of the negative outcomes linked with performance goals appeared to be confined to the performance-avoidance orientation (Senko et al., 2011). In summary, the Achievement Goal theory of motivation delves deeper into “the cognitive bases of the reasons people do what they do” (Middleton & Spanias, 1999, p. 72). In this chapter we will incorporate a survey instrument based on this theory to explore students’ goal orientations and how they differ across different contexts.

2

Aims

The aims of the study were to investigate: – The factorial structure of the goals items in the Australian Reframing Mathematical Futures II (rmfii) Project survey. – The existence of statistically significant differences between the derived factors (dependent variables) and the independent variable Year Level.

3

Methods

3.1 Data Source and Sample An online survey was undertaken as part of the rmfii Project, which aims to find ways to improve the teaching and learning of mathematics students in Year 7 to 10 and to develop learning progressions (trajectories) in mathematics education. The purpose of the survey being analysed here, was to examine students’ views, motivations, goals, preferences and attitudes regarding their learning experiences in mathematics. The participants came from Australian State and Catholic schools involved in the rmfii project across

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various Australian States and Territories. A total of 606 Year 7 to 9 students from nine schools across the New South Wales (nsw), Queensland and South Australia States and the Northern Territory responded to the survey. 3.2 Instrument The survey consists of 95 items and was designed by adapting items from instruments developed in prior studies (Dweck, Chiu, & Hong, 1995; Frenzel, Goetz, Pekrun, & Watt, 2010; Midgley et al., 2000; PISA, 2006; Watt, 2004, 2010; Wyn, Turnbull, & Grimshaw, 2014; You, Ritchey, Furlong, Shochet, & Boman, 2011). The survey examined the following constructs: Mathematics Learning Climate, Friends Perceptions of Mathematics, Perceptions of the Australian National Assessment Program – Literacy and Numeracy (naplan), Homework, Mathematics Motivations and Perceptions, Gender Perceptions of Mathematics, Personal Goals in Mathematics, Mathematics Mindset, Perceptions of School, Perceptions of Mathematics Teaching, and Mathematics Career. For the purposes of this chapter only the 2016 Personal Goals in Mathematics item responses have been analysed. There are 14 Personal Goals in Mathematics items in the survey, which have been adapted from the Patterns of Adaptive Learning Scales (pals) developed by Midgley et al. (2000). 3.3 Data Collection A link to the online survey was provided to participating students by their teachers from September 2016 and it was completed either in the students’ own time at home or during class time. The survey was anonymous and students and their respective parents were made aware of the purpose of the survey.

4

Results

An initial data screening was carried out to test for univariate normality, multivariate outliers (Mahalanobis’ distance criterion), homogeneity of variancecovariance matrices (using Box’s M tests) and multicollinearity and singularity (tested in the manova analysis). Descriptive statistics normality tests (normal probability plot, detrended normal plot, Kolmogorov-Smirnov statistic with a Lilliefors significance level, Shapiro-Wilks statistic, skewness and kurtosis) showed that assumptions of univariate normality were not violated. Mahalanobis’ distance was calculated and a new variable was added to the data file. There were fewer than twenty outlying cases, which is acceptable in a sample of 606 students. These outliers were therefore retained in the data set. Box’s M Test of homogeneity of the variance-covariance matrices (which

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tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups) was not significant at the 0.001 alpha level and we therefore concluded that we have homogeneity of variance. The questionnaire items were subjected to an Exploratory Factor Analysis (efa) by using spsswin. Reliability tests were also conducted. A Multivariate Analysis of Variance (manova) test was used to investigate statistically significant differences between the three derived factors and the independent variables Year Level. 4.1 Exploratory Factor Analysis (efa) Given the exploratory nature of the study an exploratory factor analysis using data from 606 complete students’ responses to the 14 items forming the questionnaire, indicates that the data satisfy the underlying assumptions of the factor analysis and that together three factors (each with eigenvalue greater than 1) explain 75.3% of the variance, with 45.5% attributed to the first factor – Performance Approach Goal Orientation (Table 10.1). Further, according to Coakes and Steed (1999), if the Kaiser–Meyer–Olkin (kmo) measure of sampling adequacy is greater than 0.6 and the Bartlett’s test of sphericity (bts) is significant then factorability of the correlation matrix is assumed. A matrix that is factorable should include several sizable correlations. The Kaiser–Meyer–Olkin (kmo) measure of sampling adequacy in this study is greater than 0.90 and the Bartlett’s test of sphericity (bts) is significant at 0.001 level, so factorability of the correlation matrix has been assumed. Reliability analysis yield satisfactory Cronbach’s alpha values for each factor: Factor 1, 0.97; Factor 2, 0.93 and Factor 3, 0.88. These values indicate a strong degree of internal consistency in each factor. The naming of the three factors was guided by the relevant literature and the nature of the questionnaire items associated with each factor. This resulted in the following three factors (F1–F3): – F1: Performance Approach Goal Orientation – F2: Mastery Goal Orientation, and – F3: Performance Avoidance Goal Orientation. All items were adapted from the Patterns of Adaptive Learning Scales (pals) developed by Midgley et al. (2000). We will discuss each factor in detail, in what follows. Factor 1: Performance Approach Goal Orientation The first factor consists of five items, which examine goals centred around a performance-approach orientation in mathematics. As summarized by Midgley et al. (2000), the main focus for this type of goal is to demonstrate competence

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table 10.1  Rotated factor matrix (varimax rotation)

1 Factor 1: Performance Approach Goal Orientation One of my goals is to look smart in comparison to the other students in mathematics It is important to me that other students in my class think I am good at mathematics One of my goals is to show others that mathematics work is easy for me. It is important to me that I look smart compared to others in mathematics One of my goals is to show others that I am good at mathematics Factor 2: Mastery Goal Orientation It is important to me that I thoroughly understand my mathematics It is important to me that I improve my skills in mathematics this year It is important to me that I learn a lot of new concepts this year in mathematics One of my goals in mathematics is to master a lot of new skills this year Factor 3: Performance Avoidance Goal Orientation It is important to me that my teacher doesn’t think that I know less than others in mathematics One of my goals in mathematics lessons is to avoid looking like I have trouble doing the work It’s important to me that I don’t look stupid in mathematics lessons One of my goals is to keep others from thinking I am not smart in mathematics classes.

Factor 2 3

.870 .841 .819 .792 .757 .885 .863 .835 .817

.832 .824 .804 .789

(i.e., to look smart in front of others). These items were the only items, which examined a performance-approach goal orientation in Midgley et al.’s (2000) pals study and have thus loaded consistently within this analysis. Factor 2: Mastery Goal Orientation The second factor consists of four items, which examine goals centred around a mastery-approach orientation in mathematics. Unlike performance

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approach goals, which focus on demonstrating competence, mastery goals aim to develop competence as students seek to extend their understanding (i.e., striving to improve) (Midgley et al., 2000). As with the items in Factor 1, the items in Factor 2 have loaded consistently within this study when compared to the pals study from which the items were adapted from. Note that one item from the pals study was not included within this questionnaire due to its similarity to the third item within this factor (“to learn as much as I can” is similar to “to learn a lot of new concepts”). Factor 3: Performance Avoidance Goal Orientation The final factor consists of four items, which examine goals centred around a performance-avoidance orientation in mathematics. According to Midgley et al. (2000), the main focus for this type of goal is to avoid demonstrating incompetence (i.e., looking stupid in front of others). These items were the only items, which examined a performance-approach goal orientation in Midgley et al.’s (2000) pals study and have thus loaded consistently within this analysis.

Analysis of Variance (anova) 4.2 The existence of statistically significant differences on each of the three derived factors by Year Level was investigated by conducting an Analysis of Variance (anova) statistical test. The dependent variables (DVs) were the three factors derived from the efa and the independent variable was Year Level. Pillai’s Trace criterion was used to test whether there are no significant group differences on a linear combination of the dependent variables. Pillai’s Trace criterion is considered to have “acceptable power and to be one of the most robust statistics against violation of assumptions” (Coakes & Ong, 2011, p. 158). Pillai’s criterion was not violated and we have significant univariate main effects for the following variable: – Factor 2: Mastery Goal Orientation [F(2, 491) = 2.98, p < 0.05, η2 = .01] Effect sizes were calculated using eta squared (η2). In our interpretation of effect sizes we have been guided by Cohen, Manion, and Morrison’s (2007) proposal that: 0.1 represents a small effect size, 0.3 represents a medium effect size, and 0.5 represents a large effect size. For this analysis, the effect size was .01 (small effect). Post hoc tests The purpose of a post hoc tests is to determine which Year Levels are statistical significant different from each other. A Games-Howell post hoc multiple

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comparisons test was performed in order to explore the differences for each factor. The Games-Howell test has been used because the Year Level sizes differ. It was found that Year 8 and Year 9 scores had significantly different mean values (p < 0.001) for Factor 2. The Year 8 students’ scores had a mean score higher than both the Year 7 students’ and the Year 9 students’ mean scores and the Year 7 students’ scores had a higher mean than the Year 9 students’ mean scores.

5

Discussion

In order to investigate the factorial structure of the goals items of the rmfii Project student survey an Exploratory Factor Analysis (efa) was conducted. The results from this analysis found that together three factors, explained 75.3% of the variance. The three identified factors were consistent with the study from which the items were adapted from (Midgley et al., 2000) and were labelled as follows: Performance Approach Goal Orientation, Mastery Goal Orientation, and Performance Avoidance Goal Orientation. An Analysis of Variance (anova) was conducted to examine if statistically significant differences existed between the three identified factors and the independent variables Year Level. Results showed that statistically significant differences were evident for the variable Year Level and that there were interaction effects between Factors 1 and 2 and the independent variable Year Level. Post hoc analysis showed that The Year 8 students’ scores had a mean score higher than both the Year 7 students’ and the Year 9 students’ mean scores and the Year 7 students’ scores had a higher mean than the Year 9 students’ mean scores. In other words, Year 8 students valued mastery orientation, such as, extending their understanding, more than the Year 7 and the Year 9 students and the Year 7 students valued a mastery orientation approach more than the Year 9 students did. There is limited research exploring changes in goal orientation over time, however, an earlier longitudinal study conducted by Anderman and Midgley (1997) found that students in the fifth grade (elementary school) were more oriented towards mastery goals than when they moved into the sixth grade (middle school). The authors attributed these findings to the transition between school environments (elementary to middle school) noting that “for many children, the nature of the learning environment changes in a negative way during early adolescence” (p. 291). In a similar manner, Year 7 students are also in a period of transition. This is the first year of high school for students in Australia and there are many changes in class structure, assessment,

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and student cohort (i.e., students from various different primary schools). Students may be more inclined to value performance at this stage in order to “look smart” in front of their new peers and teachers. By Year 8, students may have adjusted to this change in environment and may begin to value the development of competence over its demonstration. Further investigation involving different classroom environments in other countries could provide more insights into how student perceptions of their classroom environment can orient them towards particular goals.

6

Conclusion

This study presented in this chapter set out to explore the factorial structure of the goals items of the rmfii Project student survey and to determine if there were statistically significant differences between the identified factors and the variables Year Level. The results have shown that the three identified factors were consistent with the study from which the items were adapted from (Midgley et al., 2000) confirming the continued validity and reliability of these items in a mathematics education context. Statistically significant differences were also found between Year Levels for the factor Mastery Goal Orientation. Students in Year 8 were more oriented towards mastery goals than Year 7 and Year 9 students and students in Year 7 were more oriented towards mastery goals than Year 9 students. The findings of this study have highlighted avenues for further investigation. There is limited research exploring goal orientations in mathematics education across different year levels and how they may change over time. A study examining goal orientations across different year levels could provide more insights into students’ learning experiences in mathematics and examination of different contexts in different countries could highlight the importance of the learning environment in the development of different types of goals.

References Ames, C. (1992). Classrooms: Goals, structures, and student motivation. Journal of Educational Psychology, 84(3), 261–271. Anderman, E. M., & Midgley, C. (1997). Changes in achievement goal orientations, perceived academic competence, and grades across the transition to middle-level schools. Contemporary Educational Psychology, 22, 269–298.

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Coakes, S. J., & Steed, L. G. (1999). SPSS: Analysis without anguish. Australia: John Wiley and Sons Ltd. Di Martino, P., & Zan, R. (2010). ‘Me and maths’: Towards a definition of attitude grounded on students’ narratives. Journal of Mathematics Teacher Education, 13(1), 27–48. Dweck, C. S. (1986). Motivational processes affecting learning. American Psychologist, 41(10), 1040–1048. Dweck, C. S., Chiu, C., & Hong, Y. (1995). Implicit theories and their role in judgements and reactions: A word from two perspectives. Psychological Inquiry: An International Journal for the Advancement of Psychological Theory, 6(4), 267–285. Frenzel, A. C., Goetz, T., Pekrun, R., & Watt, H. M. G. (2010). Development of mathematics interest in adolescence: Influences of gender, family and school context. Journal of Research on Adolescence, 20(2), 507–537. Goldin, G. A. (2002). Affect, meta-affect, and mathematical belief structures. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education (pp. 59–72). Dordrecht, The Netherlands: Kluwer Academic Publishers. Hannula, M. S. (2006). Motivation in mathematics: Goals reflected in emotions. Educational Studies in Mathematics, 63(2), 165–178. Middleton, J. A., & Spanias, P. A. (1999). Motivation for achievement in mathematics: Findings, generalizations, and criticisms of the research. Journal for Research in Mathematics Education, 30(1), 65–88. Midgley, C., Kaplan, A., & Middleton, M. (2001). Performance-approach goals: Good for what, for whom, under what circumstances, and at what cost? Journal of Educational Psychology, 93(1), 77–86. Midgley, C., Maehr, M. L., Hruda, L. Z., Anderman, E., Anderman, L., Freeman, K. E., … Urdan, T. (2000). Manual for the patterns of adaptive learning scales. Retrieved from http://www.umich.edu/~pals/PALS%202000_V13Word97.pdf Nicholls, J. G. (1984). Achievement motivation: Conceptions of ability, subjective experience, task choice, and performance. Psychological Review, 91(3), 328–346. Patrick, H., Ryan, A. M., & Kaplan, A. (2007). Early adolescents’ perceptions of the classroom social environment, motivational beliefs, and engagement. Journal of Educational Psychology, 99(1), 83–98. PISA. (2006). Student questionnaire for PISA 2006. Retrieved from http://www.oecd.org/ pisa/pisaproducts/database-pisa2006.htm Senko, C., Hulleman, C. S., & Harackiewicz, J. M. (2011). Achievement goal theory at the crossroads: Old controversies, current challenges, and new directions. Educational Psychologist, 46(1), 26–47. Tabachnick, B. G., & Fidell, L. S. (1996). Using multivariate statistics (3rd ed.). New York, NY: Harper Collins.

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Tulis, M., & Ainley, M. (2011). Interest, enjoyment and pride after failure experiences? Predictors of students’ state-emotions after success and failure during learning in mathematics. Educational Psychology, 31(7), 779–807. Watt, H. M. G. (2004). Development of adolescents’ self-perceptions, values, and task perceptions according to gender and domain in 7th- through 11th- grade Australian students. Child Development, 75(5), 1556–1574. Watt, H. M. G. (2010). STEPS: Study of transitions and education pathway. Retrieved from http://www.stepsstudy.org/ Wolters, C. A. (2004). Advancing achievement goal theory: Using goal structures and goal orientations to predict students’ motivation, cognition, and achievement. Journal of Educational Psychology, 96(2), 236–250. Wyn, J., Turnbull, M., & Grimshaw, L. (2014). The experience of education: The impacts of high stakes testing on school students and their families. Sydney: Whitlam Institute. You, S., Ritchey, K., Furlong, M., Shochet, I. M., & Boman, P. (2011). Examination of the latent structure of the psychological sense of school membership scale. Journal of Psychoeducational Assessment, 29(3), 225–237.

Epilogue Mike Askew

Park planners in Finland make good use of the first snowfall each year, not to rush out because they want to build a snowman, but to observe where people choose to walk through the snow and then use that data to inform planning future trails. In a warmer climate, University of California, Berkeley, is one of several educational institutions reported to have purposely waited to put down campus paving until after seeing where foot traffic suggested it should go. These are two examples of ‘desire paths.’ Rather than planners predetermining what they think are the best routes and putting down paths accordingly, they lay paths after the people moving around have decided on the best ways to get from A to B. Desire paths arise from where people do walk, rather than presumptions of where they should walk. Metaphors for learning are replete with references to walking, or more generally to travelling, journeying – indeed such metaphors are at the core of this book. Of course, a key difference between actual walking and student learning is that a walker usually knows her destination in advance – she makes her path with an end already in mind. A learner cannot, by virtue of their lack of knowledge, know what their destination is, so a guide, a teacher, is important in helping them set out along the path, either, traditionally, guiding the learner into following an already existing path, or helping to lay a path as they travel forward together. This volume presents a welcome move in this latter sense of setting out some desire paths in mathematics education by looking at how educators might lay down paths that are, in part at least, determined by the learner (walker) as much as by the teacher (planner). Inevitably, research as rich as that reported on here raises more questions, and rather than trying to synthesise the various strands of this book, in these final pages I reflect on some of the questions raised for me in my reading.

1

Trajectories: Development versus Instruction

A core theme running through the research reported on here is the attempt to better model and understand the relationship between student development and the instruction in mathematics that they experience. Samara and Clements (this volume) exemplify the stance taken here in stressing the importance of instruction being based on learning and consistent with ‘developmental © koninklijke brill nv, leiden, 2019 | doi:10.1163/9789004396449_012

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progressions.’ The first question this raises, for me, is to what extent are developmental progressions somehow ‘natural’? Are there possibly near universal ways in which mathematical understandings unfold? Whilst substantial research now suggests that we are all born with an innate propensity towards recognising and distinguishing quantities (Dehaene, 1999), beyond counting, how much of mathematics itself is ‘natural’? As the famous quote (commonly attributed to the mathematician Kronecker) puts it ‘God made the integers, all the rest is the work of man’ (and woman!). Perhaps that is why, as Samara and Clements suggest, there is better understanding of hypothetical learning trajectories that are based in psychological development in early number than there are for the later years of schooling when the mathematical content encountered becomes more important more important than innate understandings. This observation fits with the point made by Confrey and colleagues (this volume) that differences in prior teaching and learning on concepts such as fractions, decimals, ratio and proportion, may ‘interfere’ with subsequent learning about percents. In a similar vein, Watson, Callingham and Oates (this volume) report on research showing that learning outcomes related to statistics were related to teachers’ statistical pedagogic content knowledge, again pointing to later learning trajectories being ‘buffeted’ by the teaching students are subjected to. A key issue here is the degree to which learning trajectories, particularly as students move up through school, are a result of opportunities to learn as much as any ‘natural’ order of coming to understand something. In other words, does, and if so when, the ‘cultural’ overtake the ‘natural’ in a learning trajectory? To what extent do trajectories unfold in the way that the trajectory of a ball thrown through a vacuum is so predictable that a robust model of its path can be created, or to what extent are trajectories a result of external forces and so less predictable, in that way that the journey of a bottle thrown into the sea is going to be nigh on impossible to predict? Are learning trajectories models arising from highly predictable, complicated systems, or is learning a complex system and so less predictable? You might say, does it matter? If there is a good fit between teaching and learning, and if hypothetical learning trajectories are, by definition, linking teaching and learning, then why worry? I think the question is worth exploring as it opens up to examining to what extent research such as that reported on here raises possibilities for new practices or new curricula, or whether it merely helps us work more effectively within existing parameters. For example, multiplicative reasoning is traditionally taught in a particular order, in many curricula following on from prior teaching of additive reasoning, with an emphasis on multiplication as repeated addition and working with groups of

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discrete counting objects. If that is the introduction to multiplicative reasoning for most students, then the teaching experiences are bound to influence the learning outcomes, and, hence, any learning trajectories. The authors here, whilst all acknowledging the role of context and the social, vary in the extent to which they foreground this. Siemon acknowledges taking the view of learning as involving both individual construction and social enculturation, going so far as to rename Rasch levels as ‘hierarchical zones,’ the metaphor of zones carrying the connotation that movement – learning – can occur in a multitude of directions, both within and between zones. Similarly, although Samara and Clements explicitly ground their research in a cognitive, constructivist paradigm, they do acknowledge that any curriculum is rooted in ‘culturallyvalued’ content, thus implicitly acknowledging the socio-historical nature of expected learning outcomes. Learning trajectories for them not only set out a route through stages of thinking and learning but also need to provide collections of teaching tasks to guide learners along the route in the direction of specific learning outcomes. Linked to this Samara and Clements also draw on the construct of learning, in one reading of Vygotskian theory, as mediated by tools and signs – a tool-for-result position. But an alternative reading of Vygotsky does not position tools and signs ‘between’ the known and the unknown but instead sees them as actually creating and shaping learning – the tool-and-result position. In formulating tool-and-result Vygotsky’s gaze was on the discipline of psychology itself, but his identification of the paradox at the heart of the psychology – that psychology creates the very objects that it investigates – can also be applied to mathematics education. As Holzman (1997) puts it: As “simultaneously tool-and-result,” method is practiced, not applied. Knowledge is not separate from the activity of practicing method; it is not “out there” waiting to be discovered through the use of an already made tool. … Practicing method creates the object of knowledge simultaneously with creating the tool by which that knowledge might be known. Tool-and-result come into existence together; their relationship is one of dialectical unity, rather than instrumental duality. (p. 52) Models of teaching and learning based on ‘mediating means’ (Cole, 1996) or scaffolding are largely predicated on a tool-for-result perspective rather than tool-and-result: which perspective one takes has implications for how you see the development of learning trajectories, either as relative stable across time and cultures, or the result of particular practices in particular cultures. In other words, would not different teaching trajectories lead to different learning trajectories? An early years’ curriculum focused on working with

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continuous quantities and measures, in contrast to discrete quantities and counting, as would be in evidence in a curriculum based on Davydov’s theory (1990), would surely lead to distinctly different learning trajectories (particularly for multiplicative reasoning – see for example the argument put forward by Schmittau, 2003). While it is welcome that the research presented here points to findings that teaching what we have traditionally taught can be done better it leaves open the answer to the thorny question of whether or not we might be better off doing something completely different. So, which comes first – learning or teaching? If, as Samara and Clements argue, a learning trajectory comprises, firstly, a specified learning outcome (goal), second, hypothesised learning stages and levels towards that goal and, third, instructional tasks, then to what extent can the second and third aspects be interchanged? Might a carefully structured set of learning tasks be designed to bring about a goal and so create, rather than follow, the developmental progression? Simon’s (1995) original definition of hypothetical learning trajectories does have that reversal, seeing them as being ‘made up of three components: the learning goal that defines the direction, the learning activities, and the hypothetical learning process – a prediction of how the students’ thinking and understanding will evolve in the context of the learning activities’ (p. 136). Simon’s original definition would seem to fit with what scholars working with variation theory do: through careful analysis of the desired learning outcomes, analysis based on epistemology rather than learner thinking, tasks are designed to bring about the desired development in thinking (for an extended account of this theory see Huang & Li, 2017). This interchanging of the second (learning stages) and third (learning tasks) is how I understand Sumara and Clements’ distinction between ‘instructional design based on task analysis,’ based on identifying the ‘expert’ performance, and learning trajectories that present models of learning first. In actuality, I think there is going to be a blurring of these distinctions, especially if the goals are determined by curriculum standards. From a strong socio-cultural neo-Vygotskian position these alternatives may dissolve altogether. It would be interesting for conversations to start between researchers working within the trajectories paradigm and those working with variation theory and so begin to tease out the possibilities of some synthesis between the two.

2

The Impact on Teaching

Samara and Clements make the important observation that there is a paucity of research examining the impact, if any, of teaching that draws upon learning

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trajectories to teaching that does not. This volume starts to address that gap, although the authors are, rightly, cautious about making strong claims. A key issue here is reconciling research into students’ individual learning trajectories and the implications of this for teaching in schools. For example, a central aim of Tzur’s work reported here is to precisely delineate between students’ current schemes and the schemes which may arise from the transformation of these extant schemes. Tzur’s detailed analysis of individuals, their schemes and the transforming of these is powerful, but how might this knowledge be applied in classroom contexts, where a single teacher is having to deal with multiple individual trajectories? Although not couched in the language of learning trajectories, research, carried out by Hazel Denvir and Margaret Brown (Denvir & Brown, 1986a, 1986b) showed the difficulties of linking learning outcomes to teaching inputs, even under conditions more ideal than afforded by most classrooms. Denvir developed a framework setting out the paths that students might take in their understanding developing from pre-counting through to the addition and subtraction of two-digit numbers. Close assessment of students revealed that the model provided a good fit with understanding – students demonstrated ‘clusters’ of understandings that broadly fell into hierarchical bands within the framework. In terms, however, of providing guidance that closely specified next teaching steps the framework was less helpful: targeting what might be the best next constructs in the framework towards which to guide students was no guarantee that those constructs were what was learned, although learners did progress largely within the broad bands that had been identified. Such findings fit with the research reported here on ‘zones’ of development. A focus on zones shows promise for bridging the gap between attending either to the learning trajectories of individual students or to taking a class along a path in a lock-step fashion. If the teaching within zones is based, as suggested upon ‘rich tasks’ then there is the potential for students to engage in these at different levels of mathematical activity. Allied to this is the advice that teaching tasks, within zones, either focus on consolidating and establishing fluencies and concepts that are on the borders between zones, or on introducing and developing ideas to prepare movement into the next zone. The research thus begins to offer concrete advice to teachers. A further strength to zones is the treatment of them as dynamic. Taken together, the three projects Siemon reports on effectively build upon each other and link teaching and learning, through establishing a hierarchy of zones of understanding but not treating these as fixed. Changes as a result of teaching led to a further refinement of the zones which then, cyclically, further informed teaching.

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Confrey and colleagues use the language of learning maps, made up of ‘relational learning clusters’ comprising collections of constructs, each of which has its own learning trajectory. It would be interesting to know more about the similarities and differences between ‘relational learning clusters, and the construct of zones as developed in the Australian research. A question raised by these different ‘grain levels’ of a learning trajectory is what level of detail is likely to be more helpful to teachers? Too fine detail and there is a danger that the markers on a trajectory become the specific teaching objectives, too coarse and they cease to become helpful for guiding teaching. But it may be that the details themselves are not that important. As Confrey and her colleagues note, when working with their digital learning system, teachers who were learner-centred engaged with the data about learners’ trajectories differently from those teachers who treated the data simply as providing evidence of aspects of the mathematics that needed to be re-taught. These authors note that as teachers moved away from a view of the information provided about learners ‘heat maps’ as indicating gaps to fill, so they began to treat items that were used to assess levels of learner understanding as simply exemplar indicators rather than particular skills, and developed a broader view on the learning outcomes. This points to an important role in the use of LTs in supporting teachers’ pedagogical content knowledge. Many of the chapters report the important finding that teachers who consistently attend to learning, or are student-centred in their orientation towards teaching, improve the quality of their teaching through working with learning trajectories. Perhaps it is not be the hypothetical learning trajectories themselves itself that makes the difference, but the shift in attention from teaching to learning that they prompt.

3

The Universality of Trajectories

Tzur raises the important question of whether hypothetical learning trajectories transfer to different cultures. Reporting on a study of four 4th grade learners in China, their available schemes appeared to fit with the learning trajectories developed in the United States, but apparently the students were struggling as such schemes did not seem to fit with the mathematics they were being taught. I take Tzur to be implying here that the curriculum was not a good fit with the hlts. But to what extent are the tasks set for students in creating models of learning a consequence of the expectations set by the local curriculum? The U.S. hypothetical learning trajectories were presumably informed by the curriculum expectations, so an hlt developed in China would

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presumably be informed by the content and order of the curriculum there? Would an hlt developed locally have provided a better fit? Siemon reporting on three large-scale projects notes the identification of a seven-year range in mathematics achievement in the middle years of schooling, a finding echoing, some 35 years later, the U.K.’s Cockcroft report (1982) which reported a seven-year gap in mathematical attainment at age eleven. This continued evidence for such a gap is important, particularly given much of the current discourse around attainment arguing that a greater majority of students should be more highly attaining in mathematics. Whether such a spread in attainment is inevitable and a result either of some underpinning spread of ability, or due to differences in the rates at which students come to understandings, suggests a need for further research. It would be interesting to know whether such a spread of attainment would also be in evidence with students in, say, Shanghai or Japan, jurisdictions which, if we are to believe the evidence of pisa and timms, already have narrower ranges of attainment. One suggestion common across many of the chapters is that pedagogical practices involving classroom dialogue that elicits learners’ thinking can help teachers design a better fit between instruction and learning. Whilst agreeing with the need for classroom dialogue I think we need to be careful about not assuming that this is a necessary condition for teaching. In many jurisdictions around the world, of which South Africa is typical, the culture and context of teaching is a long way from such a pedagogic style: the culture still being one of the teachers as the ‘expert’ in the classroom and the context being one of large, under-resourced classrooms.

4

Conclusion

Confrey and colleagues argue that, in the US plans for teaching, and the subsequent lessons, are frequently based on working through standards (curriculum objectives): such an approach is not unique to the US. In South Africa, for instance, year-long curriculum planners set out teaching objectives and the order in which to teach them. The inspection services then focus on whether or not teaching is ‘on-track’ (in the sense that what is being taught at a point in time is in line with the overall plan), a judgment made in the absence of little, if any, attention to whether or not learning is on-track. I agree with these authors’ claim that LTs have the potential to help teachers better understand patterns in and models of learning – an understanding that may help better coordinate teaching plans. As Pete Griffin (1989) noted many years ago, teaching takes place in time whilst learning takes place over time. This volume of

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research in learning trajectories provides powerful insights into how these two differing time-lines might be reconciled.

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