Indigenous Knowledge and Ethnomathematics 3030974812, 9783030974817

Table of contents :
Introduction
Contents
Ethnography and Mathematics
Re/Creating ‘Evocative Images’ (sunannguanik iqqaigutinik): Procedural Knowledge and the Art of Memory in the Inuit Practice of String Figure-Making
1 Introduction: String Figure-Making as a Widespread Activity Involving Procedural Knowledge
2 String Figure-Making in Pre-colonial Inuit Societies
2.1 A Practice Referring to Different Spatiotemporal Scales
2.2 A Procedural Activity Relating to Memory and Knowledge Sharing
3 Symbolic Meanings Ascribed to Some Structural Features of ajaraaq (ayarr’ar, ayaqhaaq): An Insight into Mathematical Ideas as Embedded in Inuit Cosmology
3.1 String Figure- and Knot-Making: Generative/Ordered Thread Crossings Versus Tangles
3.2 Cultural Interpretations of Some Geometrical Ideas Involved in Inuit String Figure-Making
References
Modeling of Implied Strategies of Solo Expert Players
1 Introduction
1.1 My Ethnographic Fieldwork and the Object of My Research
1.2 What Are Abstract Combinatorial Games ?
1.3 Why the ``Sowing Game'' Appellation?
1.4 Presentation of the Next Sections of this Chapter
2 Sowing Games
2.1 Some Artifacts
2.2 Common Characteristics
2.3 Two Classes of Sowing Games
2.4 Geographical Distribution
2.5 Diagrams
3 The Zanzibar Bao
3.1 The Bao la Kujifunza
3.2 The Bao la Kiswahili
3.3 Notation of the Games
4 The Other solo
4.1 A 4 times4 Katro from the ``Hauts Plateaux'' of Madagascar
4.2 The 4 times8 fanga by Flacourt
4.3 The 4 times8 mraha of Mahajanga
4.4 The 4 times8 mraha of Mayotte
5 Regulated Movements and Optimized Movements
6 Modeling
6.1 Some Preliminary Definitions
6.2 An Enlightening Interview
6.3 A Modeling Proposal
7 Conclusion
References
Sand Drawing Versus String Figure-Making: Geometric and Algorithmic Practices in  Northern Ambrym, Vanuatu
1 Introduction
2 Cultural and Symbolic Aspects of String Figures and Sand Drawings: Some Elements of Comparison
3 Ethnomathematics of String Figures and Sand Drawings
3.1 Algorithmic/Procedural Aspects of String-Figure and Sand Drawing Practice
3.2 Shared Mathematical Properties
4 In Conclusion: From Epistemological to Educational Issues
References
Impact of Indigenous Culture on Education in General, and on  Mathematics Classes in Particular
Indigenous School Education: Brazilian Policies and the Implementation in Teacher Education
1 Introduction
2 Socio-historical and Demographic Panorama of the Brazilian Indigenous Population
2.1 Invasion and Destruction of Indigenous Peoples in Brazil
2.2 Protection of Indigenous Peoples
3 The Implementation of Indigenous School Education in Brazil
4 The Advances and Setbacks of Indigenous School Education in Brazil
5 An Experience of Collaboration with Xukuru of Ororubá Teachers
6 Final Considerations
References
Indigenous Mathematical Knowledge and Practices: State of the Art of the Ethnomathematics Brazilian Congresses (2000–2016)
1 Introduction
2 The Brazilian Ethnomathematics Congresses
3 Methodology
4 Results
4.1 Works on Indigenous Themes
4.2 Ethnicities Represented in the Works
4.3 Indigenous Authorship
4.4 Research Themes
4.5 Anthropology and Ethnography in the Works
5 Indigenous Ethnomathematics and Anthropology
6 Final Considerations
References
Subverting Epistemicide Through ‘the Commons’: Mathematics as Re/making Space and Time for Learning
1 Epistemicide
2 Land and Locals: Tracing Signs of Epistemicide
3 The Commons: A Matter of Learning to Subvert Epistemicide
4 Learning and ‘The School’: Α Radical Pedagogy for ‘The Commons’
5 Mathematics: Re/making Space and Time for Learning
References
Meta-studies
The Tapestry of Mathematics—Connecting Threads: A Case Study Incorporating Ecologies, Languages and Mathematical Systems of Papua New Guinea
1 Introduction
2 Threads of the Tapestry
2.1 Caveat on This Perspective of the Tapestry
3 The Diversity of Cultures
3.1 Time and Place
3.2 Time and Language
3.3 Migrations and Languages
3.4 DNA Assessments
3.5 Displacement and Migration
3.6 Proto Languages
3.7 Social Values, Sociopolitics, and Language
4 The Tapestry Section on Number Systems—A Window into Diversity
5 Mathematics in Cultural Activities
5.1 Trade
5.2 Group Decision Making and Displays
5.3 Valuing Culture in the Mathematical Language
5.4 Intergenerational Relationships
6 The Need for Large Numbers for Cultural Reasons
7 The Tapestry Section on Time and Work Patterns
8 The Tapestry of Transactions
9 The Tapestry of Mathematics in Art
10 Discussion
11 Conclusion
12 PostScript
References
Indigenous Mathematics in the Amazon: Kinship as Algebra and Geometry Among the Cashinahua
1 Introduction
2 Note on the Cashinahua
3 Cashinahua Name-Sake Classes
4 The Group Structure of Same-Sex Cashinahua Kinship Terms
4.1 Definition of a Group
4.2 The Klein Group of Cashinahua Alliance: Same-Gender Kinship Terms
5 Cashinahua Structure of Eight Name-Sake Classes
5.1 The Cashinahua Eight-Sections Group
5.2 Different Representations of the Dihedral Group Structure
6 Conclusions
References
The Western Mathematic and the Ontological Turn: Ethnomathematics and Cosmotechnics for the Pluriverse
1 Introduction
2 Multiple Technics, Multiple Natures, and Multiple Ontologies
3 Mathematics, Technology and the Technoecological Condition
4 The Ontological Turn and Technodiversity
5 Computational Mathematics: Singularity Versus Pluriversality
6 Ethnomathematics and Cosmotechnics
References
Conclusions. Some Possible Lines for Further Research in Ethnomathematics
More Ethnography of Diverse, Local Mathematical Knowledge
More Anthropological Research
Mathematics Education and Ethnomathematics
References

Citation preview

Eric Vandendriessche Rik Pinxten   Editors

Indigenous Knowledge and Ethnomathematics

Indigenous Knowledge and Ethnomathematics

Eric Vandendriessche · Rik Pinxten Editors

Indigenous Knowledge and Ethnomathematics

Editors Eric Vandendriessche National Centre for Scientific Research Paris, France

Rik Pinxten Cultures and Languages Ghent University Ghent, Belgium

This work contains media enhancements, which are displayed with a “play” icon. Material in the print book can be viewed on a mobile device by downloading the Springer Nature “More Media” app available in the major app stores. The media enhancements in the online version of the work can be accessed directly by authorized users. ISBN 978-3-030-97481-7 ISBN 978-3-030-97482-4 (eBook) https://doi.org/10.1007/978-3-030-97482-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

1. Preliminary Remarks The present volume aims to offer some more insights on the relationships between indigenous knowledge and ethnomathematics. These relationships have a history, as some of the texts remind us (e.g. Almeida, Vandendriessche). We think it is important to elaborate on the possible links between both domains because of the richness involved, and because mathematics education on a global level is likely to benefit from such lines of research. In the conclusions to this volume, we indicate some points which may have potential for the future. It is good to question old views in both disciplines: many mathematicians will regard their thinking as context or culture-free, on the one hand, while some anthropologists will claim that the ethnoscience studies of recent decades are uncritical types of reductionism, stereotyping thinking in other traditions as particularistic and weaker instances of what was developed in Western history. Although classification studies and the later ethnoscience approaches can be arguably described as emanating from such frames in the 20th century, the combined interdisciplinary work by mathematically literate anthropological researchers in the present volume questions this sort of type casting. What was called ethnomathematics in the ‘70s and ‘80s of the past century has since matured, mainly thanks to an indepth collaboration of knowledgeable researchers in both disciplines. It appears that the mere evaluation of the relationships between Indigenous Knowledge and what is nowadays called Ethnomathematics cannot be reduced to a “culturalist” bias on native knowledge (as it was to some extent in the early days of the American ethnoscience scene: Descola 2015). It is the hope of this book’s editors that the contributions to this volume will allow for a renewed and substantially deepened discussion on these issues. Most of the chapters in this volume were the subject of a presentation in a symposium, as part of the “18th International Union of Anthropological and Ethnological Sciences (IUAES) World Congress”, Florianopolis, Brazil, 2018.1 However, 1

The idea of elaborating this book on “Indigenous Knowledge and Ethnomathematics” has emerged during the Panel “Indigenous Mathematical Knowledge and Practices: (Crossed-) Perspectives from Anthropology and Ethnomathematics”, coordinated by Eric Vandendriessche, Rik Pinxten, and v

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the texts have been thoroughly reworked in order to become substantial contributions to the field we work in. A few texts are new and were invited afterwards: M. Baker, A. Chronaki, L. Tiennot, S. Oliveira et al. The editors added a comprehensive introduction and conclusive remarks. The structure of the book has been discussed thoroughly among the editors, since a book is, of course, altogether different from mere proceedings at a conference.

2. The Book’s Structure Part I “Ethnography and Mathematics” This part presents papers focusing on particular ethnographic studies, in order to reveal the obvious and visible, but also the hidden or deep structural mathematical features in particular and diverging cultural phenomena. Céline Petit offers the first ethnographic text under the title Creating ‘Evocative Images’ sunannguanik iqqaigutinik: Procedural Knowledge and the Art of Memory in the Inuit Practice of String Figure-Making. This study begins with the amazing fact that string figure making can be found around the world. It proves to be a widespread activity of procedural knowledge. The chapter focuses primarily on the Inuit tradition. The meanings of String Figure-Making are diverse, ranging from cosmological and ecological referencing to more strictly social roles and uses: indeed, it is often found as a means of socializing, in the guise of storytelling while making the figures. Petit analyzes string figure-making as a kind of procedural activity, which is important in knowledge transfer, memory training and to frame cultural learning processes. Moreover this tradition has symbolic meaning in different groups: the figures can be a hindrance during hunting endeavors, causing trouble during the hunt. Petit then goes on to reveal the structuring and recurring patterns and relations, which can be identified as mathematical relations. Especially geometric forms with their hidden or explicit symmetry relationships are highlighted. In the final paragraphs, the impact of the recognition of this tradition for mathematics teaching is explained. In Modeling of Implied Strategies of Solo Expert Players, Luc Tiennot makes an in-depth analysis of several forms of sowing games. These games were probably invented in Africa, but traveled to a larger area around the world, throughout Eastern South America, and Southern Asia in particular. They are combinatorial board games for two players that consist in moving seeds—organized in the board’s “cups”—, thus Céline Petit, as part of the 18th International Union of Anthropological and Ethnological Sciences (IUAES) World Congress”, Florianopolis, Brazil, 2018. The organization of this panel and thereafter the implementation of this book project would not have been possible without the financial support of the French National Research Agency (“ETKnoS” Project, 2016–2021).

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“sowing and capturing” the latter seeds through precise rules and procedures. The chapter concentrates on a particular complex form of these sowing games, called Solo, that are played from Central to Southern Africa, including Madagascar, where the author has carried out field research. Among the different kinds of Solo, Tiennot studies the sowing games bao, originating from Zanzibar, and the different variants of this game that he has observed throughout Madagascar. The different forms of the game show the complex use of mathematical operations. The author manages to show how systematic ethnographic work on these games can reveal the mathematical algorithms which are hidden in them. To this end, Tiennot also introduces different modeling tools (symbolic writing, and mathematical graphs in particular) for encoding and analyzing the procedures implemented by the players during the game. This modeling approach allows us to gain a better understanding of how the sowing game experts can play so fast, by anticipating the outcome of each move (without “overloading their short-term memory”) through the implementation of “optimized movements”. In Sand Drawing Versus String Figure-Making, Eric Vandendriessche compares two activities of the Northern Ambrym Islanders in Vanuatu, which are both seen as forms of “writing” in the local knowledge. One is the making of string figures with hands, feet and/or mouth. The other is drawing a continuous line in a loop with the finger in the sand (or dusty ground). Both express and record knowledge on either particular mythic entities or specific environmental phenomena in the area. In the second place, and most importantly, the author investigates what notions, terms and concepts of operation, procedure, sub-procedure, symmetry and iteration can be found in both types of practices and how these reveal shared geometric and algorithmic properties. In this way, ethnographic detail and in-depth analysis on mathematical thinking in these practices are offered as material for the basic discussion in this volume, going from a particular ethnographic context to the general interdisciplinary field of investigation the editors have in mind.

Part II “Impact of Indigenous Culture on Education in General, and on Mathematics Classes in Particular” Here the reader will find two Brazilian studies. Both rightly claim that Brazil has a particular and very interesting history with regard to the impact of native cultures on educational programs. Apart from these two articles, a chapter is included that focuses on projects in the north of Greece. In Indigenous School Education, Sérgia Oliveira, Liliane Carvalho, Carlos Monteiro and Karen François first give a synthetic overview of the history of Indigenous peoples in Brazil, from the beginning of the invasion/colonization (from 1500 on), when these peoples have been decimated by the invaders, and thereafter subjected to colonial authorities and religious institutions. After independence (1822), different educational programs have attempted to acculturate these populations. It will take

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more than 150 years, through different Indigenous movements’ struggles, to see public policies emerge in Brazil, and the creation of the Indian’s National Foundation—FUNAI, finally granting civil and political rights to Brazilian indigenous people, including legal regulations of their lands, education, culture, and health. Our authors show that the development of local educational systems, progressively taking into account the cultures of the various Brazilian Indigenous societies, has been done through a long process which finally led to legal recognition. Recently, these political and educational advances have been deliberately dismantled, by the federal government, seeking to “deconstruct the rights and guarantees of social policies aimed at indigenous peoples. Education was the hardest hit by these constraints.” However, various Brazilian indigenous communities continue to struggle for an intercultural education based on the respect of diversity. It is indeed the case of the Xukuru of Ororubá people settling in different villages in the state of Pernambuco. They “strengthen their ethnic identity and reinforce their struggles through a sociopolitical organization”. In particular, the Council of Xukuru Indigenous Teachers of Orurubá which seeks to discuss issues related to intercultural education, helping indigenous teachers to entwine the official curriculum with their own local culture. It is in collaboration with the latter council that our authors have undertaken a collaborative research program related to the teaching of statistics. The chapter ends with the description of this experiment: At first, Oliviera’s team carried out a participantobserver ethnographic research, in order to better understand the Xukuru cultural context. Thereafter, a collaborative working group was formed with both indigenous teachers and our researchers, in order to determine how teaching statistics could be correlated with the Xukuru’s concerns, while implementing a (locally based) course plan on statistics (related to the use and preservation of water resources in these communities). Indeed, the results of this local experiment suggest that such pedagogical projects (when carried out in collaboration with local educational institutions) can enhance the relationship between the local curriculum and indigenous communities’ challenges. The role of so-called non-western worldviews and formal reasoning procedures in the general educational landscape of a large and intensely diverse country like Brazil illustrates a point that will have much relevance in the future, for many countries in this progressively mixed world. In Indigenous Mathematical Knowledge and Practices, Cecilia Fantinato and Ketio Leite focus on a particular topic: within the context of Brazil’s studies of mathematics and education, one finds a series of six consecutive congresses (2000–2016). The authors analyze formats and presentations during these nationwide conferences and indicate how they got to be progressively more interested in and marked by native perspectives in mathematics education. They show how the political focus allowed for this shift over the years, combined with the successful development of cohorts of “native” teachers. Brazilian ethnomatheticians Maria Cecilia Fantinato et Kecio Leite give an overview of the research in ethnomathematics carried out in Brazil over the past 20 years, dealing with issues related to indigenous knowledge and culturally based education. Since the seminal works by Ubiratan D’Ambrosio in the 1980s, the ethnomathematics research field has significantly developed in this country (as

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nowhere else in the world) and institutionalized in many Brazilian Universities. In order to draw a picture of the impressive Brazilian production in ethnomathematics, the authors first analyze the different topics addressed throughout the five editions of the Brazilian Congress of Ethnomathematics (CBEm), from 2000 to 2016. Notably, this brings to light an increasing interest—over this period—for research devoted to indigenous mathematical practices and knowledge and their pedagogical implications. The chapter demonstrates that the reasons are threefold; first, there are over 300 different indigenous societies in Brazil. This great linguistic and cultural diversity in this country offers invaluable opportunities for carrying out such research, which few ethnomathematicians have undertaken since the 1980s. Secondly, in 2008, a law was adopted, establishing the guidelines for National Education, while including in the official curriculum the mandatory topic “History and Afro-Brazilian and Indigenous Culture”. Consequently, this led to an increase of the number of works on indigenous mathematical knowledge and their possible uses for the elaboration of culturally based curricula. Thirdly, the development of Indigenous Intercultural Licentiate University Courses implies an increasing “capillarization of Ethnomathematics in teacher training courses” from undergraduate to graduate education. As a consequence, Brazilian academic research in ethnomathematics is conducted by indigenous researchers who belong to the societies/communities under study. The chapter thus raises some issues regarding the research methodology implemented by these new generations of researchers arguing that they do not need to carry out genuine ethnographies—as non-native researchers did—since they are dealing with their own cultures. In Subverting Epistemicide Through ‘The Commons’: Mathematics as a re/making space and time for learning, Anna Chronaki and Eirini Lazaridou focus on the global—and the Greek—push towards detached and presumed universal mathematics and mathematics educational programs, emanating from the neoliberal think tanks of our time. With an in-depth ethnographic research on alternative developments in one particular region of Greece (bordering on Macedonia) the authors characterize two opposing educational lines of development. The first trend yields what is called “epistemicide”, resonating with critical studies within mathematics education and schooling in general. The term epistemicide refers to the destruction of local, but also of situated learning processes and contents, replacing them with so-called global universally powerful alternatives. The authors describe what this trend allows for and is producing in the local community: the destruction of knowledge (hence the specific term “epistemicide”) and the uprooting of the local people who are enduring this trend. The second trend is observed as a local alternative and goes by the name of “radical pedagogy of commons”. It is linked to a profound criticism of mathematics education, as it is emerging in for example South America, with an ontological shift towards a “pluriverse” instead of the western-dominated so-called universal knowledge. In that sense the study relates to the project of this book, and it prefigures some of the theoretical critiques in Baker’s chapter (part III). Also, it exemplifies what was precedingly referred to in this field as “street mathematics”, which may be interpreted as ethnomathematics in western cultural contexts. The authors’ study shows how the explicit explorations of

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mathematical knowledge in and through local places, objects, experiences and ways of dealing with the wide-ranging local world of experience allows for emancipatory alternatives: mathematics learning through walking, producing and other human and human-nonhuman phenomena.

Part III “Meta-studies” The former parts deal with particular cases or cultural traditions, introducing the reader to a lot of empirical research, mostly by ethnographers or by ethnographically informed mathematicians and educators. In Part III we introduce three texts that take a “bird’s-eye view” on the subject of this book. Both texts aim to sketch the broad panorama of knowledge, education and the urgent question of decolonization, which becomes particularly relevant in mathematics education today. In The Tapestry of Mathematics, Kay Owens affords a synthetic picture of the (ethno-) mathematics that has been developed for several millennia in Papua New Guinea, Melanesia, and South Pacific. This work has been carried out through the analysis of different kinds of sources borrowed from anthropology, archeology, linguistics, and social geography in particular, as well as ethnomathematics, and more recent personal fieldwork observations. Over time, the migratory routes and “intense language contacts” in this geographic area, would presumably have changed the characteristics of the proto-language counting systems (associated with either Austronesian or non-Austronesian language families). As a result, Papua New Guinea shows an incredible diversity of counting systems (still in use among the more than 700 hundred different Papuan indigenous societies). The concepts of “base” and “cycle” (introduced by Z. Salzmann in 1950, and later used by late ethnomathematician Glenn Lean (1992)) enable us to highlight/analyze the amazing and unique tapestry of counting systems from Papua New Guinea. Owens thus brings to light the ingenuity of the human mind, for creating different ways for quantifying the world. This analysis further shows that the latter counting systems are deeply linked to cultural patterns, as well as a “tapestry of transactions” related to a long history of trading. Consequently, numbers generally do not exist “in isolation.” They are primarily used for quantifying and qualifying “relations between people, objects and other entities”. Besides the mathematics embedded in counting and measuring systems, this chapter brings us into the “tapestry of mathematics in art”, reminding us of the large number of different Papuan activities with a geometric character, such as carving, tapa painting, string figures, etc. related to intellectual work on curves as well as ordered sequences of spatial operations. Finally, this chapter makes the point that the extremely diverse and old indigenous formal thinking in Papua New Guinea may open up the discussion on what is mathematics and what should be on the school curricula anew. In Indigenous Mathematics in the Amazon: Kinship as Algebra and Geometry Among the Cashinahua, Mauro de Almeida takes a bird’s-eye view on the question of mathematical skills and concepts, while considering both the older discussions (from the perspective of mathematicians and philosophers) and the more recent

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ethnographic studies on mathematical practices in different cultures. His contribution reveals the hidden or embedded mathematics in a large set of ethnographic material, principally dealing with kinship. The chapter is deeply rooted in previous works in mathematical anthropology on kinship systems (Weil 1949, Courrège 1965, Lévi-Strauss & Guilbaud 1970). It is moreover in line with Marcia Ascher’s ethnomathematical studies (1991) on the Australian Aboriginal Warlpiri and Ni-Vanuatu Northern Ambrymese kinship rules, arguing that the implementation of such complex kinship systems (that modern algebra helps to analyze) can be seen as genuine indigenous mathematical practices. Here, de Almeida presents in detail the Cashinahua terminological system of kin relationships, where order is established along the criteria of generation, moiety and sex. The argument, which is painstakingly developed in this contribution is based on the isomorphism in the system of kinship relations, on the one hand, and the mathematical structure of dihedral groups in algebra on the other hand. The detailed explanation of this isomorphism between both an indigenous ordering system (i.e. the Cashinahua kinship structures) and an established part of pure mathematics (i.e. group theory) is innovative and reformulates some of the older discussions in anthropology. Finally, de Almeida demonstrates that the underlying algebraic structures that regulate kin relationships are also at the heart of basketry weaving practices, through the transformations of basic patterns and designs implemented in such technical accounts. In his The Western Mathematic and the Ontological Turn: Ethnomathematics and Cosmotechnics for the Pluriverse, Mike Baker sketches a broad meta-discussion: he states that we are presently wrestling to overcome the singular world ontology of modernity with a pluriversal ontological politics of knowledge and education. It is ontology and politics: de-westernization goes together with political shifts in the world. Reference to the movement under Walter Mignolo’s postcolonial critique is central to this appreciative reasoning. The analysis is meant to be appreciated as a deep intellectual critique, since Baker places himself in the company of those philosophers and postcolonial thinkers who want to promote an ontological turn. The relationships between the human knower and his/her metaphysical view on reality (humans versus nature, or holism, etc.) is the focus of this paper, together with the socio-political turn of our time under the guise of the de-westernization of knowledge. This critical in-depth analysis of the status and role of formal thinking in general and mathematics in particular leads the author to situate ethnomathematics in a pivotal positioning in the profound upheaval he is witnessing. He invites the reader to look at a deeper level of ontological processes in order to discuss this shift at the level of explicit knowledge. The link between a knowledge specialism (i.e. mathematics) and the broad politically laden field of education in a decolonizing world offers unforeseen horizons. The founding of these processes in the context of shifts in anthropology-cum-political contexts is innovating in a field that is presently deeply stirred. The strong relationship the author develops between present-day critical anthropology and mathematical education is very up to date. This link is yielding what the author refers to as “the ontological turn” in philosophy and some scientific

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disciplines, thus relating to a contemporary epistemological and political discussion, with a major role for ethnomathematics.

Conclusions The book formulates conclusions that go beyond the materials presented through detailed studies. The field of mathematics education, in as far as it allows for indigenous knowledge to be integrated in the course materials and in the teaching procedures, has been maturing over the past half-century. The pioneers came either from anthropology or from mathematics. The former had little competence in mathematics as such, while the latter did poorly on research with living human beings whose cultural background they did not share. Over the years we found more and more convergences between both, with a clear growth for genuine first-hand ethnographic material (such as presented here, especially in Part I). The volume’s editors begin with this observation to sketch possible future avenues of interesting and emancipative work in this domain. They launch open invitations to future scholars in this fascinating field. Eric Vandendriessche Rik Pinxten

Acknowledgement The Editors, Eric Vandendriessche and Rik Pinxten, warmly thank Swapna Mukhopadhyay, Kay Owens, Alan Bishop, Brian Greer, Frédéric Keck, David Lancy, Roger Miarka, Jean-Paul Van Bendegem, and Claude Karnoouh, who generously participated in reviewing and proofreading the chapters of the present book.

References Ascher, M. (1991). Ethnomathematics: A multicultural view of mathematical ideas. Pacific Grove, California: Brooks and Cole Publishing Compagny. Courrège, P. (1965). Un modèle mathématique des structures élémentaires de parenté. L’homme, 5(3): 248–290. Descola, P. (2015). Par-delà nature et culture. Paris: Gallimard. Lean, G. (1992). Counting systems of Papua New Guinea and Oceania. Unpublished Ph.D. Thesis, Lae, Papua New Guinea: PNG University of Technology. Lévi-Strauss, C. & Guilbaud, G.-Th (1970). Système parental et matrimonial au Nord Ambrym. Journal de la Société des Océanistes, 26: 9–32. Salzmann, Z. (1950). A method for analyzing numerical systems. Word, 6: 78–83. Weil, A. (1949). Sur l’étude algébrique de certains types de lois du mariage (Système Murgin). In : C. Lévi-Strauss, Les structures élémentaires de la parenté (Appendix, pp. 279–295), Paris: Presses Universitaires de France.

Contents

Ethnography and Mathematics Re/Creating ‘Evocative Images’ (sunannguanik iqqaigutinik): Procedural Knowledge and the Art of Memory in the Inuit Practice of String Figure-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Céline Petit Modeling of Implied Strategies of Solo Expert Players . . . . . . . . . . . . . . . . Luc Tiennot Sand Drawing Versus String Figure-Making: Geometric and Algorithmic Practices in Northern Ambrym, Vanuatu . . . . . . . . . . . . Eric Vandendriessche

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Impact of Indigenous Culture on Education in General, and on Mathematics Classes in Particular Indigenous School Education: Brazilian Policies and the Implementation in Teacher Education . . . . . . . . . . . . . . . . . . . . . . . 121 Sérgia Oliveira, Liliane Carvalho, Carlos Monteiro, and Karen François Indigenous Mathematical Knowledge and Practices: State of the Art of the Ethnomathematics Brazilian Congresses (2000–2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Maria Cecilia Fantinato and Kécio Gonçalves Leite Subverting Epistemicide Through ‘the Commons’: Mathematics as Re/making Space and Time for Learning . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Anna Chronaki and Eirini Lazaridou

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Meta-studies The Tapestry of Mathematics—Connecting Threads: A Case Study Incorporating Ecologies, Languages and Mathematical Systems of Papua New Guinea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Kay Owens Indigenous Mathematics in the Amazon: Kinship as Algebra and Geometry Among the Cashinahua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Mauro W. B. Almeida The Western Mathematic and the Ontological Turn: Ethnomathematics and Cosmotechnics for the Pluriverse . . . . . . . . . . . . . 243 Michael Baker Conclusions. Some Possible Lines for Further Research in Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Rik Pinxten and Eric Vandendriessche

Ethnography and Mathematics

Re/Creating ‘Evocative Images’ (sunannguanik iqqaigutinik): Procedural Knowledge and the Art of Memory in the Inuit Practice of String Figure-Making Céline Petit Abstract This paper examines how certain procedural and structural features of string figure-making as practiced in Inuit societies fit into the cosmology and knowledge systems of these peoples. Drawing on data stemming both from classic ethnography in the Inuit area at large and from fieldwork carried out in Inuit communities of the eastern Canadian Arctic since the 2000s, it focuses chiefly on the symbolic and cultural interpretations of some mathematical ideas involved in this practice (aya’rraq, ayaqhaaq, ajaraaq). After a brief consideration of the importance ascribed by pioneering anthropologists to the study of the methods of making string figures (notably among the Inuit, then called “Eskimo”), particular emphasis is put on Inuit vernacular concepts and expressions that refer to this practice as a procedural activity relating to knowledge encoding and sharing. The re/creation of evocative images rooted in the ancestors’ experiences and perceptions is notably mentioned as a major principle underlying the mnemonic and cognitive values associated with such a practice, along with the storytelling performance that would often accompany the making and animation of these ephemeral artifacts. Furthermore, the analysis of the ritual rules and the performativity that characterized the practice in pre-Christian Inuit societies sheds light on the symbolic prevalence of several structural features that situated string figure-making in a significant relation to sila, as a cosmological principle referring to the universe, the outside, the weather, and by extension a principle at the basis of certain forms of cognitive ability. Examining the symbolic efficacy that was formerly ascribed to Inuit string figure-making also highlights the spatial conceptualization of this practice (perceived as upwards and outwards oriented) and its display at several spatiotemporal scales, while providing clues on particular meanings attached to topological and geometrical ideas embedded in this practice. Among the practice’s procedural and geometrical properties whose significance appears as most salient in Inuit cosmologies, thread-crossing (or knotmaking), symmetry, transformation and iteration are more specifically considered here. The bilateral symmetry that prevails in a significant part of the iconography comprising Inuit string figures is notably analyzed as featuring a notion of pair that C. Petit (B) SPHERE, UMR7219, University of Paris Cité & CNRS, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_1

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relates either to the anatomical field (of human and animal bodies) or to intraspecies relationships. In conclusion, considerations on building on Inuit traditional string games for studying and teaching mathematics in a culturally responsive curriculum in contemporary Inuit communities are developed. Keywords Inuit · String figures · Knowledge transmission · Mnemonic tool · Mathematical ideas

1 Introduction: String Figure-Making as a Widespread Activity Involving Procedural Knowledge The practice commonly known as “cat’s cradle” or “string game” consists in producing a figure with a loop of string, by carrying out a succession of operations on this loop, using mostly fingers. Observed in many societies, this practice has interested anthropologists and a number of mathematicians since the late nineteenth century. Several anthropologists initially saw, in studying the corpuses of string figures, a method of collecting clues on the phenomena of cultural diffusion, migratory routes and inter-group contacts (Paterson, 1949; Tylor, 1879), whereas quite early on a few mathematicians suggested the mathematical character of this procedural practice, without necessarily backing up this premise (Ball, 1911). The importance of precisely collecting the series of operations involved in making each string figure was clearly suggested or emphasized by famous pioneering anthropologists, but generally with no explicit claim regarding the heuristic value of such a collection. Whereas the American anthropologist Franz Boas appears to have been the first to publish a procedural description of—indigenous—string games (as part of his monograph on the “Central Eskimo”, i.e. Inuit of the South Baffin area in the Canadian Arctic, 1888a: 569–570, 1888b: 229–230, Fig. 1), the urge to systematically record the methods of making string figures was expressly put forward by the Cambridge anthropologists Alfred Cort Haddon and William Rivers, in a paper presenting for that purpose a nomenclature based on definite operations (1902). In this paper aspiring to “induce field workers to pay attention to the subject”, the ethnological relevance of documenting the sets of operations involved in the string games is mainly suggested by referring to the existence of data indicating that “at least some of these string figures have, or have had, a definite significance or utility amongst several peoples” (1902: 147). The elaboration of a methodology intended to be applicable anywhere was, furthermore, underlaid by the anthropological project of promoting the comparative study of the string games collected worldwide, from a perspective later mentioned by Haddon in the following terms: “it is only when the comparative method is applied to [cat’s cradle] that we begin to discover that it, too, has a place in the culture history of man” (Haddon 1905, in Jayne, 1906: xi). In line with Rivers and Haddon’s statement emphasizing that “making string figures is so widespread an amusement that it deserves special attention” (1929: 323), the French anthropologist Marcel Mauss advocated the close study of cat’s cradle as “the most widespread game, reported worldwide”, by recommending to precisely give an

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Fig. 1 The figure of “the deer” (The Central Eskimo, 1888a: 569). A first description of its method of making by Franz Boas: “Wind the loop over both hands, passing it over the back of the thumbs inside the palms and outside the fourth fingers. Take the string from the palm of the right hand with the first finger of the left and vice versa. The first finger of the right hand moves over all the parts of the thong lying on the first and fourth fingers of the right hand and passes through the loop formed by the thongs on the thumb of the right hand; then it moves back over the foremost thong and takes it up, while the thumb lets go the loop. The first finger moves downward before the thongs lying on the fourth finger and comes up in front of all the thongs. The thumb is placed into the loops hanging on the first finger and the loop hanging on the first finger of the left hand is drawn through both and hung again over the same finger. The thumb and first finger of the right and the thumb of the left hand let go their loops. The whole is then drawn tight” (ib.: 570)

account of the moves applied to the string ([1947] 2002: 135). This requirement may be regarded in the light of his preceding development that suggests considering how games fit into specific cultural—notably artistic and religious—systems. Although observed worldwide, string figure-making appears as a practice which has been more significantly developed in “indigenous” societies characterized by the prevalence of an oral tradition. Major corpora of string figures were more particularly documented through collections made in societies from Oceania and the Americas since the late nineteenth century. While comparative analyses of the sequences of operations involved in these corpora revealed contrasted schemes—between different cultural areas—in the combinatorics of mostly similar operations, studies of the sets of “procedures” pertaining to a given corpus highlighted its systemic organization, notably through a principle of arborescence (Vandendriessche, 2015). Considering these structural features along with the available ethnographic data—on the socio-ritual contexts and symbolic meanings locally associated with string figuremaking—led to the elaboration of the research project ETKnoS (“Encoding and Transmitting Knowledge with a String”), which aims to shed light on some mathematical ideas involved in this practice as embedded in particular cosmologies and knowledge systems. Building on Marcia Ascher’s understanding of mathematical ideas as “ideas involving number, logic, spatial configuration, and, more significant, the combination or organization of these into systems and structures” (Ascher, 1991: 3), this research more specifically examines how this procedural practice is or was associated with the expression and transmission of different types of knowledge (mythological, cosmological, ecological, ritual, etc.), as suggested by the ethnographic data previously collected in Oceanian and Native American societies. Within

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Fig. 2 Map of Inuit Nunangat, Inuit regions of Canada © Inuit Tapiriit Kanatami

that project, further ethnographic research has been carried out in Melanesian societies (Ni-Vanuatu, Trobriand and Awiakay peoples of Papua New Guinea) as well as in societies from South America (Guarani-Ñandeva in Paraguay, Nivakle and Enlhet in the Chaco) and North America (Canadian Inuit).1 This paper provides an insight into the part of the research regarding string figure-making in Inuit societies. While referring more largely to data collected among peoples of the “Inuit” area (whose languages are part of the Eskimo-Aleut linguistic family), the information presented below mainly comes from participant observation and interviews with Inuit elders in the communities of Iglulik, Iqaluit and Panniqtuuq in Nunavut (a Canadian federal territory) and in Kuujjuaq and Inukjuak in Nunavik (Arctic Quebec, Canada), Fig. 2. Inuit testimonies describing string figure-making as an activity relating to memory and knowledge sharing will be first considered here, notably with a view to highlighting vernacular concepts associated with procedural and structural features of this practice. The intertwining of its structural and religious/cosmological aspects will be more closely examined in a second part, by focusing on symbolic meanings associated with some mathematical ideas involved in Inuit string figure-making (Fig. 3). 1

For a detailed presentation of ETKnoS (ANR-16-CE27-0005-01), see: http://www.sphere.univparis-diderot.fr/spip.php?rubrique153&lang=en.

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Fig. 3 Playing String Game, Acrylic by Hannah Tooktoo (1995–), Kuujjuaq, Canada, 2020. Some parallels have been drawn by several Canadian Inuit women artists between certain designs of the traditional—female— tattoos (tunniit) and some shapes involved in Inuit string figures

2 String Figure-Making in Pre-colonial Inuit Societies 2.1 A Practice Referring to Different Spatiotemporal Scales Since the late nineteenth century, string figures have been collected in various societies of the “Eskimo” or Inuit cultural area at large (including Yupiks from eastern Siberia and southwestern Alaska, Inupiat from northern Alaska, Inuit from Canada, and Kalaallit, Inughuit and Tununiit from Greenland). After the preliminary collections made by the American anthropologists Franz Boas (1888a, 1888b), Alfred Kroeber (1899) and George Gordon (1906) in particular, extensive work was carried out in the 1910s by the Canadian anthropologist and archaeologist Jenness (1924), in Alaska and the western Canadian Arctic. Studies aimed at documenting all the string games known by a particular Inuit group were also undertaken later on (Hansen, 1975; Victor, 1940), for some part in a comparative perspective at the scale of the Inuit area (Mary-Rousselière, 1969; Paterson, 1949). These collections indicate that many string games were common to different Inuit societies, which constitutes one of the clues suggesting that the knowledge of a particular repertoire of string figures spread out over the American Arctic with the diffusion of the Thule culture from northern Alaska to eastern Greenland between the thirteenth and fifteenth centuries (cf. Friesen & Arnold, 2008; Mary-Rousselière, 1969; Paterson, 1949). Identified as a “Paleo-Eskimo” element by the Canadian archaeologist Kaj BirketSmith (1953: 192, see also Mary-Rousselière, 1969: 165), string figure-making is generally described by Inuit elders as a practice which was known by their earliest ancestors. Although mainly mentioned as a game (pinnguarut, pinnguarusiq), it is also presented by some elders as an activity whose purpose was “beyond mere amusement” in the past, before Christianization took place. One argument to that effect is the common mention of the former practice consisting in traveling to potentially

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distant camps, in order to re-learn string figures that had been forgotten in one’s own regional group, among Inuit of the central and eastern Canadian Arctic (MaryRousselière, 1965: 14; Petit, 2009: 261–262, ajarausisuqtuq). Inuit elders’ testimonies on string figure-making in the “pre-modern” nomadic period more generally suggest the prime importance ascribed to the ability of mastering the string games seen as inherited from distant ancestors, which seems to be also expressed by the homogeneity of the Inuit string figures’ repertoires collected from the nineteenth to twenty-first centuries. Furthermore, other data relating to Inuit practices of string figure-making in the pre-Christian period indicate that symbolic efficacy—referring to impacts at different spatiotemporal scales—was ascribed to this activity in various contexts. In the Inuit area as a whole, string figure-making was indeed embedded in a ritual system of prescriptions and prohibitions, notably linked to the seasons’ alternation and the course of the sun. This practice was said to be forbidden when the sun was visible or ascending, and more specifically in the septentrional regions, when it would return—or when its reappearance was announced by the particular position of a constellation including Altair and Tarazed—in the late winter (Birket-Smith, 1953: 104; Hawkes, 1916: 121–122; Jenness, 1922: 184, 1924: 181; Lantis, 1946: 217; Rasmussen, 1929: 183, 1931: 167, 1932: 52; Stefansson, 1914: 244, etc.). In Inuit societies of the eastern Canadian Arctic, this prohibition was related to the idea that the sun’s rising would be hampered by the string loops’ crossings induced by that practice: a well-known tale in these regions relates how Sun (Siqiniq, conceived of as a female being) was led to hide after tripping over—and getting cut from—the string loops held by playing humans while she was engaged in an ascending path above the horizon (Holtved, 1951: 11; MacDonald, 1998: 125–126; Petit, 2009: 252; Saladin d’Anglure, 2003: 80–81). Besides causing bad weather or storms (a collective punishment), wrongdoers were said to be exposed to sanctions such as getting cuts on the thighs or being tickled to death by the Sun or by the (male) “spirit of the string figures”, who was himself described as eager to compete in making string figures with his own intestines (Jenness, 1922: 184, 1924: 182; Petit, 2009: 252; Rasmussen, 1931: 391, 1932: 52, etc.).2 As a rule, the right period for playing string games was thus during the late autumn and the first part of the winter (known as the “great darkness period”, tauvigjuaq, in some areas). At least among some Inuit groups of Canada, such a practice was, moreover, prescribed in that period, either to “catch the [south-going] sun in the meshes of the string, and to prevent its disappearance” (Boas, 1901: 151, after G. Comer), or to hasten its return and encourage its quick rising after the solstice (MacDonald, 1998: 123; Petit, 2009: 253–254) (Fig. 4).3

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In Inuit societies, string figure-making was generally performed with a thread made of a tendon, ivalu[k], most often of braided caribou sinew (and in some cases of beluga or sea-mammal sinew). String games were also played with thin thongs of seal skin, and in some areas of southwestern Alaska, with a two-ply grass string (Lantis, 1946: 217). 3 When the sun would return (or would start an ascending path) and string games were thus forbidden, another game, of cup-and-ball (referred to as ajagaq), had to be performed in some Inuit societies of the central and eastern Canadian Arctic, with the same goal to stimulate the sun’s rising (cf. Boas, 1901: 151; Rasmussen, 1929: 183, etc.).

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Fig. 4 String Game, Lithograph by Agnes Nanogak (1925–2001), Ulukhaktok/Holman, Canada, 1975

Given these seasonal rules regarding string figure-making, the creation of configurations including crossings thus appears as central in the symbolic efficacy associated with this practice. The main performative principles involved in string figure-making in relation to non-human beings such as (the) Sun and the inner/master spirit of the string figures seem, furthermore, expressed in the stem (ayak-, ajak-) building the vernacular terms that refer to the practice throughout the Inuit area (aya’rraq, ayaqhaaq, ajaraaq). This stem indeed connotes the idea of “propping up” or “pushing (something) away, further”, which suggests a symbolic prevalence of the upwards and outwards orientations in the Inuit perception of string figure-making (although this activity also includes frequent downward and inward moves). This echoes both with the prescriptive aspect of the relation to the sun (to be propped up while on its descending path or encouraged in its rising, [upwards, qummut]) and to the prohibitive aspect regarding the link to the spirit of the string figures, said to draw his intestines out [outwards, silamut] to challenge to death whoever would play this game alone or excessively at night (Elbaum & Sherman, 2013: 203–204; Jenness, 1922: 203, 1924: 182; Johnston et al., 1979: 59; Lantis, 1946: 293; Petit, 2009: 256; Rasmussen, 1931: 248, 391) (Fig. 5). While string figure-making was sometimes performed as a competition (mostly aimed at being the fastest at executing a given figure), it was principally practiced in a family context involving several generations and often, including a storytelling

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Fig. 5 Two teams compete to play string game, Lithograph by Victoria Mamnguqsualuk (1930– 2016), Qamanittuaq/Baker Lake, Nunavut, Canada, 2006 (Feheley)

component. The narrative dimension associated with this (indoor) practice could include both codified words and personal creations. For one part, formal recitatives used to accompany the making of various string figures: perceived as very ancient, these words were—and still are—generally sung or uttered when displaying and/or moving the named figure. According to Canadian Inuit elders, the enunciation of these condensed forms of stories pertaining to the oral tradition was required to adequately perform the corresponding string figures.4 For another part, narratives rooted in more recent personal experiences were also told through string figuremaking in Inuit societies. Expressed in the meaning attributed to the term referring to this practice among Siberian and Alaskan Yupiks and North Alaskan Inupiat (cf. ayaXaaq-, ayaRaaq-: “to play string game, tell string story”, Fortescue et al., 1994: 59), this narrative component, however, seems to have been mostly part of the insertion of string figure-making within a symbolic continuity to ancestors’ experiences and knowledge. Considering the marked predominance of string figures that represent animals (or depict hunting activities through various instruments or body techniques) in the Inuit corpora, these experiences and knowledge revolved for a significant part 4

The idea that a particular song or story was at some time associated with each string figure of the Inuit repertoires was suggested by several anthropologists (cf. Birket-Smith, 1953: 104; Gordon, 1906: 87; Spencer, 1959: 240, 384). Most of my informants emphasized for their part that the words (uqausiit, uqausinnguat) or songs (inngiusiqtait, inngiusinnguat) that traditionally accompany some of the string figures (ajarausiit) “originate from ancestors who really knew how to speak the (Inuit) language”, and must be thus re-produced even when their full literal meaning appear now unclear. The names associated with each string figure (generally ending either with the morphemes (a)rjukor -tsiaq/-ttiaq in Inuit societies of the central and eastern Canadian Arctic) are seen themselves as including old linguistic forms inherited from forefathers.

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Fig. 6 String game, Lithograph by Luke Anguhadluq (1895–1982), Qamanittuaq/Baker Lake, Canada, 1973

around the observation of and relation to animals –said to have shared the same language as humans in former times (cf. Rasmussen, 1929, 1931: 208, etc.) (Fig. 6). Mythological and non-human beings, anatomical parts (of humans or animals), social and ritual interactions (scenes of daily life in the camp, antagonistic relationships, games and drum-dancing festivities, shamanistic rituals), tools and housing, as well as features of the physical environment (notably constellations)5 are further themes dealt with in these corpora (Jenness, 1924; Mary-Rousselière, 1969; Paterson, 1949, etc.). A closer examination of some vernacular concepts drawn from Canadian Inuit elders’ testimonies on string figure-making as a practice meant to produce evocative images shall now contribute to highlighting some links between procedural—or embodied—knowledge and memory.

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An “ancient conceptual link between [constellations] and [string-figure depictions]” (MacDonald, 1998: 16) seems to be suggested by both linguistic and mythological data collected in various Inuit societies (ib.; Petit, 2015: 216–219). While the practice of making string figures was ritually related to a particular constellation (consisting of a pair of stars, aagjuuk, Altair and Tarazed) among the Nuataagmiut Inupiat of Alaska, knowledge about the constellations’ movements—and timereckoning—was involved in the performance of some string figures in Inuit groups of the central and eastern Canadian Arctic (Mary-Rousselière, 1969: 94–95, 122; Petit, ib.).

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2.2 A Procedural Activity Relating to Memory and Knowledge Sharing The idea of a procedural activity seems to be expressed in the general term used— beyond dialectal variations—to refer to string figure-making in Inuit societies. This term (ayarr’ar-, ayaqhaaq-, ajaraaq-) may indeed be analyzed as being constituted by the stem ayak-/ajak- ([ayaγ-]: to prop up, to push away or further) and the suffix -qhaaq/-raaq, which connotes a multiple or repeated/prolonged action, or an action occurring in distinct stages, one after another (Fortescue et al., 1994: 423; Harper, 1979: 63; MacLean, 2012: 922; Schneider, 1979: 90). The reference to a process involving ordered series of operations is, furthermore, discernible in some descriptions of string figure-making by Canadian Inuit elders, through the use of the verb aaqqiksuk- (to put in order, to arrange over a period of time or in a series of operations, to coordinate, Schneider, 1985: 3; Spalding, 1998: 1). This notion of ordered combination—which is at the basis of the Inuktitut terms referring to activities such as jigsaw puzzles, dominoes or solitaire card games (aaqqiksugait, aaqqisugait)— is used in particular by Ootoova (2000: 231) to refer to the manipulation of the loop of string in the game of ajaraaq.6 Besides this idea of ordered sequence, the main concepts that appear generally involved in Canadian Inuit elders’ definitions of ajaraaq deal with the making of an image that represents and reminds of something, someone, or some event. This image-making process is often expressed by a term aggregating the morpheme -liuq- (“construct”) either with the stem ajji- (that conveys an idea of likeness: copy, imitation, reflection, picture) or with a noun including the morpheme -nnguaq- (that indicates resemblance or representation: e.g. uumajunnguaq = uumajuq [animal] + -nnguaq, the image of an animal). In some cases, the idea of likeness associated with the figure produced through ajaraaq is merged with one of relevance or truth telling expressed by the morpheme miksi-, as in the definition of the sinew-thread “ajaraaq” string provided by Qumaq (1988: 80): “a sinew string used as a playful means for making relevant images of animals and any kind of things (ivalu pinnguarutik uumajunnguanik qanuittutuinnanik miksiliurutik)”. Considering other testimonies, this notion indicates not only adequacy in the resemblance (i.e. realism), but also refers to the accuracy of the representation as rooted in knowledge or direct observation. The ability to create recognizable (string) images and give them meaning (tuki, “sense, direction”) [tukitaarti-] is often mentioned as a skill which was highly characteristic of the ancestors (said to have been very talented in that matter, pisitiktualuut&utik, pisitimmariit). The relevance or veracity perceived in the string figures “created by ancestors” is mostly associated with a notion of evocative power or mnemonic efficacy inherent to them. According to several elders, the string figures that used to be performed had indeed a cognitive purpose in that they were meant to “call to mind” or “keep in mind” (irqaiguti-, irqaumatsiuti-) things relating to what was represented. Testimonies from elders stressing that some of the stories and words that traditionally accompanied string games were used to 6

Some Iglulingmiut elders also used the term aaqqikpalliallugu (“to fix/adjust gradually”) in reference to the making of a string figure.

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remember the figures themselves suggest that the mnemonic process was also based on significant links between the enactment of a particular sequence of operations on a string loop and the “traditional” utterances associated with it.7 The emphasis put on the necessity to learn and systematically perform the oral expressions associated with the string figures (as experienced by many Canadian Inuit elders in their youth) may further be seen as a recognition of the cognitive value of the intertwining of gestural procedures, image production and orality in string figure-making, for both encoding and sharing knowledge (Fig. 7). Learning “through string games” was a common practice for Inuit children in the past, as emphasized by elders who were born between the 1910s and the 1930s: “We were led to learn a lot through ajaraaq. This was a learning tool [qaujimavallirutaulluni]” (L. Ukaliannuk, Iglulik), “This was our way of entertaining ourselves, but this was to keep the memories of events too. It was our parents’ and ancestors’ way to teach the children about their past, their culture” (L. Weetaluktuk, Inukjuak).8 The children’s practice of string figure-making appears, however, to have been valued not only from a perspective of knowledge transmission but also with respect to skill development. Testimonies from Canadian Inuit elders generally refer to ajaraaq as an activity building or reinforcing fine motor and coordination skills, as well as observation, memory, and more largely cognitive skills. When string figure-making is in some cases explicitly described as a practice involving or fostering thinking abilities expressed as isuma, it is most commonly associated with the promotion of a form of intelligence known as silatuniq, that refers to the ability to accurately

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While trying to remember the construction method of a given string figure during the interviews, some elders sang its accompanying chant several times, which may echo with George Gordon’s observation on the apparent value of these rhythmical recitations in the ability to achieve elaborate string figures for some Yupiit of the Yukon-Kuskokwim delta in Alaska: “An old man at Mamtrelich village in developing some of his more elaborate figures accompanied his motions by a kind of incantation or rhythmical recitation (…). By way of experiment I tried to induce the old man (…) to develop one of his string figures without recitation. His attempt to do so seemed to result in confusion and he had in each case to begin again and finally to repeat his formula” (Gordon, 1917: 130–131). Observations made during fieldwork with Canadian Inuit practitioners suggest, furthermore, that the procedural knowledge involved in string figure-making mainly builds here on the memorization of sequences as a whole (or as particular sets of movements more than a mere succession of particular moves). Most of the practitioners were indeed rapidly lost when trying to demonstrate the procedure slowly or step-by-step, in a similar way as what was observed by Margaret Lantis among the Yupiit/Cup’ig of Nunivaaq (cf. “An attempt was made to record the procedures for making the figures. But they always were made quite rapidly and in every case, no matter who was asked to demonstrate, the person became so self-conscious and confused when asked to slow down the movements that his or her fingers became thumbs, the string became tangled, and we had to give up”, 1946: 217). 8 Some Inuit elders drew a parallel between string figure-making and “writing” (titirarniq, lit. “to make repeated markings”) as symbolic systems involved in knowledge sharing, while several stressed that learning how to make traditional string figures was of major importance in the past “since there was no writing system” (titirausiqannginami) like the one later introduced by Christian missionaries.

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Fig. 7 Composition (Bear attached and String Game), Drawing by Victoria Mamnguqsualuk (1930–2016), Qamanittuaq/Baker Lake, Canada, ca. 1980 (Feheley)

interpret—and adapt to—one’s environment or the universe, sila (Petit, 2009: 263).9 The fact that talented male ajaraaq practitioners were said to be the best hunters in several Inuit groups of the eastern Canadian Arctic (Mary-Rousselière, 1969: 123; Petit, ib.) also suggests that the abilities demonstrated in string figure-making were perceived as echoing those expressed in hunting or navigating—on the sea ice or the tundra—in particular. Relating adequately and fruitfully to the outside/world (sila) is seen as requiring notably significant visuospatial and visuoconstructive skills whose development was conceived as involved at some level in string figure-making. Several elders mentioned in that sense that, as a youth, they were eager to learn and master the most string figures of the traditional repertoire and would do their best to become “savant” (ilittiarasuk&uniuk, ilisimatsiarasukpalauqtugut) in that domain, even if string figure-making was primarily regarded as a female practice (Petit, 2009: 262–264) (Fig. 8).10

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Being silatujuq (lit. “imbued with sila”, aware, wise) involves notably the ability to carefully observe or examine and notice, ujjiqsuttiaqtuq (cf. Ootoova, 2000: 541). On silatuniq and Inuit socialization, see Annahatak (2014). 10 Although string figure-making is generally described—both in the classic ethnography and in contemporary elders’ testimonies—as being mainly or symbolically a female game in pre-colonial Inuit societies (Boas, 1888a: 569; Hall, 1865: 129; Hawkes, 1916: 121–122; Lantis, 1946: 217; Low, 1906: 175; Murdoch, 1892: 383), the ethnographic data gathered since the nineteenth century clearly indicate that Inuit and Yupiit men would also indulge in such a practice.

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Fig. 8 Silarjuaq, the whole world, Lithograph by Teevee Ningiukulu (1963–), Kinngait/Cape Dorset, Nunavut, Canada, 2006. The universe seems depicted here as a loop of string that contains images of various beings pertaining to different—yet interconnecting—cosmic spaces or levels: underwater, terrestrial and celestial. Sanna, the female master spirit of the sea mammals, is represented at the bottom

When the cognitive aspects associated with ajaraaq are often mentioned by contemporary Inuit elders, their comments—as well as previous ethnographic data— on this practice in pre-colonial times also reveal particular symbolic interpretations regarding some of its structural features that refer to mathematical ideas or properties. Some cultural meanings ascribed to procedural and topological principles involved in string figure-making will be more particularly considered in that they shed light on the expression of mathematical notions in various fields of Inuit cosmology.

3 Symbolic Meanings Ascribed to Some Structural Features of ajaraaq (ayarr’ar, ayaqhaaq): An Insight into Mathematical Ideas as Embedded in Inuit Cosmology 3.1 String Figure- and Knot-Making: Generative/Ordered Thread Crossings Versus Tangles A significant part of the ritual prohibitions and prescriptions that used to concern string figure-making in Inuit societies indicates that this practice was symbolically

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associated with knot-making in general. The creation of repeated crossings or knots was actually considered from an ambivalent perspective, its efficacy being either seen as generative (especially when involving an idea of ordered or regular knots comparable to those required in net-making) or disruptive (when knots take the form of tangles that impede a process) (Fig. 9). As mentioned above, making string figures in the fall—and possibly some of them in particular according to some elders—was said to be a means to keep the sun longer above the horizon. Whereas some ethnographic data stress that the idea was “to catch the sun in the meshes of the string” (Boas, 1901: 151; Low, 1906: 170, for the ‘Iglulik Inuit’ or Aivilingmiut of the eastern Canadian Arctic), or “to tangle the legs of the sun (…) to delay its disappearance” (Birket-Smith, 1953: 104, for the Sugpiat-Alutiit or ‘Chugach Eskimo’ of southwestern Alaska), testimonies collected since the 1990s among elders from the Iglulik area refer to the aim of favoring a quicker return and rising of the sun after its disappearance: they rather suggest the idea of keeping the sun from descending too far during the dark period (which echoes with one literal meaning of ajak- as “to prop up”), with the goal to hasten—or enable the proper fulfillment of—the sun’s cyclical course “around the universe” (silamik kaivallainiq). On another hand, when the sun was known to be on an ascending path in the late winter, string figure-making had to cease (as a rule in all the Inuit societies), since it would otherwise hinder the sun’s progress and cause

Fig. 9 String Ravens, Drawing by Germaine Arnaktauyok (1946–, Iglulik), Canada, 2017. A possible view of the dialectical relationship between generative thread-crossing and entanglement underlying the making of string figures

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bad weather (with clouds’ aggregation in particular). These unfavorable conditions would thus jeopardize the success of the hunt that was supposed to take place again. Such a conception of string knots as creating obstacles to the fulfillment of a positive or generative process was also discernible in other prohibitions regarding string figure-making. The thread-crossing action involved in ajaraaq was and for some part still is perceived as entailing a risk of entanglement that could be detrimental to the hunter or the mother to be. String figure-making was thus said to be forbidden or restricted for boys and young men, as well as for pregnant women. The first (and more particularly those who had not caught a member of each of the local sea mammal game species yet) would otherwise risk getting drawn under water or sea ice because of the entanglement of their harpoon line around their hand(s) when they would harpoon a sea mammal later in life, while pregnant women would be threatened with the entanglement of the umbilical cord around the unborn child’s neck (Boas, 1901: 161; Lantis, 1946: 204, 217; Petit, 2009: 254–255; Rasmussen, 1929: 177; Saladin d’Anglure 2003: 79).11 Tangles as unorganized knots disrupting a vital or subsistence process were, furthermore, expressed in the Canadian Inuit cosmology through the belief that the long hair of the female master spirit of the sea mammals (Sanna, Takannaaluk) would get entangled whenever humans would infringe ritual rules. The game animals were said to be then trapped in these knots, until a great shaman would visit Sanna and manage to untangle her hair (and then potentially braid it i.e. order the strands through repeated regular crossings), thus appeasing Sanna and making game available to humans again (Holtved, 1951: 22–23; Rasmussen, 1929: 127; Saladin d’Anglure 2006: 164–165; Smith, 1894: 209–210). The ritual necessity to undo the knots (before possibly engaging into generative crossings) in order to maintain an order favorable to humans echoes here with certain symbolic meanings associated with the final operation involved in each string figure-making procedure. In Inuit societies, a major value appears to be generally ascribed to the ability to undo the created figure in one move, without creating any knot on the string loop (which implies knowing in many cases the particular segments to be seized or released in the end): the immediate return to the original loose loop of string (nu˙glu, nurlu[k]) is mentioned indeed as the last stage required for properly completing every string game, that thus seems to be conceived as a circular process (Petit, 2014).12 11

Iglulingmiut elders stressed that the boys were told not to play ajaraaq “too much” in the past since they would have to harpoon (swift) sea mammals as hunters and would have to be very fast at pulling the harpoon line then: this rope would more easily tangle around their fingers and they would thus be pulled out by their hands (aggait qiluktallutit) if they had been making string figures too often. Interestingly, the perception of string figure-making as an outward/forward-oriented action appears also involved in analogical connections made between the string used for this game, the harpoon line and the intestines. The following statement by a Yup’ik elder may further shed light on that propelling or stretching aspect: “When the harpoon is shouting through the air, as [its line] plays out, it appears (…) as though a person’s entrails were flowing out from him” (cited in Jacobson, 2012: 442, under nenge-: to stretch, extend). 12 It is noteworthy that the Inuktitut term referring to the string loop (nurlu) is also the one used to refer to the geometrical form of the “circle” in the Nunavut School Mathematics Glossary (Allen, 2015: 7).

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Fig. 10 A bound shaman, before freeing himself and flying to the invisible world with the help of his familiar spirits (after Jenness, 1924: 133–134, Fig. 176). (Photo © C. Petit)

For some string games in particular, success in undoing in one jerk the tangle or multiple central crossings created in making the figure was, furthermore, interpreted either as the fulfillment of a ritual action of the represented character (the shaman extricating himself from his ties with the help of his familiar spirits, Jenness, 1924: 133–134, cf. Fig. 10), or as a good omen for the weather conditions to come (Lantis, 1946: 221–222, ma’taγ iy˘at figure from the Yupiit/Cup’ig of Nunivaaq, Alaska). In this last case, the negative significance of the remaining tangle was conversely expressed by the belief that “if the figure does not come apart, it is a sign of a storm” (ib.: 222). The efficacy associated with the creation of crossings (or knots) as a structural feature of string figure-making was, moreover, discernible in the “weather prediction games” played by Alaskan Inuit, possibly with the objective to favor the coming of the expected meteorological conditions. Involving sets of similar crossings and patterns (or translational symmetry), the final configurations obtained through these string games—such as the “range of mountains” or the “fish net” (Figs. 11 and 12)—were indeed said to provide a visual clue to some traits of the weather (sila) to come: “good visibility” was to be expected if the final patterns—and crossings— appeared regular or even and distinctly continuous, whereas “poor visibility” was supposed to prevail if they appeared uneven, with irregular intervals and staggered lines or heaped string segments (Johnston et al., 1979: 59, and see Jenness, 1924: 156). Some data suggests that other symmetrical string figures characterized by a series of similar designs and/or regular crossings (like those depicting the sun between the mountains, the flames of the oil lamp, or the Pleiades constellation referring

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Fig. 11 The “range of mountains”, a figure known as “the flames of the oil lamp” (ikumarattiaq, ikumaquattiaq) in Inuit societies of the eastern Canadian Arctic. Presented by Sheepa Ishullutaq, Iqaluit, Nunavut (photo © C. Petit)

Fig. 12 The “fish net” (after Jenness, 1924: 56–57, XLVII). (Photo © C. Petit)

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Fig. 13 The “breast-bone” or the “constellation” sakiattiaq. (Photo © C. Petit)

to a [caribou’s] breast-bone) may have been part of this playful weather prediction practice related to a cosmology—and a form of analogical thinking—connecting several spatiotemporal scales, the micro- and the macrocosm (Fig. 13).13 The symbolic meanings associated with the formal structures of these string artifacts (resulting for some of them from the iteration of a particular set of operations, such as the one termed as the “Inuit Ending” in Wirt et al., 2009) invite to further consider certain cultural interpretations regarding geometrical properties of various Inuit string figures.

13

The making of such string figures was also part of “divinatory games” that were aimed at hastening the sun’s rising (siqiniq sukkaniqsautijut) in the late winter, at least in the North Baffin area (Canada). An Iglulingmiut elder thus mentioned that, during the fall, when the sun had just set, children would make one of these string figures and present it above the horizon: looking through the figure as if it was a window, they would position it and predict where the sun would come out the next day. If the figure was accurately made (and not twisted), it was said to have a positive effect on the sun’s course. See Petit (2021), https://stringfigures.huma-num.fr/items/show/279 and https://stringfigures.humanum.fr/items/show/278 for an insight on the Iglulingmiut figures referring to the flames of the oil lamp, without and with a support.

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Fig. 14 Kiligvagjuk, the mammoth (by Jaikopi Atami, Iqaluit, Nunavut, photo © C. Petit). The making of this figure—referred to as “the spirit of the lake” among the Inupiat (Alaskan Inuit)— was there accompanied by words recalling a ritual prescription: the offering of water to this spirit when in a particular place (cf. Jenness, 1924: 43)

3.2 Cultural Interpretations of Some Geometrical Ideas Involved in Inuit String Figure-Making Asymmetrical figures and transformational geometry The “complicated asymmetrical picture-making” (Compton, 1919: 205) or the “frequency of asymmetric and one-sided figures” (von Hornbostel [1933] in Sherman, 1989: 36) was often presented, in the first decades of the twentieth century, as a characteristic feature of the “Eskimo” string figures contrasting with the geometric patterns such as “diamonds and nets” most commonly observed in the string figures’ repertoires of other indigenous societies (from Oceania and Africa in particular). This feature was in many cases analyzed as the expression of a “predilection for realistic representation of animals and (…) objects” (von Hornbostel, ib.) or as the result of a “very high degree [of development of] the art of string figures” associated with the ability to make “realistic” patterns (Haddon, 1930: 13–14),14 while the symmetrical figures “of purely geometrical character” collected among Oceanian peoples were sometimes considered as expressing a stronger tendency to “abstraction”, or even to a more “abstract” thinking (Compton, 1919: 206) (Fig. 14). 14

The complexity of Inuit string figures and the “cleverness” expressed by the practitioners were generally asserted by the first (western) ethnographers who observed that practice: cf. “I never before witnessed such a number of intricate ways in which a simple string could be used” (Hall, 1865: 129–130); “The women are particularly fond of making figures out of a loop, a game similar to our cat’s cradle (…). They are, however, much more clever than we in handling the thong and have a great variety of forms” (Boas, 1888a: 569–570); “The women (…) make a number of complicated figures with the string, many of which represent various animals. One favorite figure is a very clever representation of a reindeer (…)”, Murdoch, 1892: 383.

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Fig. 15 Tukturjuk, the caribou (by Niriungniq, Iglulik, Nunavut, Photo © C. Petit)

Regarding geometrical concepts, it is however noticeable that several of them are involved in Inuit string games characterized by the presentation of an asymmetric figure. Among these figures which mostly portray an animal (caribou, hare, fox, wolf, wolverine, musk ox, brown bear, black bear, lynx, squirrel, ermine, beaver, polar bear, beluga, bowhead whale, seal, seagull, duck, goose, raven, ptarmigan, snowy owl, crane, brant, swan, etc.) or a mythical being, many are for instance put into motion in such a way that they are subjected to a geometrical translation and/or reduction. These are mainly terrestrial mammals and legendary characters that are thus depicted as running away until vanishing (Jenness, 1924: 40, 46, 77, 97, 116, 118; Mary-Rousselière, 1969: 17, 24, 47, 125, etc., Fig. 15). The representation of their gradual disappearance expresses the variations in the visual perception of the witness (or hunter), thus rendering the perspective. Produced mainly through a swaying movement of one hand, the sliding move of the figure often refers to a behavioral characteristic of game animals (said by definition to flee from the hunter who has to pursue them): considering Canadian Inuit elders’ testimonies, the subject’s final disappearance is then interpreted as the animal escaping or on the contrary being caught (shot). The depiction of animals’ horizontal move thus connotes a significant aspect of the human relation to (game) animals which was at the core of the traditional nomadic way of life in Inuit societies.15

15

In the School Mathematics Inuktitut Glossary edited by the Nunavut Arctic College (Allen, 2015: 52), the idea of translation is expressed by nuuttiniq, a term built on a stem (nuu-) typically used to refer to a change of (dwelling) place (moving to a certain distance, from one place to another, cf. Fortescue et al., 1994: 237; Schneider, 1985: 229).

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Some string games also represent an animal which is made to move repeatedly up and down in reference to its alternative appearance/disappearance, thus illustrating a dual principle of the visual perception (visible/hidden) also central in Inuit relations to game animals (e.g. a seal surfacing to breathe in a crack and then diving, an Arctic ground squirrel popping up out of its hole and returning inside, Jenness, 1924: 85–86; Mary-Rousselière, 1969: 103–104; Paterson, 1949: 40, 43–44). Another way of rendering the variations in the visual perception depending on spatial configuration or perspective is to be found in the inclusion, in Inuit repertoires, of various string figures depicting the same animal from different angles. Several terrestrial mammals—caribou, fox, brown bear, wolverine, etc.—are in particular identified as portrayed either standing or upside down through similar or quite similar patterns that result from different sequences of operations or procedures (either related through variations from a common subset, or very distinct). Such an expression of alternate perspectives echoes with the representation of the same scene from different viewing points that constituted a significant feature of many Inuit drawings collected since the nineteenth century (cf. Blodgett, 1986; Boas, 1888a; Rasmussen, 1929, 1931, etc.) and which may have more largely been an aspect of pre-Christian Inuit graphic art (cf. Hoffmann, 1897: 920). When focusing on the single patterns that refer to animals in the Inuit corpora of string figures, it is also notable that quite a number of them are involved in a transformational “series”, in that their making is directly followed or preceded by the presentation of another figure (without returning to the preliminary loose position of the string loop). While each string game implies successive transformations of string configurations, the succession of figures organized as a series illustrates the transformation of shapes induced by one or only a few operations on the loop of string. The named figures successively shown in these series typically refer to beings or elements conceived in relation through a particular story.16 In some cases, the direct succession of two animal figures is or was interpreted as representing an interspecies transformation (such as an old squaw duck becoming a fox, Jenness, 1924: 116), in accordance with a major theme in Inuit mythology and oral tradition (fluidity of bodily forms in early times). In most cases however, such a succession provides an insight on different protagonists associated with the narrative. Beyond some variations in the stories, accompanying recitatives or interpretations prevailing in the different Inuit societies, one of the significant topics of the series depicting animal figures appears to be predation, either between animal species or from humans to animals (cf. Jenness, 1924; Mary-Rousselière, 1969: 45–47, 50–52, 55–56). Whereas the recitatives associated with some of the short series involve the prey’s perspective (cf. Jenness, 1924: 32, 153–154…), those accompanying some of the longer series include an enumeration of the game animals caught by a young hunter (while suggesting a parallelism between his development and theirs), as in “the brown bear cub” or the “caribou” series (Jenness, 1924: 59–61; Mary-Rousselière, 1969: 50–52, 55–57). In this last case, the transformation from one figure to another is implemented by iterating the same set of two operations (Figs. 16 and 17). 16

See for instance Petit (2021), https://stringfigures.huma-num.fr/items/show/78.

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Figs. 16 and 17 Two figures that are part of the “caribou” series, presented by Herve Paniaq, Iglulik (Photos © C. Petit). (Fig. 16) Nanua, “his” polar bear in an upside-down position. This figure is transformed into tuktua, “his” caribou, in a similar position (Fig. 17)

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Bilateral symmetry and the importance of pairs While a few “purely geometrical” forms appear as named string figures in the Inuit repertoires (e.g. triangular shapes of the “oil lamp” or the “duck spear”, Jenness, 1924: 45, 163; the rectangular “platter”, Jenness, 1924: 100; the “burbot” as two diamonds, Jenness, 1924: 49–50), such shapes (and parallelograms in particular) are most often involved as intermediate and unnamed string configurations or as parts of named figures. Beyond the significant number of asymmetrical final figures, another major part of these corpora is, however, constituted by dual patterns characterized by a bilateral symmetry. Here again, many of these dual figures refer to animals, either as a representation of “two of a same kind which form a pair” (iglugiik, illugiik) or as one focusing on dual anatomical attributes of such beings. The reference to the conceptual division of the animal and human body in two longitudinal halves (quppariik) actually underpins the symbolic meanings associated with a large number of these dual figures. Besides the fact that the structure of these figures is seen as the result of the mirror activity of both hands—also “two of a pair”—on the string loop, it is indeed noticeable that many of them are interpreted as depicting anatomical pairs or extensions of them (Fig. 18). This appears in continuity with the very frequent use of dual marks in the Inuit (and notably Inuktitut) anatomical vocabulary, clearly suggesting that “[t]he Inuit body is shaped by duality” (Bordin, 2003: 44–45). The focus on anatomical bilaterality expressed in the denomination of various string figures concerns both the human body (e.g. two big eyes, Jenness, 1924: 46; the [two] arms [and shoulder blades], ib.: 41–42; the [two] scapulae, ib.: 48–49; the two hips, ib.: 88–89; the two thighs, ib.: 108; the pair of long trousers, the pair of breeks, the pair of mittens, ib.: 102–104; his [two] snow shoes, ib.: 91; [two] foot-pads [used for standing when sealing], ib.: 94; [two] boots hanging up, ib.: 95; the smile [or the two corners of the mouth], ib.:

Fig. 18 Talirjuk, two arms (and shoulder blades), by Jiini Alivaktuq, Panniqtuuq, Nunavut (Photo © C. Petit)

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Fig. 19 Nurrattiak, two caribou fawns, by Tauki Qaappik, Panniqtuuq, Nunavut (Photo © C. Petit)

112; two toy labrets, ib.: 33–34; the phlegm [coming out from two nostrils], ib.: 90, etc.) and animals’ bodies (e.g. a seal’s kidneys, ib.: 82, etc.): in several cases, even when the name of the figure refers to the animal as a whole (or to a single part), the selection of dual attributes as most salient feature is evident (e.g. the walrus [head with two tusks], ib.: 72; the seagull, the butterfly [with their two legs and two wings outspread], ib.: 42, 129; the head of a caribou or musk-ox [two antlers or two horns], ib.: 56, etc.).17 In a few cases, the duality expressed in the depiction of a game animal’s body (or body part) refers, however, to a butchering process and thus connotes a successful catch (e.g. the cutting-up of the whale or the whale carcass, ib.: 77; a caribou’s stomach [in two halves], ib.: 70) (Fig. 19). As mentioned previously, the Inuit string figures presenting a dual pattern involve a notion of pair that goes beyond the reference to anatomical duality, since a significant part of them are seen as portraying two objects or beings (and most often animals) of the same kind.18 Regarding the animals thus depicted in pairs, a recurrent feature is their (perceived) position with their backs turned to each other (cf. two brown bears, ib.: 13–14; two mammoths, ib.: 44; two polar bears, ib.: 57; two musk oxen, ib.: 67–68; two bull caribou, ib.: 68–69; two fawns, ib.: 89–90; two wolves, ib.: 79–80; two brown bears issuing from two caves, ib.: 74; two lemmings and their burrows, ib.: 81; two mountain sheep, ib.: 70; two mice, ib.: 134; two butterflies, ib.: 84; two 17

In the interest of clarity, the references to Inuit string figures are here mainly taken from Jenness’ extensive collection made mostly in Alaska and the western Canadian Arctic in the 1910s. While some of these figures bear similar names in various Inuit groups of the eastern American Arctic, the interpretations associated with others vary according to the areas. The comments on pairs and bilateral symmetry are however generally relevant for all the Inuit regional repertoires. 18 It is noteworthy that bilateral or reflective symmetry (involving mostly animals and spirits) is a significant feature of many drawings—and lithographs—made by contemporary Canadian Inuit artists (see for instance the works by Kenojuak Ashevak and Saimaiyu Akesuk, and more widely the Cape Dorset Print Collections). It also seems to have been a recurring pattern of the multi-figure sculptures created by Canadian Inuit in the 1950s-60 s (cf. Graburn, 1972: 168).

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wolverines, Mary-Rousselière, 1969: 57–58; two jaegers, Mary-Rousselière, 1969: 34, etc.).19 Such postures might have been interpreted in relation to stories from the oral tradition or to behavioral traits in some cases (the two musk oxen being represented in a typical defensive position faced with the hunters, as suggested by one elder). The main comments collected from Inuit practitioners (as expressed in the available ethnographic data of this last century) revolve, however, around the similar action of two protagonists—mostly terrestrial mammals—going away in opposite directions. The depiction of such a mirror (opposite) action also concerns human pairs in several cases, but here often as a final move (outwards reflectional slide) preceded by the representation of a mutual action from protagonists perceived in face-to-face positions. This symmetrical inward action refers mostly to conflictual relationships, by portraying a fight before the separation (or disappearance) of the characters: cf. “the two pulling each other’s hair” and “the two enemies” (Jenness, 1924: 51–52; Mary-Rousselière, 1969: 91–94; Paterson, 1949: 43, also interpreted as the two who fight for a piece of frozen meat in the Iglulik area, and possibly in a former reference to the mythological pair of women causing thunder and lightning, said to quarrel about a dried seal skin); “two men” arguing then fighting and finally being eaten or carried off by a spirit (Jenness, 1924: 63, the ending appearing as a punishment for breaking the rule that forbids quarreling about animals), etc. The reflectional or bilateral symmetry that characterizes most of the Inuit string figures of human pairs also refers, in some cases, to a parallel action connoting a partnership (“two men sealing”, Jenness, 1924: 91; Mary-Rousselière, 1969: 86; “two men at play swinging on tight rope”, Comer, 1908; “two men carrying water buckets”, Jenness, 1924: 66, etc.). Some string figures depicting two men involved in a competing partnership (or with fictive kin ties) include for their part radial or rotational symmetry, either precisely (e.g. two-dimensional figure “two men having a tug-of-war in a dance house”, Jenness, 1924: 21) or imperfectly (e.g. 3-D figures “the meeting of two brothers-in-law”, ib.: 49 and “the two illuriik”, Mary-Rousselière, 1969: 90–91). It is worth noting that some complementary pairs (aippariik or piqatigiik) of two different beings are also depicted (“the shaman and the brown bear” [his helping spirit], Jenness, 1924: 17; “the young man and his dog”, ib.: 29; “man and woman”, ib.: 96–97, etc.). In their great majority though, the pairs represented in Inuit string figures involve bilateral symmetry, which is generally expressed by Canadian Inuit elders as iglugiik ajjigiik(uti), lit. “with two [patterns] of a pair which are alike” or “two (of a pair) similar to each other on each side”.20 The stem aki- that conveys a spatial notion of opposite position (“on the other side”) —and more largely notions of equivalence or opposition—is also involved in some of the Inuit elders’ descriptions of the string figures’ dual patterns, especially when the two (animal) beings are 19

See for instance “two brown bears” https://stringfigures.huma-num.fr/items/show/77 and “two brown bears with a line caught between their legs” https://stringfigures.huma-num.fr/items/sho w/249. 20 The notion of “axis of symmetry” is similarly rendered by the words illugiik ajjigiik akunninnguanga (lit. “the in-between space/middle of two of a pair/on each side which are alike”) in the Nunavut Arctic College’s School Mathematics Glossary (Allen, 2015: 4).

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Fig. 20 Musk ox and string game, Drawing by Jessie Oonark (1906–1985), Qamanittuaq/Baker Lake, Canadian Museum of History

portrayed with their backs turned to one another (akiliriik, “two which are opposite one another”; akiliriittuuk, “they form a pair as two, each on his own side”) (Fig. 20). Regarding the prevalence of the notion of a pair expressed in the meanings ascribed to many Inuit string figures, it may also be considered in relation to cosmological or mythological features associated with the emergence of string figure-making. Among the Nautaagmiut Inuit (Inupiat), string figures were indeed said to originate from Aagjuuk, two stars—and previously two human ancestors—forming a constellation (which is depicted in a dual string figure of the central and eastern Canadian Inuit, Comer, 1908), while the inner “spirit of the string figures” (Tuutannguarjuk, Tuutannguaq or Tuutarjuk, known in various Inuit groups) was more particularly related to the figure depicting a pair of labrets (cf. Jenness, 1924: 182–183; MacDonald, 1998: 49–50; Petit, 2009: 256; Rasmussen, 1931: 248) (Fig. 21). Single, dual and multiple animal figures: some relations in consideration of procedural aspects As suggested previously, the Inuit string figures’ repertoires include various representations of a same type of animal, which is portrayed either as single or in a pair, and in a few cases, as multiple. Some patterns referring to a single animal or to a pair “of the same kind” are also identified as appearing with variations in different string figures. While the patterns interpreted as portraying the same animal (within the same regional corpus) result for some part from very distinct methods of execution, they appear in other cases as the result of related procedures, whose variations are worth analyzing from mathematical perspectives (focusing on induction or algorithms in particular). Among the animal figures that share common subsets of operations on the string loop and appear related in their symbolic interpretation (through a variation between a generic form and one with a peculiarity), there

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Fig. 21 The figure known as tuutannguaq (in reference to the spirit of the string figures) in Iglulik, Nunavut, Canada. By Herve Paniaq (photo © C. Petit)

are single-pattern ones—such as “the caribou” and “the caribou in the willows” or “on the hill” (Jenness, 1924: 34–35; Mary-Rousselière, 1969: 7–8), “the hare” and “the hare with a collar” (Mary-Rousselière, 1969: 17–18), “the wolf” and “the wolf with its tail pushed by the wind” (Mary-Rousselière, 1969: 27–28)21 —and some including a dual pattern (“the two brown bears” and “the two brown bears linked through the rectum”, “the two lemmings linked through the rectum” and “the two lemmings coming out from their burrows”, Mary-Rousselière, 1969: 13–14, 29–30, etc.). Depending on the figures, the variation occurs either in the first movements of the procedures, or after a common preliminary set of operations. Regarding now the procedural relations between single-pattern and dual-pattern figures referring to a same animal type, they vary significantly according to the degree of similarity of the patterns in single and dual forms. Identical or closely similar patterns—from the single to the dual form—are induced in some cases by performing with both hands, on each “side” of the string loop and in a reflective way, a same subset of operations as the one implemented by a single hand (iglupiarmut, illuinnarmut) when making the single-pattern figure (e.g. “a mammoth” and “two mammoths”, Jenness, 1924: 43–44; Mary-Rousselière, 1969: 42–44—for the first method observed among the Arviligjuarmiut, another method with more significant variations being also known there; “a fawn” and “two fawns”, Jenness, 1924: 70–71; Mary-Rousselière, 1969: 10–11). In other cases, when the single and the dual animal figures are only characterized by quite similar patterns, the methods for making them can be unrelated (e.g. “a fox” and “two foxes”, Mary-Rousselière, 1969: 18–20). 21

Methodological variations regarding the creation of a single animal pattern appear particularly developed in the case of the wolverine figure, as known by the Arviligjuarmiut Inuit of Canada in the 1960s: it was/is indeed considered as being formed through ten making processes, that are related for some of them and lead to the presentation of the animal in different spatial positions (left/right, standing/upside down, etc.), cf. Mary-Rousselière, 1969: 52–55.

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When considering the processes of making the string figures that portray several animals of the same species, some links may appear between the principles of transformation (of a named figure) and iteration (of an operation or a set of operations) underlying several of these string games, and the idea of generative or reproductive process suggested by the names of the final figures. The iteration of a same pattern through the repetition of a same set of operations on the string loop (i.e. on a string configuration topologically similar despite some continuous change) is for instance a principle at the basis of the transformation of “the killer whale” (aarluattiaq) figure into one referring to “several killer whales” (aarluattiat) or “a family of killer whales”, according to its interpretation in the Iglulik area (Fig. 22, cf. “children cycle” in Jenness, 1924: 130–133; Mary-Rousselière, 1969: 96–97). The creation of further patterns (not necessarily similar but related to the original ones) through the iteration of a same set of operations is also involved in the transformation of dual animal figures into figures representing the parent animals with their offspring, such as “the two brown bears and their cubs” (Jenness, 1924: 16–17; Mary-Rousselière, 1969: 16) or “the two mammoths and their two/four cubs” (issued from the “two mammoths” figure, the even number of cubs being only limited by the string length, Mary-Rousselière, 1969: 44–45). In these cases where a reproductive process is symbolically represented, the two similar animals initially depicted may be seen as male and female, and thus forming a complementary pair or aippariik (Fig. 23). However, the generative process underpinned by the transformation of a single or dual animal figure including iterative sets of operations is not always interpreted in the field of reproduction or kinship: it can also refer to added attributes (cf. “the fox” transformed into “the fox with ears”, Jenness, 1924: 120–121, “the two brown bears” followed by “the two brown bears with ears”, Mary-Rousselière, 1969: 16–17, etc.) or to the emergence of animals from another species (from the “two brown bears” to the “two mountain sheep”, then to “the two caribou with antlers”, Jenness, 1924: 15–16), the succession of different animal forms being in this last case a possible reference to

Fig. 22 Aarluattiat—(a family of) killer whales, by Herve Paniaq, Iglulik, Nunavut, Canada (Photo © C. Petit)

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Fig. 23 Kiligvaraariit—two mammoths and their cubs (Mary-Rousselière, 1969: 44)

the prevalence of body changes in the early times described in Inuit mythology. As a whole, a brief examination of some structural relations involved in the processes of making Inuit string figures representing animals suggests the relevance of further investigating the mathematical ideas expressed in string figure-making in association with their cultural interpretations and enactment (Fig. 24). (In conclusion): String figure-making as a tool in mathematics education in Inuit communities?

Fig. 24 Nuluaq, the “net”, a contemporary figure presented by a young Kuujjuamiuq, Nunavik. (Photo © Céline Petit)

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According to various testimonies from elders who experienced the pre-colonial Inuit nomadic way of life, string games used to carry a mnemonic significance relating to the ancestors’ experiences and knowledge, and were as such performed as a common means for teaching children about mythology, ecological knowledge and sociocultural values. Besides the connection suggested between “cultural memory” transmission and procedural knowledge involved in string figure-making (among Inuit as in other indigenous societies, cf. Métraux, 1940),22 Canadian Inuit elders’ testimonies reveal that this practice was regarded as promoting coordination and visuoconstructive skills—as well as subtle abilities in interpreting relations in space at various scales—among the young. Since the transition to “modern life” induced by the colonization processes (involving Christianization, formal schooling and the creation of permanent “communities”) in the American Arctic, the knowledge associated with the traditional Inuit repertoires of string figures appear, however, to have decreased significantly, and seems now to fade away (nungupalliaqtuq, “it disappears gradually” as one Iglulingmiut elder put it). In Canada and in Alaska at least, some string games are nevertheless performed by contemporary Inuit children (mostly girls), usually as a peer-group practice involving an element of competition. While including the interactive “cat’s-cradle” practice commonly known in many cultural areas (but which was not familiar to Canadian Inuit elders in their youth), these games also refer to some traditional string figures that appear to have been learned partly in a school context. In the past decades, string figure-making has actually been promoted sporadically within “culture classes” at the elementary and secondary school levels, as one of the games pertaining to Inuit heritage: Canadian Inuit elders have been invited to school now and then for that purpose. As suggested in this paper, various data drawn from both classic ethnography and interviews with contemporary Inuit elders (from Nunavut and Nunavik, Canada) indicate that further crossing the analysis of the procedural “threads” of the Inuit string games with that of the sociocultural patterns and symbolic interpretations of their enactment (since the nineteenth century at least) could prove to be fruitful from an ethnomathematical perspective, specifically with the view to promoting culturally rooted mathematics education. A few works have already suggested the relevance of studying or teaching mathematical concepts by building on string figure-making as a cultural practice known by the “indigenous” students or by their ancestors. Whereas some chiefly recommended exploring geometrical notions and relations discernible in the created string figures (among the Brazilian Tapirapé, Paula & Paula, 1986; or in South African societies, Mosimege, 1998 for instance), others focused on procedural features while suggesting that string games—as played by the Navajo and more widely North American Indians—involve(d) “complex mental operations” (Emerson, 1988: 1) 22

About the Rapa Nui (Easter Islanders), the anthropologist Alfred Métraux thus mentioned: “The natives state that string figures were used to memorize popular chants and to recall tales. I was told that children were taught to make them and were obliged to memorize the accompanying chants before they were instructed in the sacred lore of the tribe and the art of carving tablets” (1940: 354). From a close perspective, it is worth noting that among the Brazilian Tapirapé (whose string figures are for some part similar to those of the Inuit), the word referring to the string used for making figures literally means “instrument for learning” (Paula & Paula, 2002: 187).

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or “cognitive processes associated with mathematical thought” (Moore, 1988: 25) that should also be further highlighted for educational purposes.23 The underlying idea of enhancing indigenous students’ cultural pride and confidence in their ability in mathematics—by bringing out the mathematical knowledge expressed in one of their ancestors’ practices such as string figure-making—appear actually supported in some way by some Inuit elders’ comments on the remarkable skills of their forebears who “could create and memorize many ways of making figures out of a mere sinew thread string (ivalutuinnarmik)”. Since the late 1990s, the cognitive issues associated with the promotion of “mathematical enculturation” and/or the elaboration of culturally responsive math curriculum and pedagogy have also been stressed in reference to Inuit and Yup’ik students (cf. Bacon & Rajotte, 2016; Berger et al., 2009; Lipka, 1994; Lipka et al., 2005; Pallascio et al., 1998, 2002; Poirier, 2007, etc.). Several traditional Inuit or Yup’ik games have been considered in that perspective for their potential use in teaching math skills—relating mainly to measuring, counting/adding, evaluating/probability (see for instance Ilutsik, 2002; Savard et al., 2014), and in some cases to topology and motion geometry (Lipka et al., 2001). Inuit practices of string figure-making would also be worth including in a culturally based educational program in mathematics and the database created as part of the ETKnoS project aims to provide resources for that purpose. Reclaiming the complex cultural knowledge associated with the making of string figures could fit into a wider project of decolonizing science education (cf. Aikenhead & Elliott, 2010). Acknowledgements I am deeply grateful to each of the elders (inullariit, inummariit) who accepted to share some of their knowledge during interviews held in Iglulik, Panniqtuuq, Iqaluit, Inukjuak or Kuujjuaq. I sincerely thank them for their patience and benevolence when teaching me how to make traditional string figures. I am greatly indebted to them for the present work. I also wish to thank the reviewers and editors of this paper for insightful comments and suggestions. Fieldwork—carried out in Inuit communities as part of the ETKnoS project—was made possible by the financial support of the French National Research Agency (ANR).

References Aikenhead, G., & Elliott, D. (2010). An emerging decolonizing science education in Canada. Canadian Journal of Science, Mathematics and Technology Education, 10(4), 321–338. Allen, D. (dir.). (2015). School mathematics glossary (English-Inuktitut). Nunavut Arctic College. Annahatak, B. (2014). Silatuniq: Respectful state of being in the world. Etudes/Inuit/Studies, 38(1– 2), 23–31. Ascher, M. (1991). Ethnomathematics: A multicultural view of mathematical ideas. Brooks and Cole Publishing Company. Bacon, L., & Rajotte, T. (2016). Réflexions portant sur le développement d’un programme de mathématiques pour les communautés d’Ivujivik et Puvirnituq au Nunavik. Etudes/Inuit/Studies, 40(2), 71–91. 23

On teaching math skills through the study of procedural features involved in (indigenous) North American string figures, also see Murphy (1999).

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Ball, W. W. R. (1911). Mathematical recreations and essays (5th ed.). Macmillan and Co. Berger, P., Angutiqjuaq, R., Attagootak, R., & Audlakiak, J. (2009). Finding Inuit math: The mathematical knowledge embedded in traditional practices in Nunavut. In Communication at the 17th International Inuit Studies Conference. Université du Québec en Abitibi-Témiscamingue. Birket-Smith, K. (1953). The Chugach Eskimo. Nationalmuseets Skrifter, Etnografisk Række 6. Blodgett, J. (1986). North Baffin drawings. Collected by Terry Ryan on North Baffin Island in 1964. Art Gallery of Ontario. Boas, F. (1888a). The Central Eskimo. Sixth Annual Report of the Smithsonian Institution, 1884– 1885. Boas, F. (1888b). The game of cat’s cradle. International Archiv für Ethnographie, 1, 229–230. Boas, F. (1901). The Eskimo of Baffin Land and Hudson Bay from notes collected by Capt. George Comer, Capt. James S. Mutch and Rev. J. Peck. Bulletin of the American Museum of Natural History, XV (1). Bordin, G. (2003). Lexique analytique de l’anatomie humaine. Analytical lexicon of human anatomy. Timiup ilangitta atingit Nunavimmilu Nunavummilu. Peeters. Comer, G. (1908). Eskimo string figures collection. Museum of the University of Pennsylvania. https://www.penn.museum/collections/accessionlot.php?irn=41 Compton, R. (1919). String figures from New Caledonia and the Loyalty Islands. Journal of the Royal Anthropological Institute, 49, 204–236. Elbaum, K., & Sherman, M. (2013). Some string figures and tricks from Rankin Inlet. Bulletin of the International String Figure Association, 20, 188–224. Emerson, L. (1988). NA’AT’LO—Dine string games as an educational process. Unpublished manuscript. (Cited by Stevenson, G., & Murphy, J. (2002). Teaching mathematics skills with string figures. In W. Secada, J. Hankes, & G. Fast (Eds.), Changing the faces of mathematics: Perspectives on indigenous peoples of North America (pp. 187–200). National Council of Teachers of Mathematics). Fortescue, M., Jacobson, S., & Kaplan, L. (1994). Comparative Eskimo dictionary (with Aleut Cognates). University of Alaska Fairbanks. Friesen, M., & Arnold, C. (2008). The timing of the Thule migration: New dates from the Western Canadian Arctic. American Antiquity, 73(3), 527–538. Gordon, G. B. (1906). Notes on the Western Eskimos. Transactions of the Department of Archaeology, Free Museum of Science and Art, University of Pennsylvania. Gordon, G. B. (1917). In the Alaskan Wilderness. John Winston Co. Graburn, N. (1972). A preliminary analysis of symbolism in Eskimo art and culture. In Proceedings of the 40th Congress of Americanists 1972 (Vol. 2, pp. 165–170). Tilgher. Haddon, K. (1930). Artists in string. (String figures—Their regional distribution and social significance). Methuen & Co. Ltd. Hall, C. F. (1865). Arctic researches and life among the Esquimaux. Narrative of an expedition in search of Sir John Franklin in the years 1860, 1861 and 1862. Harper and Brothers Publishers. Hansen, K. (1975). String figures from West Greenland. Folk (Dansk Etnografisk Tidsskrift), 16–17, 213–226. Harper, K. (1979). Suffixes of the Eskimo dialects of Cumberland Peninsula and North Baffin Land. National Museum of Man, Mercury Series n° 54. Hawkes, E. (1916). The Labrador Eskimo. Anthropological Series No. 14, Memoir 91, Government Printing Bureau. Hoffman, W. (1897). The graphic art of the Eskimos. Smithsonian Institution, Government Printing Office. Holtved, E. (1951). The Polar Eskimos, Language and Folklore. Meddelelser om Grønland, 152. Ilutsik, E. (2002). Kakaanaq. A Yup’ik Eskimo Game. In W. Secada, J. Hankes, & G. Fast (Eds.), Changing the faces of mathematics: Perspectives on indigenous peoples of North America (pp. 145–149). National Council of Teachers of Mathematics. Jacobson, S. (dir.). (2012). Yup’ik Eskimo dictionary (Vol. 1, 2nd ed.). Alaska Native Language Center, University of Alaska Fairbanks.

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Jayne, C., (Furness). (1906). String figures. A study of cat’s-cradle in many lands. Charles Scribner’s Sons. Jenness, D. (1922). The life of the Copper Eskimos. Report of the Canadian Arctic expedition 1913–1918 (Vol. XII, Part A). F.A. Acland. Jenness, D. (1924). Eskimo string figures. Report of the Canadian Arctic expedition 1913–1918 (Vol. XIII, Part B). F.A. Acland. Johnston, T., Nicolai, M., & Nagozruk, K. (1979). Illeagosiik! Eskimo string-figure games. Music Educators Journal, 65(7), 54–61. Kroeber, A. (1899). The Eskimo of Smith Sound. Bulletin of the American Museum of National History, 12. Lantis, M. (1946). The social culture of the Nunivak Eskimo. Transactions of the American Philosophical Society, 35(3). Lipka, J. (1994). Culturally negotiated schooling: Toward a Yup’ik mathematics. Journal of American Indian Education, 33(3), 14–30. Lipka, J., Wildfeuer, S., Wahlberg, N., George, M., & Ezran, D. (2001). Elastic geometry and storyknifing. A Yup’ik Eskimo example. Teaching Children Mathematics, 7(6), 337–343. Lipka, J., Hogan, M., Webster, J., Yanez, E., Adams, B., Clark, S., & Lacy, D. (2005). Math in a cultural context. Two case studies of a successful culturally based math project. Anthropology and Education Quarterly, 36(4), 367–385. Low, A. (1906). Cruise of the Neptune. Report on the Dominion Government expedition to the Hudson Bay and the Arctic Islands on board the D.G.S. Neptune 1903–1904. Government Printing Bureau. MacDonald, J. (1998). The Arctic Sky. Inuit astronomy, star lore and legend. Royal Ontario Museum & Nunavut Research Institute. MacLean, E. A. (2012). Iñupiatun Uqaluit Taniktun Sivunniu˙gutiŋit. North Slope Iñupiaq to English Dictionary. University of Alaska Press. Mary-Rousselière, G. (1965). Eskimo string figures. Eskimo, 70, 9–15. Mary-Rousselière, G. (1969). Les Jeux de Ficelle des Arviligjuarmiut. Musées Nationaux du Canada, Bulletin 233. Mauss, M. ([1947] 2002). Manuel d’ethnographie. Payot. Métraux, A. (1940). Ethnology of Easter Island. Bernice Bishop Museum Bulletin, 160. Moore, C. G. (1988). The implication of string figures for American Indian mathematics education. Journal of American Indian Education, 16–26. Mosimege, M. D. (1998). Culture, games and mathematics education: An exploration based on string games. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 279–286). University of Stellenbosch. Murdoch, J. (1892). Ethnological results of the Point Barrow expedition. 9th Annual Report of the Bureau of American Ethnology 1887–1888. Government Printing Office. Murphy, J. (1999). Using string figures to teach math skills. Part 3: The North American net system. Bulletin of the International String Figure Association, 6, 160–211. Ootoova, E. (2000). Uqausiit tukingit. Inuktitut Dictionary (Tununiq Dialect). Baffin Divisional Education Council. Pallascio, R., Allaire, R., Lafortune, L., Mongeau, P., & Laquerre, J. (1998). Vers une activité mathématique inuit. Etudes/Inuit/Studies, 22(2), 117–135. Pallascio, R., Allaire, R., Lafortune, L., Mongeau, P., & Laquerre, J. (2002). The learning of geometry by the Inuit. A problem of mathematical acculturation. In W. Secada, J. Hankes, & G. Fast (Eds.), Changing the faces of mathematics: Perspectives on Indigenous Peoples of North America (pp. 57–68). National Council of Teachers of Mathematics. Paterson, T. (1949). Eskimo string figures and their origin. Acta Arctica, 3, 1–98. Paula, L. G. de, & Paula, E. D. de (1986). Xema’eawa, Jogos de Barbante Entre os Índios Tapirapé. Unicamp.

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Paula, L. G. de, & Paula, E. D. de (2002). Xema’eãwa. String games among the Tapirapé Indians of Brazil. Bulletin of the International String Figure Association, 9, 186–202. Petit, C. (2009). Jouer pour être heureux. Pratiques ludiques et expressions du jeu chez les Inuit de la région d’Iglulik (Arctique oriental canadien) du XIXe siècle à nos jours. thèse de doctorat en ethnologie, Université Paris Ouest Nanterre (France) & Université Laval (Canada). www.theses. ulaval.ca/2011/26829/26829.pdf Petit, C. (& Claassen, S.). (2014). Aperçu d’un jeu de ficelle inuit (et de ses variantes) à partir d’un fragment de film de Jean Gabus (1938/39). Ethnographiques.org, 29. http://ethnographiques.org/ 2014/Petit,Claassen Petit, C. (2015). Arctic string astronomy. Bulletin of the International String Figure Association, 22, 216–219. Petit, C. (2021). String Figures. Inuit Corpus (Arctic Collection), Database of the ETKnoS Project. https://stringfigures.huma-num.fr Poirier, L. (2007). Teaching mathematics and the Inuit community. Canadian Journal of Science, Mathematics and Technology Education, 7(1), 53–67. Qumaq, T. (1988). Sivulitta piusituqangit. Inuksiutiit Katimajiit, Université Laval. Qumaq, T. (1991). Inuit uqausillaringit (Ulirnaisigutiit). Inuksiutiit Katimajiit/Institut culturel Avataq. Rasmussen, K. (1929). Intellectual culture of the Iglulik Eskimos. Report of the Fifth Thule Expedition 1921–1924 (Vol. 7, no. 1). Gyldendalske Boghandel. Rasmussen, K. (1931). The Netsilik Eskimos: Social life and spiritual culture. Report of the Fifth Thule Expedition 1921–1924 (Vol. 8, no. 1–2). Gyldendalske Boghandel. Rasmussen, K. (1932). Intellectual culture of the Copper Eskimos: Social life and spiritual culture. Report of the Fifth Thule Expedition 1921–1924 (Vol. 9, no. 1–2). Gyldendalske Boghandel. Rivers, W. H., & Haddon, A. C. (1902). Torres Straits: String figures. A method of recording string figures and tricks. Man, 2, 146–153. Rivers, W. H., & Haddon, A. C. (1929). A method of recording string figures and tricks (Abridged version). In Notes and queries on anthropology (pp. 323 and f). B.A.A.S (British Association for the Advancement of Science). Saladin D’Anglure, B. (2003). String games of the Kangirsujuaq Inuit. Bulletin of the International String Figure Association, 10, 78–199. Saladin D’Anglure, B. (2006). Être et renaître inuit (homme, femme ou chamane). Gallimard. Savard, A., Manuel, D., & Wan Jung Lin, T. (2014). Incorporating culture in the curriculum: The concept of probability in Nunavik Inuit Culture. E in Education, 19(3), 152–171. Schneider, L. (1979). Dictionnaire des infixes de l’esquimau de l’Ungava. Études de langue esquimaude, Ministère Richesses naturelles/Direction générale du Nouveau-Québec. Schneider, L. (1985). Ulirnaisigutiit: An Inuktitut-English Dictionary of Northern Quebec, Labrador, and Eastern Arctic Dialects (with an English-Inuktitut Index), Québec, Presses de l’Université Laval. Smith, H. (1894). Notes on Eskimo traditions. Journal of American Folk-Lore, 7(26), 209–216. Spalding, A. (1998). Inuktitut. A multi-dialectal dictionary (with an Aivilingmiutaq base). Nunavut Arctic College. Spencer, R. (1959). The North Alaskan Eskimo. A study in ecology and society. Bureau of American Ethnology, Smithsonian Institution, Bulletin 171, Gov. Printing Office. Stefansson, V. (1914). The Stefánsson-Anderson Arctic expedition of the American Museum. Preliminary ethnological report. Anthropological Papers of the American Museum of Natural History (Vol. 14, Part 1). Tylor, E. (1879). Remarks on the geographical distribution of games. Journal of the Royal Anthropological Institute, 9, 23–30. Vandendriessche, E. (2015). String figures as mathematics? An anthropological approach to string figure-making in oral tradition societies. Springer. Victor, P.-E. (1940). Jeux d’enfants et jeux d’adultes chez les Eskimo d’Angmassalik: Les jeux de ficelle. Meddelelser Om Gronland, 125(7), 1–212.

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von Hornbostel, E. (& Sherman, M.). (1989). Kwakiutl string figures: A preface. String Figure Association Bulletin, 16, 25–48. [From a manuscript written by E. von Hornbostel in the early 1930s]. Wirt, W., D’Antoni, J., Darsie, R., Read, R., Pollock, M., Claassen, S., & Sherman, M. (2009). Jenness’s Eskimo string figures. Bulletin of the International String Figure Association, 16, 91–297.

Modeling of Implied Strategies of Solo Expert Players Luc Tiennot

Abstract My research relies on the study of a class of sowing games largely unknown in Europe and, more broadly, in almost all the Northern Hemisphere: the solo. This chapter focuses on an ethnography carried out in my fieldwork: the part of the southwest of the Indian Ocean where these games are very common. Through interviews with inhabitants of the concerned countries (Tanzania and especially the island of Zanzibar, the Comoros archipelago, Madagascar and Mozambique) and video sequences that I filmed in Madagascar and Mayotte, I deliver several analyses and models. First, all the solos derive from the bao of Zanzibar, -the most complex type-, following the trade routes of the dhows in the region, to give the mraha of Mayotte, the various katro of Madagascar, and other solo. Secondly, the seemingly illegal and incomprehensible moves of expert players can be understood by their desire to optimize seeds freight and loading / unloading operations. Thirdly, the extremely fast moves of these same expert players can be modeled by the knowledge of a set of simple graphs, the vertices of which are the successive connected configurations of seeds. Finally, as a tutor for teachers, I give some ideas of a didactic exploitation of the sowing games when teaching mathematics from nursery school to high school. Keywords Ethnomathematics · Sowing games · Solo · Southwestern indian ocean · Modeling of expert players strategy

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-030-97482-4_2. The videos can be accessed individually by clicking the DOI link in the accompanying figure caption or by scanning this link with the SN More Media App. L. Tiennot (B) La Réunion University, Saint-Denis Cedex, France e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_2

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1 Introduction 1.1 My Ethnographic Fieldwork and the Object of My Research High-quality works on mathematical activities in Africa are well known: historically (Gerdes & Djebbar, 2007), or more ethnomathematically (Zaslavsky, 1973). In Africa, where there are plenty of games (Béart, 1955), various sowing games (see Sect. 1.3) are played (Culin, 1894; Popova, 1976). On the East coast of the mainland and some of the nearby islands, in the Indian Ocean, people play particular sowing games, the solo (see Sect. 2), and in particular the more complex of these abstract combinatorial games: the Zanzibar bao, (see Sect. 3). In my study area, there is a tradition of ethnomathematics research related to teacher training: Paulus Gerdes in Mozambica (for example Gerdes, 1994, 2009, 2014), Dominique Tournès in Reunion Island (Tournès, 2012), some research has begun in Mayotte Island. Since the appearance of mathematical thought, two types of intellectual constructions have complemented rather than opposed each other. On the one hand, a hypothetico-deductive approach, the origin of which is attributed to the ancient Greeks, whose manifestations are necessarily written. On the other hand, an algorithmic approach, much earlier from the historical point of view, since preceding the first writings (Smith, 1958; Keller, 2006), and more extensively geographically but do not necessarily present a written record. Mathematics teachers, and all those who remember their mathematical education, know well that it is the first form that has long been favored, in particular, in France, from the middle of the Secondary School. Things are changing with the appearance

Fig. 1 Two solo players in Mahajanga (Madagascar) (Tiennot, 2014c)

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of computer-related algorithms in school curricula in recent years, but also with the older awareness of the role of game in early learning, a role now also asserted in these same curricula. The practice of games has a clearly anthropological aspect and, at least for games involving the respect of precise rules, therefore algorithms, interests ethnomathematics. For years, there has been a lot of confusion about sowing games. In almost all African countries there are several vernacular languages in addition to the language of communication or official language, usually of its former colonizers, and there might be as many different names for a given variety of sowing games. There might be also a large number of varieties of sowing games in the same country, with the same name, as in Madagascar or Mozambique. This has led to erroneous recensions of the names or rules attached to a sowing game. Moreover, awélé seems to be the only sowing game really known in Europe. Its rule can be found in the Marc Chemillier’s book (Chemillier, 2007) or that by Jean Retschitzki (1990). Italy is doing a little better, thanks to the popularisation work around bao game, led by Nino Vessella (2011) and Luca Cerrato (2020). My research is that of a mathematics teacher trainer, from primary to high school, teaching at the University of Reunion Island, in a region of the world, the SouthWest of the Indian Ocean, where there is a class of abstract combinatorial games (see Sect. 1.2), and more precisely of sowing games, the solo, which is still little explored from an ethnomathematic point of view. It is characterized by an ethnography, which can be qualified as a safeguard, for the lesser known of these games, by modeling certain aspects of these variants, see Sects. 5, 6.1, 6.2, 6.3, and by an ongoing reflection on their didactic interest in the numerical field.1

1.2 What Are Abstract Combinatorial Games ? Among the immense variety of games invented by mankind, with a varied mathematical interest, there is a category of abstract combinatorial games, as referred to by specialists, that we simply call “combinatorial games” in the following, characterized as follows: 1. 2. 3. 4. 5.

chance does not occur; neither the physical strength nor the skill of each player plays a role; all the elements that can influence the course of the game are known; exactly two players are opposing each other; these players take turns.

Point 1 excludes games of chance. Point 2 excludes sports games. Point 3 means that there is a rule of the game setting the starting position, the allowed shots and 1

The necessarily constrained format of this chapter does not allow to give an extensive bibliography of sowing games. The reader may refer to (Tiennot, 2017).

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Fig. 2 Diagrams showing the start position of three combinatorial games

those that are prohibited throughout the game and how to determine the winner. Point 4 excludes games involving alliances. Point 5 excludes games where players play simultaneously, without waiting for the other to have finished his turn. In China and Japan, Go game is a well-known combinatorial game (see Fig. 2a). In Russia, Europe and North America, it is the same for Chess (see Fig. 2b). Similarly in Africa, and along some coasts and on some islands of the Indian Ocean, sowing games, as described later in this chapter, are practiced a lot. They are of two types: wari and solo.2 The most complex of them, is a solo called Bao la Kiswahili3 (see Fig. 2c). All these games, in areas where they are practiced, are described in the form of diagrams, in the specialized journals, but also in the general information press. These diagrams present the state of the game at a moment clearly identified, usually below the diagram. In solo diagrams, the numbers indicate how many seeds are inside the corresponding cup. The required equipment for these games are characterized by a fixed board on which are placed or moved the moving elements. The board is a grid of 19 × 19 lines, thus determining 192 = 361 intersections, called Go-ban at the Go game. The Chess board, called chessboard, is a square table of 82 = 64 boxes. Sowing games require a rectangular table of n l × n c cavities called cups in sowing games. n l is the number of lines and n c the number of columns, they vary according to the games. For example, the bao la Kiswahili is played on a 4 × 8 board (see Fig. 2c). The Go moving pieces are called “stones”. They are two colors, one by camp, and can only be placed, one by one, on the Go-ban, or removed in numbers that can vary. Therefore their mobility is limited. In a game without handicap, between equal force players, the Go-ban is initially empty. The Chess moving pieces are simply called “pieces”, they are also two colors, one by camp, they are all set on the chessboard at

2 3

These two names are invariable and do not take the plural mark. That means “The Swahili Bao” in the Swahili language.

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the beginning of the game and, at every move, only one (or two if there is a capture or a castling), is moved or removed. The moving parts of sowing games are called “seeds”, they are of one color, and one move can displace a large number of them, even all.

1.3 Why the “Sowing Game” Appellation? The games we study are combinatorial games, a term we have just reminded earlier. They are called “sowing games”4 because they are characterized by sowing a cup. Which means taking all the seeds of an authorized cup, called eligible, and sowing the content, with a seed each time, in the following neighboring cups, without jumping any and stopping when the last seed is sown in the cup, called “end of sowing cup”. Eventually, a capture follows, depending on the type of sowing game. Let us consider Fig. 1 and represent the line of cups that is in front of the player on the right by a very incomplete diagram of this position of the game. Figure 3 illustrates what a sowing is. All seeds in cup of rank 5 (a) can be removed and sown, either to the right (b) or to the left (c).

1.4 Presentation of the Next Sections of this Chapter In the second section, we will specify what a seed game is. We will present the two types of sowing games: wari and solo, present their geographical distribution, and give the types of diagrams and marking of each type. In the third section, we will present the most complex of the solo: the Zanzibar bao. We will give the main characteristics of that of beginners and that of the experts. In the fourth section, we will quickly describe four other solo. A very simplified solo, one of the solo practiced on the Hauts-Plateaux of Madagascar; a solo of historical interest, the first description in the literature of a solo, which we found; a Mahajanga solo very similar to the Zanzibar bao la Kiswahili; finally a close solo, the only sowing game practiced traditionally on a European territory: the mraha of Mayotte. In the fifth section, we will propose to analyze the seemingly illegal moves of expert players. For this, we will introduce the notion of regulated movement, the one that would result from the strict observation of the rules, as do the beginner players, and optimized movement, the one who is actually observed, when it comes to expert players. In the sixth section, we will introduce some mathematical tools to explain the very fast sequence of moves made by expert players. 4

In the literature, we sometimes find the name “mancala” , but it is also the name of certain sowing games and we will not use this term here.

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Fig. 3 Two sowing of cup number 5

In a last section, we will conclude this chapter by summarizing its content and evoking the educational developments that our work allows.

2 Sowing Games 2.1 Some Artifacts Artifacts can be true works of art. Examples are visible on a dedicated page of the Elliott Avedon Virtual Museum of Games (University of Waterloo, Ontario, Canada).5 But they can also be mere holes dug into the soil or charcoal lines plotted on it, which is more the case with the less complex games (Fig. 4). Examples of artifacts of both types are present in large numbers in the collections of the British Museum. Photographs, in particular archaeological objects, are also present on the Museum’s online website.6

2.2 Common Characteristics We call a sowing game, any game having the following characteristics, detailed in (Tiennot, 2017): 1. A sowing game is an abstract combinatorial game, the two players are called North and South. South starts. 2. The fixed element is a board with n l × n c cups, arranged in n l rows and n c columns. 3. The mobile elements, called seeds, are or should be considered as indistinguishable.

5 6

http://healthy.uwaterloo.ca/museum/VirtualExhibits/countcap/pages/index.html. https://www.britishmuseum.org/collection, with keywords “mancala” and “board”.

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Fig. 4 Three solo artifacts in Madagascar

4. The board may also have one or two cavities or boxes called granaries, either to store the seeds that have not yet entered the game (and then there may be one or two storage granarie(s)) or the seeds won by each player (and then there are two harvest granaries, one for each player). 5. Seeds can only be deposited (sown) in the cups or the granary(ies). 6. Each player has a determined set of cups on the board, this set is called the player’s camp, the camps define a partition of the set of cups on the board. 7. A player must start a sowing by a cup of his camp. This cup must also be eligible for a sowing, this eligibility condition may vary according to the sowing games. 8. A move consists of taking the contents of a cup or a single seed from a granary and sowing this seed(s), one by one, in the following cups, until exhaustion of the seeds in hand, this movement is called sowing, the sowing is either simple or iterative and it can be followed by a harvest whose conditions depend on the sowing game. 9. The aim of the game is either to block the opponent (in the sowing games of the bao family) or to capture as many seeds as possible (in all other sowing games).

2.3 Two Classes of Sowing Games The relevant distinction, the one that will give quite different rules, between wari and solo, is based on the number of cycles. In the two-lines wari, North and South sow according to the same cycle, in a determined direction or not, while in the four lines solo, North and South sow in two different cycles, in a fixed sense or not (Fig. 5).

2.4 Geographical Distribution Linguists call the Bantu Line the border of more or less constant latitude, crossing all of Africa and passing roughly south of the Sahara. It is called this way because

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Fig. 5 Wari and solo numbers of cycle(s)

all the languages of the Bantu family lie on the south of this line, although there are a few non-Bantu languages in this region. In 1977, the mathematician André Deledicq and the ethnologist Assia Popova published Wari et solo, le jeu de calculs africain (Deledicq & Popova, 1977) and proposed two maps giving a geographical distribution of the sowing games in two classes: the solo, two-cycle sowing games, all of which are south of the Bantu Line, and the wari, single-cycle sowing games, all of which are north of the Bantu Line. We reproduce these maps Figs. 6 and 7. In Fig. 6, the authors offer a map of the African continent and Madagascar, omitting the other neighbouring islands, and in particular Zanzibar and the Comorian archipelago, where solo are very present. The boundary between the wari and solo zones more or less covers the Bantu Line. They also propose a diffusion of the solo that would have stopped at the western half of Madagascar, whose small satellite islands are also omitted. This distribution seems to suggest a diffusion from the Bantu zone of the continent towards Madagascar. It could be supported by the fact that there are no practices, testimonies or artifacts of sowing games in the Mascarene Islands, east Madagascar. In reality, solo are present everywhere in Madagascar, as my ethnographic research has brought to light: I have observed them, after Étienne de Flacourt (1661, 2007) who did it in the South-East of Madagascar, in various places in the eastern part and even on Sainte-Marie Island east of Toamasina, the large Malagasy port on the east coast. But, despite much research in Reunion Island and Mauritius, I confirm that there is no historical presence of sowing games in these two islands. In Fig. 7, the authors propose a map of the distribution in accordance with the literature. The solo have remained “stuck” by the natural border constituted by the Sahara and the oceans, more than by the linguistic border of the Bantu Line, as shown by the diffusion of wari on at least three continents, which is not linked to any linguistic proximity. But it is true that the rules of the wari are globally simpler, can be mastered in a few minutes from exemplifying situations without the need for a complex discourse commonly understood.

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Fig. 6 The distribution of sowing games between wari and solo on the African continent and in Madagascar, according to (Deledicq & Popova, 1977)

The wari is spread along the well-known land routes of the extra-African migrations of mankind to Asia, which later became the land silk routes, to Asia Minor, with probably an intrusion into Europe. That was at least in the Balkans where ancient artifacts that may resemble wari boards have been found. It has spread also to China, where sowing games exist. This diffusion followed also the coasts of South Asia, where later the counters of the maritime silk routes were found, as far as in Southeast Asia, then in Indonesia, at a time when even rudimentary boats made it possible to reach Borneo and the neighboring islands because the level of the ocean was much lower than today but still prohibited passage to Australia and New Zealand and, a fortiori, to the Pacific islands. This migration had been lasting for several centuries and it can be thought that the arrival in Borneo and the neighboring islands of the wari was posterior to the

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Fig. 7 The diffusion of wari outside Africa, according to (Deledicq & Popova, 1977)

beginning of the second millennium. Indeed, Asian populations from Borneo would go and settle on the Malagasy Highlands with their language, which would become (with Bantu loans from the coastal populations) the current Malagasy, their customs, the rice culture. However, if wari do exist in Indonesia, no trace of them can be found today, neither in the rare artifacts of archaeological origin, nor in the rules of the Malagasy solo. The wari also migrated to the plantations of the eastern coasts because of the deportation of Black people from the west coast of Africa during Slavery. The ethnomathematics maintaining a sympathetic relationship with ethnomusicology, which is not very present in sowing games, let us quote here Georges Paczinski, jazz drummer and percussionist, musicologist and historian of this music, in Une histoire de la batterie de jazz, tome 1: des origines aux années swing (Paczinski, 1997). In the slave trade practiced from Africa to America until the beginning of the 17th century, there were 200,000 people, then one million of Black people was deported during that century and 4 million during the 18th century. This trade took place from Benin, Dahomey and the Gold Coast. For four centuries, the destination was mainly Brazil for a third of the slaves, the West Indies for half, the North of South America and the United States for the rest.7 The slave trade continued until 1870. During four centuries, 11 million people were forced to leave Africa and at least 9 million of them landed in America. During endless journeys, many died from disease and exhaustion (Paczinski, 1997: 5)—my translation.

7

So, for a sixth.

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Fig. 8 The spread of the solo from Zanzibar to the southwestern Indian Ocean (Tiennot, 2017)

As Paczinski reminds us, the Africans deported to the Americas are all from the wari zone and it is not surprising that wari are found in the coastal areas of western South America and the Caribbean indicated by Deledicq and Popova but no solo. The sowing games practiced in my study area (see Fig. 8) are all of the solo type, and the most complex of the sowing games. Since at least the second half of the 17th century, when Flacourt observed (de Flacourt, 1661), these games were disseminated to the ports and trading posts most of them concerned by the ArabSwahili dhows8 trade (Gueunier et al., 1992). That is to say, from the commercial 8

Dhow is a traditional sailing vessel with one or more masts with settee or sometimes lateen sails, used in the Red Sea and Indian Ocean region.

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center of Zanzibar to mainland Tanzania, on the mainland coasts of the Mozambique Channel, in the Comorian archipelago, on the Malagasy coast. Thus, they seem to have been practiced by relatively educated population, often traders, who were not a priori those reduced to slavery and sold to European slave traders. The most complex forms present in Madagascar are still observed in the large historical port of Mahajanga on the Mozambique Channel and in coastal areas, including in the south-east of Madagascar, linked to the existence of small trading posts practicing the spice trade. Thus, on the island of Mayotte, there is a frequently practiced solo, the mraha, with rules identical to the bao practiced in the mainland of Tanzania, dispensing with some of the subtleties of that of Zanzibar. It is the only European territory where a sowing game is practiced. In Madagascar, when you go up to the “Hautes Terres”, in merina and betsileo country, the practiced solo, called katra or katro, have increasingly simple rules. The same phenomenon exists in Mozambique, when one moves away from the coast, according to the surveys I conducted with Mozambican pedagogical advisers on language courses at my university.

2.5 Diagrams Several rating systems are used in the literature. The most interesting are the ones that highlight the central symmetry of the game, they are the only ones I used at that time, inspired by the official awélé notation for the wari and the official bao notation for solo and it would be interesting to standardize these notations for any sowing game. For two-line wari, the cups will be marked starting from the left of each player with the letters of the Roman alphabet. In upper-case letters for South and lower-case for North. For all wari with 2 rows and 6 columns, the marking is identical to that of awélé. For all 2-line wari with a number of columns different from 6, the marking given in Fig. 9 can be easily adapted by adding or removing the appropriate letters of the Roman alphabet. For the solo, we will mark the lines starting from the inner line, the one closest to the border between the two camps, by the letters of the Roman alphabet. Always in upper-case letters for South and lower-case for North. For the four-line solo, from North to South, the lines are thus b, a, A, B, and for the very rare six-line solo, from the Betsileo cultural area, mentioned by Alexander de Voogt (de Voogt, 1999), what I described in (Tiennot, 2017), the lines are c, b, a, A, B, C. The columns of the solo are marked by their number, taking as the first column the one to the left of each player. For all 4-rows and 8 columns solo, the marking is the same as for the bao. For all 4-rows solo with a number of columns other than 8, the marking given in Fig. 10 can be easily adapted by adding or removing the necessary numbers.

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Fig. 9 Coordinate system of the wari cups - Example of awélé

Fig. 10 Coordinate system of the solo cups - example of bao

3 The Zanzibar Bao Two games of different complexity are played in Tanzania, the bao la kujifunza, “the bao of beginners”, and the bao la Kiswahili, “the bao of the Swahilis” which is the real bao. Both are played with 64 seeds and on the same board 4 × 8.

3.1 The Bao la Kujifunza Here are a few essential points of this simplified form, which is an entry to learn the most complex form, but whose characteristics are also found in the simpler solo, especially in the Malagasy Highlands, where bao la Kiswahili is unknown. The starting position is given in Fig. 11, it is uniform and all the seeds are in the cups from the beginning of the game.

Fig. 11 Diagram of the beginning of the game of bao la kujifunza

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Some cups have a specific name because they play a particular role, explained as follows: ◦ A5 is the Southern nyumba. The nyumba is represented by a square in the official diagrams, I use here. It is often shaped as a square on the artifacts, particularly in Zanzibar. ◦ A1 and A8 are the Southern kichwa. ◦ A2 and A7 are the Southern kimbi. These names are the same for the North camp. Just change the upper-case of the coordinates in lower-case.

3.1.1

Sowing

The player who starts is usually drawn, or invited to start by his or her opponent, following the usual method and usage of sowing games. The player must choose a direction of sowing and a cup eligible for sowing, (i.e. it must be in his camp and contain at least two seeds): (1) resulting in a harvest, if a harvest opportunity exists. Otherwise, he will have (2) to sow without harvesting, from the inner row, if possible, otherwise (3) from the outer row. In bao, the condition for iteration of a sowing without harvest is that the finishing cup is not empty before the sowing ends inside it. In an iterated sowing without harvest, the direction of sowing is fixed by the first sowing, chosen by each player at the beginning of their move, and cannot change until the end of the move. If, at the end of a sowing without harvest, (1) the iteration condition is satisfied, the player carries out a new sowing whose starting cup is the cup following the cup of arrival of the sowing which has just finished. Otherwise, (2) the movement of this player is finished and it is up to the other player to play. The end of the round is called kulala, a Swahili word meaning “to sleep”. The verb mandry, having the same meaning in their language, is used by Malagasy players. A player cannot play in such a way as to voluntarily empty his internal line, from the beginning of his move. If a player starts a sowing without harvest by sowing the only non-empty cup of the internal line, he must therefore sow it in such direction that he is not emptying this internal line. If a move starts with a sowing without harvest, then no harvest can take place in all this move, it is a kutakata or kutakasa (“purification” in Swahili). A movement with harvest is a mtaji (“capital” in Swahili). This word also designates a phase of bao la Kiswahili. A player has lost the game as soon as one of the following two conditions is met: 1. his inner line is empty (even momentarily, during a move). 2. he cannot make any moves. These few rules will help to understand the only filmed game of bao la kujifunza available on the Internet: (Tembo, 2007).

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Harvesting

In bao, as in the other solo, the player, before any movement, and sometimes during the movement of the opposing player, begins by examining which of his cups are likely to allow him a harvest. These are those, which we call eligible for a harvest, which check points 1, 4 and 5 below. The player then examines whether he can sow seeds checking points 2 and 3 from the seeded cup to the harvested cup. The harvested cup must : 1. 2. 3. 4. 5.

be in the internal line; be that of the end of sowing; be reached after sowing no more than 15 seeds; contain, before sowing, at least one seed; face in front of a non-empty cup of the opponent.

Concerning the item 3, notice that a sowing of exactly 16 seeds could not in any case allow a harvest because it would end in the sown cup, thus empty just before the arrival of the last seed. The harvesting consists then, for the harvesting player, to take all the seeds of the opposite cup and to sow them, always going towards the center of his internal line, and taking for starting cup of sowing the one of his kichwa: 1. which will allow to keep the sowing direction of this move if the harvest takes place in one of the four central cups and is of at least two seeds. 2. which would make it possible to keep the sowing direction of this move if the harvest took place in one of the four central cups and was of at least two seeds while only one seed was captured. 3. which is closest to the captured cup if it is a kichwa or a kimbi. This is therefore the only case where the direction of sowing can change during the same movement.

3.2 The Bao la Kiswahili We only give a few points allowing the reader to understand the documents quoted at the end of the section. Three fundamental differences with the bao la kujifunza lie in (1) the existence of two phases kunamua and mtaji, the seeds being gradually introduced into the first, (2) a different starting position and (3) the particular role played here by the nyumba, this cup is subject to specific rules, called privileges. As for it, the mtaji phase is almost identical to the bao la kujifunza. The starting position is given in Fig. 12. The complete rule of bao la Kiswahili, as I gave it in (Tiennot, 2017), with examples allowing to understand the complexity of this game, and the order in which the various rules must be applied, would occupy the whole of this chapter, the reader

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Fig. 12 Diagram of the start of bao la Kiswahili

may refer to Nino Vessella site (Klubo Internacia de Bao-Amantoj, 2022), which provides in particular: • an abbreviated rule in English, Italian, French downloadable and foldable to form a small booklet. • a page allowing to see lessons and many filmed game parts. Reading the first item above, after reading the section devoted to bao la kujifunza in this chapter, should make it possible to understand the video sequences of the last of these items, which are much more numerous than for bao la kujifunza.

3.3 Notation of the Games Finally, let us finish with the standardized notation of bao games (Vessella & Cerrato, 2011), that I now recommend to generalize to all solo. A game transcription starts with a title containing information about the game (players, place, date), when it is a real game and when these elements are known. Each line starts with the number of the double-move, i.e. the South and North moves, a colon, a space, the South move, a space, the North move and a semicolon ending the line. After the semicolon it is possible to add a comment. Movements are described by the line (A, B, a, b ) and the number (1-8) of the cup from which the movement starts. The direction of the move is given by > if it is to the right of the player (kushoto) and by < if it is to the left of the player (kulia). 1. 2. 3. 4.

< means to the left; > means to the right; + indicates that you are sowing the seeds of a nyumba (see Sect. 3.1); ∗ indicates a kutakata movement (see Sect. 3.1.1);

During the kunamua phase (see Sect. 3.2), one can omit the indication of the line. If the harvest cup is a kichwa or a kimbi, > or < can be omitted. The direction of movement is the one followed by the hand of each player, after sowing the first seed and beginning its sowing in the kunamua phase, or after harvesting in the mtaji phase. Furthermore, in a harvest in the kunamua phase, the direction,

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to the left () indicates which kichwa was set as the starting point. Examples of this are: A5 > + means that a seed is put into the fifth cup from the left in the inner line, the seeds are harvested from the cup opposite and reseeded from the right. A3 > ∗ means that the seeds in the third cup from the left in the inner row are sown to the right, without harvesting. Finally, one can use the symbols of the chess game to indicate the value of a move: 1. 2. 3. 4. 5.

!: good move; !!: very good move; ?: weak move; ??: very weak and decisive move; !?: move whose value is uncertain.

4 The Other solo It is not possible to detail here the numerous kinds of solo described in the literature or during my fieldworks, so we will choose to illustrate this simplification of the rules with a few video sequences, to be watched in connection with Fig. 8, and the chronological order of which will be edited to move from the simplest solo to the most complex.

4.1 A 4 × 4 Katro from the “Hauts Plateaux” of Madagascar I wrote, in 2014, at the request of the CIJM (Comité International des Jeux Mathématiques), for its review Maths Express, a short article entitled “Les jeux de semailles à Madagascar” (Tiennot, 2014b) presenting a 4 × 6 katro from the Malagasy Highlands, and provided the rules which are identical to the 4 × 4 katro in the following video footage. This article still uses a different historical notation for solo parts from what I often use and now recommend, for rating all games, but it is explained. The sequence titled Séquence vidéo 2013-08-10 MG Antananarivo (Tiennot, 2013a) was shot on August 10, 2013, at the Alliance Française de Tananarive, where Mr. Manitra Harimisia Razafindrabe, a Go game champion of Madagascar, organized the XVIIIth parlor games fair at the Alliance Française of Tananarive, offering registered children a large number of combinatorial games and a large katro tsotra tournament; which means “simple katro”, as it is indeed the simplest solo version I have observed. It makes a proper use of the recommended notation and the first two moves are briefly commented on.9 9

The complete game, still with the old notations can be viewed in the article entitled “Á la recherche de jeux de semailles de type solo à Madagascar”, Special Issue “Ethnology and mathematics”, Online journal ethnographiques.org (Tiennot, 2014).

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There remains the geometry of the board which is indeed of the solo type: the young ladies start a sowing by taking a seed in their camp, harvest in the other, practice the iterated sowing, the condition of stopping, and the end of game.

4.2 The 4 × 8 fanga by F LACOURT The earliest and most accurate description that has come down to us is dated from 1661 and is about a solo in Madagascar, and not from 1694, as it is believed by Murray in his book A History of Board Games Other Than Chess (Murray, 1952), p. 206, citing the book De ludis orientalibus of Thomas Hyde (Hyde, 1694), p. 232. In his book Histoire de la Grande Isle Madagascar (de Flacourt, 1661), Étienne de Flacourt, “head of the french colony” of Fort-Dauphin, devoted his chapter XXIV to the “Jeux, Passetemps, Chansons, et Danses” of the Malagasy people he met near the first French establishment in Fort-Dauphin. Over two and a half centuries before the first ethnographers, in an unusual way for people of his time and of his social class, he displayed remarkable precision in the presentation of his observations of the popular activities announced in the title of the chapter10 (see Fig. 13). Flacourt called this game fifangha (it would be spelled fifanga today if the word existed in modern Malagasy, but the current spelling, in Roman characters, of Malagasy did not exist at the time of Flacourt). He describes a game which is clearly more complex than the katro tsotra, like the bao: the 4 × 8 geometry of its board, its number of seeds and the essentials of the rules, referring specifically to the particular role played by certain cups. Intrigued by the fact that this much older attestation of a solo occurred in that very region where solo is considered non-existent, as described in (Deledicq & Popova, 1977) (cf. Fig. 6); I thus carried out a field investigation in the south-eastern region of Madagascar to try to find Flacourt’s “fifangha”. In a region rich in the production of cloves, it was only necessary to seek, in the villages not too far from the coast, the existence of fady (“prohibitions”) linked to the consumption of pork or alcohol. This could be the sign of a coastal trade, even ancient, with the dhows of the Islamized Arab-Swahili traders who would have introduced the bao to local farmers. Apart from the rules that also exist in the katro tsotra, we see Fig. 13, that the board is of the 4 × 8 type, like that of the bao, the game played does not (or no longer) take up all the complexity of Flacourt’s description, which is that of the bao la kujifunza, but a linguistic argument must also be taken into account. Given the precision of Flacourt’s observations, one can think that he does not make an error in reporting the phonetic spelling fifangha and that this form has simply disappeared from the modern lexicon, because it has become, by apheresis, the one that I have noted. Indeed, I found traces of this game near Ankazofatatra, in a house in the forest

10

For an analysis of this text, see (Tiennot, 2017).

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Fig. 13 The solo board of “fifangua” by Flacourt, 1661 ((de Flacourt, 1661), pp. 108–109)

north of Toalagnaro ( Flacourt’s “Fort-Dauphin”) which is even located slightly in the north of Manakara, the players still call it “fanga”. This is the only occurrence that I have found of this word to refer to a Malagasy solo and it therefore seems to be related to this region. When you leave the paved road from the National 7 to Manakara, and cross vast areas of deforestation, there is a small crossroad with a refreshment bar, as it is common on Malagasy roads. Some players were playing a solo near this refreshment bar, but they called it katro and did not know the word fifanga nor the word fanga. You can take a “taxi-brousse’’ there that will take you to Ankazofatatra, about ten kilometers away. This village is on the edge of the forest, of about an hour’s walk away, and where I met two young ladies whose games I filmed: Séquence vidéo 2013-08-17 MG Ankazofatatra fanga (Tiennot, 2013b). Their family lives isolated in this area (Fig. 14).

4.3 The 4 × 8 mraha of Mahajanga This game, Séquence vidéo 2014-08-02 MG Mahajanga mraha (Tiennot, 2014c), filmed on August 2, 2014, near the mosque, in the Abattoir district in Mahajanga, was performed in a very joyful atmosphere, which is often the case during solo games in Madagascar. Most of the spectators are Malagasy. They are very expressive, and

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Fig. 14 Scan with “SN More Media App”, or use this link: (Tiennot, 2013b) ( https://doi.org/10.1007/000-7re)

Fig. 15 Scan with “SN More Media App”, or use this link: (Tiennot, 2014c) ( https://doi.org/10.1007/000-7rd)

do not hesitate to comment and joke with one another. However, one may find a few Comorians, who remain silent most of the time, among the audience (Fig. 15).11 The statement of the game is: 1 : A6 < ∗ a5 < ; 2 : A3 > a5 < ; 3 : A5 > a3 > ; 4 : A8 < ∗ a2 < ; 5 : A6 < a8 < ; 6 : A2 < a5 < ; 7 : A7 > a1 > ; 8 : A4 < a4 > ; 9 : A6 > a8 < ; 10 : A6 < a6 > ; 11 : A1 > a3 > ; 12 : A3 < a3 < ; 13 : A4 > a6 < ; 14 : A6 > a2 < ; 15 : A6 > a5 < ; 16 : A6 < a2 < ; 17 : A5 < a3 > ; 18 : A4 > a7 > ; 19 : A5 < a6 > ; 20 : A6 > a7 > ; 21 : A7 > a3 < ; 22 : A7 > a4 > ; 23 : A7 < ∗ b7 > ; 24 : B8 < ∗ ; 25 : B1 > ∗ ; 26 : B2 > ∗ b8 > ; 27 : B3 > ∗ a1 > ; 28 : B7 > b1 A7(3) A4(1) Again, regulated movement and optimized movement differ slightly (Fig. 18). 0:30 North begins his movement while South has not finished his. He plays 2 : A3 > a5

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Fig. 18 Situation at the end of double move 2

Fig. 19 Situation at the end of double move 3

The full notation of the movement is: 3 : A5(8 + 1) × a4(1) > A8(1) – The observed movement is the regulated movement. 0:36 North plays 3 : A5 > a3 > The full notation of the regulated movement equivalent to the observed movement would be (Fig. 19): 3 : a3(1 + 1) × A6(1) > a8(1) 0:37 South plays 4 : A8 < ∗ 0:39 North plays 4 : A8 < ∗a2 < The full notation of the regulated movement equivalent to the observed movement would be: 4 : a2(1 + 1) × A7(1) < a1(1)

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Fig. 20 Situation at the end of double move 4

Here again, it is the translation loaded over a shorter distance that makes the difference (Fig. 20).

4.4 The 4 × 8 mraha of Mayotte Here, we will only present two moves of the game of mraha filmed in Mayotte, at Mtasangamouji, on May 29, 2015: these are Séquence vidéo 2015-05-29 YT Mtsangamouji mraha 1 (Tiennot, 2015a) and Séquence vidéo 2015-05-29 YT Mtsangamouji mraha 2 (Tiennot, 2015b). The reader will see that rules are the same as of the bao la Kiswahili without the refinements specific to Zanzibar, as for the mraha practiced in Mahajanga. Contrary to what happens in Madagascar, there is no assembly making wise, deliberately ironic, or even earthy comments. I was the only spectator and unfortunately, only environment sound is provided by car traffic by the road, where the game is taking place, as often in Mayotte. Once more, both players appear to have a good level (Figs. 21, 22). We are currently trying with my fellow mathematics trainers, or teachers, in Mayotte to develop the learning of this game with teachers training for the interest of mental arithmetic, but also to involve more schoolchildren’s parents or grand-parents to enhance their knowledge and show that it is interesting for the school. Indeed, this is one of the benefits of ethnomathematics.

Fig. 21 Scan with “SN More Media App”, or use this link: (Tiennot, 2015a) ( https://doi.org/10.1007/000-7rc)

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Fig. 22 Scan with “SN More Media App”, or use this link: (Tiennot, 2015b) ( https://doi.org/10.1007/000-7rf)

5 Regulated Movements and Optimized Movements We saw, in the previous section, examples of optimized movements, which seem, at a given moment, to contradict the complex rules of these games but which leave a final position identical to that which would be observed if the player had scrupulously respected the rules. We called this last movement a regulated movement. However, expert players seem to make it a point of honor to perform optimized movements, which seriously complicated the readings that I made of these rules because when I believed to be able to induce a rule of repeated observations of repeated situations, a new observation, in a new situation that seemed similar, but played out by expert players, provided a counterexample to this rule. This couldn’t be a mistake, since at least in Madagascar, spectators are often present during a game and do not hesitate to comment, as we have seen, but also express their disapproval of an illegal move, even if it means being themselves disowned by another part of the audience, as I have regularly observed. However, spectators here weren’t critical of his seemingly illegal moves. Furthermore, the level of the players, obviously high, made the rules of movement or harvest unlikely to be ignored. For the attentive observer, the existence of these optimized movements, under the conditions that we have seen, has an obvious interest: it clearly shows that, despite the complete upheaval of the position of the seeds on the board, due to the existence of iterated sowing, typical of solo, and of complex rules, typical of these solo from the southwest of the Indian Ocean, which can even change the direction of rotation during the same move, an expert player knows, from the start in turn, what the final arrangement of the seeds at the end will be. We will denote Mi (resp. m i ) the regulated movement of South (resp. North) during turn i the regulated movement and Mi (resp. m i ) the observed movement of South (resp. North). Let us take again the differences between the adjusted movements and the optimized movements of the previous section. We will describe these differences with the following modeling: sn (...) (resp. rn (...)) means that the player sows (resp. collects) n seeds in the cup whose coordi-

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nates are in parentheses. In all the pairs of movements observed, the terms indicating a sowing or a harvest are the same two by two, occurring only in a different order at every movement. So this is not where the observed movement introduces a “gain”. tn (...) represents an horizontal translation of the n seeds in the hand of the vector player indicated in parentheses, the vector i (resp. j) is normalized by the distance between the centers of two consecutive cups, in the direction of a row (resp. a column) of the board and oriented from left to right of South (resp. North). We will call t0 an empty translation and tn , with n > 0, a translation with freight. For any strictly positive integer n, we will call cost of the translation tn of vector αi + β j, the strictly positive integer n(|α| + |β|), this cost is obviously zero for an empty translation, proportional to the number of seeds transported and increases with the norm 1 of the vector of the translation, or with the associated Manhattan distance, that is to say, very naively, with the length of the path. We will denote by C p (resp. c p ) the cost of the regulated movement of the move p of South (resp. North) and C p (resp. cp ) the cost of the movement observed of the move p of South (resp. North). Finally we will denote by  p (resp. δ p ) the gain, that is to say the difference between the cost of the regulated movement and the observed movement of South (resp. North) during the move p. Let’s go back to the different couples of movements from the previous section. Let’s start with move 1 from North:  m 1 = s1 (a5) t0 (− j) r1 (A4) t1 (4i + j) s1 (a1) m 1 = s1 (a1) t0 (−4i − j) r1 (A4) t1 ( j) s1 (a5) The two movements contain, in different positions, the following sowings and harvest s1 (a5), r1 (A4), s1 (a1). The empty translations being of zero cost, with our definition, m 1 contributes to the cost only with the translation with freight t1 (−4i − j), so c1 = 4 + 1 and m 1  so c1 = 1. contributes to the cost only with the translation with freight t1 (−i), Finally, δ1 = 4 − 1 = 3 > 0. Similarly, with move 2 from South, we have successively: ⎧ M2 = s1 (A3) t0 ( j) r2 (a6) t2 (5i − j) s1 (A8) t1 (−i) s1 (A7) r3 (A7) t3 (−i) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s1 (A6) t2 (−i) s1 (A5) t1 (−i) s1 (A4) ⎪ ⎪ ⎪ ⎪  ⎪      ⎪ ⎨ M2 = r2 (a6) t2 (− j) s1 (A3) t2 (4i) s1 (A7) t1 (i) s1 (A8) t0 (−i) r3 (A7) t3 (−i) s1 (A6) t2 (−i) s1 (A5) t1 (−i) s1 (A4) ⎪ ⎪ ⎪ ⎪ C2 = 2(5 + 1) + 1 + 3 × 1 + 2 × 1 + 1 × 1 = 19 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C2 = 2 × 1 + 2 × 4 + 1 + 3 × 1 + 2 × 1 + 1 × 1 = 17 ⎪ ⎪ ⎩ 2 = 19 − 17 = 2 > 0 Then move 2 from North:

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⎧ m 2 = s1 (a5) t0 (− j) r1 (A4) t1 (4i + j) s1 (a1) t0 (− j) r1 (A8) t1 ( j) s1 (a8) r3 (a8) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t3 (−i) s1 (a2) t2 (−i) s1 (a3) t1 (−i) s1 (a4) ⎪ ⎪ ⎪ ⎪  ⎪      ⎪ ⎨ m 2 = s1 (a5) t0 (− j) r1 (A4) t1 (4i) r1 (A8) t2 ( j) r1 (a1) t3 (−i) s1 (a2) t2 (−i) s1 (a3) t1 (−i) s1 (a4) ⎪ ⎪ ⎪ ⎪ c2 = 5 + 1 + 3 + 2 + 1 = 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c2 = 4 + 2 + 3 + 2 + 1 = 12 ⎪ ⎪ ⎩ δ2 = 0 Here, with δ2 = 0, there is no gain for the movement. But no loss either. And there are 5 loads and 6 unloads, for the regulated movement. But only, respectively, 3 and 4 for the optimized movement. Finally, with move 4 from North: ⎧ m 4 = s1 (a2) t0 (− j) r1 (A7) t1 (i + j) s1 (a1) ⎪ ⎪ ⎪ ⎪ ⎨ m 4 = s1 (a1) t0 (−i − j) r1 (A7) t1 ( j) s1 (a2) c4 = 1 + 1 = 2 ⎪  ⎪ ⎪ =1 c ⎪ ⎩ 4 δ4 = 2 − 1 = 1 > 0 This simple modeling makes it possible to understand why expert players attach so much importance to optimize moves, they produce a lower “cost” of the freight carried. Should we still see there a distant link with the origin of these complex solo: you have to take goods in given ports, sell them in others similarly determined, from Zanzibar to the Comoros and to the counters of Madagascar or the East coast of Africa by optimizing movement with goods? Even if navigation in the Indian Ocean and the Mozambique Channel has little to do with the checkerboard formed by the streets and avenues of Manhattan ... This “naive” distance has only allowed us to have a simple expression for distance, without harming the generality of minimizing the distance traveled with the goods. At least two questions are now arising. The first is how these players do not overload their short-term memory with the anticipation of the representation of the particular seed arrangements occurring at the end of each of the elementary sowing constituting, most often, an iterated sowing? The second is, how expert players manage to play so fast? To answer these questions, I will introduce some modeling tools.

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6 Modeling 6.1 Some Preliminary Definitions Therefore, I propose here various mathematical objects allowing to study the iterated sowing specific to the solo, independently of the organization of the cups: linéa, arrangement of seeds, configuration of seeds, composition of seeds and weight of an arrangement of seeds, finally all the compositions of weight n. These objects generalize, in order to adapt them to the solo, different notions already introduced by (Chemillier, 2007, chap. 3).

6.1.1

Linéa

The idea of introducing this object came to me while reading (Chemillier, 2007)14 : This rule assumes that the seed pattern is placed on an infinite row of boxes.15 Of course, this does not correspond to the reality of the awélé board. But it is convenient to make this kind of hypothesis highlight certain regularities in order to understand the causes and to propose a mathematical model (Chemillier, 2007: 88)—my translation.

Definition 1 We will call linéa the representation of an infinite board, the centers of the cups of which, represented by circles, occupy the natural abscissa points of the half-line of the positive reals, these natural ones serving to index the cups. Notice that this is simply a non-traditional graphic representation, but adapted to seed sets, of a sequence (u n )n∈N , with values in N. We will therefore designate a linéa, when the contents of the cups will be defined, with the shortened notation of a sequence by u, v, w, ... On a linéa, the indexing set is always N, it is no longer necessary to specify it.

6.1.2

Seeds Arrangement

Definition 2 We will call seeds arrangement, or quite simply arrangement, any non-zero sequence u whose terms are zero from a certain rank. 14

We chose this term for two reasons. The first is that the phrase that defines it (see def. 1) is very long, that “ligne”, which could have been appropriate, is already very polysemous whereas the Latin root was available in French. The second, linéa, has the original meaning of “flax” in Latin, a plant from which threads are made and which can be woven, however, as we will see, the movement of an arrangement of seeds on the linéa, when reposition the head of the group in the first cup, translating the whole arrangement to arrive at what we call a configuration of seeds, resembles that of the clapper tamping the threads on the loom. Finally, we accentuate the “e”, fix the gender to masculine and mark the plural with a final s, by analogy with the French word “alinéa” which has the same etymon. 15 Marc Chemillier calls box (case in Fresnch) what we call cup (cupule).

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Fig. 23 Representation, on a linéa, of an arrangement of seeds

Definition 3 We will call the head of the arrangement, (resp. tail of the arrangement) or simply head (resp. tail ), and we will denote α (resp. ω) the smallest (resp. greatest) index of a disposition corresponding to a non-zero term. Note that the existence of these two characteristic values is evident from the def. 2. Definition 4 We will call length of an arrangement of head α and tail ω, the natural  = ω − α + 1. In Fig. 23, the length of the arrangement is  = 4; its head is worth α = 2 and its tail is worth ω = 5. We have 4 = 5 − 2 + 1. We will usually represent a layout on a linéa, but can also write it as a list implicitly indexed by the coordinates on the linéa, stopping writing the list at the term whose index is the tail of the distribution. Thus the arrangement of Fig. 23 can be written as [0, 0, 3, 1, 0, 2].16 Definition 5 We will call a connected arrangement of seeds, or quite simply a connected arrangement, a disposition of seeds in which all the terms between the head and the tail are not zero.

6.1.3

Arrangement Translated from an Arrangement

Definition 6 For any integer h, for any arrangement u, of head α, and of tail ω we will call translated arrangement (or simply translated) of h cups of u, with h  −α, and we will denote th (u) the arrangement u  defined by: ∀i ∈ / [[α + h, ω + h]], u i = 0 and ∀i ∈ [[α + h, ω + h]], u i = u i−h 6.1.4

Seed Configuration

We can see a seed configuration in the form of two mathematical objects, which we will identify. On the one hand, as a special arrangement. Figure 24 shows the configuration associated to Fig. 23. Definition 7 We will call seed configuration, or simply configuration, associated with an arrangement u of seeds, head α and tail ω, the translated arrangement of −α cups of u. 16

This is the notation for the Python language (https://www.python.org/). Also, as in this language, indexing starts at 0.

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Fig. 24 Translated arrangement, on a linéa, of −2 cups of the arrangement of Fig. 23

On the other hand, as a finite sequence or a list of natural numbers. Definition 8 We will call seed configuration, or simply configuration, associated with an arrangement u of seeds, head α and tail ω, the list ordered according to the increasing indices of all the terms from u α to u ω . We will use the notation (u 0 , u 1 , ..., u −1 ), or even u 0 u 1 ... u −1 , as (Chemillier, 2007), to indicate this configuration ( − 1 = ω − α is indeed the last non-zero term of this sequence). The configuration associated with the arrangement of Fig. 23 is thus (3, 1, 0, 2) or 3 1 0 2.

6.1.5

Composition Associated with a Connected Configuration

Definition 9 We will call composition associated with a connected configuration of seeds, or quite simply composition, the list ordered in descending order of the terms of this configuration. Thus, the composition associated with the connected configuration (2, 3, 1, 2) is (3, 2, 2, 1).

6.1.6

Weight of an Arrangement

Definition 10 We will call weight of an arrangement or weight of a configuration of seeds, or quite simply weight, associated with this arrangement u of seeds, or with this configuration, of head α and tail ω, and we will denote  (u) the following sum:  (u) =

ω 

ui

i=α

The weight therefore simply refers to the total number of seeds in an arrangement or given configuration. For example, the weight of the connected configuration (2, 3, 1, 2) is 8. Definition 11 We will call Cn the set of connected configurations of weight n.

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Sowing a Configuration

Definition 12 We will call s (sowing initial) the application which to any element u of Cn associates the element u  of Cn , obtained by sowing the head cup of u. Sow the head cup of u meaning add the arrangement (1, ..., 1) to (u α+1 , ..., u ω , ...u α+u α ),   

 u α terms

u α terms

replace u α by 0, then translate the obtained configuration of −1 cup, so that the new head cup checks always α = 0 on the linéa. The function s is indeed an application whose starting set is Cn , since the head cup of any configuration, connected or not, can be seeded. The target set is indeed Cn since the result of sowing a connected configuration is obviously connected.

6.1.8

Iterated Sowing

It can also be interesting to sow a layout, in this case we will not perform the last translation of the previous definition and the head of the new layout will be translated by one cup. Exemple Let’s sow the arrangement [3, 2, 0, 1], it becomes the connected configuration (3, 1, 2) and continue, with successive single sowing of the head of the configuration. We successively obtain the configurations, always connected, (2, 3, 1) ; (4, 2) ; (3, 1, 1, 1) ; (2, 2, 2) ; (3, 3) ; (4, 1, 1) ; (2, 2, 1, 1) then, finally, (3, 2, 1) which is a walking group (cf. Sect. 6.1.9) and this configuration will therefore no longer change in the following sowing. By definition, a configuration is given by starting always with the non-zero content of its first cup, but to calculate these configurations, it may be easier to work on the groups of seeds, moving, over the sowing, on a linéa that is to say on arrangements. We give the steps in the Table 1.

6.1.9

Walking Groups, Periodicity

In the work cited above, Marc Chemillier indicates that the publication of one of the first reviews of the rules of awélé, by Captain Robert Sutherland Rattray (1881– 1938), an anthropologist of the British government, dating from 1927 in one book of which a chapter, entitled “wari”,17 was written by the Cambridge mathematician Geoffrey Thomas Bennett, who did not content himself with giving the rules of awélé, but introduced also the notion of slow movement, useful for the ends of the awélé game, and of a walking group. Definition 13 We call walking group any configuration [n, n − 1, . . . , 2, 1] of length n. 17

I do not use this game name because of the possible confusion with a class of sowing games with several games other than awélé.

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Table 1 Repeated sowing of the head of the arrangement (3, 2, 0, 1), with displacement on the linéa Semis Dispositions no 0 1 2 3 4 5 6 7 8 9

3

2 3

1 2

1 2 3 4

1 2 3

1 2

1 2 3

1 2 3 4

1 2

1 2 3

1 2

1 1

By sowing its head, such a configuration reproduces itself in a similar way and it is the only configuration of length n that has this property. Notice that a necessary condition for the existence of a walking group for a connected configuration is that the weight of this one is a triangular number and the condition becomes moreover sufficient if the associated configuration is the attached decreasing additive decomposition to these numbers.

6.2 An Enlightening Interview When I filmed the first complex solo games I encountered, around the mosque of the “Abattoir” quarter in Mahajanga, in 2014, I could not fully understand the rules of the games that I was observing the whole afternoon. I went back to this place several times once I knew the full rules of the bao and after understanding that deviations from these rules could be interpreted by means of the optimized movements. However, since 2014, what struck me the most was the speed at which the moves were executed. After my last afternoon of shooting, when my camcorder batteries were empty, I was able to strike up a conversation with some viewers, obviously experts, and express them my surprise to see how quickly expert players appeared to anticipate their moves. Then, one of them suggested me to follow him in his courtyard, after the game, in order to show me how an expert player like him proceeded. This gentleman, of Comorian origin, spoke his language better than Malagasy language, and he only spoke a few words of French, making a lot of effort. Rather than explaining verbally the strategy he was using, he proceeded into a “demonstration” with a few pebbles collected in his yard. I no longer had the possibility to film, it was the end of the day anyway and the dim light coming from his tin hut would

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Fig. 25 Three pages from one of my field notebooks: successive configurations

not have allowed it. As I had to return to Antananarivo the next morning, my “taxibrousse” seat being booked in advance, I could not return to see him again. I noted precisely in my notebook what this gentleman was showing me. Figure 25 contains three successive right-hand pages of this notebook. The first page on the left of this sequence (which is not showed below), before the page in sub-figure (a), where it was mentioned that this was happening on August 2, 2014, in Mahajanga, in the courtyard of his house, which was made of wood and sheet metal, occupied by Mr. Ibrahim, a 52-year-old man who has been playing this game for more than 25 years. The moving material is pebbles from his sandy yard, which is shown at the top of each page. They are placed on the ground, without discretization of this one. The circle surrounding the pebbles is drawn only once with the index finger, probably to indicate the weight of the patterns each time. The player then takes the pebbles out of this circle and arranges them in aligned piles, each containing the number of pebbles indicated in the diagram below. On each page, the top drawing indicates that the numbers are represented by a collection of these pebbles, as it is always the case on artifacts. Here, there is no attempt to delimit the cups, the limits between them being obvious since the small piles of stones showing the numbers are separated by distances much greater than those existing between the stones of the same pile. My interlocutor first took out two pebbles from the small heap he had collected, and told me the corresponding number, in French language. He then placed one of the two pebbles to the right of the other one, looked at me, then returned that pebble to its previous location, told me to look at it again, then placed that same pebble to the other one’s right again. He made a cyclical gesture with his right hand, which was not difficult to interpret to mean that, the same causes producing the same effects, this alternation of two configurations was to be repeated indefinitely. This gesture was reproduced during the presentation of the descendants of the pile of 4 pebbles

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at the first repetition indicating the existence of a cycle (arrow at right angles going up towards the occurrence previously observed). What followed showed me that when he looked at me furtively, after each movement of pebble(s) it was not only to make sure that I understood, but also to mark a discretization of the time decomposing the seedling of a connected arrangement of seeds in elementary sowing. The horizontal axis describes the spatial arrangement of the piles, while the vertical axis indicates the passage of time, and it is implicitly discretized as described previously. Not any sowing movement is made. The dotted arrow foot indicates which pebble is taken from the pile that will be added to the pile that arrow points to. After a first phase, when each of these movements was performed for about ten seconds, a second phase took place, when these successive configurations of weights 2, or 3, or 4 were repeated several times at high speed, nearly one second for the last four weight configurations 4, those which recur periodically, as shown by my rolling up arrow in solid line. This is a sign that the person who carried them out was not recalculating anything, but had saved his configurations, and their evolution, in his memory. In fact, he did not perform any gesture resembling a seedling, but simply went from one configuration to the next by minimizing the movement of stones.

6.3 A Modeling Proposal Everything therefore happens as if this expert player had recorded graphs for each of the first weights (2, 3 and 4) corresponding to each configuration sequence. The vertices of each graph are the elements of Cn , for n ∈ {2, 3, 4}. This graph is oriented, an arc exists, from a first to a second vertex, if and only if a sowing allows to pass from the first to the second configuration. A question arises: do the described graphs contain all the vertices of Cn , for n ∈ {2, 3, 4}?

6.3.1

A Little Counting

Number C(n) of connected configurations of weight n Let n ∈ N∗ be this weight and C(n) ∈ N∗ the number of connected configurations of weight n. C(n) is easy to determine. Imagine that we align the n seeds. Each item in the configuration contains at least one seed. We must therefore place from 0 to n − 1 separations between two consecutive seeds, these separations will define each configuration. For example, in Fig. 26, two separations have been placed and the configuration will start with (2,1, …). This configuration of weight n  4 will be length 3 if there are no other separation(s).

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Fig. 26 Number of connected configurations of weight n

For the separations, there are n − 1 possible locations between two consecutive seeds. In each of these locations, we have a separation or not, so there are two possible choices for each location and, in all, 2n−1 connected configurations of weight n. C(n) = 2n−1

Number C(n, p) of connected configurations of weight n and length p As we have just seen, to define a configuration, it suffices to give, among the n − 1 possible locations, those of the separations in the single file of n seeds. By assumption, this configuration is of weight n. Let us now look at the number of these separations, to obtain a configuration of length p, it is necessary to make exactly ( p − 1) separations in this queue. The number of possible choices, of p − 1 unordered objects without possible repetition, among a set of n − 1 is the number of subsets with ( p − 1) elements of a set with ( p − 1) elements, therefore

n−1 (n − 1) ! C(n, p) = = ( p − 1) !(n − p) ! p−1 Inventory of connected configurations For each weight n  2 (n = 1 corresponds to a trivial walking group), we will follow the same algorithm: 1. Calculate the number C(n) of connected configurations 2. For each length between 1 and n, calculate the number of configurations connected of weight n and length p 3. List the compositions associated with each of these connected configurations of weight n and length p 4. Deduce the list of connected configurations corresponding to each of these components 5. Give the table of the application s by ordering the compositions by increasing length then decreasing lexicographic order 6. Draw the graph corresponding to this table 7. Explain any deviations from the expert player’s knowledge-in-practice graph.

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Table 2 Table of the application s of C2 in itself

Fig. 27 Graph of the connected configurations of weights 2

6.3.2

Related Configurations of Weight 2

1. The number C(2) of connected configurations is C(2) = 22−1 = 2. 2. There are two possible lengths: 1 and 2. The number of connected configurations of length 1 is:

2−1 1 C(2, 1) = = =1 1−1 0 and the number of connected configurations of length 2 is: C(2, 2) =



2−1 1 = = 1. 1 2−1

3. The only composition of weight 2 and length 1 is obviously (2) and the only composition of weight 2 and length 2 is, just as obviously, (1, 1). 4. The list of connected configurations of weight 2 is therefore limited to (2) and (1, 1). There are indeed 2. 5. The application s table is given in Table 2. 6. This graph (see Fig. 27) is the only graph of connected configurations which is cyclic. 7. This graph is a perfect illustration of the knowledge-in-practice of the expert player (cf. Fig. 25a).

6.3.3

Related Configurations of Weight 3

1. The number C(3) of connected configurations is C(3) = 23−1 = 4. 2. There are three possible lengths: 1, 2 and 3. The number of connected configurations of length 1 is

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Table 3 Table of the application s of C3 in itself

C(3, 1) =

3−1 2 1! = 1, = = 0 !(1 − 0) ! 1−1 0

the number of connected configurations of length 2 is

3−1 C(3, 2) = =2 2−1 and the number of connected configurations of length 3 is

2 3−1 C(3, 3) = = = 1. 2 3−1 3. The only composition of weight 3 and length 1 is obviously (3), the only composition of weight 3 and length 2 is, just as obviously, (2, 1) and the only composition of weight 3 and length 3 is (1, 1, 1). 4. The list of connected configurations of weight 3 is therefore limited to (3), (2, 1), (1, 2) et (1, 1, 1). There are indeed 4. 5. The table of the application s is given in Table 3. Note the existence of a cross on the descending diagonal, which indicates the existence of the walking group (2, 1). Note also that the existence of two crosses in the same column, which proves that s is not injective for C2 . Neither this graph, nor any of the following ones can be cyclic, in fact the vertex (1, n − 1) cannot have an ascendant for n  2, which means that s is not surjective, not only for C2 , but also from C2 . 6. The graph corresponding to this table can be found in Fig. 28. 7. The presentation of the expert player does not say anything about the vertex (1, 2) (cf. Fig. 25b) probably because this one, without any ascendant, cannot be observed during the propagation of a connected configuration of weight 3.

6.3.4

Related Configurations of Weight 4

1. The number C(4) of connected configurations is C(4) = 24−1 = 8.

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Fig. 28 Graph of connected configurations of weights 3

2. There are four possible lengths: 1, 2, 3 and 4. The number of connected configurations of length 1 is

4−1 3 C(4, 1) = = = 1, 1−1 0 the number of connected configurations of length 2 is C(4, 2) =

4−1 3 = = 3, 2−1 2

the number of connected configurations of length 3 is

4−1 3 C(4, 3) = = = 3, 3−1 2 and the number of connected configurations of length 4 is C(4, 4) =

4−1 3 = = 1. 4−1 3

3. The only composition of weight 4 and length 1 is obviously (4), the two compositions of weight 4 and length 2 are (3, 1) and (2, 2), the only composition of weight 4 and length 3 is (2, 1, 1), the only composition of weight 4 and length 4 is (1, 1, 1, 1). 4. The list of connected configurations of weight 4 is therefore (4), (3, 1), (2, 2), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2), (1, 1, 1, 1). They are 1 + 3 + 3 + 1 = 8. 5. The table of the application s is given in Table 4. We notice, here also, the existence of two crosses in the same column, indicating that s is not injective for C4 . 6. The graph corresponding to this table can be found in Fig. 29. 7. Here too, the presentation of the expert player does not say anything about the orphan vertices (1, 3), (1, 2, 1), (1, 1, 2), (see Fig. 25c), confirming the hypothesis that this is because these ones, devoid of ascendant(s), cannot be observed during the propagation of a connected configuration of weight 4.

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Table 4 Table of the application s of C4 in itself

Fig. 29 Graph of connected configurations of weights 4

Everything happens as if, for example, when a player sows a seed in the head of the configuration (1, 1, 1) of the graph of connected configurations of weight 3 (see Fig. 28), he would recognize the vertex (2, 1, 1) of the graph of connected configurations of weight 4 and deduces the following configuration. Here, there is no disturbance by other seeds, since the length of the resulting configuration is strictly less than that of the initial configuration. This kind of knowledge-in-practice, relating to simple graphs, may help explain the speed at which expert players manipulate the seeds and have a representation, from the start, of the final arrangement of the seeds. It requires, in a way, the memorization of the first graphs. Moreover, this modeling provides research activities in high school classes concerned by the graphs.

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7 Conclusion My field surveys in the South-West of the Indian Ocean have shown that the extreme variety of solo-type sowing games could be explained by a gradual simplification of the rules from the most complex of them, the bao from Zanzibar, down to the simplest katro, those practiced by young children in the Malagasy Highlands. The deviation from the most complex rule being all the more reduced as the Arab-Swahili trade relations of previous centuries between Zanzibar and the ports or trading posts of the Mozambique Channel have been strong. This is particularly visible in Madagascar where, in the large historic dhows port of Majunga, a practically identical variant of the bao is played. On old seasonal counters, linked to the spice trade, which were the subject of less regular relations, there are simplified forms but practiced on a set of the same size, such as the one noted by Étienne de Flacourt from the second half of the seventeenth century. The iterated sowing, characteristic of the solo, are played with great speed by expert players, limited only by the agility of the hand sowing the seeds and the sowing result is anticipated, as evidenced by the existence of optimized movements. Thus, the complex rules of the solo games practiced in the southwest of the Indian Ocean are modified by the expert players, without modifying the final state of the move, in order to optimize the movement of the seeds, which seems to be more elegant to them. Then the knowledge-in-practice of what we have called graphs of related configurations allows them to play the moves very quickly, at least when they involve few seeds. These skills present an interest for numerical activities, from nursery school and primary education (subitizing,18 global perception, estimation, instantaneous additions) to grammar school (algorithmic, graphes). We know today that digital skills to build from nursery schools on and before the manipulated numbers become too large, passes through access to counting not limited to enumeration, but using the capability of the human brain to instantly recognize small quantities, even in disorder, as are the seeds in a cup, and by the additive calculation of small numbers (I will have 3 seeds in my cup of end of sowing and can capture the 4 seeds of the cup of the opponent’s cup, I will have 4 + 3 = 7 seeds to sow in an iterated sowing, as practiced in a solo). On the other hand, in secondary education sowing games provide algorithmic activities unplugged, or not, and activities on graphs properties, described in the previous section.

18

The learning psychologists and some didacticians in French-speaking countries specialized in the construction of numbers, in particular, in France, Rémi Brissiaud (Brissiaud, 1989) thus call “subitizing”, that is to say the capacity that humans and some animals recognize immediately, without enumeration, the cardinal of a very small collection of objects regardless of their spatial arrangement.

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This may be of interest to the countries concerned by the dissemination of these solo games (Tanzania, Comoros, Madagascar, Mozambique), but also other African countries where sowing games exist), as well as the two French overseas departments of the Southwest of The Indian Ocean, with an aim of mathematics activities open to the regional cultural context. In the coming years, I hope to be able to reinforce ties with my colleague from Mayotte, Jean-Jacques Salone, on the educational applications of the mraha, a local solo, as well as with other researchers interested in this, still in the area, but further away from my study area (Ismael, 2001). Besides, I also hope to be able to initiate school meetings between pupils from Zanzibar, where this game is widely practiced in schools, those from Madagascar, Reunion and Mayotte to strengthen the cultural and educational links between our islands.19

19

The expected results will be available at least on the IREM of Reunion Island website (IREM, 2022).

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References Books and Articles Brissiaud, R. (1989). Comment les enfants apprennent à calculer : le rôle du langage, des représentations figurées et du calcul dans la conceptualisation des nombres. Retz. ISBN: 978-2-7256-2232-3. Béart, C. (1955). Jeux et jouets de l’ouest africain (vol. 2, pp. 475–516). IFAN (Institut Français d’Afrique Noire, today Inst. Fondamental d’Afr. Noire). Chemillier, M. (2007). Les jeux de stratégie. In: Les mathématiques naturelles (Chap. 3, pp. 69–110). Odile Jacob. ISBN: 978-2738119025. Culin, S. (1894). Mancala: The national game of Africa. In: Report of the National Museum (Philadelphie) (pp. 597–611). Deledicq, A., & Popova, A. (1977). Wari et solo, le jeu de calculs africain. Cedic. ISBN: 2-71240603-6. de Flacourt, E. (2007). Histoire de la Grande Isle Madagascar, édition annotée, augmentée et présentée par Claude Allibert (1re éd. 1995, 712 p). INALCO, Karthala. ISBN: 978-2-86537578-3. de Flacourt, E. (1661). Histoire de la Grande Isle Madagascar (471 pp). Clouzier, Gervais. https:// gallica.bnf.fr/ark:/12148/bpt6k1047463/f1.item Gerdes, P., & Djebbar, A. (2007). Les Mathématiques dans l’histoire et les cultures africaines. Une bibliographie annotée. Gueunier, N. J., Hébert, J.-C., & Viré, F. (1992). Les Routes maritimes du canal de Mozambique d’après les routiers arabo-swahili. In: Taloha (Musée d’Art et d’Archéologie, Tananarive) (vol. 11 , pp. 77–120). Gerdes, P. (2009). L’ethnomathématique en Afrique. CEMEC, lulu.com. Gerdes, P. (2014). Ethnomathematics and education and Africa. Mathematical and educational explorations (2nd edn, 284 p). http://www.lulu.com/spotlight/pgerdes. Av. de Namaacha 188, Belo Horizonte, Boane (Mozambique): Instituto Superior de Tecnologias e Gestão. Gerdes, P. (1994). On mathematics in the history of sub-saharan Africa. Historia Mathematica, 21, 345–376. Hyde, T. (1694). De ludis orientalibus. (vol. 2). Oxford. https://books.google.com/books? id=mp3orHNIu7sC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage& q&f=false Ismael, A. (2001). An ethnomathematical study of Tchadji—about a Mancala type board game played in Mozambique and possibilities for its use in mathematics education. Ph.D. thesis. University of the Witwatersrand. Keller, O. (2006). La figure et le monde. Une archéologie de la géométrie. Peuples paysans sans écriture et premières civilisations. Vuibert. Murray, H. J. R. (1952). A history of board games other than chess (268 pp). Oxford University Press. ISBN: 0-19-827401-7. Paczinski, G. (1997). Une histoire de la batterie de jazz, tome 1: des origines aux années swing. Outre Mesure. Popova, A. (1976). Les mankala africains. Cahiers d’études africaines, 16(63–64), 433–458. Retschitzki, J. (1990). Stratégies des joueurs d’awélé. L’Harmattan. ISBN: 2-7384-0617-3. Smith, D. E. (1958). History of mathematics (vol. 1, p. 596). Dover Publications Inc. Tiennot, L. (2014a). À la recherche de jeux de semailles de type solo à Madagascar. In: ethnographiques.org 29. http://www.ethnographiques.org/2014/Tiennot Tiennot, L. (2014b). Les jeux de semailles à Madagascar. In: Moyon, M., Pestel, M-P., & Janvier, M. (Eds.), Maths Express au carrefour des cultures. CIJM. Tiennot, L. (2017). Ethnomathématique des jeux de semailles dans le sud-ouest de l’océan Indien. Ph.D. thesis. Université de la Réunion.

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Tournès, D. (2012). Ethnomathématique dans l’océan Indien : les lambroquins à la Réunion. CultureMath. Vessella, N., & Cerrato, L. (2011). Il libro quasi completo del gioco del bao (2nd ed). Onlus Changamanos. de Voogt, A. J. (1999). Distribution of mancala board games: a methodological inquiry. Board Games Studies, 2, 104–114. Zaslavsky, C. (1973). Africa counts. Prindle, Weber & Schmidt inc. ISBN: 1-55652-350-5.

Sites and Video Sequences Cerrato, L. (2020). Il Fogliaccio degli Astratti. http://www.tavolando.net/FdA.html. Visited on 11 May 2020. IREM. (2022). IREM de la Réunion. https://irem.univ-reunion.fr/. Klubo Internacia de Bao-Amantoj. https://www.kibao.org/. Tembo, M. (2007). Séquence vidéo Bao game. https://www.youtube.com/watch?v=i2YiWzaQmv4 Tiennot, L. (2013a). Séquence vidéo 2013-08-10 MG Antananarivo. https://youtu.be/ LZQNujTTeNc Tiennot, L. (2013b). Séquence vidéo 2013-08-17 MG Ankazofatatra fanga. https://youtu.be/ IFsCTLS1dpg Tiennot, L. (2014c). Séquence vidéo 2014-08-02 MG Mahajanga mraha. https://youtu.be/ PwXT5nryfQc Tiennot, L. (2015a). Séquence vidéo 2015-05-29 YT Mtsangamouji mraha 1. https://youtu.be/ Yq0gP-rphrE Tiennot, L. (2015b). Séquence vidéo 2015-05-29 YT Mtsangamouji mraha 2. https://youtu.be/D3O9qWP6io

Sand Drawing Versus String Figure-Making: Geometric and Algorithmic Practices in Northern Ambrym, Vanuatu Eric Vandendriessche

Abstract This chapter aims to compare—through an ethnomathematical approach—two activities carried out by the Northern Ambrym Islanders (Vanuatu, South Pacific), and locally termed using the same vernacular verb tu (lit. “to write”). These practices consist in making a figure, either with a loop of string (“string figuremaking”, using fingers and sometimes feet and mouth) or by drawing a continuous line in the sand with one finger (“sand drawing”). Initially, we examine the cultural and symbolic aspects of both practices, bringing to light that the making of string figures and sand drawings are both means of recording and expressing knowledge relating to particular mythological entities, rituals, or environmental elements, in Northern Ambrym society. Secondly, by focusing on concepts such as operation, procedure/algorithm, sub-procedure, symmetry, transformation and iteration, we demonstrate that both practices share geometric and algorithmic properties. The chapter ends by providing an overview of a pedagogical experiment, aiming at bringing both string figure-making and sand drawing practices into a local mathematics classroom. It thus contributes to the discussions that are currently occurring in Vanuatu, related to the development of culturally based curricula. Keywords Ethnomathematics · Mathematical practices · Sand drawing · String figure-making · Vanuatu

1 Introduction For a long time, indigenous societies have been excluded from the field of the history of mathematics (D’Ambrosio, 1985, 2001). Until a few decades ago, historians and Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-030-97482-4_3. The videos can be accessed individually by clicking the DOI link in the accompanying figure caption or by scanning this link with the SN More Media App. E. Vandendriessche (B) National Center for Scientific Research (CNRS), Paris Cité University, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_3

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philosophers of science have indeed left out of their field of study small-scale and/or indigenous societies—frequently endowed with an oral tradition. The prevalence of the evolutionist (Tylor, 1871) and “prelogical thought” (Lévy-Bruhl, 1910) theories, arguing that these peoples had a lesser ability to abstract and generalize than ours, appears to have durably impeded the recognition of genuine mathematical practices carried out in the various indigenous societies worldwide (Vandendriessche, upcoming-2022). At the turn of the second half of the twentieth century, however, a significant epistemological change occurred on this issue, fostered through the work of anthropologist Claude Lévi-Strauss in particular (1962). The latter epistemological rupture seems to have prompted the development of studies—in the 1970s—that are now generally considered to be seminal works in the establishment of ethnomathematics (Vandendriessche & Petit, 2017). The current development of this nascent interdisciplinary field of research contributes to further widen our perspective on mathematical knowledge and its history, while including in the picture all activities displaying mathematical characteristics carried out in social groups/societies within which they are (or were) usually not recognized as such. In the various indigenous societies of the planet, mathematics has not commonly appeared as an autonomous category of knowledge. However, as many ethnographies on “traditional” societies have demonstrated—throughout the twentieth century—particular forms of reasoning/logic occur(red) within various practices of theirs, such as the making of calendars or ornaments, the establishment of camps and dwellings, textile production, navigation, games, the implementation of kinship relations, etc. (Austern, 1939; Deacon & Wedgwood, 1934; Desrosiers, 2012; Galliot, 2015; Gladwin, 1986; Lévi-Strauss, 1947; MacKenzie, 1991; Pinxten et al., 1983; Rivers & Haddon, 1902 …). A major epistemological issue tackled by ethnomathematicians is therefore to determine to which extent some of these practices relate to mathematical activities, and how. In order to avoid being constrained by “Western connotations of the word mathematics”, Marcia Ascher (1935–2013), one of the founders of ethnomathematics in the 1990s, introduced the concept of “mathematical ideas.” Mathematical ideas are defined as any idea in which are involved “numbers, logic, and spatial configurations, and particularly the arrangement of these ideas into systems or structures” (Ascher, 1991: 3). Ascher has developed a methodology, based on the use of modeling tools, aiming to bring to light these systems and structures, related to various activities practiced in “small scale or indigenous cultures”. However, like most ethnomathematicians, Ascher’s work consists in the analysis of second-hand ethnographic sources. By contrast, in the last decades, some ethnomathematicians have suggested that an ethnological approach should play a central role in ethnomathematics, while applying an ethnographic participant-observer methodology in particular (Chemillier, 2011). This should enable the collection of new data concerning activities that involve mathematical ideas/reasonings/logics. Meanwhile, long-term ethnographic fieldwork should lead to a better understanding of the cognitive acts that underlie these activities, and also the way in which they are embedded in the social organization and symbolic systems of the societies in which they are practiced.

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Fig. 1 Left: Map of Vanuatu. Right: Ambrym Island, Vanuatu

With this theoretical and methodological framework in mind, I have undertaken ethnographic research that aims to compare two activities, described as string figure-making and sand drawing—both implying mathematical ideas—carried out in Northern Ambrymese society, on Ambrym Island, Vanuatu,1 South Pacific (Fig. 1). String figure-making consists in applying a succession of simple gestures to a string— knotted into a loop (made—in Vanuatu—of a slice of pandanus leaf), using mostly fingers and sometimes feet, wrists or mouth. This succession of operations, which is generally performed by an individual and sometimes by two individuals working together, is intended to generate a “final figure” whose name refers to a specific being or thing. For over a century, this practice has been observed by anthropologists in many regions of the world, especially in oral tradition societies (Braunstein, 1992; Evans-Pritchard, 1972; Jenness, 1924; Rivers & Haddon, 1902). In Vanuatu (ex-New Hebrides), string figure-making has first been documented by anthropologist Lyle Dickey, who published in 1928 the instructions for making nine string figures collected from New Hebrides immigrants—mostly Erromango Islanders—living in Hawaii. Until very recently, no other collection of Ni-Vanuatu 1

Called the New Hebrides by the British navigator James Cook in the 1770s, this archipelago— located 1,750 km east of northern Australia—was managed through a Colonial French & English Condominium from 1906, until its independence in 1980 and the foundation of the Republic of Vanuatu.

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Fig. 2 Left: Mata (village of Fona) displaying the string figure mel (fruits of the tree limel “trunk of mel” or nakatambol in Bislama, Dracontomelon vitiense), Vanuatu. Right: Buwekon (village of Topol) drawing fanwochepu (name of a village of the past) © Vandendriessche

string figures had been published so far. In 2014, the lost collection of string figures collated by the anthropologist Arthur B. Deacon in the 1920s on Malekula Island finally reappeared and was edited (Deacon & Sherman, 2015).2 The practice of sand drawing consists in making a figure by drawing a continuous line with the finger—either in the sand or on dusty ground—drawn through the framework of a grid made of either perpendicular lines or dots, without retracing any part of the drawing, ending it (most of the time) where it has begun. Like string figures, the making of sand drawings has been observed in Vanuatu since the early twentieth century—notably by anthropologist Deacon (1934) and Layard (1942)— and is still practiced widely in the center of the archipelago and in Northern Ambrym specifically (Fig. 2). Previous studies have shown that string figure-making practices are widespread throughout Melanesia (Hornell, 1927; Landtman, 1914; Maude, 1978; Noble, 1979). 2

Arthur Bernard Deacon (1903–1927) was a student of Cambridge anthropologist Alfred Cort Haddon. He died of blackwater fever (an acute form of malaria) on Malekula Island, New Hebrides, where he had carried out fieldwork in the years 1926–1927. His ethnographical field notes were published in 1927 and 1934 by another student of Haddon, anthropologist Camilla H. Wedgwood (1901–1955) (Deacon, 1927, 1934; Deacon & Wedgwood, 1934). We had evidence that Deacon focused on both string figures and sand drawings: an editorial footnote in Deacon’s monograph on Malekula (Deacon, 1934) indicates that Haddon and Wedgwood had hoped to publish his research on string figures (alongside his records of sand drawing) in a separate volume. Unfortunately, while the sand drawing records were published shortly afterward (Deacon & Wedgwood, 1934), the string figures have never appeared. The worst of it is that Deacon’s field notes on string figures had never been found. Recently, Mark Sherman, the president of “International String Figure Association” http://www.isfa.org/), received a message from a stranger, explaining that he found a collection of documents on string figures from all over the world, through his late father’s effects, and looking to “give them a good home” (Deacon & Sherman, 2015). These documents were sent to Sherman, and Deacon’s field notes was one of those. This collection made of 32 string figures collected in the 1920s in the Islands of Malekula, Santo, Malo, Ambrym is of great historical interest, and shall be put in perspective—in upcoming works—with the whole corpus of string figures that I personally collected throughout the archipelago.

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In contrast, and although we have scant evidence of such a practice in Papua New Guinea (Bell, 1935), sand drawing in the Melanesian area has mainly been observed in central Vanuatu. Accordingly, this practice has become emblematic of Vanuatu cultural tradition. Moreover, it is noteworthy that sand drawings from Vanuatu have been listed, since 2008, by the UNESCO. Nowadays, both sand drawing and string figure-making practices are recognized—in the societies concerned and beyond—as a traditional graphic art with a mnemonic dimension. Involved in the recollection of ritual, mythological, geographical, as well as ecological knowledge, these two practices are considered as belonging to the so-called kastom; “a word that people in Vanuatu use to characterize their own knowledge and practice in distinction to everything they identify as having come from outside their place” (Bolton, 2003: xiii). Although, in Vanuatu, both sand drawing and string figure-making practices appear as a transcultural activity, it nevertheless includes—in the Center/North of the archipelago—significant local characteristics, identified as genuine cultural markers (VKS, 2009). As noticed by a few anthropologists, string figure-making and sand drawing practices are sometimes linked to one another, in the local mythology of different NiVanuatu societies notably. About a century ago, anthropologist Layard observed the invention of new sand drawings in Malekula and suggested their relation to string figures: Indeed, now that they are used as games of skill this tendency, as with cat’s cradles [string figures], with which they are probably allied, now serves to stimulate the creative ingenuity of the natives, who, while certain designs, of course, become traditional and are copied over and over again, constantly invent new ones as an intellectual pastime (Layard, 1942: 654).

More recently, anthropologist Margaret Jolly collated a story of the Sa people, South Pentecost, which explicitly connects the two practices. The story is about a man, Barkulkul, who “put a taboo on his wife,” before paddling to the nearby Ambrym Island. He confined “his wife, placing two string figures, one on the door of the hut and the other on the woman’s genitals.” Despite these precautions, a man, Marelul, invited by Barkulkul’s wife, came into the hut. “They made love and the man left, and as he left he twisted the string figure which was across the woman’s genitals. He turned it around. He also twisted the string figure on the door. He turned it around.” “The husband returned home” and thought that “someone has come already.” Thereafter, Barkulkul asked Marelul to come with him to the shore, and “made drawings in the sand […] the man who went into the hut made sand drawings and it was the right design [i.e. the same as the string figure]”. Barkulkul thus understood what happened and thereafter killed the adulterer3 (Jolly, 2003: 199–200). In North Ambrym, however, I was not told about such mythological links between string figure-making and sand drawing practices. Nevertheless, both activities are locally named using the same vernacular term. This suggests that both activities are 3

A variant of this myth had already been collated in South Pentecost by missionary Tattevin (1929). More recently, in the 2000s, Stephan Zagala collated a version of this story on Ambae (typewritten letter entitled “String Figures”, Archives of the Vanuatu National Library).

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perceived by the Ambrymese as conceptually linked to one another. The comparative analysis of Ambrymese string figures and sand drawings developed in this paper is based on a long-term ethnography aiming at collecting various types of data: (1) the procedures involved in the two practices under study, forming a corpus of 52 sand drawings and 60 string figures, (2) the vernacular (technical) terminology linked with these activities, and (3) the oral texts and/or discourses which are sometimes associated with the latter. In the following, I will first describe the context in which are practiced both string figure-making and sand drawing among the Northern Ambrymese. The second part of this paper will be devoted to the comparative analysis of algorithmic and geometric properties of both practices, while illustrating how mathematical practices can be revealed in and through such ethnographic research. Finally, we will show how this study shall contribute to the ongoing debates conducted in Vanuatu for about a decade, regarding the use of traditional knowledge to improve the local mathematics curriculum, taking more account of the various cultural and linguistic contexts of the country.

2 Cultural and Symbolic Aspects of String Figures and Sand Drawings: Some Elements of Comparison The Ambrymese refer to both string figure and sand drawing practices using the expressions “tu en awa” and “tu en tan” respectively. “en awa” and “en tan” mean “on a rope” and “on the ground” respectively, whereas “tu” is a verb translated by “writing”, “drawing”, “representing”. This suggests that these two practices are locally conceptualized as belonging to their “writing tradition”. At the same time, “tu” also means “discussing/debating/palavering” an issue in order to figure it out step by step, while organizing certain ritual events such as weddings or baptisms in particular. In the context of the practice of sand drawing and string figure-making, this literal interpretation of the verb tu seems to echo the procedural aspect of this activity (cf. Sect. 3). This seems here to indicate a vernacular conceptualization of the making of these figures as procedural activities. String figure-making is described by Northern Ambrymese as a female activity, whereas sand drawing is said to be a male activity. However, they are indeed practices shared by many people on the Island; almost everyone, including children, is able to perform a few of these figures. According to Ambrymese elders, string figure-making was generally practiced in the family, mothers or grandmothers often playing with their children. As for sand drawings, they were generally applied on the dance ground

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Fig. 3 Left: Sand drawing perrlio (magic wood columns). Right: Sand drawing pelaperaoù (murderous device)

(har, or nasara in Bislama4 )—where men used to meet at the end of the afternoon after a working day in the gardens. Nowadays, as far as I have been able to ascertain, only a few women in each North Ambrym village are generally considered string figure experts. By contrast, it is mostly but a few men, either elders or young adults who are considered sand drawing experts. Furthermore, the practice of sand drawing seems to be more highly valued than the practice of string figure-making, the latter being often described as a game or playful activity. Nevertheless, both practices have been a means of recording and expressing knowledge relating to specific mythological entities, rituals or activities involving humans, non-humans, or environmental elements. For instance, the sand drawing perrlio represents one of the wood columns (perr) made from the trunk of the tree named lio (“red wud” in Bislama, Bischofia javanica) of the mel “men’s house” (nakamal in Bislama) in the village of Olal (cf. Fig. 3a). It is a magical wood which has the diabolical power to catch and eat anyone who stands near to it. Similarly, pelaperaoù (name of a murderous device) is a highly complex drawing representing a (“bad”) person lying inside an underwater machine armed with long blades, cutting off the legs of people who swim on the beach of the North Ambrym village of Ranon (cf. Fig. 3b). It is noteworthy that none of these entities are generally represented using both techniques. There are a few exceptions, though. For instance, the string figure and sand drawing named bulbul algon (literally “canoe lizard”) are both related to the story of Yaulon, one of this society’s mythical heroes (Guiart, 1951) (Fig. 4). Bulbul designates this hero’s canoe, while algon (lizard) refers to the symbol of one of the fourth grades (sagran) of the chieftainship system (Patterson, 1981), whose conception is attributed to this local mythological hero. Another example is the representation of yams (cf. Fig. 5); tubers harvested in the Ambrymese gardens, and that have a key role in the goods exchange system which is at the heart of many rituals in Ambrym and more generally in Melanesia (Coupaye, 2013; Lanouguère-Bruneau, 2000; Rio, 2007). 4

The (pidgin English) Bislama together with French and English are the three official languages of the Republic of Vanuatu. This pidgin functions as a lingua franca, as there are about 120 vernacular languages in the country, for about 300,000 people. However, there is currently a creolisation of Bislama in Ni-Vanuatu’s cities, where speakers of many different linguistic areas are mixed. Consequently, Bislama has become the mother tongue of many Ni-Vanuatu children.

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Fig. 4 Sand drawing and string figure bulbul algon © Vandendriessche

Fig. 5 Left: sand drawing rem (yam). Right: String figures tapli rem (the body of yam) © Vandendriessche

Some string figures and sand drawings are explicitly related to different ritual practices. The string figure malyel mweter lon turereoù as well as the sand drawing temarr ne luan are one of those. Malyel designates a young man to be circumcised. Mweter “to look” and lon tu rereoù (literally “in a hollow little banyan”), refers to the position of the young boy, during the circumcision ritual, looking at a fixed point, in order to focus his attention elsewhere (Fig. 6). Temarr ne luan (spirit of luan) also refers to the young men’s initiation ritual bearing the name luan “hiding”, which still took place in Northern Ambrym about two decades ago. A few young men were isolated from the community, for a couple of months, to be secretly enlightened. This initiation is described as rigorous, bringing

Fig. 6 Left: String figure malyel mweter lon tu rereoù (young man looking in a hollow tree). Right: Sand drawing temarr ne luan (spirit of luan) © Vandendriessche

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Fig. 7 Yam growth and entanglements © Vandendriessche

prestige to the participants. The drawing temarr ne luan was painted on a piece of wood and placed at the entrance of the house (mel) sheltering the participants. The latter representation of the spirit (of the secret society) of luan carries a taboo and only a luan insider can make this artifact. These two activities were part of the same prescription/prohibition system. A few decades ago, they were still preferably performed during the yam harvest, from February to July, while their practice is prohibited outside this period, the making of such figures being perceived as having a negative impact on the growth of the plant’s stem winding around the stake: it would favor the entanglement of the stem, slowing down the plant’s growth. Sand drawings and string-figures can both be analyzed as “entanglements” or “knots”. This prohibition seems to indicate that the Ambrymese perceive a link between the making of these “entangled figures” and the possible vegetal knots that can arise during the development of the stem. Therefore, one can thus hypothesize that the procedures leading to either sand or string patterns can interfere in the normal course of the vital process of growing (Fig. 7).5 In Northern Ambrym, as well as in the other societies where I carried out fieldwork, many people know how to perform numerous string figures, and I often had the opportunity to work with genuine “experts”.6 Nevertheless, I have never met anyone with the ability or even the desire to invent a new string figure. Furthermore, it is generally asserted by the Ambrymese practitioners that string figures have been invented by their ancestors “a long time before”. In contrast, a number of sand drawings are assumed to be of recent invention. For instance, the sand drawing poar 5

Such a ritual efficacy attributed to both sand drawing and string figure-making practices was confirmed by practitioners from different Northern Ambrym villages. Nevertheless, it seems that this phenomenon does not concern the whole region. In some places, these practices are related to mourning: they would have been authorized only during the five days following the death of high rank people. Making such figures in the house of the deceased person would retain his/her spirit in his/her place for a while before going to the land of death. 6 By experts, I mean people who are recognized as the most knowledgeable in either string figuremaking or sand drawing practices by the other members of the community. These practitioners generally know almost all the procedures known in the village and are able to perform them slowly, step by step, operation after operation.

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Fig. 8 Left: Sand drawing poar (white sea bird). Right: Sand drawing lengkon & paul © Vandendriessche

(white sea bird, “Pacific golden polver”, Puvialis fulva, cf. Fig. 8a) would have been invented/created a century ago by the Ambrymese RirrRirr (village of Wow), inspired by a number of white sea birds sitting in a trench used for planting taro plants (sitan) in his garden. Furthermore, a few young people do sometimes create novel sophisticated sand drawings after having learned the traditional ones from the elders. Since 2006, several knowledgeable sand drawing Ambrymese practitioners showed me the drawing called lengkon & paul (cf. Fig. 8b), asserting that the latter has been created some years ago by a young man living in Northern Ambrym. Many years later (in 2019), I have finally met the man in question, Buwekon from Topol village, who confirmed that he imagined this new drawing, while thinking of it mentally for a whole night and finally succeeded in making it after a working day.7 As the ethnomathematical study of both Ambrymese sand drawings and string figures suggests, it is most probably through an algorithmic practice, aiming at exploring the combinatorics of basic patterns that these complex procedures have been created in the past in central Vanuatu. Furthermore, as we will see in the next section, these two practices indeed share algorithmic and geometric properties, involving mathematical concepts such as operation, procedure, sub-procedure, iteration and symmetry.

7

Buwekon and myself have undertaken a (long term) collaboration aiming at studying how he actually proceeded to invent this new sand drawing and the logical system involved in such creation processes in particular. Indeed, this sand drawing seems to have been inspired by some other drawings (fanwochepu and perrlio shown in Figs. 2 and 3 in particular), sharing some characteristic patterns with these traditional drawings. However, the latter patterns are combined in a different way, and basic motifs transformed into other ones (cf . Sect. 3.2, Transformation). Further (cognitive) research should be carried out to clarify this point.

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Fig. 9 The Opening A

3 Ethnomathematics of String Figures and Sand Drawings 3.1 Algorithmic/Procedural Aspects of String-Figure and Sand Drawing Practice String figure procedures A string figure can be analyzed as the result of a “procedure” (or geometric algorithm8 ) consisting of a succession of “elementary operations”—, insofar as the making of any string figure can be described by referring to a certain number of these operations. Most of these elementary operations can be defined as “geometric” (or “topological”) operations which modify a given string configuration, transforming it into another (Vandendriessche, 2015). The string passing around a finger forms a “loop”. Any string figure procedures of the corpus begin by a series of movements (or opening) aiming at making loops on fingers. Some of them are associated with technical vernacular expressions. For instance, hu pokopreo (pricking indices) corresponds to the movement that has been observed in many societies around the world, and named “Opening A” (cf. Fig. 9) by anthropologists Rivers and Haddon (1902).

8

We borrow the expression “geometrical algorithm” (“algorithme géométrique”) from ethnomathematician Paulus Gerdes (1952–2014), thus indicating that the operations at work in the algorithm are of geometrical/topological nature (Gerdes, 1995).

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The following instructions/pictures illustrate the making of the Ambrymese string figure wayu (hairy yam)9 which starts with the opening A.10 1. 2.

Opening A (picture a)11 . Release thumb loops12 (picture b).

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With right thumb, hook down both right index loop and near little finger string. With the right index hook up the far-right little finger string, and return to position (picture c). Insert the left index into the upper right index loop, and return to position (palms facing each other) (pictures d-e).

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9

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Pass thumbs under the index loops, insert them into little finger loops, from below, return with near little finger strings (picture f). Insert thumbs from below into the upper index loops, pick up the upper near index string (picture g).

See also the description of this string figure on the website “String Figures: Cultural and Cognitive Aspects, Construction Methods, Modelling” (database of the project “Encoding and Transmitting Knowledge with a String: a comparative study of the cultural uses of mathematical practices in string figure-making (Oceania, North & South America)—ETKnoS” Project, 2016–2020): https://string figures.huma-num.fr/items/show/19. The string figure wayu can be found (with little variations) in many published collections of string figures from Oceania (Vandendriessche, 2015 : 373–375). To my knowledge, it has been first recorded as “Ten Men” in 1902 in the Caroline Islands by anthropologist William Henry Furness, and published in 1906 by his sister, Caroline Furness Jayne (1962: 150–156). 10 From the first scientific studies on string figure-making practices—from the nineteenth century and onwards—a number of anthropologists/mathematicians have attempted to devise suitable and efficient nomenclatures to record the procedures involved in the making of such figures, cf. (Vandendriessche, 2015, 2019). The instructions of the procedure wayu are here adapted from the description of the string figure “Ten Men” made by Jayne (1962). 11 The black square on the thumbs indicates the release of the string from the finger. 12 A loop is made of two strings: the near string and the far string, with respect to the practitioners’ viewpoint, fingers pointing up and palms facing each other. When a finger carries two loops, we differentiate them by using the expressions “upper loop” and “lower loop”.

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Navaho the thumbs13 (pictures h–i).

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Release upper index loops (picture i). Transfer thumb loops to indices (pictures j–k).

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Repeat (5, 6, 7) (pictures l-n).

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Pass middle fingers, over far index strings, insert them from below, into lower index loops, return with lower near index strings (pictures o–p).

When two loops lie on the same finger the “Navaho” operation is implemented on this finger by passing the lower loop over the upper one, and then, over the fingertip where it is released. This movement is indicated on the Pictures with capital letter N. To my knowledge, the term “Navaho” (or more precisely the verb “Navahoing”) was first used in this context by Kathleen Haddon (A. C. Haddon’s daughter), in her book Cat’s Cradle from Many Lands (1911).

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(o) 11.

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Release little fingers (picture p), then extend turning palms away from you with fingers spread out (picture q).

(q) Sand drawings As mathematician Marcia Ascher has first emphasized in the 1980s—while analyzing the sand drawings collected by Deacon (1934)—the making of a number of these figures consists in the implementation of an ordered sequence of “elementary transformations” applied to one or more patterns or “tracing procedures” (Ascher, 1988, 1991: 30–65). The latter “tracing procedures” result from the organization of “basic patterns” into procedures (cf . Fig. 11). If A and B denote two such tracing procedures, Ascher notes 1/ AB the procedure obtained by executing “A followed by B”, and 2/AT the procedure obtained by transforming A into another procedure through a given geometric transformation T. For example, if, starting from the end point of procedure A, one repeats the same drawing procedure, the “process” is identity, the resulting procedure will still be A, and the global procedure “A followed by A”, formalized by the expression AA. When each movement of procedure A undergoes a 90° clockwise rotation, the process is the 90° rotation, the new procedure is symbolized by A90 , and the global procedure is denoted by AA90 . For instance, Ascher (1991: 52–54) analyzes the sand drawing nimbingge (a variety of yam), collected by Deacon on Malekula Island, in terms of “tracing procedures” and “processes”: the choice of the initial procedure A allows us to describe the global procedure as the iteration of the same pattern A, through the series of processes AA90 A180 A270 (cf. Fig. 10).

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Fig. 10 (a) Sand drawing nimbingge (Deacon & Wedgwood, 1934: 148). (b) Tracing procedure A. (c)–(e) The sequence of processes A90 A180 A270 involved in the making of the sand drawing nimbingge © Vandendriessche & Da Silva14

The latter drawing nimbingge, known as peng “green turbo”15 by the contemporary Ambrymese practitioners (cf. Fig. 24), like many other sand drawings in the same region, can thus be formalized/analyzed as an ordered sequence of “elementary transformations” (or “processes”) applied to one (or more) given pattern(s) (or “tracing procedures”). This conceptualization has indeed proven to be relevant in the analysis of a number of sand drawings (cf . sand drawings fanwochepu, Fig. 2 and perrlio, Fig. 3, as well as the sand drawing konang lipang, Fig. 16 below) that I have collected on Ambrym over the years. Technical vernacular terminologies Some vernacular expressions are used by practitioners to refer to the basic movements/operations involved in both activities. Although these vernacular terms usually 14

Alban Da Silva is currently carrying out doctoral research under the author’s supervision. His ethnomathematical research aims at gaining a better understanding of the mathematical aspects of the practice of sand drawing in Vanuatu, and in North Pentecost Island in particular. As part of this project, Da Silva has created a digital modeling tool in order to rewrite the sand drawing procedures and display the final figures on a screen (Da Silva, upcoming-2022). 15 Green turbo (Turbo marmoratus) is a marine gastropod snail from the turbinidae family.

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(a)

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Fig. 11 The basic patterns: (a) itel. (b) li. (c) kuvür. (d) hupeng. (e) hu

differ from one activity to another, they are action verbs in both cases. For sand drawings, five verbs are thus used to designate the making of the (basic) patterns represented in the figure below (Fig. 11). Although, the verbs li “to make a knot”, kuvür “to turn”, and hu “to prick” (something with a stick) are used in other contexts, the two verbs itel and hupeng seem specific to the practice of sand drawing. The etymology of the verb itel would derive from the term tel “rope”, used—in particular—to tie breadfruits together with a knot for carrying them. As for the verb hupeng, it is a combination of the verb hu “to prick” and the noun peng “green turbo”. Some other verbs are used to designate particular drawing movements (cf . Fig. 12). For instance, the verb tukor “to cross” (a house or a garden for example) is used to indicate that the continuous line crosses one grid square diagonally. In other cases, the latter line sometimes has to pass outwardly through one corner of the grid, making a curve aiming at re-entering the grid again at the symmetrical corner. This movement is described by the verb hoùre, which designates the action of going out and then re-entering the same space (a house or a village, for example). The expression roù

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Fig. 12 (a) Movement tukor. (b) Movement hoùre. (c) Movement roù lipi. (d) Movement likete © Vandendriessche

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Fig. 13 (a)–(b) “picking up a string” (pokolam hu pokopreo “thumb pricking index”); (c)–(d) “twisting of a loop” (pokopreo ro si kilhi “turning the index returning towards me”); (e)–(g) “releasing a loop” (tororrro pokopreo “release index”) © Vandendriessche

lipi (to wrap the trunk of the tree pi16 ) describes a sinusoidal line crossing several consecutive nodes of the grid. Finally, the verb likete “to maintain” refers to the movement consisting in joining several little loops (li) drawn in a row on the same line of the grid. This suggests that the latter drawing movement is perceived as a link holding together the different lines (yen/yeroù) of the grid.17 The verb hu “to prick” is also used while making string figures. For instance, the sentence pokolam hu pokopreo (lit. “thumb pricking index”) means that the thumb(s) pick(s) up (one of) the strings carried by the indices. The verb tororrro is used to indicate that a finger must release its loop (for instance, tororrro pokopreo “release index”). As for the expression si kilhi (lit. “turning/rotating, returning towards me”, it indicates a twist of a finger (or loop) (Fig. 13).18 Some other expressions permit the designation of more sophisticated operations (made of several consecutive elementary operations). For instance, cherrhururu is a term specific to string figure-making practices referring to an operation known as “navaho” in the anthropological literature devoted to the topic (cf. note 13). While the verb wehe “to kill” is used when the final figure is spread out through a specific movement called “Caroline extension” in the same literature (cf . Fig. 14). All of these technical expressions associated with the practice of sand drawing and/or string figure-making in North Ambrym are used spontaneously by practitioners in action, in instances of transmission from one person to another in particular, 16

The tree pi belongs to the croton tree family (Codiaeum variegatum). It has asymmetrical leaves; one of their edges having a sinusoidal curved shape. 17 The horizontal lines of the grid (from the practitioner’s point of view) are named yeroù, which also means “beams”, “lintels”, or “purlins” of traditional houses. The vertical lines are named yen “leg”, which also refers to the wooden posts supporting the houses. The use of these vernacular terms suggests that the grid is perceived as a framework supporting the drawing. 18 We use the spelling introduced by linguist Michael Franjieh (2012), specialist of Ambrymese vernacular languages. The trill is noted /rr/ and the retroflex (used in the area of Fona, instead of the tap/flap heard by Franjieh in the area of Ranvetlam) is noted /r/. When these two s are consecutive, /r/ always follows /rr/. Therefore, we will simply note ‘rrr’ this succession of s. As for the sound [*] and [y], they will respectively be noted /ù/ and /ü/.

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(a)

(b)

(d)

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Fig. 14 (a)–(c) Sub-procedure cherrururu. (d–g) Sub-procedure wehe © Vandendriessche

although not consistently. In both cases, the procedures are indeed often taught or shown without any technical comments. However, the knowledge of this technical terminology seems to be shared by most Ambrymese, as it can be seen when the audience guides a hesitant practitioner. Finally, the existence of these expressions is an indicator of the perception by the actors of orderly sequences of operations, suggesting a local perception of the notion of elementary operations/basic patterns revealed by ethnomathematical analysis.

3.2 Shared Mathematical Properties Cyclicity The making of sand drawings, as well as that of string figures, seems to be perceived by the Ambrymese as cyclical processes. As often pointed out by practitioners, using the specific term paritu, literally “beginning” (pari) of the “action of drawing” (tu), it is indeed essential to end a sand drawing where it has begun. Similarly, a string figure is usually only shown for a few seconds, before being undone by the practitioner who thus obtains the original loop from which he started the procedure. This seems to indicate that a string figure is seen simply as a stage of the procedure. Moreover, the undoing of the figure appears to be included in the whole process of performing a string figure. Symmetry Most of the figures drawn either in the sand or with a string have at least one symmetry axis. In the practice of sand drawing, the term tahitu (lit. “drawing [it] on the other side”) refers to the action of successively making a given pattern on one side of the drawing and the symmetrical pattern on the other side (Fig. 15). Remarkably, in the practice of string figure-making, when the same ordered sequence of operations has to be performed symmetrically on both hands, the practitioners most often proceed in two stages, performing the operations on one side, and

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Fig. 15 Expressing symmetries: tahitu “drawing [it] on the other side” © Vandendriessche

Fig. 16 Left: konan lipang (banyan taboo). Right: rom dance, North Ambrym © Vandendriessche

then on the other. Either the expressions tahitu or tahite “the other side” are actually used to refer to this operational process. The ubiquity of symmetries is an important feature in both corpora of figures, and sometimes underlined by either the names given to the figures or the stories (or myths) which accompany the making of some of them. For instance, the sand drawing konan lipang (lit. “taboo banyan”) is accompanied by a narrative describing the birth of the Northern Ambrym rom dance.19 The story starts with two women, sitting down on either side of a banyan tree, both weaving a mask for their children. Even though they could not see each other, the two women thereafter realized that they had made exactly the same mask (Fig. 16). In the case of string figure-making, the symmetry of the final figure is sometimes related to the representation of the same object or being on both sides of the figure. This feature can be further emphasized through the vernacular name given to the figure. For instance, the string figure be mkor be (lit. “shark hunting shark”) represents two sharks symmetrically: a male and a female swimming away from each other (cf. Fig. 17). Sub-procedures/algorithms Our comparative analysis of both corpora of figures has brought to light numerous “sub-procedures” i.e. ordered sets of “elementary operations” or “basic patterns” 19

For further information about the rom dance, see (Guiart, 1951: 68–69).

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Fig. 17 String figure be mkor be (shark hunting [another] shark) © Vandendriessche

either iterated within a given procedure or repeated identically within several different procedures of the same corpus. For string figure-making, these sub-procedures aim at making either basic string patterns or complex crossings (cf. Fig. 18), whereas for sand drawings, they allow the practitioner to draw different motifs made of several basic patterns (cf. Fig. 10) or to implement a specific algorithm for completing the grid (cf. Fig. 20). The currency of such sub-procedures in both corpora suggests comparable operational practices in the creation of sand drawings and string figures, which would have consisted in the organization of these elementary operations/basic patterns into procedures. In a comparable way in both corpora of string figures and sand drawings from Northern Ambrym, some sub-procedures/algorithms have been used as a basis for creating new procedures. For instance, the three string figure procedures olol tamre (lit. “coconut germ upward”), warmupta (bread fruit stem), and rralngien burbur

(a)

(b) Fig. 18 (a–b) Two basic string patterns. (c) Complex (string) crossing © Vandendriessche

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Fig. 19 (a–d): Same sub-procedure involved in the string-figure procedures olol tamre (coconut germ upward, fig. (e)), warmupta (bread fruit stem, fig. (f)), and rralngien burbur (hearing [a little animal with a] tuft of hair/feather, fig. (g)): Opening A. Pass both thumbs under all intermediate strings and insert them into little finger loops (fig. (a)), return to position with the far little finger string (fig. (b)). Release little finger loops and extend (fig. (c) and (d))

Fig. 20 Algorithm K: first part of the sand drawing kil “grasshopper”. Scan the link with “SN More Media App” ( https://doi.org/10.1007/000-7rh) © Vandendriessche

(lit. “ears of burbur” [a little animal with a] tuft of hair/feather”),20 all start with the same sub-procedure implemented after the “Opening A”, as described in Fig. 19. As for the practice of sand drawing, the same algorithm, say K, is involved in the making of different drawings. For the sand drawing kil “grasshopper” notably, this algorithm allows drawing the part which represents the body of the animal (cf. Fig. 20). This procedure—carried out here on a rectangular grid of 4 horizontal lines 20

See the ETKnoS project database: https://stringfigures.huma-num.fr/items/show/82; https://str ingfigures.huma-num.fr/items/show/18; https://stringfigures.huma-num.fr/items/show/110.

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and 3 vertical lines (4 × 3)—requires following the diagonal lines of the grid; when the line goes out of the grid, it is re-entered through the nearest grid node; except if the line exits the grid through one of its four corners, and must, in this case, be re-entered inside the grid through the symmetrical corner (with respect to the vertical symmetrical axis of the grid—cf. movement called hoùre, mentioned above). The same principle is used for the making of the sand drawings vyu “turtle” and awuyil (liana yil, Epipremnum pinnatum). Made on a 5 × 3 grid, the sand drawing vyu starts with the tracing of two short lines (cf. Fig. 21a). The figure is then completed by implementing (part of) the algorithm K, while deforming the upper big arc downward (movement hoùre) to draw the carapace of the turtle (Fig. 21a, b). About a hundred years ago, on Ambrym, Deacon actually collated a variant of the latter sand drawing vyu, for which the complete algorithm K was implemented (cf . Fig. 21c, Deacon & Wedgwood, 1934: 154). As for awuyil—made on a 4 × 5 grid—this algorithm occurs in the last phase of the drawing. In the latter case, although the tracing principle is the same as for the sand drawing kil, the arcs—connecting the nodes outside of the grid—are replaced

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(c)

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Fig. 21 (a–b) Algorithm K implemented in vyu “turtle”. (c) Sand drawing hi “turtle”, adapted from (Deacon & Wedgwood, 1934: 154). (d) Algorithm K implemented in awuyil (a liana) © Vandendriessche

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Fig. 22 String figure meya (inedible orange fruit), resulting from the iteration of a same pattern © Vandendriessche

by broken lines (cf . Fig. 21d); movement designated by the vernacular term hu “to prick”.21 Iteration The concept of “iteration” is ubiquitous in both corpora. In many string figures and sand drawings, we can analyze the iteration of a pattern as a consequence of the iteration of a sub-procedure: for string figures, an ordered sequence of operations, and, in the case of sand drawings, an ordered sequence of basic patterns. Furthermore, vernacular expressions—uniform in both activities—explicitly express the iteration of a pattern or a sub-procedure. As seen above, the sand drawing peng can be analyzed as the iteration of one pattern or tracing procedure (cf . drawing nimbingge, Fig. 10, and Fig. 24 & accompanying videos). Indeed, practitioners sometimes explain the making of this drawing by describing first the pattern to be iterated, and then making the three other patterns while saying pe pa ru “this one twice”, pe pa sul “this one three times”, pe pa wir “this one four times”. As with many other string figures of the corpus, the making of the string figure wayu (described above, Sect. 3.1) is also based on the iteration of a sub-procedure. We have seen that a series of elementary operations (cf. Steps 5, 6, 7) is repeated twice in the same way on different substrata/configuration of strings during this string figure process. Furthermore, as in the case of the making of sand drawings (cf. peng or konang lipang), the impact of an iterative sub-procedure is sometimes to iterate the same motif. For instance, the three “doubled lozenges” in a row of the string figure meya (inedible orange fruit) result from the iteration of the same (iterative) sub-procedure (cf. Fig. 22).22 21 The algorithm underlying the making of kil, vyu, and awuyil has been called “turtle algorithm” and further studied by Alban Da Silva, cf. (Da Silva, upcoming-2022). He has first observed this pattern as part of the sand drawing hi “turtle” already recorded by Deacon (Deacon & Wedgwood, 1934: 154). Consequently, I noticed that the same procedure is actually involved in different North Ambrym sand drawings. 22 See the ETKnoS project database: https://stringfigures.huma-num.fr/items/show/83.

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Fig. 23 (a–b) String figures ao mwiti lowon (crab which defends itself with its claws). (c–d) String figure blaol (empty coconut). (e) String figure silimri (a tree). (f) String figure lopevi (name of an island)

Transformation Some examples suggest that string figures and sand drawings have circulated throughout the North Ambrym area (and beyond) while undergoing transformations at the level of procedures and/or final figures. First, some Ambrymese string figures suggest that practitioners have elaborated some procedures—or paths—leading to identical or very similar final figures. For instance, the string figure ao mwiti lowon (crab which defends itself with its claws) is the mirror symmetry of the string figure blaol (empty coconut). This property results from the exchange of the role of the thumbs and little fingers throughout the procedures, thus implying symmetrical movements of the loops. As for the procedures silimri (a tree) and lopevi (name of an island) they both lead to similar final figures made of four lozenges in a row (cf . Fig. 23e–f). Similarly, in the case of sand drawings, it can be shown that different paths have sometimes been created to obtain exactly the same final drawing, varying the order in which the various segments are drawn. For instance, there are at least two procedures used to draw the sand drawing peng: the procedure described above (cf. Fig. 10) and another procedure starting from the center of the grid and taking another route (cf . Fig. 24).23 23

See these two variants of the sand drawing peng: cf . Fig. 24 and its accompanying videos. In North Ambrym, peng actually consists in two parts. The first part is a simple continuous drawing and the second part is the drawing analyzed by Ascher.

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Fig. 24 Two different routes for making the sand drawing peng. Scan the link with “SN More Media App” ( https://doi.org/10.1007/000-7rg) ( https://doi.org/10.1007/000-7rj)

Fig. 25 Transformation of konang lipang. Left: konan lipang of Bahaltalam © Vandendriessche. Right: rom of Ranvetlam, variant of konan lipang © Michael Franjieh24

This brings forward the interest that some practitioners had in procedures: if they had been only interested in the final figures (made either in the sand or with a rope/string), they probably would not have bothered devising different procedures, which display identical figures. In some cases, however, it is rather the final figure that one has sought to transform. Sand drawing procedures have thus been altered by adding and/or removing certain patterns from a given drawing. For example, the konan lipang procedure (cf. Fig. 16 above) that I collated in Bahaltalam (a village near Fona), has also been recorded by linguist Mike Franjieh in the southern village of Ranvetlam. While the two drawings are very similar and have the same meaning, the Ranvetlam drawing shows two additional symmetrical patterns (cf. Fig. 25)— which represent the two women sitting on either side of the banyan tree—that are absent from the previous sand drawing. The same phenomenon occurs for string figures. For instance, the procedure lipwi’m lalao mwecherkete wayu (the roots of the tree lalao seize the hairy yam) is 24

Pictures extracted from (Franjieh, 2018) with the kind permission of Michael Franjieh.

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(a)

(c)

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Fig. 26 (a–b) Transformation of the starting position of the procedure meya. (c) titiman liseseo (“The devil’s breasts”- represented by two hanging loops), variation on meya (red fruit). (d) Final figure of lipwi’m lalao mwecherkete wayu, variation on the string figure wayu © Vandendriessche

obtained by adding the picking up of two strings (the lower far index strings) within the iterative sub-procedure involved in the making of the string figure wayu described above (cf. Sect. 3.1).25 This alteration of the procedure leads to the transformation of the final figure (cf. Fig. 26d). The string figure titiman liseseo (devil liseseo’s breasts) can also been analyzed as the transformation of the string figure meya (inedible orange fruit). However, in this case, it is done through the alteration of the starting position (cf . Fig. 26a, b), aiming at adding two long hanging loops (representing the breasts) to the final figure of meya (cf. Fig. 26c).

4 In Conclusion: From Epistemological to Educational Issues We have seen that the Ambrymese string figure and sand drawing practices are both locally conceptualized as a way of representing things or beings, and are/were embedded in the same system of prescription/prohibition, related to their negative impact on the growth of yams. On another level, their creation very likely required an intellectual task of selecting “elementary operations” or “basic patterns”, and 25

More precisely, Step 5 of the procedure wayu becomes “Pass thumbs under the index loops, insert them into little finger loops, from below, return with both near little finger and lower far index strings”.

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organizing them in procedures. Both string figures and sand drawings thus appear as the result of genuine algorithms. Moreover, these practices are both of a “geometrical” order, insofar as they are based on investigations into complex spatial configurations. Several recurrent phenomena confirm this point: specifically, the organization of basic operations in sub-procedures, the use of iterations, the transformation of final figures, and the search for different paths leading to the same figure, are ubiquitous in both practices. As we have seen throughout this chapter, the comparison of the cultural and cognitive aspects of different practices comprising a geometric and algorithmic character (such as string figure-making and sand drawing) should lead to a better understanding of how these practices are locally conceptualized as well as correlated with one another in various ways, in their terminology for instance or through other cultural patterns. At the same time, the comparison of the combinatorial properties at work in these activities contributes to an improved understanding of the nature of mathematical knowledge and practices involved in these activities. At another level, this ethnomathematical approach shall contribute to the discussions that currently occur in many indigenous societies worldwide, requiring a local education system and the development of a culturally based curriculum (Pinxten, 2016; Vandendriessche, 2017). For about a decade, the Republic of Vanuatu has undertaken a reform of the National Curriculum (inherited from the colonial period), taking more into account the various local cultures i.e. the 120 different vernacular languages as well as traditional knowledge and practices (such as sand drawing and string figures in particular) prompting their integration in teaching programs (VNCS, 2010). These new educational directives have already been implemented in the elementary schools where pupils now learn how to read and write in their mother tongue (vernacular languages in rural contexts and Bislama26 in towns). It is in this context of the revision of the National Vanuatu Curriculum (still in progress) that the “Vanuatu Cultural Center”, the local institution working for the preservation and the promotion of different aspects of Vanuatu’s culture—drawing on my ongoing research on string figure-making and sand drawing—has given its (mandatory) assent for this project in ethnomathematics, provided it leads to pedagogical applications. As I have pointed out elsewhere (François et al., 2018), beyond this institutional incentive, another motivation for undertaking such educational research is that the Northern Ambrymese communities welcome with interest the idea of using their traditional practices in the local curriculum. Whereas the purpose of ethnomathematical (theoretical) research is generally not clearly perceived as vital as we think it is, indeed, its educational valorization makes sense for these people. Aware of these practices’ decline in their community, and asserting that young people are no longer interested in traditional knowledge, they consider this valorization as a way of preserving their local culture. However, the elaboration of teaching programs related to social and cultural concerns is still the subject of discussions in the country’s educational circles. Both the Ministry of Education and the Vanuatu Cultural Center actually prompt 26

Cf . footnote 4.

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Fig. 27 Bringing string figure-making and sand drawing into the classroom, Topol High School, North Ambrym, Vanuatu © Vandendriessche

Ni-Vanuatu as well as foreign researchers to take part in the debate, through their participation in conferences/workshops/trainings organized in the capital Port Vila in particular.27 Regarding the teaching of mathematics, Georges Tauanearu, Professor of mathematics at the “Vanuatu Institute of Teacher Education”-VITE teacher training, has been giving a course in ethnomathematics for about a decade. This course allows students to study seminal works in ethnomathematics, while being encouraged to use—as pedagogical materials—the local mathematical practices that they would be able to collate in the cultural/linguistic areas where they will be sent to teach in local secondary schools, throughout the archipelago. In 2016, I was invited by Norbert Napong, the Head of Topol (Francophone) High School, North Ambrym, to present my research in ethnomathematics to the students and their teachers. On this occasion, I met Ambrymese mathematics teacher Frederick Worwor. Trained at the VITE, in ethnomathematics specifically, he had already introduced the practice of sand drawing as a mathematical activity into the classroom. Generously, he showed me his students’ productions and we planned to further collaborate in the development of teaching materials on both string figures and sand drawings. This collaboration was undertaken in 2019 through the elaboration of a sequence of three teaching sessions to be implemented in 6th and 10th year classrooms (Fig. 27). We first agreed that this pedagogical sequence on both string figure-making and sand drawing should be elaborated in an attempt to experiment the use of these (ethnomathematical) practices “as such”, in and of themselves.28 27

For example, a workshop in ethnomathematics (co-organized by Georges Tauanearu, Professor of mathematics at the “Vanuatu Institute of Teacher Education”-VITE, Pierre Metsan, Principal Education Officer-Higher Education division, and myself) was held at the VITE on Dec. 9–12, 2019, in order to discuss, with different education actors (Ni-Vanuatu mathematics teachers, teacher students at the VITE, PhD student in mathematics education at the National University of Vanuatu), the various issues raised by the elaboration of a culturally based mathematics curriculum in Vanuatu. 28 A major critique of ethnomathematics (and its pedagogical implications) refers to the use of local/cultural artifacts as pedagogical tools which often serve as little more than a motivation or an “illustration” (Pais, 2011) when learning the official mathematics curriculum (using a particular pattern woven on a basket for instance to introduce Pythagoras’ theorem, cf. Gerdes, 1992). The connection between such a pedagogical practice and cultural reinforcement is thus questionable. To

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Furthermore, their mathematical aspects shall not be isolated from the other forms of local knowledge embedded within these activities, with the intention of bringing into the classroom the cultural and cognitive complexity of these practices. In that perspective, the Ambrymese students would thus be incited to use their vernacular language for expressing the various operations involved in the making of these figures as well as the stories that often accompany them (cf . Sect. 2). Ni-Vanuatu teachers use a standard preparation sheet in order to plan their pedagogical sequences. This document lists the objectives of the lesson, and divides it into different periods (introduction, tasks, results/conclusion, exercises to be done for the next session). Frederick Worwor and myself devoted a full day to create three such preparation sheets related to our project entitled “Jeux de ficelle, dessins sur le sable: des activités mathématiques ?” (String figure-making, Sand drawing: mathematical activities?) Frederick suggested organizing the sessions in the classroom, in order to inspire the students to perceive the inclusion of these traditional activities in a more formal context. For the making of sand drawings, a couple of wood trays had been prepared and some ashes spread on them. As for string figures, long loops of pandanus leaves were ready to be woven. Basically, the sequence was carried out over a three month period: the first session consisted of 1/explaining the project while discussing the idea of grouping the two practices in a single project 2/working in groups of four or five individuals, where they have been asked to remember the making of 3–5 figures of each type, as well as their meanings (vernacular names, stories, etc.), making sure that all members of the group were able to do so by themselves, 3/choosing one sand drawing and one string figure in the list, while describing the procedures/patterns as well as the symmetries and iteration of motifs in their mother tongue. For the second session, we decided to concentrate on the concept of iteration at work in the making the sand drawing peng (cf. Sect. 3.1) and the string figure meya (cf. Sect. 3.2). The students were thereafter asked to identify other string figure/sand drawing processes based on the same principle. As for the third session, it was devoted to the analysis of the sand drawing poar and the string figure wayu (and its variation lipwi’m lalao mwecherkete wayu, cf. Sect. 3.2), bringing to light the use of the concept of alteration in both practices. In the first case, the students tried to implement the procedure poar starting on grids with a different number of vertical lines (cf. Fig. 28). This led them to conjecture that the procedure works properly if and only if the later number of lines is an odd number.29 In the string figure case, the students readily realized that it is through the addition of a “picking up” (hu) operation that the final figure wayu is transformed into another figure. Consequently, they were able to find other examples of such phenomenon (alteration/transformation) occurring in the Ambrymese string figure corpus. overcome this contradiction, our proposal is thus to introduce these different “ethnomathematics” (or culturally-related mathematical practices) “as such”, in and of themselves (e.g. practicing string figure-making as well as sand drawing in “math class”, cf. François et al., 2018), bringing into the mathematics classroom the cultural and cognitive complexity of these practices. 29 The mathematical proof of this property will be published in (Da Silva, upcoming-2022).

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Fig. 28 Variations on the sand drawing poar

Concerning the homework given to the students at the end of each session, they were asked to share all the procedures that had been demonstrated during the class. Furthermore, they had to identify the experts in string figure-making and/or sand drawing in their own villages, with the objective of learning more from these people about “tu en tan” and “tu an awa”, and bring their findings back to the classroom. During this experiment, we quickly realized that although some students were well acquainted with one or both of the practices under study, others knew almost nothing about them. According to Frederick Worwor and Norbert Napong, an increasing interest for these traditional activities was perceptible on campus: some students spontaneously demonstrating (out of the classroom) the making of sand drawings and/or string figures collated in their own villages. Although such an experiment obviously increases the transmission of these declining local activities to the younger generation, the impact of this pedagogical practice in the mathematics classroom remains an open issue (François et al., 2018; Pais, 2011). The students involved in this educational experiment have readily connected both string figure-making and sand drawing with mathematical practices through different analogies, comparing such practices with geometrical constructions or calculations that must be carried out with attention, step by step. Therefore, these sessions have emphasized that particular forms of mathematical practices/reasoning can be found in their close environment. The next step should be to evaluate whether a regular practice in the mathematics classroom, say during a year, of local activities with mathematical characteristics, “as such”, in and for themselves, might have an impact on the students’ skills in “academic” mathematics—as a few previous studies have suggested (Moore, 1988; Murphy, 1998, Lipka, 1994, 2004). In other words, the issue would be to evaluate whether practicing/valorizing activities defined as kastom practices in Vanuatu—such as sand drawing, string figure-making, but also mat-making, basketry, housing construction, etc.—in a formal educational context could serve as a lever for increasing both the Ni-Vanuatu students’ interest and their abilities in practicing mathematics. The Vanuatu Educational Institutions currently promote such investigations notably by supporting doctoral research in mathematics

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education, carried out in the recently created National University of Vanuatu.30 Given the exceptional linguistic and cultural diversity of Vanuatu, this archipelago seems to be the perfect place to undertake such empirical research whose outcomes would be of fundamental importance for the entire ethnomathematics research community. Acknowledgements I warmly thank the North Ambrymese string figure and sand drawing practitioners, Malikli and Lengkon (village of Fantan), Juliano Napur (village of Wow), Buwekon (village of Topol), Mata Koran (village of Fona), Rachel (village of Bahaltalam) who have generously shared their expertise. I would also like to thank mathematics teacher Frederick Worwor for our discussions and collaboration, Camille Etul for the vernacular terms spelling check, Alban Da Silva for our collaboration and the digital sand drawing illustrations, and Philippe Bordaz for his careful proofreading of this chapter. This research would not have been possible without the consent of the Vanuatu Cultural Center and the financial support of the French National Research Agency (“Encoding and Transmitting Knowledge with a String: a comparative study of the cultural uses of mathematical practices in string figure-making (Oceania, North & South America)— ETKnoS” Project, 2016–2021) and the “Institut des Sciences Humaines et Sociales” (InSHS) of the French National Center for Scientific Research (International Mobility Support program, 2019, “String figure-making, sand drawing, and mat making: a comparative study of (ethno-)mathematical practices from Vanuatu” project).

References Ascher, M. (1988). Graphs in cultures: A study in ethnomathematics. Historia Mathematica, 15, 201–227. Ascher, M. (1991). Ethnomathematics: A multicultural view of mathematical ideas. Brooks and Cole Publishing Company. Austern, L. (1939). The seasonal gardening calendar of Kiriwina, Trobriands Islands. Oceania, 9, 237–253. Bell, F. (1935). Geometrical art. Man, 35, 16. Bolton, L. (2003). Unfolding the Moon: Enacting women’s Kastom in Vanuatu. University of Hawai’i Press. Braunstein, J. (1992). Juegos de hilo de los indios maká. Hacia una nueva carta étnica del Gran Chaco, III, 24–81. Chemillier, M. (2011). Fieldwork in ethnomathematics. In N. Thieberger (Ed.), The Oxford handbook of linguistic fieldwork (pp. 317–344). Oxford University Press. Coupaye, L. (2013). Growing artefacts, displaying relationships. Yams, art and technology amongst the Nyamikum Abelam of Papua New Guinea. Berghahn. Da Silva, A. (upcoming-2022). Une étude ethnomathématique du dessin sur le sable du Vanuatu. De l’ethnographie à la modélisation mathématique, regards croisés sur la pratique des Uli-Uli chez les Raga de Nord-Pentecôte. PhD in History and Philosophy of Science. Université Paris Cité. D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44–48. 30

Pierre Metsan (Principal Education Officer-Higher Education division) is currently doing a Ph.D. entitled “Teaching and Learning Mathematics in Vanuatu: Assessing the impact of pedagogical uses of sand drawing practices on students’ interest and performance” under the supervision of Pr. Catherine Ris (University of New Caledonia). The latter Ph.D. is one of the first two doctoral investigations carried out in partnership with the (recently created) National University of Vanuatu.

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D’Ambrosio, U. (2001). Ethnomatematics: Link between tradition and modernity. Sense Publishers. Deacon, A. B. (1927). The Regulation of Marriage in Ambrym The Journal of the Royal Anthropological Institute of Great Britain and Ireland, 57, 325–342. Deacon, A. B. (1934). Malekula: A vanishing people. Routledge. Deacon, A. B., & Sherman, M. (2015). Some string figures from Vanuatu. Bulletin of the International String Figure Association, 22, 146–212. Deacon, A. B., & Wedgwood, C. (1934). Geometrical drawings from Malekula and other Islands of New Hebrides. Journal of the Royal Anthropological Institute, 64, 129–175. Desrosiers, S. (2012). Le textile structurel: Exemples Andins dans la très longue durée. Techniques et culture, 58(2), 82–103. Dickey, L. A. (1928). String figures from Hawaii, including some from New Hebrides and Gilbert Islands (Reprint Edition 1985, Germantown, New York ed., Vol. Bulletin n˚54). Bishop Museum. Evans-Pritchard, E. E. (1972). Zande String figures. Folklore, 83, 225–239. François, K., Souza Mafra, J. R., Fantinato, M.-C., & Vandendriessche, E. (2018). Local mathematics education: The implementation of local mathematical practices into the mathematics curriculum. Philosophy of Mathematics Education Journal, 33, 1–18. Franjieh, M. J. (2012). Possessive classifiers in North Ambrym, a language of Vanuatu: Explorations in Semantic classification. Ph.D. thesis, SOAS, University of London. Franjieh, M. J. (2018). The languages of northern Ambrym, Vanuatu: An archive of linguistic and cultural material from the North Ambrym and Fanbyak languages. SOAS, Endangered Languages Archive. Retrieved December 10, 2020, from https://elar.soas.ac.uk/Collection/MPI1143013 Galliot, S. (2015). Le tatouage samoan et ses agents: Images, mémoire et actions rituelles. Gradhiva, 21, 159–183. Gerdes, P. (1992). Pitágoras Africano: Um estudo em cultura e educação matemática. Instituto Superior Pedagógico. Gerdes, P. (1995). Une tradition géométrique en Afrique. Les dessins sur le sable, tome 1. L’Harmattan. Gladwin, T. (1986). East is a big bird. Navigation and logic on Puluwat atoll. Harvard University Press. Guiart, J. (1951). Sociétés, rituels et mythes du Nord Ambrym (Nouvelles-Hébrides). Journal de la Société des Océanistes, 7, 5–103. Haddon, K. (1911). Cat’s cradles from many lands. Albert Saifer Publications. Hornell, J. (1927). String figures from Fiji and Western Polynesia (Reprint edition 1971, Germantown, New York: Periodicals Service Co ed., Vol. 39). Bishop Museum. Jayne, C. F. (1962). String figures and how to make them: A study of cat’s cradle in Many Lands. Dover Edition. (A reprint of the 1906 edition entitled “String Figures”, published by Charles Scribner’s Sons.) Jenness, D. (1924). Eskimo string figures. Report of the Canadian Arctic expedition 1913–1918, Part B (Vol. XIII). F.A. Acland. Jolly, M. (2003). Spouses and siblings in Sa stories. The Australian Journal of Anthropology, 14(2), 188–208. Landtman, G. (1914). Cat’s cradle of the Kiwaï Papuans, British New Guinea. Anthropos, 9, 221– 232. Lanouguere-Bruneau, V. (2000). L’igname, ni-hnag, une « nourriture sociale et affective » à Mota Lava (îles Banks - Vanuatu). Journal d’agriculture traditionnelle et de botanique appliquée, 42, 81–106. Layard, J. (1942). Stone men of Malekula. Vao. Chatto & Windus. Levi-Strauss, C. (1947). Les structures élémentaires de la parenté. Mouton de Gruyter. Levi-Strauss, C. (1962). La pensée sauvage. Plon. Levy-Bruhl, L. (1910). Les fonctions mentales dans les sociétés inférieures. F. Alcan. Lipka, J. (1994). Culturally negotiated schooling: Toward a Yup’ik mathematics. Journal of American Indian Studies, 33(3), 1–12.

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Lipka, J. (2004). Some evidence for ethnomathematics quantitative and qualitative data from Alaska. In Proceedings of the 10th International Congress of Mathematics Education-ICME (pp. 87–98). MacKenzie, M. A. (1991). Androgynous objects: String bags and gender in Central New Guinea. Hardwood Academic. Maude, H. (1978). Solomon island string figures. Homa Press. Moore, C. G (1988). The Implication of String Figures for American Indian Mathematics Education. Journal of American Indian Education, 28(1), 16–26. Murphy, J. (1998). Using String Figures to Teach Math Skills. Part 2: The Ten Men System. Bulletin of the International String Figure Association, 5, 159–209. Noble, P. D. (1979). String figures of Papua New Guinea. Institute of Papua New Guinea Studies. Pais, A. (2011). Criticisms and contradictions of ethnomathematics. Educational Studies in Mathematics, 76(2), 209–230. Patterson, M. (1981). Slings and arrows: Rituals of status acquisition in North Ambrym. In M. Allen (Ed.), Vanuatu: Politics, economics and ritual in Island (pp. 189–236). Academic Press. Pinxten, R. (2016). Multimathemacy: Anthropology and mathematics education. Springer International Publishing Switzerland. Pinxten, R., Van Dooren, I., & Harvey, F. (1983). Anthropology of space. University of Pennsylvania Press. Rio, K. M. (2007). The power of perspective. Social ontology and agency on Ambrym Island, Vanuatu. Berghahn Books. Rivers, W. H. R., & Haddon, A. C. (1902). A method of recording string figures and tricks. Man, 2, 146–153 (The Royal Anthropological Institute). Tattevin, P. E. (1929). Mythes et Légendes du Sud de l’île Pentecôte. (Nouvelles Hébrides). Anthropos, 24, 983–1004. Tylor, E. (1871). Primitive culture: Researches into the development of mythology, philosophy, religion, languages, art and customs (Vol. 1). John Murray. Vandendriessche, E. (2015). String figures as mathematics? An anthropological approach to string figure-making in oral tradition societies. Studies in history and philosophy of science (Vol. 36). Springer. Vandendriessche, E. (2017). Des pratiques algorithmiques et géométriques propres à des sociétés autochtones : quels usages pour un enseignement des mathématiques culturellement situé ? In A. Adihou, J. Giroux, D. Guillemette, C. Lajoie, K. & M. Huy (Eds.), La Diversité des mathématiques : dimensions sociopolitiques, culturelles et historiques de la discipline en classe (pp. 11–27), Actes du colloque du Groupe de didactique des mathématiques du Québec 2016. Université d’Ottawa. Vandendriessche, E. (2019). A new symbolic writing for string figure procedures. Bulletin of the International String Figure Association, 26, 182–208. Vandendriessche, E. (upcoming-2022). Lévy-Bruhl’s theory on prelogical mentality and its influence on the field of history and philosophy of science: Léon Brunschvicg (1869–1944)/Abel Rey (1873–1940). In K. Chemla & A. Keller (Eds.), Writing histories of ancient mathematics– Reflecting on past practices and opening the future, Col. “Why the Sciences of the Ancient World Matter”. Spinger. Accected for publication. Vandendriessche, E., & Petit, C. (2017). Des prémices d’une anthropologie des pratiques mathématiques à la constitution d’un nouveau champ disciplinaire: l’ethnomathématique. Revue d’histoire des sciences humaines, 31, 189–219. VKS. (2009). Introdaksen blong Sandroing. Documentary, Vanuatu Kaljoral Senta Productions, Vanuatu National Cultural Council & United Nation Educational, Scientific and Cultural Organisation. VNCS. (2010). Vanuatu National Curriculum Statement. Ministry of Education.

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Multimedia References String figures: Construction methods, modelling, cultural and cognitive aspects. https://stringfig ures.huma-num.fr/ String Figures: Cultural and Cognitive Aspects, Construction Methods, Modelling. https://stringfig ures.huma-num.fr/ Vandendriessche, E., & Henri, A. “WAYU, fairy yam”, String figures. Retrieved March 26, 2021, from https://stringfigures.huma-num.fr/items/show/19. Vandendriessche, E., & Henri, A. “OLOL TAAMRE—OLOL TAFEAN, coconut germs upwards— coconut germs downwards”, String figures. Retrieved March 26, 2021, from https://stringfigures. huma-num.fr/items/show/82 Vandendriessche, E., & Henri, A. “WARMUPTA, bread fruit stem”, String figures. Retrieved March 26, 2021, from https://stringfigures.huma-num.fr/items/show/18 Vandendriessche, E., & Henri, A. “RRALNGIEN BURBUR, ears of burbur”, String figures. Retrieved March 26, 2021, from https://stringfigures.huma-num.fr/items/show/110 Vandendriessche, E., & Henri, A. “MEYA, a red fruit”, String figures. Retrieved March 26, 2021, from https://stringfigures.huma-num.fr/items/show/83

Impact of Indigenous Culture on Education in General, and on Mathematics Classes in Particular

Indigenous School Education: Brazilian Policies and the Implementation in Teacher Education Sérgia Oliveira, Liliane Carvalho, Carlos Monteiro, and Karen François

Abstract This chapter describes the history of Brazilian colonization and suppression of indigenous peoples as a necessary context to understand the Brazilian policies on indigenous education and the implementation in teacher education. It summarizes the history of education of indigenous people right up to the present, and it describes a particular intervention in teacher education, involving close collaboration among researchers, indigenous teachers, and a community. The chapter provides a clarification of the use of the term “indigenous” and of the researchers’ status as respectful outsiders and allies. Our place of speech, our research and our voices are of nonIndian researchers in the field of education, who are respectful and concerned about an educational program that considers the cultural diversity of Brazilian original inhabitants. Our reflections are not representative for Brazilian indigenous peoples; our analyzes have limits and do not want to overlap the narratives of indigenous peoples who have greater authority in reflection, in the sense that they are authors, and they have authorship of their history and their struggles for the right to have education. What we—as non-Indian researchers—can add to the debate is to provide an empirical evidence (literature) review and results of our empirical investigations. Our reflections are based on aspects related to the social function that school education associated with an intercultural perspective can contribute to ensure better living conditions considering the realities of each indigenous community. Our focus corresponds to sociopolitical aspects and legal frameworks that are linked to educational issues and their impacts on indigenous peoples. In this chapter we first have a brief contextualization about the original inhabitants of Brazil and their trajectory S. Oliveira · L. Carvalho · C. Monteiro Federal University of Pernambuco, Recife, Brazil e-mail: [email protected] L. Carvalho e-mail: [email protected] C. Monteiro e-mail: [email protected] K. François (B) Vrije Universiteit Brussel – Free University Brussels, Evere, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_4

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to conquer the right of formal education which respects their history and cultures. Secondly, we present aspects of the trajectory of indigenous school education in Brazil based on a literature review, including an overview of contents and ideas related to official policy regulations concerning indigenous school education, and the principles of intercultural education. We also discuss some implications from the recent regulations of Brazilian curriculum. Finally, we will provide an example of an empirical investigation on indigenous school education developed in the period 2014–2016 on the teaching of statistical topics in indigenous schools of the Xukuru of Ororubá people. Keywords Indigenous peoples · Indians · Xukuru of Ororubá people · Brazilian curriculum · Statistics teaching

1 Introduction Our proposal of this chapter is to analyze aspects of indigenous school education as organized by Brazilian public policy and implemented as a result of struggles, organizations and mobilizations of social movements and indigenous communities to having their civil rights recognized and implemented.1 We will use the term indigenous peoples, which contains a plural notion of collective identity and ethnic diversity. The use of this term is not to homogenize nor relativize all subjects and groups to the same dimension, as it is impossible to include all indigenous peoples in one single category, due to the pluricultural existence that different peoples have (Maracci, 2012). Even recognizing that some authors (Minaya & Roque, 2015) from other parts of the world also use the term originary peoples, in this chapter we will use the terms indigenous peoples and Indians, considering the construction of meaning for these words throughout the trajectories of these peoples’ struggles in Brazil. In this chapter we first have a brief contextualization about the original inhabitants of Brazil and their trajectory to conquer the right of formal education which respects their history and cultures. 1

Our place of speech, our research and our voices are of non-Indian researchers in the field of education, who are respectful and concerned about an educational program that considers the cultural diversity of Brazilian original inhabitants. Our reflections are not representative for Brazilian indigenous peoples; our analyzes have limits and do not want to overlap the narratives of indigenous peoples who have greater authority in reflection, in the sense that they are authors, and they have authorship of their history and their struggles for the right to have education. What we—as nonindian researchers—can add to the debate is to provide an empirical evidence (literature) review and results of our empirical investigations as explained below. This way we hope to contribute to the survival and the cultural preservation of ethnic groups in Brazil. Our reflections are based on aspects related to the social function that school education associated with an intercultural perspective can contribute to ensure better living conditions considering the realities of each indigenous community. Our focus corresponds to sociopolitical aspects and legal frameworks that are linked to educational issues and their impacts on indigenous peoples.

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Secondly, we present aspects of the trajectory of indigenous school education in Brazil based on a literature review, including an overview of contents and ideas related to official policy regulations concerning indigenous school education, and the principles of intercultural education. We also discuss some implications from the recent regulations of Brazilian curriculum. Finally, we will provide an example of an empirical investigation on indigenous school education developed in the period 2014–2016. The study aimed to analyze the teaching of statistical topics in indigenous schools of the Xukuru of Ororubá people. The investigation was set up in collaboration with the local community, especially as to how the method was implemented in the school practice (Oliveira, 2016; Oliveira et al., 2018).

2 Socio-historical and Demographic Panorama of the Brazilian Indigenous Population 2.1 Invasion and Destruction of Indigenous Peoples in Brazil It is estimated that around the world there are about 370 million people who self-declare as Indian, distributed among approximately 5,000 different indigenous peoples (Verdum et al., 2019). They represent less than 5% of the world population (ONU, 2019). In terms of their socioeconomic situation, indigenous peoples are classified in the group that refers to 15% of the poorest in the world, which situate them as a socially vulnerable group. Several studies from different knowledge areas have addressed and reflected on different aspects of education related to Brazilian indigenous groups (e.g. Rosa & Coppe de Oliveira, 2020; Silva, 2019). In this chapter, we could not approach all those aspects related to the education of indigenous peoples in Brazil over these 520 years since the first European invaders arrived in Brazil. However, we need to mention some sociohistorical elements in order to contextualize how formal education in Brazil was developed. In 1500, when Portuguese invaded the territory that is now known as the Federative Republic of Brazil, there were over 1,000 ethnic groups, which had a total population estimated between 2 and 4 million of inhabitants. There were about 1,200 different languages (Rodrigues, 2005). The original peoples began to be called índios in Portuguese language, due to the denominations related to expeditions to conquest the West and East Indies. The term índio was utilized to name a “new world” person. The “discovering” of Brazil was also a starting point to a genocide of millions of original people. Several expeditions of European businesspeople, including religious groups, established a colonial model that extracted natural resources and imposed the instruction of European languages and cultural practices. Their population decreased due to constant exterminations, armed conflicts, and epidemics between the sixteenth and twentieth centuries. In the 1650s, that number was reduced to around 700,000

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indigenous people and in the 1950s, it reached 70,000 (Ribeiro, 1957). In the mid1970s, it was believed that indigenous communities would inevitably be susceptible to disappear, due to the tragic historical context. The last national census indicated 817,000 Brazilians classified themselves as indios. There is consensus that first experiences of schooling Brazilian indigenous people were developed by Jesuit priests, who aimed to convert them into their indoctrination. According to Monteiro (1999), the educational action developed by the Jesuits in Brazil followed three guidelines: the conversion of the main indigenous leaders, the elimination of their pajés (shamans) associated with the imposition of priests’ guidance, and the indoctrination of youths. This colonial project makes explicit that school education was introduced to Brazilian indigenous people as an instrument of cultural domination. The main objective was to exercise the colonizer’s power and the cultural exclusion of dominated people, revealing a practice to promote acculturation and loss of ethnic identity. However, the indigenous peoples resisted the annihilation of their identity. The colonial process of globalization was associated with new ways of social organization in power structures which allow very few people to amass greater wealth, to retain knowledge and, in some cases, to dominate other people. D’Ambrosio (2007) argues that the fundamental process for conquest of an individual is to keep him/her inferior, weakening her/his roots, removing her/his history from her/him, eliminating her/his mother tongue, and his/her religion. This process excludes dominated people’s survival strategies and replaces them with dominator’s culture. Indigenous peoples have for a long time experienced an actual domination that excluded their culture and their people. D’Ambrosio (2007) emphasises that indigenous communities that survived from the process of domination, remained as excluded groups with their cultures disguised in the incorporation of the dominator’s culture. A dramatic example is related to the extinction of over one thousand indigenous languages. Nowadays, it is estimated that there are 188 living languages with 155,000 speakers in total (IBGE, 2010). According to Carvalho (2003), ethnic identity corresponds to the individual recognition that each one has an individual identity. This ethnic identity reflected at the same time the own social identity. Castro (2015) argues that an Indian can be any member of an indigenous community, recognized by the latter as such. An indigenous community is any community founded on kinship or co-residence relations between its members, who maintain historical-cultural ties with pre-Colombian indigenous social organizations. During the colonial period, there were no formal education institutions in Brazilian territory. Until the nineteenth century only the wealthiest population who lived in main urban centers had access to formal education. Brazil became “independent” from Portugal in 1822, when it began a monarchy until 1889. Then a military cup founded a republican period. It is important to mention that until 1889 slavery was legal in Brazil. To that end, millions of black people joined thousands of indigenous people, lower classes of white people, and rural populations had almost no access to formal education and citizenship.

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Silva (2000) explains that in the middle of nineteenth century, when Brazil began to build itself as a nation, the indigenous people were not considered part of society, since they resisted accepting the domination of the “non-indigenous”. After the Constituent Assembly of 1823 it was ruled that the indigenous people needed to be integrated into the Brazilian people’s way of life through education, moral and religious instruction, and work. Félix (2008) argues that the educational policy for indigenous peoples had an integrationist objective, which in that context aimed at eliminating indigenous customs from the educational environment, in order to make them submissive to the ideal of life which the national state reserved for them.

2.2 Protection of Indigenous Peoples During the twentieth century, indigenous peoples experienced a new scenario with regard to education policies. Félix (2008) explains that the Brazilian State evaluated that unstable coexistence with indigenous people would result in the extinction of these people. Therefore, official institutions were established to intervene in the indigenous population. Oliveira and Freire (2006) exemplify that in 1910 the Service for the Protection of Indians and Location of National Workers—SPILTN was created, which later was called Indian Protection Service (SPI). The SPI had the following objectives: to guarantee the physical survival of indigenous peoples; to promote that Indians gradually adopt “civilized” habits; to enable access or production of economic goods in the indigenous lands; to use indigenous workforce to increase agricultural productivity; to strengthen the indigenous feeling of belonging to the Brazilian nation. According to Félix (2008), the SPI implemented schools that were called “Indian houses”. Initially, these institutions attempted to convert the indigenous people through religious education, in several Brazilian states. In a second phase these “schools” emphasised a professional training curriculum. Oliveira and Freire (2006) state that in order to achieve its objectives, the SPI sought to remove the church from the responsibility of educating the indigenous people and sought to adopt educational techniques to try to control the indigenous, transforming them into workers. Therefore, the country became legally responsible for managing indigenous people’s lands and lives. However, instead of protecting them, the national state collaborated in the invasion of their territories. As Félix (2008) emphasised, such an educational approach showed how Brazilian public policy aimed at the disappearance of indigenous cultures, because the state did not respect their cultural backgrounds, and the indigenous people had to incorporate the type of life which was imposed on them. Oliveira and Freire (2006) explain that SPI faced several accusations on collaboration with the genocide of indigenous people and on corruption. In the mid-1960s, a Parliamentary Commission of Inquiry found evidence for the crimes and decided that the SPI was to be terminated. In order to continue the Brazilian state tutelage of indigenous people, in 1967 the federal government created the National Foundation of Indian—FUNAI, which initially worked with the same principles of the extinct SPI.

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Ferreira explains that under FUNAI actions, the statute of Indians was implemented in 1973, which gave civil and political rights to Brazilian indigenous people, as well as the possibility of legal regulations of their lands, education, culture, and health. The statute of Indians allowed the organization of institutional education as bilingual schools when located in indigenous territories (using the native language of the group and Portuguese) in order to alphabetize Indians. However, at those indigenous schools the pedagogical practices were remarkably similar to those of non-indigenous schools, since educational targets aimed at the integration of national values for indigenous people. The indigenous schools in Brazil began to have small improvements with the support of the Indigenous Missionary Council—CIMI, an organization created in 1972 by the National Conference of Bishops of Brazil, CNBB—, which fought together with the Indians in order to guarantee their rights, including the access to a differentiated school which suits the different cultures of indigenous peoples (Félix, 2008). During the 1970s, throughout the country, indigenous peoples mobilized intensely to implement their rights, initiating what we call the indigenous movement. The different indigenous groups organized meetings in which they were able to promote political decision-making and share inter-ethnic experiences (Aguilera, 2001). The national indigenous movement was associated with several regional indigenous organizations, such as: the Union of Indigenous Nations—UNI; Federation of Indigenous Organizations of the Upper Rio Negro—FOIRN, and the Association of Indigenous Peoples of the East, Northeast, Minas Gerais and Espírito Santo—POINME (Oliveira & Freire, 2006). All of them claimed the demarcation of lands and the autonomy of indigenous communities to manage their activities, because at that time the FUNAI tutored these peoples. Félix (2008) argues that the increasing articulation between Brazilian indigenous groups and non-governmental organizations empower them to confront dominant society, but they were also persecuted by squatters of indigenous lands, which resulted in many deaths. The struggle of the indigenous movement promotes recognition for civil society of the problems that these communities faced. The indigenous people, who according to the official narrative were doomed to disappear, moved into a period of strengthening. Aguilera (2001) states that one of the meetings which gathered indigenous peoples had a specific aim to discuss issues related to education. This resulted in a publication entitled “Indigenous Education and Literacy” (Meliá, 1979). This book helped to confirm for the Brazilian society how much these communities were re-elaborating educational processes for the development of their people, based on their own cultural practices. The people were aware that indigenous school education can be an instrument of resistance and power. The struggles of the indigenous movement strengthened public policies, such as proposals for the right to have access to intercultural education.

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3 The Implementation of Indigenous School Education in Brazil The indigenous movement strengthened throughout the country and paved a way to a discussion for a new indigenous school, which guaranteed cultural identity preservation, where curricular contents would be adapted to the reality of each indigenous people. However, the implementation of changes needed to have a legal status. Therefore, the Union of Indigenous Nations—UNI, allied with civil organizations, began to pressure the National Constituent Assembly representatives to propose a chapter for the new constitution which would approach the situation of indigenous peoples (Oliveira & Freire, 2006). With the promulgation of the democratic Brazilian Constitution (1988), many changes and innovations were directed towards indigenous communities. Among them, we can mention that for the first time, the Brazilian state recognizes in a constitutional text, indigenous peoples as citizens and assures them the right to full citizenship, freeing them from the tutelage of national state, as well as the recognition of their different identity. The indigenous communities were legally recognized, and therefore the Union has the obligation to respect and protect their social organization, languages, beliefs, traditions, and rights (Henriques et al., 2007). From a conceptual point of view, there is a difference between indigenous education and indigenous school education. The term indigenous education refers to the institutionalization of experiences, knowledge, and educational processes typical of indigenous societies, which, added to the scientific knowledge of non-indigenous culture, integrate the educational environment in various ways. Therefore, indigenous education represents the process by which each community, based on a particular way of being, guarantees its survival and reproduction through the internalization of the teaching and learning of values, and social and cultural standards in everyday life. Consequently, it is seen as a socialization process based on traditions, designated by all members of the community, as there is no specific institution for its realization (Scandiuzzi, 2009). On the other hand, indigenous school education is the result of struggles of these communities for the right to have an education based on an intercultural perspectives, which contributes to the strengthening and visibility of indigenous culture at school, as well as breaking with integrationist practices and thus strengthening and preserving the distinct cultures (Aguilera, 2001). Therefore, indigenous school education enabled new meanings as it could guarantee access to general knowledge, without having to deny cultural specificities (Araújo, 2006). FUNAI oversaw indigenous schools in collaboration with the states and municipalities. The Presidential Decree nº 26 (1991) designated the Ministry of Education (MEC) to manage all levels and modalities of indigenous school education, having the legal responsibility to carry out the provision of this public policy (Félix, 2008). In order to expand the dialogue and define actions, MEC instituted a National Committee for Indigenous Education, which aimed to open a space for dialogue

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with representatives of indigenous communities, universities and non-governmental organizations (Henriques et al., 2007). The indigenous school education was included in Law 9.394 (1996), which establishes the Guidelines and Bases of Education (Lei de Diretrizes e Bases da Educação—LDB). This law recognized the sociocultural and linguistic diversity of indigenous communities, affirms freedom in learning, in the plurality of pedagogical ideas and conceptions. Its realization can be reached on the basis of bilingualism, which allows the use of the native language of each people in the educational process (Carvalho, 2003). LDB highlights the importance of developing autonomy in the elaboration of teaching methodologies and the use of specific didactic materials, according to the needs of each indigenous people. The LDB also provides that it is up to the Brazilian states to authorize, create, recognize, accredit, supervise, and evaluate indigenous schools. The National Curriculum Reference for Indigenous Schools—RCNEI (MEC, 1998) was published to contribute with guidelines on the development of teaching materials and actions to be carried out in indigenous schools across the country (Grupioni, 2002). Nascimento et al. (2016) emphasize that Law No. 9,394 (1996) and RCNEI (MEC, 1998) demarcated important advances in the issue concerning the effectiveness of intercultural education, transforming it into a state policy. However, these documents do not provide a definition of what should be understood as interculturality or intercultural education, nor do they guide educationalists in how it should be practiced in communities. Nascimento et al. (2016) conceptualize interculturality as a dialogic process in constant transformation, that occurs through interactions between different people and/or groups from specific cultures. The term intercultural education has been used in policies aimed at indigenous peoples to stimulate and expand the debates around the valorization of diverse knowledge that come from different Brazilian ethnic groups. Tubino (2005) explains that in indigenous school education, there are two perspectives of understanding interculturality as an educational approach: functional and critical. The functional perspective corresponds to a field that seeks social harmony through a dialogical opening between the different groups, however, it is encouraged that the “subordinate” groups assimilate the hegemonic culture, without a problematization about the power relations. The critical perspective focuses on questioning the differences and inequalities which were socially constructed and imposed. The debates involving this perspective seek, mainly, the construction of a truly democratic society, in which differences are assumed and respected through coexistence based on egalitarian relations. The agents involved in indigenous school contexts may use elements in their practices which indicate these two intercultural perspectives, depending on the local context, the sociocultural relations which happen in school daily life, and teachers’ understanding of interculturality and intercultural education (Nascimento et al., 2016).

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The demands of indigenous peoples for their right to have an intercultural education is based on a critical perspective in favor of breaking with an educational imposition, which does not consider the specificities and needs of each community. In this sense, indigenous school education is configured for different communities to be a democratic instrument of cultural resistance within the school (Aguilera, 2001). Therefore, the inclusion and adoption of intercultural educational practices makes it possible to contribute to an educational scenario that enhances the reduction of social injustices, the formation of critical people and the ethnic strengthening of each indigenous group. Murabac Sobrinho et al. (2017) state that from the end of the 1990s, indigenous school education had other achievements with the approval of favorable laws, including the creation of the category of indigenous teacher (Parecer nº 14, 1999). However, indigenous movements and other segments of society continued to fight for other achievements such as the recognized and assured right of specific courses for indigenous teachers’ education in Brazilian Universities (Medeiros, 2014). It was published in the Referential for Indigenous Teachers Education—RFPI (MEC, 2002), which provides guidelines for Higher Education. In 2005, MEC launched a program that had as main objective to support the indigenous teacher education programs implemented at federal and state public higher education institutions across the country (Medeiros, 2014).

4 The Advances and Setbacks of Indigenous School Education in Brazil Between 2003 and 2016 Brazil has developed several affirmative actions and public policies marked by democratic propositions, based on greater openness for discussions with different sectors of national society. Brazil experienced a process of improvement of formal education with public policies which promoted teacher education at different levels. Governmental programs and projects provided expansion of public higher education institutions and supported the universalization of compulsory basic education for children and adolescents from groups which were excluded from the educational system. All these governmental actions aimed to promote the inclusion of people belonging to socially disadvantaged groups to access basic and higher education systems. For example, during this period there was the official recognition of Quilombola School Education (Resolução CNE/CEB nº 8, 2012). The specific school education is to provide to quilombola populations value Afro-Brazilian and African history and culture, as well as respect for ethnic-racial relations (Monteiro et al., 2019). This name is related to the quilombos which were social organizations of resistance of slavered Africans. Depending on the reading of history, those people were considered as “fugitive slaves” or people who lived in fraternity, freedom, and solidarity (Arruti, 2008). In other parts of Latin America, the quilombola people are also called by maroons or cimarrons.

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According to the Organization for Economic Cooperation and Development (OECD, 2011), the educational programs placed Brazil at the level of a country that invested the most in this area. For example, investment in basic education increased by 121% between 2000 and 2008. Government actions to expand the schooling of indigenous people were based on debates across the country with indigenous organizations. These actions culminated in the implementation of public policies on indigenous peoples at basic and higher education levels. The expansion of indigenous school education was linked to socio-educational aspects offered by public policies that supported affirmative actions to reduce inequalities for these ethnic groups. In 2010, there was a significant average growth of 7.3% in the number of enrollments of indigenous students in all years of basic education, making a total of 246,793 enrollments. High school was the segment that had the highest enrollment rate, with an increase of 45.2% (IBGE, 2010). In comparison with the total enrollment of national basic education, the number of enrollments for indigenous school education corresponded to 0.5%. Despite appearing as a small percentage value, this index refers to education that follows norms and guidelines in favor of intercultural and bilingual education (IBGE, 2010). According to the results of the 2018 school census, it was possible to verify the existence of 3,345 indigenous schools from all over the country and the performance of 22,590 teachers in these institutions. The data showed that a total of 255,888 enrollments were registered that year (MEC, 2019). The year 2016 was marked by a moment of ruptures in the federal government that led to the seizure of power by a party coalition with a different political orientation. The discontinuity also ended a period in which indigenous peoples had several of their rights recognized and implemented. The federal government has revised and revoked various political, economic, cultural and social rights of the majority of the population, including some of the achievements of indigenous peoples. Although indigenous movements continue to be strong and active, in recent years, several Brazilian institutional forces have deliberated actions to silence indigenous voices through decisions that seek to weaken protection and assistance policies for these communities, such as noncompliance with demarcation and protection of lands, dismantling of protection agencies, reduction of personnel and budgets and rendering inactive educational programs aimed at indigenous school education. Therefore, the current discussion of indigenous school education must be accompanied by reporting on the weakening of policies to protect and to support indigenous peoples. In order to allow indigenous students to continue having the right access to school education, these communities still experience many adversities in their daily school life. The 2018 educational census pointed out that there are many basic structural problems and inequalities among indigenous schools in different parts of Brazil. For example, only 54% of schools in the North have electricity, while in the South this percentage is 100%. As for access to a sanitary sewer system, only 39% of the institutions in the North have this resource, while the South and Southeast have respectively 98% and 90%. Regarding the physical structures that support learning, it was found that only 6.8% of indigenous schools have computer labs, 0.5% have

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science teaching laboratories, 8% have a library and only 14.7% have access to the internet (MEC, 2019). In addition to the challenges specifically related to indigenous school education, many communities have been facing concrete problems, such as: territorial and environmental invasions and degradations, sexual exploitation of children and adolescents, enticement and drug use, labor exploitation, including child labor, as well as a high level of poverty and a disorderly exodus which has caused a large concentration of indigenous people in the different cities of Brazil (Verdum et al., 2019). Most worrying is that this political rupture created the opportunity for sectors of the ultra-right to consolidate publicly, creating an environment of threats for the various spheres of public policies conquered and established over 25 years ago by the most vulnerable populations. Specifically in the educational field, in a period of 2 years (2016–2018), there were major paradigm changes that took shape, from cuts in public funds, closure of public teacher training policies and reforms that do not consider the specificities and challenges of vulnerable populations, including indigenous peoples. For example, the publication of the BNCC (MEC, 2018) and the Reform in Secondary Education (Lei nº 13.415, 2017) do not systematically contemplate the interculturality that is so important for indigenous school education. These reforms do not address indigenous school education specificities. Due to the great cultural plurality of indigenous populations, the construction of BNCC and, at the same time, the reform of secondary education is presented as a great contemporary challenge to meet the right of these communities to a differentiated education. The elaboration of the BNCC did not consult the representatives of indigenous communities and in general it was imposed by the federal government. Therefore, there is no indication of how the BNCC can adapt to the plural specificities of indigenous peoples, which has an impact on guaranteeing differentiated education. From 2016, federal governments were changed and began to dismantle the public policies of the ministry of education. As a consequence of these actions, social conflicts have intensified in a negative way, involving several elements of which discussion is beyond the scope of this chapter, but which needs to be mentioned to contextualize that many social rights conquered in recent decades are being suppressed. The current federal government, supported by members of national institutions and sectors of the population, has defended actions which are deemed unimaginable in democratic countries. For example, groups and individuals have been encouraged to express their sympathy for fascist movements or for the return of the military dictatorship in Brazil. Individuals and groups feel more empowered today to act with prejudice towards more vulnerable people such as poor, black, and LGBT people. Since 2016, indigenous peoples have also been the target of attacks from groups, politicians and businesspeople who systematically disseminate factually false arguments about indigenous’ rights. Since January 2019, the current federal government has been deliberately acting to deconstruct the rights and guarantees of social policies aimed at indigenous peoples. Education was the area that suffered most from the constraints. The budget available for investments in this area was initially cut by 25%, from R$ 23.3 billion to

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R$ 17.5 billion (Prata, 2019). This financial blockade in education was the target of protest across the country and, through popular pressure, the government chose to reduce the budget freeze by using a reserve fund in the amount of R$ 5.4 billion (Prata, 2019). However, there were cuts in scholarships for higher education, Youth and Adult Education (EJA) and support for Basic Education infrastructure that have not yet been fully restored. These investment funds in education are fundamental in contributing and directing improvements and adaptations of schools across the country. Cutting them contributes to widening the gaps imposed by social injustice that leave indigenous communities, quilombolas and low-income students as the main victims of these measures. In addition to the cuts in the financial level, the current federal government charged a bill on the home schooling regime that leaves parents in charge of the education of the pupils and the National Literacy Policy, in which it prioritizes the fight against illiteracy by means of “phonic method”, which understands that children need to identify the segments of sound that form a word (Prata, 2019). These measures are sometimes not perceived by the majority of the population, but they have an evident impact on education, because these measures are elitist and do not consider the specificities of indigenous education. During the first 18 months of the current federal government, two ministers of education have already been dismissed due to measures explicitly against education, victimizing teachers and vulnerable sectors of society. These setbacks in the Brazilian educational system seem to be implemented without a clear planning, only as measures to dismantle the educational system and to reach organized sectors of the population that remained critical and resistant to an exclusionary society model. We corroborate the ideas defended by Cortinaz (2019), who emphasizes that in the educational field everything is interconnected with political issues of interest, since the act of educating is intrinsically a political action. For this author, besides the curricular definitions, school knowledge and theoretical references are never neutral, since issues of social inclusion and exclusion are strongly linked to them. For this reason, it is important to expand the debates on the policies and curricular references. Therefore, we believe that this new Brazilian socio-political panorama has been dismantling the educational area that reverberates in the organizational, financial and ideological sphere, a development that does not contribute to the development of science, nor with the education of critical citizens and widens social inequalities. This historical context demonstrates indifference, segregation, and marginalization that these communities still face. We can observe that although there is a continuous progress in the expansion of enrollments, difficulties and inequalities in the educational field continue to perpetuate. Only the opening of education offers does not solve the obstacles related to affirmative policies, since the effective guarantee of access to indigenous school education requires discussions and actions that break with selective traditions (Cortinaz, 2019). It questions, mainly, the guarantee of access and opportunities to high quality education that has curricular proposals centered on the subjects’ local issues. In the next section we will present the results of an empirical investigation of a particular case: the Xukuru of Ororubá people.

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5 An Experience of Collaboration with Xukuru of Ororubá Teachers The Xukuru of Ororubá people is a population of more than 12,000 indigenous people (Leal & Andrade, 2013) who live in a 27.555 ha (275.55 km2 ) Indigenous Territory (TI) demarcated in 2001 (Silva & Garcia, 2019). They live in 24 villages and in some regions of the municipalities of Pesqueira and Poção located in Agreste of the state of Pernambuco (Silva, 2017). According to their leader Chief Xikão, the name Xukuru of Ororubá means the Indian’s respect for nature (Almeida, 1997). The Xukuru people strengthen their ethnic identity and reinforce their struggles through a socio-political organization composed of the Chief, Vice Chief, Pajé, Council of Representatives of the Villages, Indigenous Health Council and Council of Indigenous Teachers Xukuru do Orurubá (COPIXO). The teachers’ council works to facilitate the debate on school aspects, especially those related to intercultural education that seeks to guarantee the autonomy of the Xukuru of Ororubá people. Therefore, The COPIXO acts through meetings and teacher education, outlines goals for the academic year of its 41 schools, linking the official educational curriculum with specific aspects of its culture, in order to reach the needs of the community and minimizing social contradictions (Oliveira, 2016). In the Xukuru of Ororubá people schools, pedagogical practices have enabled the visibility and preservation of their cultural characteristics through intercultural practices in educational processes. The teachers’ council discusses the school education activities considering the pedagogical project of Xukuru schools, which corresponds to the values and desires of the community. Therefore, the regular contents foreseen for Basic Education are approached, contextualizing Xukuru specific cultural knowledge. The educational processes take place in socialization spaces such as the house, churches, yards, courtyards, plantations, forests, rituals, and festivities, aiming fundamentally for students to experience what it means to be Xukuru inside and outside the schools. In order to enter the research field, it was necessary to carry out previous talks with indigenous leaders and present our research proposal in meetings with COPIXO so that they could evaluate and/or approve of the study. Upon authorization to carry out the research, we were allowed to participate in the meetings with the teachers’ council to explain our objectives and to meet teachers who volunteered to contribute to the study. In an initial stage of interlocutions and visits to the villages, we observed that the Xukuru people developed dialogical and collaborative practices within the community, since all decisions are discussed and defined widely with all people. In the field, we also understand that in the Xukuru schools there is still a demand in the area of mathematics for new investigations in order to subsidize study materials and guidance for teachers. With the observation of these particularities, our methodological option for working with a collaborative approach emerged, with a view to contributing to the continuous teacher education in the early years of elementary school. In addition

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to these aspects, in our ethnographic study, participant observation, interviews and document analysis were used as research instruments. Participant observation and semi-structured interviews contributed to the understanding of the reality of the participants and supported the discussions with the collaborative group. Regarding document analysis, we carried out examinations of teachers’ class diaries to compare to data collected in the interviews and with the notes from ethnographical investigations and observations in the field. The collaborative group was organized by the researcher as a discussion with the indigenous teachers based on the statistical investigative cycle (Cazorla & Santana, 2010; Guimarães & Gitirana, 2013; Rumsey, 2002). This method consists of five stages: Problem, Plan, Data Analysis, Conclusion, (sometimes abbreviated to the PPDAC cycle). The problem section is about a statistical question, and the decision about the data collection (what, who and why). The plan is about the method of data collection. The data section is about data management and organisation. The analysis is about exploring and analysing the data (including data representation and visualisation). Finally, the conclusion is related to the initial question posed in the problem section and based on the reasons and arguments as developed during analysis. This statistical investigative cycle is originally based on the idea of Wild and Pfannkuch (1999). The collaborative group was organized as four face-to-face meetings lasting 4 h each, in which data were collected through video recording of all meetings. Eleven teachers were involved in this collaborative group. The purpose of the teacher education meetings was to discuss aspects of statistics education related to the specific aspects of the community. At each meeting, the group of teachers decided on the content that would be worked on at the subsequent meeting. Therefore, it was decided collectively that the systematization and socialization of the activities carried out by each teacher would be developed through an implementation of a course plan on the contents of statistics, in which each teacher should also prepare a report to describe the actions that were developed with the activities carried out by the students. During the last meeting, the indigenous teachers socializing activities, in the sense that each teacher reported the work done with students and at the end an individual assessment was carried out about the development of activities. Regarding the results, three teachers gave an interview. All of them graduated in Pedagogy and stated that they teach about statistics contents and propose activities on the construction and interpretation of bar graphs. Two of these teachers reported that they never studied statistics curricular contents, when attending the undergraduate course. Only one mentioned that during the pre-service teacher education instructions about these contents, but she considered that the guidelines to teach statistics were insufficient. The analyses of data from interviews suggested that the teachers often develop their pedagogical approach through the promotion of intercultural practices and that all of them experience difficulties with regard to the construction of graphs: for example, they constructed graphs without sufficient basic elements such as axes, labels and titles. Two teachers offered their class diaries to be analyzed in the context of this study. We could detect that both participants elaborated lesson plans emphasizing, in a decreasing order, the subjects of Portuguese, Mathematics, Sciences, History,

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interdisciplinary classes, Geography and Arts. Regarding statistics topics, these are estimated in a minimalistic way, since only a few activities were found on the bulk of topics of numbers and operations, which are the most prominent contents in the curriculum of mathematics. The analysis of collaborative group meetings indicated that through discussions, readings and practical activities with the contents of statistics it was possible to enhance, throughout the process of continuing education, the statistical knowledge and literacy of the participating teachers. Teachers developed pedagogical projects with their students, based on the statistical investigative cycle (as explained above). During the planning of these projects, the teachers chose as their main theme the water resources of the Xukuru of Ororubá people territory. The theme was generated from a discussion on water problems the villages were facing, such as supply shortages and water pollution. The teachers found this issue important in order to relate the formal and technical aspects of the statistical investigative cycle (Problem, Plan, Data, Analysis, Conclusion) to the daily concerns and problems of the pupils worked with. It further enhanced the awareness about access, use and preservation of water resources in the communities. Finally, teachers prepared proposals with indications of interdisciplinary work associating the contents of statistics with other curricular subjects, such as natural sciences, geography, history, as well as analyzing the realities of each village and addressing specific elements of the Xukuru culture. The Xukuru people still preserve some elements of their ancestral religion, in which water is associated with deities. The proposal contemplated the use of a pedagogical project in statistics as an intercultural tool, as it can enhance the relationship between knowledge of curriculum content and the experiences and local challenges of indigenous communities. Specifically, regarding the experience of preparing and carrying out a lesson plan with the contents of statistics, we can say that all teachers carried out this step with relevance to the specific reality of each village. In this sense, we observed that it was a particularly important stage in motivating and engaging the teachers to carry out a teaching practice with activities of critical reflections with their students. Finally, the participants of the collaborative group assessed that the meetings were important for individual and collective teacher education, since they supported a dialogue between theoretical knowledge and community life contexts to allow critical learning. We conclude that the teachers carry out the educational processes in the schools of the Xukuru of Ororubá people through a pedagogical practice that allows a greater visibility and preservation of their cultural specificities. We believe that the existence of a Teachers’ Council helps to think about pedagogical action as an ongoing process that, being discussed and formulated collectively, helps to meet the community’s wishes. Finally, we highlight that intercultural education within this community, allows for educational processes not to be limited to the school institution. Rather, it takes place in different spaces of socialization such as the house, courtyards, plantations, rituals and festivities. All of this joins in so that socio-cultural identity is strengthened. Therefore, through teaching, learning relationships expand to the entire community (Oliveira et al., 2018).

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6 Final Considerations In this chapter we approached a brief overview of the construction of legal frameworks for indigenous school education in Brazil, and discussed how the country’s social and political dynamics intertwines with the conjuncture of struggles and demands of indigenous communities, in favor of the implementation and maintenance of the right to have an intercultural education based on respect for diversity. Therefore, we emphasized the importance of formal school education being carried out by combining knowledge and cultural specificities that each indigenous community has in order to enhance the development of critically conscious people. Due to the current Brazilian socio-political horizon, we believe that indigenous school education still faces many obstacles in favor of social justice. We understand that one of the great challenges is to break the idea of a simplistic perspective of interculturality, which homogenizes distinctive indigenous cultures. We argue that it is necessary to defend interculturality within a critical and questioning perspective about inequalities that are socially perpetuated, in a way that promotes the opening of greater dialogues between different knowledges, without approaching differences as being seen as inequalities, but as a plurality that values and respects the particularities of each community. Brazil provides an example of how a pro-active government in power at the national level can push forward the cause of the indigenous peoples, as it did in recent decades, and how an autocratic regime can turn the tables by passing antithetical legal measures and dismantling the institutional structures put in place, as it happened in the last few years. This brings to the forefront the serious issue of political power play way past the colonial history when political power slips into the hands of agents with colonial mind sets in post colonial era. A comparative study of such phenomena can provide a good background for analysis of the intertwining forces in operation in different countries. This could be part of a future research agenda on indigenous education. Although the current national context which makes social and political dynamics of school education more difficult, we understand that many positive experiences have been followed by different indigenous peoples, in an attempt to minimize social injustices and to value the multicultural wealth they have. In this context, we hope that by resisting arbitrariness, many debates, and attitudes of collective resilience to end educational inequalities can be enhanced. We foster an education perceived as a public good, a fairer educational system in which the subjects can fully develop and live their citizenships and democracies in an increasingly diverse and intercultural world. The context of ethnic and cultural pluralities requires the teachers’ new attitude and conceptualization of educational processes, not only related to the statistics curriculum contents, but about the curriculum as a whole. In indigenous schools, we believe that it is essential to think about intercultural education with respect and appreciation for the cultural diversity of students. The educational strategies of mathematics and in statistics for indigenous schools are necessary to provide students

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with an expansion of knowledge within a perspective of predominantly intercultural relations. We highlight the relevance of work with the investigative cycle for Statistical Education within the scope of Indigenous School Education. The research conducted by Oliveira et al. (2018) points out elements that could contribute to give a new meaning to indigenous education in the context of schools, as is the case of the identification of water resources in the village from the report of the older Indians and from the field classes. This approach, because it is based on a critical perspective of statistical literacy, can be enhanced by the collaborative nature of Indigenous Education developed by the teachers of the Xukuru of Ororubá people. Acknowledgements Research funded by CAPES from the Brazilian federal government. Thanks to the reviewers’ helpful comments.

References Aguilera, A. H. (2001). Currículo e cultura entre os Bororo de Meruri. UCDB. Almeida, E. A. (1997). Xukuru filhos da mãe natureza: uma história de resistência e luta. Centro de Cultura Luiz Freire. Araújo, A. V. (2006). Povos Indígenas e a Lei dos “Brancos”: o direito à diferença. MEC/UNESCO. Arruti, J. M. A. (2008). Quilombos. In O. S. Pinho & L. Sansone (Eds.), Raça Novas Perspectivas Antropológicas (pp. 315–350). EDUFBA. Carvalho, I. M. (2003). Diversidade étnica e educação indígena: políticas públicas no Brasil. Interações, 4, 85–93. Castro, E. V. (2015, January 27). No Brasil, todo mundo é índio, exceto quem não é. Instituto socioambiental—ISA. https://pib.socioambiental.org/files/file/PIB_institucional/No_ Brasil_todo_mundo_%c3%a9_%c3%adndio_old_version27012015.pdf Cazorla, I., & Santana, E. (2010). Do tratamento da informação ao letramento estatístico. Litterarum. Constitution of the Federative Republic of Brazil. (1988). http://www.planalto.gov.br/ccivil_03/ leis.htm Cortinaz, T. (2019). A construção da Base Nacional Comum Curricular (BNCC) para o Ensino Fundamental e sua relação com os conhecimentos escolares [Doctoral thesis, The Federal University of Rio Grande do Sul]. LUME digital repository. https://lume.ufrgs.br/handle/10183/ 202032 D’Ambrosio, U. (2007). Etnomatemática: elo entre as tradições e a modernidade. Autêntica. Decreto nº 26, de 4 de fevereiro de 1991. (1991). Dispõe Sobre A Educação Indígena no Brasil. Brasília. Félix, C. (2008). Entre conflitos e convívios: aspectos das políticas de educação escolar indígena no Brasil. Revista HISTEDBR, 30, 98–118. Grupioni, L. D. B. (2002). Programa Parâmetros em Ação Educação Escolar Indígena. MEC. Guimarães, G. L., & Gitirana, V. (2013). Estatística no Ensino Fundamental: a pesquisa como eixo estruturador. In R. E. Borba & C. E. Monteiro (Eds.), Processos de ensino e aprendizagem em Educação Matemática (pp. 93–132). Universitária UFPE. Henriques, R., et al. (2007). Educação escolar indígena: diversidade sociocultural indígena resinificando a escola. Ministério da Educação - Secretaria de Educação Continuada, Alfabetização e Diversidade – MEC-SECAD. Instituto Socioambiental. (2019 November 18). Quantos são? https://pib.socioambiental.org/pt/c/ 0/1/2/populacao-indigena-no-brasil

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Law nº 9,394, de 20 de dezembro de 1996. (1996). Estabelece as diretrizes e bases da educação nacional. Diário Oficial da União, Brasília. http://www.planalto.gov.br/ccivil_03/leis/L9394.htm Law nº 13.415/2017, de 13 de fevereiro de 2017. (2017). Altera as Leis nos 9.394, de 20 de dezembro de 1996, que estabelece as diretrizes e bases da educação nacional. http://www.planalto.gov.br/ ccivil_03/_ato2015-2018/2017/lei/l13415.htm Leal, C., & Andrade, L. (2013). Guerreiras: a força da mulher indígena. Centro Luiz Freire. Maracci, M. T. (2012). Povos indígenas. In R. S. Caldart, I. B., Pereira, P. Alentejano, & G. Frigotto (Eds.), Dicionário da educação do campo. Expressão Popular. Medeiros, L. M. B. (2014). Licenciatura intercultural indígena no centro acadêmico do agreste da ufpe: uma visão do egresso do curso 2009–2012 [Master Dissertation, The Federal University of Pernambuco]. Attena UFPE digital repository. https://repositorio.ufpe.br/handle/123456789/ 11804 Meliá, B. (1979). Educação indígena e alfabetização. Loyola. Minaya, G., & Roque, J. (2015). Ethical problems in health research with indigenous or originary peoples in Peru. Journal of Community Genetics, 6, 201–206. Ministério da Educação. (1998). Referencial Curricular Nacional para as escolas indígenas RCNEI. MEC. Ministério da Educação. (2002). Referenciais para a formação de professores indígenas - RFPI. MEC. Ministério da Educação. (2018). Base Nacional Comum Curricular. MEC. Ministério da Educação. (2019 April 19). MEC trabalha por avanços na educação escolar indígena. http://portal.mec.gov.br/busca-geral/206-noticias/1084311476/75261-mec-tra balha-por-avancos-na-educacao-escolar-indigena Monteiro, C. E. F., Duarte, C. G., Carvalho, L. M. T. L., Almeida, A. Q. G., & Diniz, A. M. R. (2019). Mathematics education and quilombola education: Reflections by teachers on the challenges to ethnic-racial equity. International Journal for Research in Mathematics Education, 9(1), 61–72. Monteiro, J. M. (1999). Armas e armadilhas. In A. Novaes (Org.). A outra margem do Ocidente (pp. 237–249). Cia. das Letras. Murabac Sobrinho, R. S., Souza, A. S. D., & Bettiol, C. A. (2017). Educação escolar Indígena no Brasil: uma análise crítica a partir da conjuntura dos 20 anos da LDB. Poésis, 11(19), 58–75. https://doi.org/10.19177/prppge.v11e19201758-75 Nascimento, R. N. F., Quadros, M. T., & Fialho, V. (2016). Interculturalidade Enquanto Prática na Educação Escolar Indígena. Anthropológicas, 27(20), 87–217. OECD. (2011). Education at a Glance 2011. Secretary-general of the OECD. http://www.oecd.org/ education/skills-beyond-school/48631582.pdf Oliveira, S. A. P. (2016). Educação estatística em escolas do povo Xukuru do Ororubá [Master Dissertation, The Federal University of Pernambuco]. Attena UFPE digital repository. https://rep ositorio.ufpe.br/handle/123456789/18717 Oliveira, J. P., & Freire, C. A. R. (2006). A Presença Indígena na Formação do Brasil (Vol. 2). MEC/SECAD/LACED/Museu Nacional. Oliveira, S., Carvalho, L., Monteiro, C., & François, K. (2018). Collaboration with Xukuru teachers: Reflecting about statistics education at indigenous schools. Revista de Educação Matemática e Tecnológica Iberoamericana – EM TEIA, 9(2), 1–15. https://doi.org/10.36397/emteia.v9i2. 237665 Organização das Nações Unidas. (2019, August 9). Em dia mundial, ONU defende direito dos povos indígenas a definir estratégias de desenvolvimento. https://nacoesunidas.org/em-dia-mun dial-onu-defende-direito-dos-povos-indigenas-a-definir-estrategias-de-desenvolvimento/ Parecer nº 14, de 14 de setembro de 1999. (1999). Diretrizes Curriculares Nacionais da Educação Escolar Indígena. Prata, P. (2019). Propostas para a educação: o que já foi feito pelo governo Bolsonaro? https://pol itica.estadao.com.br/noticias/geral,propostas-para-a-educacao-o-que-ja-foi-feito-pelo-governobolsonaro,70002857514

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Resolução CNE/CEB nº 8, de 20 de novembro de 2012. (2012). Diretrizes Curriculares Nacionais para a Educação Escolar Quilombola na Educação Básica. Ribeiro, D. (1957). Culturas e línguas indígenas do Brasil. Educação e Ciências Sociais, 2(6), 1–102. Rodrigues, A. D. (2005). Sobre as línguas indígenas e sua pesquisa no Brasil. Ciência e Cultura, 57(2), 35–38. Rosa, M., & Coppe de Oliveira, C. (2020). Ethnomathematics in action: Mathematical practices in Brazilian indigenous. Springer. Rumsey, D. J. (2002). Statistical literacy as a goal for introductory statistics courses. Journal of Statistics Education, 10(3). https://doi.org/10.1080/10691898.2002.11910678 Scandiuzzi, P. P. (2009). Educação Indígena X educação escolar indígena: uma relação etnocida em uma pesquisa etnomatemática. UNESP. Silva, E. (2000). Resistência Indígena nos 500 anos de colonização. In S. Brandão (Eds.), Brasil 500 anos: reflexões. Universitária/UFPE. Silva, E. (2017). Xukuru: memórias e história dos índios da serra do Ororubá (Pesqueira/PE), 1950–1988 (2nd Ed.). UFPE. Silva, E. (2019). Índios: pensando o ensino e questionando as práticas pedagógicas. Instrumento, 21(2), 168–186. https://doi.org/10.34019/1984-5499.2019.v21.27711 Silva, E., & Garcia, A. D. V. (2019). Discutindo os protagonismos indígenas na aula de história. Fronteiras: Revista Catarinense de História, 34, 61–75. https://doi.org/10.36661/2238-9717.2019n34. 11107 Tubino, F. A. (2005). La interculturalidade Critica como proyecto Ético Político: In: encuentro continental de educadores Agostinos. http://oala.villanova.edu/congresos/educacion/lima-ponen02.html Verdum, R., Lima, D., Amorim, F., Burger, L., Rodrigues, P., & Silva, V. A. (2019). Silenced genocides. IWGIA/GAPK. Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223–265. https://doi.org/10.1111/j.1751-5823.1999.tb00442.x

Indigenous Mathematical Knowledge and Practices: State of the Art of the Ethnomathematics Brazilian Congresses (2000–2016) Maria Cecilia Fantinato and Kécio Gonçalves Leite

Abstract This chapter focus on the works addressing indigenous mathematical practices and knowledge throughout the five editions of the Brazilian Congress of Ethnomathematics (CBEm). By aiming to identify theoretical and methodological trends, main themes and the representativeness of indigenous authorship, this study reflects on possible and necessary relations between ethnomathematics and anthropology. A bibliographic research of the state-of-the-art type has been developed, the time frame adopted—2000 to 2016—covers the years of the first and last edition of CBEm. Having defined indigenous as the main criterion for the selection of contributions the works have been analyzed according to five subcategories for analysis of the selected works, namely: researched indigenous ethnicities, indigenous authorship, main themes, references from anthropology, and mention of ethnography as a methodological option. From the universe of 450 contributions presented in the five editions of CBEm, 69 were selected considering the indigenous theme. Results indicate proportional growth over the years, jumping from 4% in CBEm1 to 22% in CBEm5. 32 indigenous ethnic groups had some form of representation at the events, only 10% of the entire ethnic diversity of indigenous peoples in Brazil. From the third edition of CBEm, there is a growing presence of indigenous authors in the works presented at the event, related to the expansion of ethnomathematics in Brazil, initially concentrated in graduate programs and research groups, and spreading towards undergraduate programs and especially for initial teacher training programs. The increasing presence of indigenous authors in CBEm shows an inversion in the researcher’s speech position in relation to the knowledge and practices researched with indigenous peoples in the country and has the potential to contribute to theoretical innovations of conceptions on ethnomathematics itself. From the reading and qualitative analysis of the selected sample four main themes were identified, indicated explicitly or implicitly by the authors in their productions: Indigenous (mathematical) knowledge, Teacher training, Curriculum, Culture and M. C. Fantinato (B) Fluminense Federal University (UFF), Niterói, Brazil e-mail: [email protected] K. G. Leite Federal University of Rondônia (UNIR), Porto Velho, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_5

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cultural identity. Although 63% of the analyzed works have references from anthropology, having cited 59 anthropologists from different theoretical currents, there is a tendency to reduce these references along the last two editions of CBEm. This occurs simultaneously with the increase of indigenous researchers’ presence at the events. These results allow us to suppose that, in the current phase of development of ethnomathematics in Brazil, the growing participation of indigenous researchers as authors of works that deal with their own peoples’ knowledge, generates relative independence in the results of anthropological studies, indicating a distance tendency between ethnomathematics and anthropology in the country. Besides this, it has been noted that only 18% of all analyzed works call themselves ethnographic. Thus, it can be said that the dialogue between ethnomathematics and anthropology, although existing since its origins, needs to be deepened in Brazil, and constitutes an opportunity to give greater density to ethnomathematics research and better appropriation of ethnographic research. Keywords Ethnomathematics · Indigenous knowledge and practices · Brazilian congresses · State-of-the-art · Ethnographic research

1 Introduction In this chapter, we analyze the works addressing indigenous mathematical practices and knowledge throughout the five editions of the Brazilian Congress of Ethnomathematics (CBEm). Our aim is to identify theoretical and methodological trends, and main themes in the indigenous cultures. Starting from this section of the proceedings we reflect on possible and necessary relations between ethnomathematics and anthropology. In general, ethnomathematics in Brazil is associated with a line of research in Mathematics Education, which investigates the cultural roots of mathematical ideas from the way they occur in different social groups. In this sense, ethnomathematics has interfaces with other areas, notably with anthropology and sociology, seeking to identify mathematical problems from the knowledge of the other, in its own rationality, terms and contexts. This is a relatively new area of research. It is still under construction. However, as indicated by Vandendriessche and Petit (2017), although the first publications in ethnomathematics are from the 1970s and 1980s, it is possible to trace the emergence of a mathematical anthropology among ethnological, mathematical and philosophical studies from the nineteenth century. The contact of European anthropologists with colonized countries increased the interest in the study of the counting and measurement systems of others, as well as the geometric features in cultural artefacts—baskets, vases, body paintings, etc. However, an evolutionist perspective of society and knowledge, with a positivist orientation, still prevailed at that time. Hence, these first studies were seen as markedly ethnocentric and took Western mathematics as a frame of reference. Non-Western peoples’ knowledge was seen as primitive.

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After Malinowsky (1922), systematic ethnographic fieldwork with deep immersion in the life and universe of those studied became the norm (Strathern, 1987). This line of research produced an epistemological turn, which stimulated the development of research seeking to identify the mathematical knowledge of these populations, and to understand their own mathematical rationality (Vandendriessche & Petit, 2017). As a consequence, early ethnomathematics were associated with the anthropology of mathematics. The field has developed significantly since then, worldwide. One of the first and main theorists of ethnomathematics is a Brazilian—Ubiratan D’Ambrosio (1985, 2001). He has played a significant role in the development of this line of research. Fiorentini and Lorenzato (2006) claim that this is the research area for mathematics education in which Brazil has stood out most internationally. However, the first researchers in ethnomathematics had a mathematical background, and aimed at didactic improvements. This perspective “remains at the heart of most of the works that now claim to be ethnomathematics and whose main objective is to promote mathematical education based on indigenous knowledge” (Vandendriessche & Petit, 2017: 212) and is part of an activist approach aimed at the decolonization. The ethnic, linguistic and cultural diversity of over 300 indigenous peoples in Brazil constitutes a broad field for research. This theme has been present in Brazilian studies in ethnomathematics since the 1980s. Eduardo Sebastiani Ferreira’s (1990) pioneering works in the nineties, with indigenous communities in the Upper Xingu and Amazonas, are a benchmark for all academic production in the field (Knijnik, 2004). Since then, other researchers (Amancio, 1999; Bello, 1995; Scandiuzzi, 1997) and more recently Silva (2012) and Severino Filho (2012), among others, have dedicated themselves to indigenous education. The congresses in ethnomathematics have contributed both to internal reflection in the field and to its consolidation. Such consolidation process involves the development of studies and research along different thematic axes. This chapter aims to present a brief overview. This is a state-of-the-art bibliographic research building on previous work (Fantinato, 2013; Fantinato & Leite, 2020; Leite, 2017). We start by reviewing the five Brazilian congresses on ethnomathematics, and we point to the trends that developed as a result of each congress influencing the next congress. Then, we provide some statistical surveys showing the numerical growth of the works, the proportion of works on the indigenous themes per event (congress), the central themes of the events and of the main lectures, and the elements that contribute to characterize the research movement in ethnomathematics in Brazil. In order to provide these survey results, we have followed up the slow but significant growth of indigenous authorship in this production and analyzed some factors that explain the changes over the years. We identified the anthropological references cited by the authors in order to determine the extent to which the researchers rely on these references, especially with regard to fieldwork of an ethnographic nature.

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2 The Brazilian Ethnomathematics Congresses Since the first studies in ethnomathematics, in the 1980s this field has been developing and consolidating itself in Brazil. Specific national congresses, which take place every four years, can serve as a portrait of the main research trends in this field, from a historical point of view. In this section, we present a summary of the characteristics of the five congresses held between 2000 and 2016, highlighting how the indigenous theme was present in each of these events. We also map the extent to which the relations between ethnomathematics and anthropology were addressed in their main lectures. The proceedings of these congresses form the literature analyzed in this current research study. The First Brazilian Congress of Ethnomathematics (CBEm1) was held at the Faculty of Education of the University of São Paulo, in November 2000. This Congress was not organized around a general theme, but papers had to address at least one of the following themes: rural education, indigenous education, caiçara1 education, urban education, artisanal practices, youth and adult education, environmental education, critical mathematical education, groups of professionals, and/or theoretical aspects. This thematic plurality was aiming to attract researchers from different ethnomathematics trends, as well as teachers from the education networks. In the proceedings, out of the 73 contributions only 3 treated indigenous themes. According to the general coordinator of the event, CBEm1 represented an attempt “to look at ethnomathematics in its multiple faces, as a social production of knowledge and an agent of inclusion” (Domite, 2000: 1). This concern with the multiplicity of perspectives was reflected in the organization of the lectures, addressing diverse themes, such as the theoretical aspects of ethnomathematics or the debate on concepts central to the theoretical-epistemological reflection of the field. The predominant ethnomathematics perspective in CBEm1 was that associated with the study of socio-cultural groups, seeking to give visibility to the mathematics practiced by these different groups (Duarte, 2009). The dialogue with other fields of knowledge contributing to the theoreticalmethodological delimitation of ethnomathematics—such as anthropology, history and philosophy—was one of the hallmarks of CBEm1. The theme of the Opening Lecture, The notion of culture, was debated by the ethnomathematician Eduardo Sebastiani Ferreira, the psychologist Marta Kohl de Oliveira, the philosopher Antonio Joaquim Severino, and the anthropologist Neusa Gusmão. Relations with anthropology and ethnography were provided in the lecture of Marcio D’Olne Campos (“To be here” and “to be there”: Tensions and intersections with fieldwork). The indigenous theme appeared in one of the discussion forums. The indigenous presence in this congress was still marginal, with representatives of some ethnic groups from the state of São Paulo selling their own handicrafts during breaks such as coffee breaks between the main activities. 1

Traditional inhabitants of the coast of the Southeast and South regions of Brazil, formed from the miscegenation between Indians, whites and blacks.

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All in all, CBEm1’s pioneering spirit, as the first Brazilian Congress of Ethnomathematics, represented an important contribution to the field, stimulating the approach of the different Brazilian research groups that already existed and the formation of new ones. The Second Brazilian Congress of Ethnomathematics (CBEm2) took place in April 2004, on the campus of the Federal University of Rio Grande do Norte, in Natal. The introductory text of the proceedings highlights its role as “one more step in the consolidation of Ethnomathematics as an area of knowledge” (Morey, 2004: 7). Unlike the previous event, the CBEm2 structure prioritized the meeting of all participants of the event in plenary sessions, instead of myriad lectures. Of the five round tables, four were dedicated to the dimensions of ethnomathematics, according to D’Ambrosio (2001): the political—Ethnomathematics and political issues—, the epistemological—Ethnomathematics and Epistemology—, the methodological— Ethnomathematics and fieldwork—and the educational dimension—Ethnomathematics and teacher education. The indigenous issue had a prominent place in the fifth round table, which was called Indigenous Ethnomathematics. The CBEm2 opening lecture was given by Arthur Powell, from Rutgers University. The presence of this black researcher, critical of the Eurocentrism of mathematical education (Powell & Frankenstein, 1997), converged with the indication, during the closing plenary of this event, of the need to expand studies on ethnomathematics and Africanity. CBEm2’s proceedings register 56 papers, with 7 of them on indigenous themes. No main lecture addressed the relationship between ethnomathematics and anthropology specifically, but some guests at the round table on fieldwork did discuss ethnographic research (e.g., Sílvia Regina Ribeiro, in the lecture Ethnomathematics: Methodological options for field research). During this event, different theoretical and/or methodological perspectives on ethnomathematics emerged due to the plurality of conceptions of ethnomathematics, acknowledged by researchers in the field (Conrado, 2005). Among the milestones of CBEm2, we can mention the launch of two books (Knijnik et al., 2004; Ribeiro et al., 2004), which became important references in the field. “From a political point of view, the decision of the final assembly of CBEm2, to create the Brazilian representation of the International Study Group on Ethnomathematics (ISGEm), deserves to be highlighted” (Fantinato, 2013: 151). The Third Brazilian Congress of Ethnomathematics (CBEm3) took place in March 2008, at the Faculty of Education of the Fluminense Federal University, in Niterói. Its general theme was Ethnomathematics: new theoretical and pedagogical challenges. Seven thematic axes were defined for submission of contributions: mathematical education in different cultural contexts; ethnomathematics and teacher training; ethnomathematics and ethnosciences; ethnomathematics and its theoretical foundations; ethnomathematics research; ethnomathematics and the classroom; and/or ethnomathematics and history of mathematics. From the 97 papers presented, 12 addressed indigenous themes.

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In its structure CBEm3 maintained the CBEm2 plenary round tables, each addressing one of the significant axes of research in the field. During the discussion forums participants were divided into subgroups, mediated by an experienced researcher. The relations between ethnomathematics and anthropology were addressed, in a way, in the opening lecture of the congress, Ethnoscience, ethnography and local knowledge, given by Marcio D’Olne Campos. At the roundtable on teacher training the indigenous issues were brought up both by Jackeline Rodrigues Mendes (2009), and by the indigenous school teacher João Lira Guarani. The growth in the number of contributions to CBEm3 seems to reflect the formation of new research centers in Brazil, as well as the growth in scientific output (Fantinato, 2013). This congress led to an important political landmark: the creation of the Brazilian Association of Ethnomathematics (ABEm), with Maria do Carmo Santos Domite as president. The Fourth Brazilian Congress of Ethnomathematics (CBEm4) took place at the Federal University of Pará, in the city of Belém, in 2012. The theme of the congress—Culture, Mathematical Education and School—highlighted the relationships between mathematics, knowledge and practices of cultural groups and their role in the school context. The context was the backdrop of the diversity of the Brazilian socio-cultural wealth. In addition CBEm4 wanted to take the educational dimension of ethnomathematics as an important focus (D’Ambrosio, 2001): the teaching/learning process, teacher training, school formation, and new educational perspectives in different socio-cultural groups. This congress also highlighted the relationship between mathematical education and Amazonian culture. 114 papers were presented at the event, of which 23 reported research with indigenous peoples. At CBEm4, presentations were organized along four thematic axes: Axis 1: Ethnomathematics and Education of the Peoples of the Forest; Axis 2: Ethnomathematics and Rural Education; Axis 3: Ethnomathematics and Relations between Trends in Mathematics Education; and Axis 4: Ethnomathematics and Education for Inclusion. The indigenous theme had a prominent place in CBEm4. The discussion on the anthropological concept of culture in ethnomathematics was prioritized in this event, with lectures by Maria da Conceição Xavier de Almeida (Notes on the concept of culture: Contributions to ethnomathematics), Jane Felipe Beltrão (Culture, school education and ethnomathematics: Possibilities in/for Pan-Amazon) and Alexandre Pais (The limits of culture). The Fifth Brazilian Congress of Ethnomathematics (CBEm5) was held at the Federal University of Goiás, in the city of Goiânia, in 2016. This congress sought to bring a global vision of ethnomathematics. It gathered people “concerned not only with how to teach and learn mathematics, but with how to conceive it around different contexts, times, cultures and inter- and intra-articulations”.2 At CBEm5, presentations were held in four groups, namely: WG 1—Ethnomathematics, educational practices and teacher training; WG 2—Ethnomathematics theoretical and philosophical foundations; WG 3—Ethnomathematics in different sociocultural contexts; and WG

2

https://cbem5.ime.ufg.br/p/12821-apresentacao. Accessed on October 13, 2020.

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4—Ethnomathematics research methodology. A total of 110 papers were presented at the event, 24 of which reporting research on/with indigenous peoples. One of the differentials of this congress was the participation of representatives of indigenous and quilombola3 people movements, either as guests or as researchers in training. They presented research related to indigenous school education. The round table Quilombola and indigenous knowledge in dialogue with Ethnomathematics and Ethnomusicology, e.g. with the participation of indigenous researcher Gilson Ipaxiaawyga Tapirapé, was one of the representative spaces of the indigenous presence at CBEm5. The participation of many students of Indigenous Intercultural Licentiate degrees from the Midwest region also distinguished the congress from the previous events, in terms of presence and indigenous authorship. During CBEm5, the ethnomathematicians community voted for the Brazilian representative of the Brazilian sector of the Latin American Ethnomathematics Network (RELAET). Olenêva Sanches was elected. Researchers from several countries in Latin America started RELAET-Brazil, which replaced the Brazilian Association of Ethnomathematics. The Sixth Brazilian Congress of Ethnomathematics (CBEm6) was scheduled for May 2020, but due to the COVID-19 pandemic, it was postponed with no date set as yet. CBEm6 will take place in the city of Palmas, State of Tocantins, in the northern region of Brazil. In this region there is a large proportion of traditional populations, whether indigenous or quilombola. Accordingly, the program of this congress already foresees in two round tables about the study subjects and demands of these peoples: Ethnomathematics and the mathematical knowledge of indigenous peoples and Ethnomathematics and quilombola mathematical knowledge. Submissions to CBEm6 must cover one of four thematic axes, the first of which is Ethnomathematics and School Education of Native and Traditional Peoples. Expectations are that research on indigenous ethnomathematics will be substantial.

3 Methodology The present chapter is an instance of bibliographic research (Moreira & Caleffe, 2006) of the state-of-the-art type, focusing on Brazilian research in ethnomathematics with indigenous peoples. This is a complementary study to research of the same type conducted by the authors (Fantinato, 2013; Fantinato & Leite, 2020; Leite, 2017). Besides ours, we also refer to the existence of other surveys on Brazilian ethnomathematics, such as Conrado (2005), Costa (2012), Martins and Gonçalves (2015), Rosa and Orey (2018), Oliveira (2018), and Fantinato and Silva (2019). State-of-the-art research provides an opportunity to map and systematically analyze academic production in a given field of knowledge, trying to identify theoretical, methodological and thematic trends that stand out in different times, spaces and institutions. To this end, they are based on “an inventive and descriptive methodology 3

Descendants of runaway Afro-Brazilian slaved people.

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of academic and scientific production on the theme that it seeks to investigate, in the light of categories and facets that are characterized as such in each individual work and the set of them, under which the phenomenon starts to be analyzed” (Ferreira, 2002). We define indigenous as the main criterion for the selection of contributions. We also adopt five subcategories for analysis of the selected works, namely: researched indigenous ethnicities, indigenous authorship, main themes, references from anthropology, and mention of ethnography as a methodological option. The time frame adopted (2000–2016) covers the years from the first to the last edition of CBEm. As a source of data, we use printed (CBEm1 and CBEm2) and digital (CBEm3, CBEm4 and CBEm5) Proceedings, as well as the institutional websites of the events. To identify the works with the indigenous category, we proceeded to read the printed proceedings and to scan the digital ones using the search tool developed by Adobe Systems for documents in Portable Document Format (PDF), in an open standard maintained by the International Organization for Standardization (ISO) and read by the free Acrobat Reader software. We identified 69 contributions in the indigenous category, out of a total of 450 presentations in all. To map the subcategories in the contributions, we prepared a spreadsheet with the following data: type of work, names of the authors, ethnic identity of the authors (indigenous or non-indigenous), title of the work, CBEm edition, indigenous ethnicities cited in the work, main research theme, anthropology references cited in the work, and mention of ethnography in the method section. The data for each of these subcategories were obtained from the reading and filing of each of the selected works. For those authors who did not explicitly mention their indigenous identity, we consulted the respective curricula of the Lattes Platform of the National Council for Scientific and Technological Development (CNPq) and the enrollment lists of indigenous students in intercultural licentiate degrees, available on the institutional websites of the universities offering such programs. The research theme dealt with in some of the papers was identified immediately from reading the abstracts or in the body of the texts. However, some of the studies analyzed included more than one research theme or did not indicate the theme explicitly. In these cases, we looked for the kinship of the work with others and classified them accordingly. Thus, it was possible to identify four main research themes in the set of works that make up this section of CBEm’s Proceedings: Indigenous mathematical knowledge, Teacher training, Curriculum, and Culture and cultural identity. The results were mapped in tables, including a reflection on the relationship between ethnomathematics and anthropology.

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4 Results This section brings some statistical results of the bibliographic research carried out, referring to each of the analytical subcategories listed for the work. From the data organized in tables, it is possible to identify some relevant trends for ethnomathematics in Brazil, especially regarding research with indigenous peoples.

4.1 Works on Indigenous Themes From the universe of 450 contributions presented in the five editions of CBEm, 69 were considering the indigenous theme. Table 1 shows the general number per event (Congress), the number on indigenous themes in each modality per event, and the percentage with an indigenous theme in each of the five CBEm editions. From Table 1, we can see that researches on indigenous knowledge, or indigenous education, have occupied a significant place in the events, showing proportional growth (jumping from 4% in CBEm1 to 22% in CBEm5). This result corroborates Leite (2017), showing the growth in Brazil of research on Indigenous School Education, “caused by an increasing capillarization of Ethnomathematics in teacher training courses, in a movement that goes from graduate to undergraduate education” (Leite, 2017: 12). Another factor that may have contributed to the expansion of academic production registered at CBEm from 2000 to 2016 is the consolidation of new research groups focusing on ethnomathematics as the main theme or line of research. The very locations of the CBEm editions may have contributed as well to this trend: the North and Midwest regions of Brazil hold the largest concentration and the greatest diversity of indigenous peoples in the country. Finally the creation of new intercultural licentiate degrees for training of indigenous teachers in Brazilian public universities, precisely in the period from 2000 to 2016, added to this trend. Table 1 Works in CBEm’s Proceedings from 2000 to 2016 Event

Works Works on indigenous themes Communication Poster Report Table Forum Total Percentage of all papers (%)

CBEm1

73

1

0

0

1

1

3

4

CBEm2

56

4

0

0

3

0

7

13

CBEm3

97

10

1

0

1

0

12

12

CBEm4 114

18

5

0

0

0

23

20

CBEm5 110

12

7

4

1

0

24

22

Total

45

13

4

6

1

69

15

450

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4.2 Ethnicities Represented in the Works We found that 32 indigenous ethnic groups had some form of representation at the events, with the number of works carried out with each ethnic group indicated in parentheses: Guarani Kaiowá (10), Guarani (6), Tapirapé (6), Kaingang (4), Krahô (4), Tupinikim (4), Apinayé (3), Xerente (3), Rikbaktsa (3), Ticuna (3), Guató (2), Javaé (2), Kadiwéu (2), Karajá (2), Xavante (2), Ashaninka (1), Bororo (1), Enawene Nawê (1), Kalapalo (1), Karajá-Xambioá (1), Karipuna (1), Wajana (1), Krahô Canela (1), Paiter (1), Paresi (1), Pataxó (1), Ramkókamekra (1), Tapuia (1), Tembé (1), Waimiri-Atroari (1), Wari (1) and Xacriabá (1). In this count, the ethnic groups explicitly mentioned in the works were considered, and sometimes the presentations refer to more than one of them or generically mention the term indigenous in a general way. We think it is significant that ethnic groups are mentioned by name in research reports, since it is a way of legitimizing such groups, instead of using the term indigenous in general. An example of a text in which the various ethnic groups were well characterized and cited by name is Monteiro and Souza Filho (2012), a work of initial training of indigenous teachers in the state of Tocantins, encompassing the Apinayé, Javaé, Karajá-Xambioá, Karajá, Krahô, Krahô Kanela and Xerente ethnic groups. Some of the works that do not specify ethnicity consist of teacher training experiences with different ethnicities together (Menezes, 2016), or are educational proposals aimed at indigenous education in general (Gonçalves & Batista, 2016), or reports of educational experiences in which indigenous and Afro-Brazilian cultures were taken into account, in compliance with Law 11645.4 Although we identified 32 indigenous ethnic groups cited in the works, this number represents only 10% of the entire ethnic diversity of indigenous peoples in Brazil. Therefore, most of the cultures, languages and knowledge of the country’s indigenous ethnic groups have not yet been included in works presented at CBEm. As the proceedings of the main national ethnomathematics event reflect, in a way, the production trends in the field itself, we can conclude that there is still a large universe of indigenous knowledge of a mathematical nature that are practically unknown in the country, which will require many new researches and studies with the respective peoples.

4.3 Indigenous Authorship In our study, we tried to register the works of indigenous authorship, either individually or in partnership, and we obtained the results in Table 2.

4

Established in March 10, 2008, law 11,645 “Changes Law No. 9.394, of December 20, 1996, amended by Law Source: http://www.planalto.gov.br/ccivil_03/_ato2007-2010/2008/lei/l11645. htm. Accessed on 30 July, 2018.

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Table 2 Works with indigenous authorship or co-authorship in CBEm’s proceedings from 2000 to 2016 Event

Communication

Poster

Report

Table

Forum

Total

CBEm1

0

0

0

0

0

0

CBEm2

0

0

0

0

0

0

CBEm3

1

0

0

0

0

1

CBEm4

2

0

0

0

0

2

CBEm5

3

7

1

1

0

12

Total

6

7

1

1

0

15

As shown in Table 2, from the third edition of CBEm, there is a growing presence of indigenous authors in the works presented at the event. From the third to the fifth edition, 15 works were identified with indigenous authors, 1 in CBEm3, 2 in CBEm4 and 12 in CBEm5. This trend of growth of indigenous authorship in ethnomathematics in Brazil is related to at least three relevant aspects. Firstly, there is a change in the researcher’s speech position in relation to the knowledge and practices researched with indigenous peoples in the country. If previously researchers external to the cultures and peoples surveyed were common, we now see researchers from indigenous societies doing the research. There are certainly theoretical and methodological implications that deserve to be better recognized and understood by these experiences of indigenous authorship. For example, indigenous researchers are less subject to the limitations of linguistic and cultural barriers commonly faced by non-indigenous researchers. There is an ethical ownership issue for non-Indigenous researchers since the research should be led and negotiated by the indigenous community. The sharing of worldviews and values is strengthened when presented by indigenous researchers to assist others, indigenous and non-indigenous, to understand epistemological aspects although anthropologists and educators might provide comparative notions for others to understand. Secondly, the increasing presence of indigenous authors in CBEm is related to a phenomenon that we have called capillarization of ethnomathematics in Brazil (Fantinato & Leite, 2020; Leite, 2017). It is a movement for the expansion of ethnomathematics, initially concentrated in graduate programs and research groups, and spreading towards undergraduate programs and especially for initial teacher training programs. Thus, in the 1980s and 1990s the works in ethnomathematics predominantly described research carried out in masters and doctoral programs, but from the 2000s onwards there was a spread through undergraduate programs, especially in teaching degrees. Certainly this phenomenon is explained as well by the change in the profile of teacher educators who make up the teaching degrees. However, there is specificity in relation to indigenous intercultural degrees in public universities across the country, created from the 2000s onwards by inducing public policies of affirmative actions. Such programs focus on the presuppositions of interculturality and the right that the original peoples have to a specific and differentiated school education of the same quality as other programs. Based on these assumptions, the education

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of indigenous teachers in these programs provided opportunities for both the presence of new researchers in academic spaces and the production of new research in various fields of knowledge, including ethnomathematics. This explains why the majority of the indigenous authors in the last editions of CBEm are students or graduates from indigenous intercultural licentiate programs. According to Fantinato and Leite (2020), indigenous intercultural degrees implemented in different regions of Brazil brought significant challenges to universities, as many teachers with indigenous licentiate degrees began to take an interest in indigenous issues and to seek training at the master and doctoral level. “On the other hand, indigenous licentiate undergraduates began to develop research projects in partnership with university professors. This entire movement increased the academic production on indigenous themes” (Fantinato & Leite, 2020: 116), and consequently provided the increasing presence of indigenous researchers in ethnomathematics in the country. Thirdly, the presence of indigenous researchers in ethnomathematics research in Brazil has the potential to contribute to theoretical innovations on conceptions on ethnomathematics itself. In this sense, the position of indigenous researchers as internal members of their own cultures, associated with the perspective of appreciating and promoting traditional knowledge as a way of strengthening identity, has triggered an epistemological inversion. It is a phenomenon observed from the discursive practices of indigenous teachers who link ethnomathematics to ethnicity, so that, through their research on traditional knowledge and practices, they invert the meaning of the description of such knowledge and practices. Thus, when linking ethnomathematics to ethnicity, the set of knowledge that was previously predominantly attributed to indigenous peoples by researchers external to the cultures of such peoples, becomes a set of knowledge claimed by indigenous researchers for their respective peoples. This inversion is possible precisely because the discursive field of research is now also occupied by subjects who belong both to the theoretically represented indigenous universe and to the theorizing instance, i.e. the academy. The figure of the academic-indigenous-professor-researcher thus brings about a necessary revision of perspectives on ethnomathematics, which is no longer an exclusive category of the discourses of non-indigenous subjects. In this sense, based on the increasing presence of indigenous authors of research in ethnomathematics, “ethnomathematics as a construct goes beyond the epistemological dimension of the totalizing character of knowledge itself and acquires political contours of identity affirmation, and therefore is linked to ethnicity because it is assumed as claimed knowledge rather than as attributed knowledge” (Leite, 2014: 332).

4.4 Research Themes From the reading and qualitative analysis of the works that make up the portion of the CBEm Proceedings that we are considering, we identified four main themes, indicated explicitly or implicitly by the authors in their productions.

Indigenous Mathematical Knowledge and Practices … Table 3 Main themes of work with indigenous ethnicities in CBEm’s proceedings from 2000 to 2016

153

Theme

Works

Indigenous (mathematical) knowledge

26

Teacher training

19

Curriculum

12

Culture and cultural identity

10

As shown in Table 3, the most common theme referred to indigenous mathematical knowledge. This category includes research on topics such as quantifiers (Silva & Caldeira, 2012), time markers (Severino Filho, 2012), geometry (Silva, 2000), or measurement systems (Pedro & Oliveira, 2016). In turn, the theme Teacher training ranks second in a number of works in the five editions of CBEm and, associated with the theme Curriculum, reflects the above-mentioned expansion of training programs for indigenous teachers in recent years. It also reflects the historically established close relationship of Brazilian ethnomathematics with researchers, research groups and institutions in education. Fourthly, there are works with the theme Culture and cultural identity. The smaller number of works in this category reflects both the tendency to reduce the presence of anthropological references in the works presented at CBEm, as will be discussed in the next subsection, and the secondary place the culture category has occupied in ethnomathematics research projects with indigenous peoples in Brazil. This result, added to the others presented in this chapter, reinforces the argument that there is a need for a rapprochement between ethnomathematics and anthropology in the country, in view of the great demand for field research yet to be carried out with the majority of Brazilian indigenous ethnicities, to which the concepts of culture and cultural identity that have been discussed secondarily from a quantitative point of view in the works presented at CBEm are therefore central.

4.5 Anthropology and Ethnography in the Works As shown in Table 4, although 63% of all the works analyzed have references from anthropology, there is a trend of reduction of these references in works on indigenous themes presented in the last two editions of CBEm. This phenomenon occurs Table 4 Presence of references from anthropology in works on indigenous themes in CBEm’s Proceedings from 2000 to 2016

Event

Works

Percentage (%)

CBEm1

3

100

CBEm2

3

43

CBEm3

11

92

CBEm4

17

74

CBEm5

9

39

43

63

Total

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simultaneously with the tendency to increase the presence of works by indigenous researchers at the event, as shown in Table 2. Such results indicate that, in the current phase of development of ethnomathematics in Brazil, the increasing participation of indigenous researchers, such as authors of works that deal with the knowledge of their own peoples, generates relative independence from the results of anthropological studies carried out by non-indigenous researchers and, therefore, external to the cultures of the peoples to which the indigenous researchers belong. As internal members of their specific cultures, indigenous researchers share epistemological, cosmological and linguistic perspectives of the researched cultures, which favors the identification, description and interpretation of knowledge and activities of their respective peoples, regardless of research with the same peoples already carried out by non-indigenous researchers. Notwithstanding these trends (reduction in anthropological references and increase in indigenous authorship), 59 anthropologists were cited in the works that make up the present portion of the proceedings of the five editions of CBEm, among which is the indigenous Gersem José dos Santos Luciano Baniwa (currently at UFAM). Table 5 presents the most cited anthropologists. The variety of theoretical references of anthropology present in the texts, which includes the structuralism of Lévi-Strauss, the perspectivism of Viveiros de Castro, the functionalism of Malinowski and the interpretativism of Clifford Geertz, suggests the absence of a theoretical or methodological unity of anthropological orientation in ethnomathematics research with indigenous peoples presented in the five editions of Table 5 Anthropologists most cited in works on indigenous themes in CBEm’s proceedings from 2000 to 2016

Anthropologists cited

Works

Mariana Kawal Ferreira

12

Clifford Geertz

9

Luís Donisete Benzi Grupioni

5

Claude Lévi-Strauss

5

Aracy Lopes da Silva

5

Julio César Melatti

4

Marcio Ferreira da Silva

3

Bartomeu Melià

3

Marta Maria Azevedo

2

Marshall Sahlins

2

Manuela Carneiro da Cunha

2

Helbert Baldus

2

Gersem José dos Santos Luciano Baniwa

2

Eduardo Viveiros de Castro

2

David Maybury-Lewis

2

Bronisław Kasper Malinowski

2

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CBEm. Therefore, there is no theoretical tendency to support on anthropology in the works, but rather different perspectives that are present in the field of anthropology itself. This theoretical variety is a common claim in texts that deal with the foundations of ethnomathematics, as a necessary condition to avoid making it a discipline or as an alert to the possibility of making it an “epistemological cage” (D’Ambrosio, 2018). The references cited in the works share the idea that the centrality of culture is a primary factor in research in ethnomathematics, considering the polysemy inherent to the term culture. In this sense, Meira’s research (2021) has sought to investigate the conceptions of this term in Brazilian ethnomathematical production. Such a fact would justify a necessary original rapprochement with anthropology, especially in research with indigenous peoples, in view of the trend identified in Table 4, regarding the reduction of anthropological references in the works presented at CBEm. Such a tendency towards distancing ethnomathematics from anthropology in Brazil becomes even more obvious when we focus on the methodological procedures described in the studies analyzed. In this regard, we identified that in CBEm1 and CBEm2, none of the 69 studies analyzed mention ethnography, with such mention appearing in 8% of the works in CBEm3, 35% in CBEm4 and 13% in CBEm5. We found, therefore, that only 18% of all works on indigenous themes presented at CBEm call themselves ethnographic. Thus ethnographic methods were being associated with ethnomathematics per se. Some other texts present data resulting from fieldwork, but without mentioning the term ethnography. For instance, Guimarães et al. (2004), in a study about the Ramkókamekra’s numerical representation, describe the methodological procedures adopted in visits to the villages, but do not refer to ethnography. On the other hand, Silva’s text (2000), despite presenting elements about the construction process of the Takãra—the house for men among the Tapirapé, does not describe how the author has obtained data but the elements are immediately associated with Western geometry. Thus, when we consider that ethnomathematics, in its origins in Brazil, was significantly based on fieldwork of an ethnographic type, especially in research with indigenous peoples, the results reported here reinforce the need for reflection on the development and future of the field in Brazil, resuming, if possible, the important relationship with anthropological theories and methods for the development of new studies.

5 Indigenous Ethnomathematics and Anthropology The data exposed here, as a representative sample of research trends in ethnomathematics with indigenous peoples in Brazil, provide a necessary reflection on some important aspects for the national development of this field of knowledge. Considering that Brazil is a multi-ethnic country, composed of more than 300 indigenous ethnicities who speak 274 different languages (IBGE, 2012) and are distributed in 723 territories (ISA, 2020), the main national event in the field (CBEm) shows that

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there is a relatively small number of groups with whom field research has been carried out in the 16 years of the first five congresses. This data points to the need to expand fieldwork in ethnomathematics with other indigenous peoples, with a view to understanding and appreciating the multiple knowledges and their respective local epistemologies not yet researched. To that end, ethnographic techniques and anthropological theories are fundamental to researchers external to the cultures of such peoples. However, they should be free of prejudice vis-à-vis the researched indigenous cultures. In this sense, it is necessary to resume the original dialogue between ethnomathematics and anthropology, reversing the historical trend of distancing between the two areas identified in the present study. The inclusion of anthropology topics in graduate programs and undergraduate courses that produce research in ethnomathematics in the country could contribute to this movement. However, for this research to have an impact on education, research should be carried out under the request and negotiation and with co-researchers from the indigenous group. Another aspect that is no less important for the development of ethnomathematics in Brazil is related to the trend identified in the last congress of CBEm: an increase in the number of indigenous researchers, mainly from intercultural licentiate programs. If this trend is sustained for a long period, which depends directly on public education policies induced from the 2000s, but under constant threat of discontinuity in the current national political scenario, there will certainly be a different qualitative increase in ethnomathematics research with indigenous peoples in the country. Such an increase includes theoretical, methodological and political aspects. From a theoretical point of view, indigenous authorship reverses the discourse position on the researched knowledge, which is no longer attributed to others other than researchers and begin to be claimed as inherent in their own internal cultures. As a consequence, a direct link between ethnomathematics and ethnicity becomes possible, when indigenous researchers claim for their respective peoples the production and mastery of their own mathematical knowledge. This movement would strengthen their identity as well. From the methodological point of view, the internal position of the indigenous researcher can provide a review or reformulation of field research procedures in ethnomathematics in Brazil, especially those of an ethnographic type, in contribution to anthropology itself, following the concepts of “endoethnography, anthropology at home, native anthropology, domestic anthropology, autoethnography, [local and] insider anthropology”5 (Ribeiro, 2018). Such methodological innovations would thus result from the historical processes of social transformations that the country went through during the last two decades of the twentieth century, which was the time of the first national research studies in ethnomathematics. These transformations involved strengthening the indigenous movement, the democratization of access to higher education, the implementation of public policies aimed at specific and intercultural

5

Research methodologies in which the researcher investigates his/her own culture, “usually by conducting fieldwork in one’s own country” (Jackson, 1987).

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educational projects, and culminated in the training of indigenous researchers who start to take over and promote studies in ethnomathematics with their own peoples. The political aspect inherent to indigenous authorship in ethnomathematics research with indigenous peoples in Brazil is directly related to the possibility of questioning historically dominant conceptions of knowledge in general. It also questions views on mathematics in particular, in so far as they are based on Eurocentric perspectives and characterize the process of cultural imposition inherent in colonialism. The figure of the autonomous indigenous researcher in ethnomathematics represents an empowerment and the possibility of breaking with the coloniality of knowledge (Lander, 2005), with a view to epistemic pluriversality (Mignolo, 2017), from the “epistemologies of the South”6 (Santos & Meneses, 2010), thus breaking with the “epistemicide” (Santos, 2007) that resulted from colonialism. The latter historically denied the rational character to all forms of knowledge that do not rely on the epistemological principles of scientific rationality characteristic of modern thought (Santos, 2010). The presence of the indigenous researcher in ethnomathematics represents, therefore, the possibility of decolonizing the production of scientific knowledge in academic pursuits, moving away from the predominance of researchers external to the cultures of the indigenous peoples producing the knowledge described.

6 Final Considerations The results of our research allow us to make some considerations about indigenous knowledge and practices in the production of Brazilian ethnomathematics congresses. The indigenous issue has been addressed with increasing representativeness, both in the form of research reports and reports of pedagogical experiences or teacher training. The initial trend, of ethnographic studies in indigenous communities, has been gradually replaced by the predominance of works focused on teacher training in intercultural licentiate degrees. Over these years, research has followed social changes and the demands of social movements. Brazilian indigenous peoples have had some legal achievements for the consolidation of specific and differentiated school education, with emphasis on teacher training in intercultural degree programs at public universities. For the continuation of this struggle, “it becomes necessary, in addition to the articulation of political movements, in which the peoples themselves emerge as protagonists, the construction of theoretical support, resulting from studies and research focused on specific themes” (Leite, 2017, p. 2). We believe that such support may come from an articulation of results already achieved by research carried out, such as those of 6

According to Santos (2007), Western domination has profoundly marginalized knowledge and wisdom that had been in existence in the global South. He contends that today it is imperative to recover and valorize the epistemological diversity of the world. Epistemologies of the South outlines a new kind of bottom-up cosmopolitanism, in which conviviality, solidarity and life triumph against the logic of market-ridden greed and individualism.

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ethnomathematics, but they may also require further studies and research. The organization and historical systematization of academic production already carried out on a given theme enables advances in the production of related knowledge. Such was our proposal in this chapter. Recently, the greater contact of indigenous populations with the culture of nonIndians brought the need to rethink indigenous education in the villages as a space to strengthen local cultures, as well as the schoolish content. The contributions in our analysis invite a constant reflection on the possible articulations and tensions between traditional and school knowledge. Another emerging trend that is still evolving, but promising, is the indigenous role in academic spaces, both in teacher training courses and in graduate school. This trend, despite the fact that the most recent production of Brazilian ethnomathematics congresses already brings some representatives of it, needs to grow. Despite the representativeness and development of research on indigenous issues, there is still a great need for more research on this topic. The dialogue between ethnomathematics and anthropology, although existing since its origins, needs to be further developed. This can be a fertile way for ethnomathematics, to give greater density to research in relation to cultural aspects and better appropriation of ethnographic research.

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Morey, B. B. (2004). Anais do II Congresso Brasileiro de Etnomatemática—CBEm2. Oliveira, M. A. M. (2018). Aproximações da etnomatemática e interculturalidade nas produções acadêmicas com a temática indígena. Anais do VII Seminário Internacional de Pesquisa em Educação Matemática. Pedro, I. S., & Oliveira, M. A. M. (2016). Práticas e saberes presentes na construção da casa de reza kaiowá: etnomatemática dos Kaiowá do Panambizinho. Anais do 5º Congresso Brasileiro de Etnomatemática. Powell, A., & Frankenstein, M. (1997). Ethnomathematics: Challenging eurocentrism in mathematics education. SUNY Press. Ribeiro, H. L. (2018). O Desafio da Endoetnografia. Ilha Revista de Antropologia, Florianópolis, 20(1), 177–205. Ribeiro, J. P. M., Domite, M. C. S., & Ferreira, R. (Orgs.). (2004). Etnomatemática: papel, valor e significado. Zouk. Rosa, M., & Orey, D. (2018). Estado da arte da produção científica dos congressos brasileiros em Etnomatemática. Ensino Em Re-Vista, Uberlândia, MG, 25(3), 543–564. Santos, B. S. (2007). Reinventar a teoria crítica e reinventar a emancipação social. Boitempo. Santos, B. S. (2010). Um discurso sobre as ciências. Cortez. Santos, B. S., & Meneses, M. P. (Orgs.). (2010). Epistemologias do Sul. Cortez. Scandiuzzi, P. P. (1997). A dinâmica da contagem e Lahatua Otomo e suas implicações educacionais: uma pesquisa em etnomatemática. UNICAMP, (Mestrado), Faculdade de Educação. Severino Filho, J. (2012). Os marcadores de tempo indígenas e a solidariedade entre o ambiente e os povos que o habitam: um olhar etnomatemático. Anais do 4º Congresso Brasileiro de Etnomatemática. Silva, A. (2000). A geometria na construção da Takãra. Anais do Primeiro Congresso Brasileiro de Etnomatemática (pp.162–169). Silva, A. (2012). Alguns aspectos do sistema de contagem A´uwê/Xavante: outros saberes, outros conhecimentos, outras soluções. Anais do 4º Congresso Brasileiro de Etnomatemática. Silva, S. F., & Caldeira, A. D. (2012). Sistema de numeração guarani: aspectos básicos. Anais do 4º Congresso Brasileiro de Etnomatemática. Strathern, M. (1987). Out of context: The persuasive fictions of anthropology. Current Anthropology, Chicago, 28(3), 251–281. Vandendriessche, E., & Petit, C. (2017). Des prémices d’une anthropologie des pratiques mathématiques à la constitution d’un nouveau champ disciplinaire: l’ethnomathématique. Revue d’histoire des sciences humaines—RHSH, 31, 189–219.

Subverting Epistemicide Through ‘the Commons’: Mathematics as Re/making Space and Time for Learning Anna Chronaki and Eirini Lazaridou

Abstract The present study considers mathematics as a matter for re/making space and time for learning in the context of a pedagogic workshop for the commons where the subversion of epistemicide becomes core. The workshop is rooted in an ecology of knowledges, histories and feelings where earth caring, artefact making and affective bodying become central in the rural village scape located in northern Greece at the southeast of Europe. The pedagogic workshop is organized by a group of people in their 30s who, instead of immigrating to the ‘West’ for a secure job, have opted to return to their place of origin striving for ‘the commons’ in their physical and cultural environment. Based on a meta-analysis of reading ethnographic data we denote: first, signs of epistemicide where erasures around local knowledge(s) with land and people make children feel insecure or intimidated; second, evidence of articulating a radical pedagogy for the commons that contributes towards subverting epistemicide through re/making space and time; and third, partial manifestations of mathematical hybridity in knowledge making practices. Keywords Epistemicide · Space and time · The commons · Anthropology · Ecology of knowledges · Mathematics science · Land · Locals · Northern Greece · Southeast Europe · Borders

A. Chronaki (B) University of Malmö, Malmö, Sweden e-mail: [email protected] A. Chronaki · E. Lazaridou University of Thessaly, Volos, Greece e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_6

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1 Epistemicide It is important for children to learn not to be afraid, to combat fear Samantha, 2018, interview data archive

Samantha, a core member of a pedagogic workshop for ‘the commons’ at the rural scape of Panagitsa village, notes through her experience how all too often children embody ‘fear’ making them feel uncertain about their ability to run their life, to know what they want or to have a voice. Sad affects like feeling insecure, unconfident or intimidated resonate with consumption habits in current postcapitalist neoliberal times as has been noted in earlier work in urban educational settings (Chronaki, 2019). Children and adults, even at the rural scape, have become detached from earthly practices of making and crafting and, instead, they approach all kinds of common goods including food, water, clothing, skills or, even, knowledge as commodities marketed with specific exchange value. This ‘modern’ way of living becomes an affective matter not only for people in urban locales but, equally, in rural areas where the internet, the mobile phone and the supermarket create a precarious layer of an assumed ‘common’ global reality. Such versions of globalization serve to erase situated ways of knowing by replacing them with commodified ways of being often detached from earth, place and body. Moreover, this becomes exemplified in hierarchies that privilege scientific knowledge (i.e. mathematical models of data handling, data visualizations, statistics etc.) capable of acting from a distance (i.e. expertise). These often work towards prioritizing science at the expense of local knowledge which is led to redundancy. This slow process of knowledge ‘murdering’ is coined as epistemicide by Santos (2016). Epistemicide, like genocide or, recently, feminicide, becomes a basic contrivance in the long era of imperialist, patriarchal, colonial and, even, decolonial struggles in our assumed as postcolonial times. Past barbaric or feudal contours of exclusion become veritable practices of neo-crypto-colonialism of the present by strategically privileging specific forms of knowledge and styles of living while, simultaneously, making ‘others’ inferior, inefficient or exceeding. The loss of natural resources, cultural heritages, local dialects, traditional skills and ways of knowing have been paradigmatic spaces configuring an assemblage of erasures that exemplify what has been called epistemicide. This has created long term oppression but also emancipative agonistic struggles for reclaiming the right to local land, language and knowledge. In particular, Santos correlates ‘epistemicide’ with ‘cognitive injustice’ exemplifying sources of inequity across ‘different forms of knowledge’ treated through hierarchical matrices (Santos, 2016: 237). Denoting the instrumental role played by scientific knowledge in creating varied injustices, he argues about the inherent limits or even failures of expert knowledge to capture not only ‘the inexhaustible diversity of the world’ but to inform the practices of emancipatory movements in the Global South (ibid: 108). One may note how critical theory scholars related to Frankfurt school such as Adorno and Marcuse have already critiqued

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science as ethically and politically responsible for the catastrophic effects of modernity in humankind. At the same time, SST scholars such as Haraway, Harding or Latour, despite theoretical differences, have interrogated the assumed ‘objectivity’, ‘neutrality’ and ‘purity’ of scientific practices and scientific knowledge. Although Santos does not argue that science is always monolithic, catastrophic or unethical in its harness, he explains that, by and large, it emphasizes a nature/culture separation by a forceful focus to produce ‘exact knowledge, based on mathematized hypotheses about nature and systematic experimental verification’ without examining its predominance in social practices (Santos, 2016: 102). Moreover, he urges, instead, for an expansive epistemic stance where scientific knowledge becomes plural and espouses empirically vindicated claims grounded in ecological systems and situated truths capable to inform emancipation practices. Santos goes further to argue that ‘there is no global social justice without global cognitive justice’ (Santos, 2016: 207, 324) This claim comes close to arguments advanced already by Donna Haraway, Sandra Harding and other postcolonial feminists who maintain that modern institutions of science along with curricular representations and education practices globally are notorious for their incapacity to consider how local actors such as women and people of color have played a crucial part for creating this ‘common’ heritage called technoscience that is always dependent and regenerated with earthly practices (see Chronaki, 2009). In this realm, there is a plead that epistemicide, a form of knowledge death, should and could be subverted by not only recognising science as being one form of knowledge among many but also by creating pedagogic spaces that exemplify knowledge diversity, cultivating equitable dialoguing through an ‘horizon of possibilities’ and acknowledging the critical social role of science in the quest for social justice. Santos and feminist postcolonial science scholars are not alone in these advocations. During the last decades, researchers in the field of mathematics education have been engaged in research inquiring about the assumed neutrality of mathematics curricula and didactic models along with practice-based research. Here, one needs to denote work in varied perspectives aiming to discuss the politics of mathematics education, to advance diverse frameworks for a critical mathematics education, to embrace ethnomathematics as ways to unfold the informal mathematical practices of indigenous or native communities, to encounter the cultural, social, cognitive or linguistic anthropological work concerning in-situ ethnographies and, also, research projects discussing social justice at the intersections of race, gender and ability. In addition, recent studies denote the ethical responsibility of mathematics educators to reconsider mathematical activity in relation to climate change and the anthropocene (Boylan & Coles, 2017) allowing us to consider the need to decenter the human. There is more and more awareness that current curricular representations of mathematical knowledge through word problems or textbook based classroom activity may limit significantly teachers and children in confronting critical societal issues of learning mathematics per se. A mathematics education that fosters openings for rethinking the subject in relation to community and nature through the perspectives of more-than-human world, could contribute to create supportive habits for mathematics as a matter of concern for social justice through sustainable ways

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of living in the world (Wolfmeyer & Lupinacci, 2017). In a similar vein, our study considers how mathematics participates in the context of a pedagogic workshop for ‘the commons’ in the rural scape and becomes reassembled with the land and the locals in sympoietic experiences. The present chapter is organised around five sections including this entry. In the next one, we discuss the ‘Land and Locals’ and we trace specific signs of epistemicide as they unfold in diverse erasures over time. This is followed by a section where agonistic struggles for ‘the commons’ are being discussed as a matter of learning to subvert epistemicide that becomes a matter of concern for the locals and especially the youth. Finally, in the last two sections, we discuss learning and mathematics as re/making space and time for learning within the pedagogic workshop for ‘the commons’.

2 Land and Locals: Tracing Signs of Epistemicide Panagitsa is a mountain village of approximately 700 people, built 700 m high at the southern side of Kaimaktsalan mountain, north of Vegoritida lake in the district of Pella and located in close proximity to the borders of Greek and North Macedonia. Known as a refugee village, it has been inhabited mostly by people who came to Greece in 1922 from East Thrace and Minor Asia and in 1924 from the area of Pontos as the result of geopolitical negotiations for Greek and Turkish population exchange. A wrecked population, forcibly expelled from their homelands, arrived to a ruined village named Oslovo, an ottoman name denoting proselytism of prior inhabitants to muslim religion, and to an already exhausted land from the Macedonian Struggle Wars (1904–1908), the Balkan Wars (1912–1913) and World War I (1915–1918). In these times and spaces, the bordering states of Greece, Bulgaria and Turkey worked through propagandas embedded in colonial and imperialist interests (i.e. European great powers and Ottoman empire) that served to fire territorial, cultural, religious and identitarian struggles. Although the turbulent period was followed by WWII, the civil war and the pain of the first immigration movement of the 60s and 70s in the US, Australia and Germany, the echo of the 1920s lasts until today. It is, still, vivid in harsh identity politics that serve to deny the presence of multicultural and multilingual histories in the broader area. Henk Driessen (1998) righteously argues how life in the territorial borders of South Europe and Meditarannean remains a challenge for anthropologists striving to sense complex histories of people who live together and apart. The Greek refugees who came in this village during the 1920s were often met with hostility or disapproval by the local population who were, by and large, Greek Orthodox Christians too and called themselves ντ o´ π ιoι (:dopioi). The ντ´oπιoι spoke a unique polyglossic dialect ντ o´ π ικα (:dopika) that was a combination of Greek, Slavic and Turkish languages. The words ντ o´ π ιoι and ντ o´ π ικα were instrumentally utilised by the ‘locals’ of that time, who are not the ‘locals’ of present time, to claim for indigeneity (i.e. origination from this land). At the same time, they aimed to

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distinguish themselves as having more rights when compared to other inhabitants of past or present times such as the Turksish inhabitants prior to the Treaty of Lausanne (Shields, 2016), the communities of Sarakatsanoi and Vlachoi or the nomad populations of Roma (Tsigganoi or Gypsy). Despite the fact that heteroglossia prevailed as part of everyday communication, commerce, leisure and cultural activities amongst people, it, still, lives in the oral practices of people in the area (see Cowan, 1998). Language became paramount for denoting not only who can enjoy the assumed privilege of indigeneity but also for negotiating in(ex)clusion practices and, even, creating spaces for racism around territorial, cultural, identity politics that are vivid up until today. In the 1920s, the language use of ντ o´ π ικα was prohibited as the Greek state was trying to establish a homogenous identity around a ‘common’ Greek language. People still feel the pain of being betrayed by the state as they were refused to speak their own languages. Tracing language loss is, perhaps, amongst the most painful signs of epistemicide. One needs to denote how epistemicide as language prevention (refuse or marginalisation) is entangled with narratives of ‘othering’ that move across local groups. They serve to denote radical shifts concerning who are, really, the ‘locals’, but also to note how these narratives change over time. Specifically, whilst the ‘locals’ of last century were the area inhabitants but without homogeneous characteristics constructing as ‘others’ the Greek refugees who arrived from Pontos and minor Asia, today ‘locals’ are considered the descendants of those Greek refugees in the 20s who used to be ‘others’ but have been by now well settled in the village. The ‘others’, today, become mainly the newly arrived immigrants or refugees. Still, borderlines around ‘local’ and ‘other’ remain undiscerned and purposefully captive to current complex cultural or identity politics, weaved with territorial struggles and prone to geopolitical battles. Beyond harsh conditions of expatriation, the newly arrived Greek refugees in the 1920s began their settling, they built a church and renamed the village into Panagitsa that literally means Panagia (αναγια´ or mother Mary). These people had high cultural capital as they valued education, faith, land respect and solidarity values helping them in survival and resilience. Samantha’s account is indicative: These people came from Argyroupoli (called today Gümü¸shane in Turkish) a city in the area of Pontos. Greek people in Argyroupoli had one of the biggest libraries at the time. They were educated. When they had to move to Greece they brought very few things with them. Amongst them were books, many geography and science books and many religious books, and ancient Greek texts. Because these were the things that were important to them. Imagine you are leaving your home place on foot and you don’t have a clue where you are going, what is going to happen and you choose to take books and two huge bells from their church. (Samantha, lines 1030–1044)

Over the years, people have embodied defeat and oppression as they learned painfully that they had no other choice but to practice silently or secretly their languages, cultures and religion. This embodied act turned them, sometimes, speechless and, other times, bitterly betrayed as they watched the national project of reestablishing a unified Greek state identity in the name of language. In this realm, the notion of local, native, autochthonous or indigenous retains its slippery and dubious meanings across time. Panagitsa’s people, who, today, advocate the identity of ‘locals’ aim

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both to revive their ancestral history (i.e. cultural origination from Pontos) and to certify their local land ownership. At the same time, people in neighboring villages, who consider themselves as indigenous Greek Macedonians, strive to identify that they belong to this land, too. Within this terrain of complex identity politics, one needs to add the current waves of refugees who, also, contribute to the already conflictual politics, economy and ideological stances across borders. Despite recent negotiations towards reconciliation, one may still find people who cannot agree, easily, on who should be identified as ‘local’ or ‘other’ making them continue to wonder who belongs where. Nowadays, people who inhabit the village are mostly farmers working in family property or seasonal workers (mainly spring and summer) with a small percentage working in the public domain or in local small-scale businesses. In addition, a newly arrived immigrant population has come from the Balkans during the last three decades, mainly Albania, working in households and in the land. At the same time, the village suffers from what we call ‘brain drain’ as most of its educated youth have immigrated to the west or to the urban cities for a secure job due to the sequential economic crises and the high unemployment rates. This is considered as a second big wave of immigration after the 60s and 70s massive mobility. In this complex historical and political context, some people, mostly well-educated and aware in matters of ecology, have opted towards returning to their homelands and have chosen to live and work with their ancestor’s land in alternative and sustainable ways. Their choice to return and work with the land in the rural scape needs to be considered as a radical political reaction to austerity measures of the recent economic crises. At the same time, it is a risky political experimentation materialized in self-organized collectivities or self-funded cooperatives. Examples include organic farms or other resourcebased cooperatives around community participation, collaboration and networking across the country rooted in initiatives for community development grounded in a philosophy of ‘the commons’. Their agonistic struggles come close to how Escobar (2016) considers ‘the commons’ as spaces towards subverting epistemicide through dialoguing across multiple worlds, diverse forms of knowledge and plural ways of making or, based on Zapatista’s worldview, maintaining the pluriverse.

3 The Commons: A Matter of Learning to Subvert Epistemicide Whilst traditionally, ‘the commons’ have been engaged with the reclaim of enclosures in the context of natural resources, such as land, forests, water and the rules to manage, organize and regulate them in solidarity within communal usage (Giousis, 2011) the term has become enormously transformed during the last fifteen years (Caffentzis, 2010). Modern ways of urban societal living denote the need to expand towards what is coined as ‘new commons’ so that to include immaterial socio-cultural resources, such as information, data, intellectual property, software and the reclaim

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of public space (Singh, 2017). Hardt (2010) adds to these that the production of ideas, images, knowledge, codes, languages, social relationships, affective histories, as well as, the shared infrastructure of public transport, electricity, post, schooling, health etc. All these and even more could be considered as ‘commons’. Today ‘the commons’ may comprise whatever could be considered as people’s products both individual or collective or what might be considered as a generation’s wealth or cultural heritage (Cobarrubias et al., 2014). This broad perspective of ‘the commons’ allows the consideration of educational commons (Means et al., 2017; Korsgaard, 2019; Pechtelides and Kioupkiolis, 2020; Berlant, 2016) as collective endeavors evolving around a double process of learning. On the one hand, educational commons may serve as spaces to learn about ‘the commons’, how to manage and regulate them. And, on the other hand, educational commons may serve as case studies for understanding how a pedagogy with ‘the commons’ is being practiced by making learning possible for a commonist citizenship. Commoners may be conceived as these new subjectivities who become agents of transformative change as they learn to practice resilient relations in solidarity with ‘others’. In this section, we will try to discern this realized need in the rural scape of Panagista in northern Greece as; first learning to work in the land with a philosophy of ‘the commons’ and second seeing ‘the commons’ as a matter of a sympoietic learning process. Concerning the first realized need, the people of Panagitsa, who, by and large, work in the domain of agriculture, have witnessed an additional deprivation to that of language or culture loss. Their land has been transformed from micro-scale family farms based on biodiversity to macro-scale agro-industrial settings through varied policies. It has been turned into a resource for the mass production of specific products (i.e. apples, cherries, peaches). A monoculture that has been recommended by specialist agriculturists following national and European guidelines. These ascribe with the rules of what Escobar (2016) calls the One-World-World (OWW) aiming to convert ‘nature’ into ‘natural resources’ and turn the inorganic or the non-human into objectified property that can be easily extracted, obtained, owned, exchanged and destroyed. These are also effects of a particular land-economy-relationship where the argument utilized, even in the context of crisis, to persuade people to change habits in farming has foregrounded discourses of accumulation, propriety and occupation. In all, the rule of OWW has abrogated the right to be “the” world, a world where only a world fits (Escobar, 2016: 15). Moreover, in the Greek rural context, certain agricultural cooperative organizations organize, sustain and mediate significant infrastructures for the local farmer industry including product packaging, storing, branding, distributing and commercializing. But they cannot compete with a trade market at the national, European and global levels. Moreover, since the 1980s, the EU based agricultural reform policy has promoted monoculture by reducing biodiversity. Such changes had immediate effects in human-nature epistemic and ontic relations. Human domination over a particular piece of land has come to privilege, today, mainly modern scientific expertise of EU-based agricultural policy. A policy that has brought onto-epistemic erasures in local traditional skills and ways of land working. And, although the area enjoys

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a rich natural water resource, high mountains and fertile fields, it faces environmental decay in the realm of intensive reforms. On the top, local farmers cannot see, perceive, articulate or resolve such matters. The recent Common Agricultural Policy (CAP for short), implemented across European countries, has caused huge concern in local municipalities as it highlights the lack of adequate skills for the assumed ‘incompetent’ farmers in modern methods and the need for their continuing training. Although, people do not deny to learn new skills, they know that simply ‘learning new skills’ is not enough. Since, globalized markets, following neoliberal norms of commerce, determine the promotion of particular agricultural products, family farms are being left exposed to wider perils for their lands, social relations and economy (Papadopoulos & Patronis, 1997). This adds to how epistemicide works towards subjecting both land and people into a deadening onto-epistemic relation. Epistemicide, moreover, tends to create geological alterations in the land resembling ontological changes as noted by Escobar (2016). This signifies a specific temporality in which human beings become themselves part of a geological force. Human activity within land as a living space has proved to be, in fact, a much more complex phenomenon that assembles with matter, times and spaces, following what Karen Barad depicts as: entanglements of spacetimemattering (Barad, 2007) a view that is closely connected with recent feminist materialist views of ‘the commons’ and ‘commoning’ (e.g. Clement et al., 2019). Trying to locate ways of subverting epistemicide, some collectives opt, during the last decades worldwide, towards creating small scale co-operative and worker-owned enterprises reviving solidarity relations of the past around ‘the commons’ (Bauwens & Kostakis, 2014; Kioupkiolis, 2020). The neoliberal capitalist system is being considered more and more as ruined and bankrupt. It becomes apparent that it is today a system under crisis that accumulates debt as a form of devastation in ecosystems, emancipatory movements, feminist protests, values and social rights all around the world (Federici, 2018; Graeber, 2014). The long term and continuing crises concerning economy, society, values and environment in Greece and worldwide have brought on the surface the importance of radical collective action for ‘the commons’ of land, air and water. Here, we could denote emancipatory struggles in Skouries for mine-exploitation, in Stagiates for free water, in Evoia, Mesochora, Ipeiros, Agrafa and Tinos for a landscape without wind turbines and the Greek environmental movement against the new Law 4782/2021 that threatens Natura protected areas. All these acts have strong alliances with social movements in Europe, the US, Latin America and the East such as the sociopolitical protest-movement in Hong Kong and farmer protests in India. In the area of rural scape of Panagitsa, a movement against the EU reforms for land was taking place during the 1980s and 1990s. Panagiotis Manikis (see at www.natural-farming. org/en/), a local environmental activist, has set a campaign informing farmers for the need to resist modern land cultivation and, instead, to work with sustainable natural farming that respects traditional ways of working with the land. In a similar vein, Ermis, Samantha, Angeliki, Marianna and ‘the workshop’ participants are also committed towards ways of working with the land in a ‘commoning’ philosophy. It is in this frame that they organize the pedagogic workshop for ‘the commons’ (see

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next section). All these ongoing agonistic struggles for ‘the commons’ are directed towards both resisting neo-colonial practices that work along neo-liberal forces and reconstructing commoning spaces that aim to reconnect with nature, culture and activism against energy transitions and natural resources occupation (Caffentzis & Federici, 2014; Escobar, 2016). Learning to strive for ‘the commons’ in the context of a local space, a matter related with the second realized need as mentioned above, weaves a relational political onto-epistemology. According to Escobar (2016) the ongoing struggles towards creating and recreating manifold discursive and material practices that help build new worlds and worldviews in a direct entanglement with land and community depend on existential views of being and becoming. This ontological concern is both relational and political because as it refuses to see land as separate from people and politics it perceives politics as a dense network where land and people coexist. For Escobar (2016: 18) it is only through this relational political ontology that emancipatory struggles for the earth could be advanced to defend land, mountains, water, forest, public spaces, languages, knowledge as ‘the commons’. In this view, ‘the commons’ are neither about imaginary accumulation of shared material or immaterial resources nor about some type of property awaiting to be occupied or freed. Within a relational political ontology for such emancipatory struggles, an epistemic shift becomes core. This shift encounters as we have also argued here a deep ontological concern for subverting epistemicide. As Santos (2016) has repeatedly argued, epistemicide is being utilized instrumentally so as to establish the assumed privileged status of specific forms of knowledge such as science and mathematics. Mathematics, in particular, can easily provide the persuasive discourse of numbers (i.e. mathematization, mathematics as a matter of ‘facts’) that can work, sometimes violently and, at other times for empowering, towards articulating, arguing, reasoning for or against certain cases concerning natural resources occupation in particular local contexts and in analogy with global neoliberal interests. In this realm, Santos (2016) argues for a sociology of absences making us attentive to how silences and unpronounceable aspirations have been constructed over periods of territorial conflicts and how they become, today, heard and accounted for. A systematic attention to such absences/presences may trace genealogies of epistemicide with or/and without science and mathematics. It can also reveal what might be other alternative moves that remain invisible due to long term oppressive acts in the realm of either violent or non-ethical response-ability of mathematizing processes in and out of schooling practices. And, this is a matter of concern for both the south and the west, the oppressed and the privileged, the marginalized and the bourgeoise. Awareness of climate change irreversibility and the global effects of covid-19 pandemic indicate how none can, really, escape the precarity of our current times. In this research study we try to create an opening for such a discussion by encountering what makes possible a relational political ontology of a learning process that is deeply concerned with subverting epistemicide. Specifically, in the realm of this pedagogic workshop for the commons, we ask what does it mean to learn mathematics with a concern for an ethical response-ability for land, nature and culture as ‘the commons’. As said before, the issue of learning with ‘the commons’ is raised

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by the pedagogical collectivity as core in any attempt to engage with emancipatory moves that aim to reverse the deadening effects of epistemicide as erasures of one’s own knowledge, skills, lands, relations and self. The words of Ermis, below, capture clearly in a few phrases the complexity of what we have tried to discuss. When he says ‘Neither adults not children can see that…’ he emphasizes the oxymoron of seeing the earth dying in front of our eyes and, yet, we cannot see it. Ermis implies that this onto/epistemology configures people unsafe, insecure and, even, unable to see and talk about their land and life. And, at the same time he places his hope on learning. A learning the encounters ‘the commons’ as a matter of learning to subvert epistemicide. There is this whole plain of fertile earth… And it doesn’t produce, now, any food for the people who live in the area. It produces only apples, peaches and cherries… Mass quantities… But these fruits will never feed anyone hungry…. We talk about a small village that produces hundred thousand tons of fruits and sends them to Dubai. Let’s say now these fruits are needed in Egypt. But, the market is set in such a way that they are interested only in specific forms of fruits. This system doesn’t seem to have a future because the fields are slowly dying due to intensive cultivation. Neither adults nor children can see that… (Ermis, lines 772-779).

4 Learning and ‘The School’: A Radical Pedagogy for ‘The Commons’ Despite knowing that this pedagogic workshop is not a formal educational context, children and adults in the area refer to it as ‘the school’ denoting that this space is about learning. The workshop can be located digitally under the name Children’s Orchard (see www.facebook.com/spiresporo) and the site-specific domain name ‘spiresporo’ translates into Greek as ‘σπε´ιρε σπ´oρo’ that literary means ‘sow seed’. This name serves to signify a commoning relation with the land by calling people to engage with habits of sustainable land farming through sharing and sowing seeds. This form of knowledge could be seen as nostalgic of ancestors working with earth in the 1920s or even before. But, it is also a critique of modern agriculture practices where genetically modified seeds, fertilizers and pesticides contribute dramatically towards environmental pollution and earth decay. It, further, creates a space of knowledge that allows people moving across historical, cultural and ecological dimensions in their lives. The pedagogic workshop is hosted at the premises of the old preschool building provided by the village’s municipality. One needs to note how recent demographic changes due to consequent economic crises with effects on persistent high rates of unemployment, low rates of child-birth and youth immigration to wealthy West states have caused brain drain in the area. Thus, the reopening of the school was seen as a revival of the place. Preparing the old school dwelling for reuse signified the opening of an entry to a process of serious restoration and renovation involving the reconstruction of both the building, the yards and the gardens. In all these, children from the village became active, from the very beginning, helping in varied domains of

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work. Specifically, three small scale glasshouses, a garden, a cob oven, a small lake, an irrigation system for their needs and a carpentry workshop were constructed over a period of 9 years. Emphasis was placed on reorganizing the space to accommodate communal life that invited children to create a sense of ownership. The focus was quickly geared towards a serious focus in making space for learning with the land and with the locals and for prioritizing ecology values and knowledge from elderly people in the village. It is by centering a relational ontology in the context of communal activity in this old preschool building where ‘learning’ becomes a matter of concern in ‘the school’ culture for the workshop and in cultivating a philosophy of living with ‘the commons’. Asking Ermis about what is ‘learning’ in ‘the school’, he accounts: It is about how children develop a kind of contact with the world around them… To sustain this natural urge for exploration that children already have. First, they express this through asking a thousand questions and, then, they have the tendency to experience by themselves. This is happening freely in younger ages… It is the same quality that is being expressed later in human relations. The children need to think about this process. It is very personal, but when it happens in the realm of a social context, an added importance takes place… It may start with the family, but we see children in larger groups where they understand that it is different being an individual and being part of a collective… It is a whole system… Many things are happening at the same time. Apart from this, the land ecosystem itself plays a huge role in their learning… Not only the locality of our workshop, but in general the relations people develop with plants and animals, and the relations animals develop with plants without the human presence. What is around us? What exists? How can we find stories from older people in the village? Stories about what animals exist in the area, or used to exist in the past? All these are separated questions. (Ermis, lines 5–16)

Reading the above, the workshop can be appreciated as a ‘crack’ where learning turns into a subject’s matter of concern for the ‘other’ both the human and the more than human others. The focus on this ‘matter of concern’ support both children and adults to experience an expansive understanding of themselves in relation to nature and the community (i.e. relational ontology). Learning with ‘the commons’ means for them first and foremost learning with the land (i.e. the earth, the plants, the seeds) and, at the same time, learning together around processes of caring, listening, sharing, respecting, inventing, making, crafting, expanding, weaving, wondering, exploring, inquiring, experimenting. As such, learning becomes a relational matter unfolding amongst human and nonhuman others where the practices of walking and making become main habits. The practice of walking involves an explicit consideration of children’s bodies within physical movement, wonder and strolling in areas both indoors and outdoors. It is about bodily activity and embodied action in the kitchen with cooking, making daw and baking cakes, in the gardens working with trees, plants and flowers, in the playroom playing diverse forms of games, music or, even, making, reading or narrating stories. But it is especially the act of walking in nature either in organized walks, wandering as flaneurs without a specific purpose that creates new existential times for children (Ingold, 2011). Ingold explains that ‘walking’ involves a focus towards ‘encounters’ with new places as we move through them and it, also, offers different ways of crafting movement itself in the world including ideas and feelings in the very act of moving within a sense of knowing our co-walkers. It engages

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participants in their environment intentionally and collaboratively in the natural flow of things. Walking the mountain routes and tracing paths is essential for the ability to understand, describe and communicate something from everyone’s experience. Exploring paths in these routes, as children experience the mountain treks of their village, involves the significance of locality while it opens for new possibilities (Pink, 2017: 150–154). At other times, they walk out to trace animal footprints or to map and to name new paths. In such adventures, Samantha explained that her sister Angeliki was, systematically, using the GPS for navigation and mapping and other methods for noting down distances or measuring the length of animal traces (Samantha, lines 496– 505). Children, also, enjoy strolling together in the familiar, yet unknown, landscape as they meet archaeological remnants of the past, abandoned warehouses or the church bells from Argyroupoli and wonder about the stories behind them. They may focus on the village’s buildings, such as the Turkish houses that were transformed, the new infrastructures that nowadays are abandoned, offering new chances for rereading the past. Visits are organized to elderly people to collect stories of their past as they contribute to research undertaken by the Museum of Pontian Hellenism on local oral history. They research local legends, creations of the local fantasies, stories, memories of disasters which are mixing with fantasy creatures and become fairy tales. At the same time, the practice of making in ‘the school’ is a habit of serious concern. Artefact making, craftwork, woodwork but also music making, food making, homework making and story making amongst many other things opened up new spaces of living in the rural scape and becoming with. A way of ‘thinking through making’, that Tim Ingold proposes (cited in Ravetz, 2017: 159) means that the processes of making are valid as means of researching and coming to know and include a process of ‘messing around’ following the same route for a pre-planned destination. Making is a process of growth, starting with a form in mind (Ingold, 2012) revealing a sense of common achievement in a moment of collective joy. The strong relations that children, adults and local community are building because of the workshop’s activities and artifacts are creating a new form of connection with land through a process of ontogenesis by coming to know more the local space, time and identity. The materiality of making brings forward social and historical contexts, emphasizes the physicality of the world around us which is a political act related to ‘the commons’. Ingold (2011: 217) mentions the story of a carpenter who lives in a mountain village. The carpenter is able to realize the time the trees need to grow, the time to let their wood dry, the time to build with them but also the process of idea and the rhythmic qualities of making. Perhaps, this story could be in analogy with how children learn to return to their environment in the realm of making artefacts and crafts in ‘the workshop’. In both walking and making, action is a matter of affective bodying with materials, techniques and others as the bodies carry their own histories concerning desires, ability, affordances and feelings (Chronaki, 2019). In this pedagogic workshop, the adults are not only responsible for the physical safety of the children but also for creating safety to express sad and joyful affective experiences. During these moments there is no clear presence of time, events of the past enter the contemporaneity of

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the present, giving time to plan for the future, through the infinity of the embodied experience (Ziogas, 2014). Having the opportunity to explore affective bodying in this collective of more-than-human others, children and adults create new relational chronotopes. Is this a promise of a return to an ethical, ontic and epistemic relation with matters of concern with land, bodies and knowledges including mathematics? The social, material, temporal and spatial dynamics are scaling up the participants’ desire to be part of existential territories of be/coming together with the more-thanhuman others. In the words of Escobar (2016) this is a matter of cultivating a relational ontology: It reveals an altogether different way of being and becoming in territory and place. These experiences constitute relational worlds or ontologies. To put it abstractly a relational ontology of this sort can be defined as one in which nothing preexists the relations that constitute it. Said, otherwise, things and beings are the relations, they do not exist prior to them (Escobar, 2016: 18)—our translation.

5 Mathematics: Re/making Space and Time for Learning ´ Under the terms μαθημα, μαθηματικα´ and μαθηματικ´oς, the ancient Greek language lexicon edited by Liedell Scott & Kωνσταντιν´ιδη, in page 75, explains that the meaning of mathematics goes beyond the art of number, arithmetic, geometry and astronomy to become learning itself. The prefix ‘mathema’ in the word mathematics signifies learning whilst the suffix -tics signifies the act of making ´ η κτ ησ ´ ις γ ν ωσ ´ εων). In this, (i.e. η μαθ ´ ησ ις ε´ιναι η π ρ αξ ´ η τ oυ μανθ ανειν, ´ learning as μαθημα is about a pedagogic process in making knowledge (i.e. τ o μαθ ´ ημα αϕ oρ α´ τ ην π αιδε´ια, γ ν ωσ ´ η και επ ισ τ ημη). ´ In this, the mathematician is a subject willing, loving and urging to learn (i.e. μαθ ηματ ικ o´ ς , η, oν αϕ oρ α´ τ o υπ oκε´ιμεν o π oυ ε´ιναι π ρ o´ θ υμo να μαθ ´ ει, o αγ απ ων ´ τ ην μαθ ´ ησ ιν, o šχ ων ´ εις τ o μανθ ανειν). ´ Whilst, today, we denote the prevailing presence ϕ ´ σ ιν δεξ ιαν of mathematics as a matter of disciplinary content knowledge in the realm of pedagogies, didactic models, curricula design, textbook use, teacher education contexts and teaching settings, a view of mathematics as making space and time for learning itself is, by and large, absent. This absence can be seen as a trace of epistemicide denoting the violent ethics of an onto-epistemic erasure of a particular form of mathematical knowledge. Samantha touches, quite poetically, this complex issue of dis/appearing mathematics (see Chronaki, 2018) when referring to mathematics as something that tends to escape human life, but, yet, lives amongst us. She says: I feel mathematics is part of our life, but, we can say, they have escaped… Do you see what I mean? I see them today as something dislocated from my life whilst they are not… they are still here in our human life (Samantha, lines 556–560).

Addressing Ermis, Samantha and Marianna with the provoking question ‘Do you engage with mathematics in the workshop?’ they felt surprised. It was an unexpected question, because, on the one hand, ‘mathematics’ was tight with the school experience of disciplinary subject areas treated as matters of ‘fact’ (see Latour, 2005:

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87) mainly detached from life. And, on the other hand, as a porous form of knowledge deeply embodied and endorsed that dissolves in your being to the extent that it becomes almost imperceptible. It is here with us, but, you do not need to speak or think about it. They explain how mathematics is not being taught or discussed explicitly since ‘mathematics exists where all living things are’ in the realm of varied activities where ‘mathematics becomes a matter of being able to observe and speak so that to be understood by others’. The varied workshop based activity contexts where mathematics is being embodied may be everywhere and can happen anytime. Whilst working with plants and animals, walking, navigating, making crafts, playing games or music. Ermis, based on his long experience and passion with traditional music in varied cultures (e.g. he was in India for a period), explains how the creation of musical soundscapes are based on skills that bring mathematics in action. For him, mathematics retains a visual nature and comes into shapes, patterns and algorithms that help him towards making music, inventing rhythms and improvising. He explains, for example, how mathematics contributes as ‘…a mechanical part of music when you need to learn to divide time in a certain way’ in a process of making learning happen within making music happen. (…) for me it (mathematics) is something visual. I see shapes in front of me… And when I started working more systematically with music (I noticed) there is a whole world (....) and also a mechanical part of music when you need to learn to divide time in a certain way. We use simple mathematics for that. And then you can work a kind of algorithm to improvise your music and find new novel ways that produce the same result. I realized that traditional music, from Greece but also from other places (like India), is based on an idea of using algorithms. Listeners do not realise them but if you are taught to play music, you engage with the rules and, then, you understand how it works. How alive it is and, at the same time, how specific it becomes. (Ermis, lines 890–905)

Next to music, Ermis discusses contexts where children are completely free to experiment with making their own constructions without guidance or support by adults. This is an important part where personal potentialities of skills in learning are taken out to their limits allowing them to explore strengths, weaknesses and desires. This becomes a space for children re/making themselves as subjectivities full of passion for encountering the process of learning itself. In this, time expands and returns to the epoch of ‘mathematicians’ who were, indeed, ‘learners’ in archaic terms. Ermis, refers to the case of Grigoris who made a Lego based motor engine that did not work properly. Yesterday, Grigoris made this little car (with Lego) and he managed to build a servo motor, meaning that this car could do only a movement. And then he worked more towards building a car door construction that opens and closes. He wanted his car to make more movements, and he managed to create his own patent but it didn’t work. I gave him the Lego booklet with proposed instructions to find a solution but he said ‘I’m going to do it on my own’. And he was trying again and again until he found a solution very similar to the one proposed in the book. And, then, I told him: ‘ok you can have a look to move your craft further and connect the front wheels with the back and move them all’. So, if we are discussing how free the children are working… It is, really, about giving them the chances and the materials to work further and further their own ideas (Ermis, lines 985–996).

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With children’s free, open and playful time into making constructions, Ermis discusses how, sometimes, the process of artefact making might need more systematic work that requires a gentle, respectful and disciplined way of working with procedural steps. The process of making, often, needs to rely on models, on tools that allow precision and on generic functions. This becomes illustrated by Ermis through woodwork with children: If you get any wood, you first, definitely, have to go through specific four stages which are punching, cutting, grinding and joining woods together (75–76)(...) In these, you have to add counting and marking on the woods as two more important parts in which mathematics is located. On the top, there is also the design process where, at the same time, there is planning on paper…. You must sketch down what you are going to do (Ermis, lines 82–84).

All aforementioned examples of music, Lego-based constructions and woodwork are only some indicative cases of thematic contexts that, by and large, we can meet in mathematics teaching and learning especially when the focus is for children to experience mathematics in action. Didactic approaches for mathematics education could comprise realistic contexts, project work, authentic situations, everyday mathematics, ethnomathematics, technomathematics, critical mathematics education, apprenticeship or in situ anthropology of material artefacts (D’Ambrosio, 2006; Pinxten, 2016). Theorizing and researching such endeavors in mathematics education has accumulated an important body of writing discussing the nature of mathematics, mathematical activity or mathematical practices that can become easily, by and large, trapped into dichotomies amongst every day, academic and school mathematics. Findings concerning the ‘mathematics’ within the present pedagogic workshop for ‘the commons’ could be reduced by cynical critical readers down to romantic discourses of the everyday of mathematics that serve to strengthen the already disordered, dichotomized and, even, polarized ways of discussing the dis/appearing of mathematics in the life of citizens as a yes and no to mathematics (Chronaki, 2018). However, by means of this study we wish to note that these emergent moments of mathematics and mathematical activity in the context of ‘the commons’ become crucial temporalities and spatialities not only for mathematics, but for learning itself. We want to place attention on acts of re/making space and time with children and by doing so to make space for rethinking mathematics with a different form of knowledge of mathematics. A diverse form of knowledge that, instead of acting from a distance (i.e. expertise), dares to take the risk to participate in the context of a radical pedagogy where mathematics is learning. Or, in line with Haraway’s notion of sympoiesis, mathematics learning, in the space and time of a pedagogy for ‘the commons’, becomes experienced as a sympoiesis of making-with-mathematics. The stress on the ‘with’ of sympoiesis comes close to how Santos’ stresses the importance of sharing knowledge, instead of commodifying, selling or exchanging. Sharing must be turned specifically the forms of knowledge that serve to destabilize images of epistemicide. As he argues, they can only be ‘efficacious only if they are amply shared’ (Santos, 2016: 145). Conceptions of space and time of the world are often based on individual experience or collective narratives offered in linear and hierarchical genres in ways that fix

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history. Redefinitions of time and space remain political and are constantly struggled over to combat oppression in discursive and material terms as happened in ‘the workshop’ we studied here. Their subversion depends largely on the availability of new socio-political imageries allowing alternative territorial, material, bodily and discursive configurations (Sheppard, 2006) that allow the production of new subjectivities. Harvey (1969) has already discussed major aspects of this political perspective such as encountering our subjective experience of space, considering how space turns into place, questioning the changing importance of place under contemporary globalization scales and, finally, emphasizing the need for a dialectic relation amongst space and time. In this spectrum, space is not a container, but a positional quality of the world shaped by the temporalities of social processes (Sheppard, 2006). The above seem to depend on the question of how diverse spatial and temporal considerations can be brought into conversation with one another and, also, within the same ‘universe’ (Harvey, 1969). The idea of creating alternative, possible worlds and worldviews as political relational ontologies (as discussed in previous sections) engages with the potential imageries of transformative social change and underlies the potential utopia of re/making space and time through and with learning itself. For some, this could easily remain idealistic if it is not grounded into real places and institutions (Sheppard, 2006). The radical pedagogy in the workshop for ‘the commons’, as we have discussed, here, strives to create new social imageries and worldviews for children grounded in the rural scape of the Panagitsa village where they all live and work. These ways counter local and global modern narratives of land and locals making some first steps towards subverting epistemicide. Such openings can help produce new subjectivities as they create the potential for dialogic relations amongst diverse forms of knowledge across diverse temporalities, spatialities and cultures. In this, the making-withmathematics becomes a new space and time of sympoiesis around the specific acts of making things, worlds and worldviews but also making routes through walking, talking and listening and inquiring. The encountered examples within the context of the workshop for ‘the commons’ indicate a process of learning that supports this complex process of subverting epistemicide. A crucial part here is the onto/epistemic shift from viewing mathematics as matters of fact to a radical different view of making mathematics as matters of concern (see Latour, 2005: 85). On the one hand, matters of fact is what formal schooling is preoccupied through textbooks, testing and performance indicators based on certain competences. On the other hand, matters of concern are always situated, specific and relational. Latour recommends a radical renewal and rethinking of technoscientific practices. Whilst, he is not advocating for scientists, including mathematicians, to dismiss of move away from matters of fact in their research, he is calling for scientists and pedagogues to invent spaces for learning where objectified facts, nature, cultures and humans become reassembled in radical ways with a profound emphasis on how they are or become matters of concern. In this realm, the radical pedagogy in the workshop for ‘the commons’ creates an expansive space and time for a purposeful and strategic troubling of ‘epistemicide’. As Santos (2016) argues ‘Only destabilizing images can give back to us our capacity for wonder and outrage’ (p. 145). Finally, the very act of mathematics as re/making space and

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time for learning itself reminds us how local, situated, hybrid, partial and impure knowledge, a form of knowledge related with lands and locals allows children and adults along with their companions to envision for a future they dream. Acknowledgements We would like to thank all participants in ‘the workshop’ sharing thoughts and ideas, but especially Ermis, Samantha, Marianna and all the children. The present project is part of a series of anthropological studies experimenting with the ambivalences of presence/absence of mathematics in the rural and urban scape during the last decade lead by Anna Chronaki at the University of Thessaly.

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Driessen, H. (1998). H περιoχη´ της Mεσoγε´ιoυ ως πρ´oκληση στην πoλιτισμικη´ ανθρωπoλoγ´ια και oι πρoκλησεις ´ στη μεσoγειακη´ εθνoγραϕ´ια. τo . κšϕoυ-Mαδιανo´ (επιμ.) Aνθρωπoλoγικη´ εωρ´ια και Eθνoγραϕ´ια. Eλληνικα´ ραμματα. ´ Aθηνα. ´ σελ. 619/646 Escobar, A. (2016). Thinking-feeling with the Earth: Territorial struggles and the ontological dimension of the epistemologies of the South. Revista De Antropologia Iberoamericana, 11(1), 11–32. Federici, S. (2018). Re-enchanting the World: Feminism and the Politics of Commons. PM Press. Giousis, K. (2011). Critical Perspectives on Slavoj Žižek’s Perception of “Commons” Post— Fordist Economy and Lukács’ Ontological Foundation of Social Classes. Paper prepared for CUA (Commission on Urban Anthropology) Annual Conference, Market Vs Society? Human principles and economic rationale in changing times, Corinth 27–29 May 2011. Graeber, D. (2014). Debt: The First 5000 years. Melville House Publishing. Hardt, M. (2010). The common in communism. Rethinking Marxism, 22(3), 346–356. https://doi. org/10.1080/08935696.2010.490365 Harvey, D. W. (1969). Explanation in Geography. Edward Arnold. Ingold, T. (2011). Being Alive, Essays on Movement, Knowledge and Description. Routledge. Ingold, T. (2012). Towards an ecology of materials. Annual Review of Anthropology, 41, 427–442. https://doi.org/10.1146/annurev-anthro-081309-145920 Kioupkiolis, A. (2020). The Alternative of the commons, new politics and ities. In S., Lekakis (ed.) Cultural Heritage in the Realm of the Commons: Conversations on the Case of Greece. London: Ubiquity Press. Korsgaard, T. M. (2019). Education and the concept of commons. A pedagogical reinterpretation. Educational Philosophy and Theory, 51(4), 445–455. https://doi.org/10.1080/00131857.2018. 1485564 Latour, B. (2005). Reassembling the Social: An Introduction to Actor-Network-Theory. Oxford University Press. Means, A., J., Ford, D., R., & Slater, G., B. (2017). Educational Commons in Theory and Practice. Palgrave Macmillan, US Papadopoulos, A. G., & Patronis, V. (1997). The crisis of Greek cooperatives in the context of the globalization process. Journal of Rural Cooperation, 25 (2), 113–126. Pechtelidis, Y., & Kioupkiolis, A. (2020). Education as Commons, Children as Commoners. The case study of the Little Tree community. Democracy & Education, 28 (1). Retrieved from, https:// democracyeducationjournal.org/home/vol28/iss1/5/. Pink, S. (2017). Drawing with our feet (and trampling the maps): walking with video as a graphic anthropology. In T., Ingold (ed.), Redrawing Anthropology, Materials, Movements, Lines. UK: Ashgate. Pinxten, R. (2016). Multimathemacy: Anthropology and Mathematics Education. Springer. https:// doi.org/10.1007/978-3-319-26255-0 Ravetz, A. (2017). ‘Both created and discovered’: the case for reverie and play in a redrawn anthropology. In T., Ingold (ed.), Redrawing Anthropology, Materials, Movements, Lines. UK: Ashgate. Sheppard, E. (2006). David harvey and dialectical space-time. In N. Castree, & D. Gregory (eds), David Harvey: A Critical Reader. Oxford: Wiley-Blackwell. https://doi.org/10.1002/978047077 3581.ch7. Shields, S. (2016). Forced immigration as nation-building: The league of nations, minority protection, and the Greek-Turkish population exchange. Journal of the History of International Law, 18, 120–145. Singh, N. (2017). Becoming a commoner: The commons as sites for affective socio-nature encounters and co-becomings. Ephemera Theory & Politics in Organizations, 17(4), 751–776.

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Wolfmeyer, M., & Lupinacci, J. (2017). A mathematics education for the environment: Possibilities for interrupting all forms of domination. Philosophy of Mathematics Education Journal, 32. Ziogas, Y. (2014). Florina’s and Prespes’ landscape as a process of actions. Walking from Pineritsa to Oriza and Agios Germanos (October 2012, 556 minutes). (in Greek To τoπ´ιo της λωρινας ´ ´ και των ρεσπων ´ ως μια διαδικασ´ια συμβαντων. ´ H πoρε´ια απ´o την ινερ´ιτσα στην Oρυζα ´ και τoν Aγιo ερμαν´o, Oκτωβριoς ´ 2012, 556 λεπτα). ´ ρακτικα´ της Hμερ´ιδας Tšχνη και Eκπα´ιδευση, λωρινα, ´ Nošμβριoς 2014.

Meta-studies

The Tapestry of Mathematics—Connecting Threads: A Case Study Incorporating Ecologies, Languages and Mathematical Systems of Papua New Guinea Kay Owens Abstract Tapestries interweave strands and colours to create beauty. Mathematics is created through time and place to provide beautiful systems of patterns and purposes. One such place of diverse ecologies and languages that spans thousands of years of intact cultures is Papua New Guinea (PNG). Its diversity of places and peoples with ancient cultures has created a mathematical tapestry. This chapter expands on a small section of this tapestry to challenge most Indo-European views of the beginnings of mathematics such as number systems and mathematical reasoning. Archaeological linguistics, local environments and ecology, sociology of economics, sociopolitical expectations, and mathematical ways of reasoning form threads to create sections of the tapestry. These pieces provide examples of ethnomathematics valuing Indigenous knowledges as mathematical and as important for all societies both within PNG and around the world. Alternative ways of understanding mathematics assist in creating new pieces of the tapestry of mathematics. Keywords Papua New Guinea · Mathematics and languages · Kula trade and art · Time, place and economy · Ethnomathematics

1 Introduction Papua New Guinea is situated in the western Pacific Ocean just below the equator and consists of the eastern half of the large island of New Guinea and many other islands, large and small. There are high mountain ranges with steep valleys and fast flowing rivers as well as large fertile valleys. There are coastal areas with large and small river systems, sago swamps, and beautiful coral reefs and beaches. There is often dense rainforest but also areas covered with tall blade-like kunai grass. Papua New Guinea’s Indigenous peoples are Melanesian. Indigenous landowners in Timor Leste, West Papua, Ambon and other places in the eastern half of Indonesia, and

K. Owens (B) Charles Sturt University, Dubbo, Australia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_7

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in Island Melanesia from the Solomon Islands to Fiji are also Melanesian although some Polynesian groups have come to these islands. Although people from south Asia may have visited over the latter half of the 40 000 years of its known inhabitancy, European travellers often passed it by until the mid-1800s. By the end of that century, the colonial power of Germany began colonising the north east of New Guinea and neighbouring islands1 while Britain colonised the south eastern area and islands (British New Guinea with administration from another colony, Queensland in Australia) with a border agreed in 1884. After Australian Federation in 1901, Australia administered the southern section as a Territory. The First World War saw the north become a Mandated Territory of New Guinea also administered by Australia. Much of the country’s north and east were occupied by the Japanese Army during the Second World War, and after the war both Territories were jointly administered by Australia until the country became Independent in 1975. Thus its colonial influence was relatively short especially given that much of the interior of the island was not regarded as safe for travel or colonisation until the late 1940s when patrols were first made into the interior ranges (Gammage, 1998). Air travel was the main form of transport before Independence and is still needed to travel from the capital, Port Moresby to Provincial capitals unless going by sea. History relevant to mathematics education is provided by Paraide, Owens, Muke, Clarkson, and Owens (forthcoming 2022). In this chapter, the idea of threads of tapestry creating a colourful image is used as a metaphor for the diversity in mathematics in Papua New Guinea creating intriguing, intricate, holistic sections of mathematics. The reason for such diversity is based on language diversity but not solely as place (ecology) and time also impact on mathematical practices. The language details are presented and explicated by looking at the diversity of counting systems. Although all Papua New Guineans are Melanesian, recognition of the differences in the languages and cultures is important as this impacts on the diversity of mathematics, evident firstly in the variety of counting systems but in many other activities requiring systematic reasoning especially those adapting to their ecologies. The chapter provides an overview of how the diversity developed by considering also migrations, cultural relationships, and some of the evidence for this diversity. Language, culture, position in PNG and relationships of different groups provides the backdrop for recognising the diversity of counting systems which have been collated, analysed, and evaluated by Lean (1992), Owens, Lean, with Paraide and Muke (2018) providing a large, textured, and colourful part of the tapestry. The current study further considers culture and ecology for they are intertwined and impact on movement on water and land, land tenure and land use over time which is itself another dimension of life. A few specific cultures are explored in this chapter with sufficient depth to illustrate the mathematics within activities. Diversity provides the texture and colour of the tapestry.

1

The Netherlands colonised the western half of New Guinea (West Papua, a name used by the Indigenous population) and much of Indonesia.

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2 Threads of the Tapestry The varied colours and textures of the tapestry of mathematics provides a picture of mathematics in this country prior to colonisation but continuing on today. To make this picture intertwining threads from the following disciplines and sources are explored: • • • • • •

Language and linguistics, Archaeology, Sociology, culture, and economics, Migration, trade, geography and ecology, Oral and written histories, and Mathematical systems and reasoning.

Some threads form sections of the picture and then putting these together gives a larger picture. Thus these categories intertwine. The result is a fascinating journey into the past with a currency today of culturally different mathematical systems and ways of thinking. From these disciplines, the threads of time, place, language, culture, thought, and mathematics are woven together to form a rich tapestry of beauty. The creativity of the Indigenous cultures has resulted in beauty but also the tapestry of interwoven knowledge creates mathematical beauty. The chapter focuses on the counting systems to illustrate the interweaving of this knowledge summarising earlier research (Owens, Lean, with Paraide, & Muke, 2018a, 2018b) before extending the arguments by studying time and work, economy, trade, travel, and art. With limited early European colonisation beginning in the age of enlightenment (second half of the 1800s) when new knowledges were respected and with the continuity of the cultures to this day with little change, the tapestry’s threads are strong. The western influence has been limited so the cultural understandings are generally intact despite the impacts of Christian ideals and European languages, especially with the relatively cloudy versions that sometimes occurred for example in churches, and the impacts of other colonialists which went from appalling by today’s standards to patronising at best. It seemed ubiquitous that students at government schools were often forbidden to speak in their own language although many church schools allowed it such as the LMS schools in Port Moresby if they had teachers who knew the language or if the linguist working with the language provided some early readers and programs in language (Paraide et al., forthcoming 2022). Language is likely to maintain the abstract concepts and ways of thinking that are often associated with mathematical reasoning. However, with good teaching especially with practical examples and materials, students would still think in their own language (Muke in Paraide et al., forthcoming, 2022). Today teachers who know one of the languages of the children will use those in various forms of codeswitching between languages (Paraide et al., forthcoming, 2022).

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2.1 Caveat on This Perspective of the Tapestry Much, but not all, of the expression of archaeological and historical development of knowledge has been influenced by western approaches to knowledge. In some ways this is fortuitous as this knowledge at least identifies difference and values the cultural ways of knowing and being. Much of the recording may have been western but it was by necessity informed by Papua New Guineans albeit dependent on the quality of the anthropology, linguistics, and the non-Papua New Guineans’ relationships with the local community. Further the anthropologists, linguists, and educated members of the community may have known the language of the group to varying degrees. When Papua New Guineans themselves have been researchers and translators working with their people and perhaps with others, mathematics may be noticed by both the Indigenous and non-Indigenous persons as mathematical reasoning and knowledge. Other areas of knowledge have developed through transcultural linguistic and cross-disciplinary work. This was the case for establishing proto languages in the region and in establishing the diversity of counting systems. Using a mathematical lens on anthropological work has also been utilised. In these cases, the research is dependent on the quality of the initial disciplinary work and its detail. Both these areas are discussed below. However, it is also acknowledged that these knowledge developments and indeed this chapter could well be viewed differently by people of the cultural group whose relationship with the artefacts, each other, the land and time may not be expressed in western terms (McConaghy, 2000; Nakata, 1998; Report from a seminar in Kárášjohka Norway, 2008). An effort is made to reflect the original expressions even if it is recorded by non-Indigenous people.

3 The Diversity of Cultures Papua New Guinea has 850 different language or cultural groups who are classified as Melanesian. The languages of these different groups are both Austronesian and non-Austronesian. The diversity of the non-Austronesian languages is exceptional coming from a range of language phyla but also diverse within the one phylum. Some, increasingly many, are endangered languages while others have over 10 000 speakers. Most languages are changing rapidly. The mountainous landscape and fast flowing rivers have not by themselves led to diversity but diversity seems to come from trade, marriage and other relationships, the sense of identity that a language provides, and the pride of people in having a language of their own and being able to speak the languages of others. Neighbours in the same valley or island can speak different languages while other languages, albeit with dialects, might cover hundreds of mountainous square kilometres. On the other hand, across vast waters, the Austronesian Oceanic languages have some similarities with words, grammars

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and counting systems although there are still differences by which they are separate languages. Capell (1969) classified the languages, broadly as verbal-oriented or nounoriented and this affects the way in which plurality is expressed. For example, a numeric adjectival would be likely in a noun-oriented language whereas a suffix, infix, or prefix on a verb root is likely in a verb-oriented language. Furthermore, counting systems have developed patterns for extending the basic counting words into higher levels of counting. This pattern of relationships and combinations of numbers is part of the tapestry on number explored later in this chapter. However, the landscape is important in the way people, who are mostly subsistence farmers, gain shelter from the available natural materials, decide where to live, how to obtain food, how to maintain ongoing community relations, relate to the land and waters, enjoy life and believe how life begins, continues, and moves to another state. Each of these aspects of life have a mathematical implication. For each, people reason to make decisions and build understanding, some of which are mathematical reasoning in a broad sense. The purpose of this chapter is to express a view of mathematics that focusses on understanding pattern, classification and relationships. The description may not be expressed in algebraic terms but in recognition of these mathematical ways of thinking. In this sense, as an example of ethnomathematics, this chapter values the Indigenous knowledges as mathematical and as important for the societies and for societies beyond, both within PNG and around the world.

3.1 Time and Place Archaeological evidence covers land usages tens of thousands of years old; DNA evidence of travel, changes in climate, seismic activity, and tools such as obsidian, pottery, and importantly single-outrigger canoes for sea voyaging. Without these, the suggestions for longevity of the languages would be limited. People inhabited the land at least 40 000 years ago and there is understood to be more than one migration across the land. Archaeological sites are spread across the mainland and on the islands. The Australian Indigenous population probably passed through during the ice-age sometime after 150 000 years ago when Sahul (New Guinea, Australia, Tasmania, and Melanesian Islands) was mostly connected. Excavations at Kuk in the Waghi valley indicated agricultural drainage as far back as 10 000 BP or earlier (Muke et al., 2007). Drainage systems require a mathematical eye for slope and straightness assisted by tools like wooden spades and string tied to pegs. Gardens are formed to cater for cold and water management with various forms of mounding and mulching. There was an ice-age finishing around 50 000 years ago and a mini ice-age about 15 000 years ago and those together with volcanic activity have changed the nature of the environment in different places. The various changes in the environment have impacted on how cultures survived, moved and interacted with each other. Some of

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these are recorded in the archaeological linguistic literature (e.g., Swadling, 1997, 2010).

3.2 Time and Language Linguistic work has also been linked with archaeology. As an example, work by Swadling (1997, 2010) summarised and analysed the archaeology of the RamuSepik inland sea that has appeared at times at different levels since 150 000 years ago with significant movement between 6 000 BP and 2 000 BP but with some continuing changes. These changes had affects on migrations and languages. For example, the related Sawos, Abelam, and Boiken languages moved north and the Boiken Austronesian language having characteristics of a Non-Austronesian language. The sea going inland allowed for marine travel and changes in diets although the water was not salty enough in some areas for marine shellfish. It meant that there was a thriving agricultural centre similar to those found in the highland valleys. The wooden artefacts (paddles or spades) survive and are similar to those used in the highlands and found in mud of the Kuk and Waghi valleys. Trade between coast and highlands was affected by the changes in the level of the inland sea. Again this affects the languages of the area with influences both ways on the Austronesian and non-Austronesian languages. Pottery remains have been found not only in the highland areas but also in the Sepik-Ramu (some with beautiful incision) and further west in West Papua predating the Lapita pottery. Pigs and other items suggesting trade with the highlands e.g. of bird of paradise plumes and possum also appear at the Akari (Eastern Ramu) site from 5 000 BP. Evidence of the trochus shell trade with findings at Akari and at Kutepa rock shelter (near Pogera, Enga by Jo Mangi) suggest that trade also encouraged loan words from Austronesian languages into the Engan family (and also in Southern Highlands and East Sepik areas). This linguistic archaeology indicates that the Sepik-Ramu Phylum is the oldest, followed by the West Papuan, East Papuan and Torricelli Phyla, and later came the Trans New Guinea Phylum (Ross, 2005; Usher, 2018; Wurm, Laycock, Voorhoeve, & Dutton, 1975). While there may be some differences in the identified languages and phyla, the overall suggestion is that these languages are quite ancient (up to 60 000 years ago for the Sepik-Ramu Phylum in particular) based on archaeological evidence and language differences. This is one reason for claiming that the counting systems are old (Foley, 2005). Furthermore, there are six minor phyla and 7 phylumlevel isolates with other isolates at the subphylum level in the New Guinea languages suggesting an existence a very long time ago.

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3.3 Migrations and Languages The Sepik-Ramu Phylum was widely spread by the inland sea in the Sepik area and it provided an easy way of trading between coast and the inland. A significant date was 4 000 BP with the inland lake between the Sepik and Ramu quickly filling in becoming swampy and the alluvial plains being lost requiring migrations (this area is still sparsely populated) (Swadling & Hide, 2005, 2010). At the same time a volcano in West New Britain erupted dispersing people to other islands together with their culture, Lapita pottery, and language. Lapita pottery trade in the New Guinea islands was associated with shell monetary trade. Obsidian trade was also becoming widespread (Spriggs, 2011). Connections in linguistic archaeology suggest migrations have been in several waves (Ross, 1988) as shown in Fig. 1. Nevertheless, migration of languages has not meant languages have been overridden but rather that the languages may have been modified including the counting systems and thus providing evidence for the history of number. Bowdler (1993, p. 66) has suggested “mainland southeast Asia, Wallacea (west of New Guinea), Melanesia and Australia colonised virtually simultaneously by modern human beings some 40 000–50 000 years ago”. Terrell et al. (1997) in fact reported that there were 9 sites in the Bismarck area (New Ireland and New Britain) with evidence of occupation 40 000–50 000 years ago including inland sites so not just for chance or brief visits from elsewhere. Trade occurred between groups based on pottery and obsidian throughout the Pleistocene period across the north coast and into island Melanesia based on intergenerationally inherited friendships (Terrell et al., 1997).

3.4 DNA Assessments Study of DNA suggests that the Polynesians have mtDNA2 of Melanesians. The Melanesians may have provided an origin but they are most likely influenced by considerable travel. Within the island Melanesian area travel was frequent over relatively short distances with currents and winds that returned the sailors to their home. For the Pacific, they also show exceptional skills for long travel across the Pacific with evidence of travel both ways (Terrell & Welsch, 1997). There is then evidence from DNA assessments that this occurred even if there were other Asian influences. JCVirus studies (Czarnecki, 2003) would support that Polynesians had a Melanesian origin although later migrations from Asia occurred. The study also suggested that the Australian Aboriginal groups were not directly derived from highland New Guinea groups and that in fact the Trans New Guinea (TNG) phylum was a later 2

mtDNA is the DNA in the mitochondria of cells and it comes down the line from mothers to sons and daughters so it can be indicative of a long genealogy.

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(Source. Lean, 1992 available from Owens, et al., 2018, p. 15). a. Early language migrations

(Source. Lean, 1992, available from Owens, et al. 2018, p. 16) b. Austronesian Oceanic migrations.

Fig. 1 Early language migrations

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(Source. Addison and Matisoo-Smith (2010, p. 8) c. Austronesian Oceanic and Polynesian language migrations.

Fig. 1 (continued)

group. Also the Oceanic groups did not penetrate inland on the north coast of New Guinea confirming that the origin of these groups was from the Bismarck area. It is interesting to consider the degree of influence of Non-Austronesian (NAN) and Austronesian Oceanic (AN) cultures and languages on the counting systems of the languages of the alternate group near the Bismarck Archipelago. Close by, Bougainville in terms of language shows considerable influence between the AN and NAN groups in terms of base 10 counting systems. A distinguishing feature of many AN languages is the use of classifiers in which a class of objects are identified by a morpheme which is combined with the counting system morphemes. Thus counting morphemes are always associated with a classifier morpheme, for example one fish, two fish, three fish, and not free-standing as in English one, two, three etc. Furthermore, some of the neighbouring NAN languages such as Nasioi have classifiers. Interestingly from the counting system point of view, the neighbouring language, Uisai, has the pattern of Manus3 Province languages with numbers 6, 7 and 8 transcribed as needing 4, 3, and 2 more respectively to make the full group (Kaleva, personal communication, 2006; North Solomons data in Appendix in Lean (1992)).4 Furthermore, islands such as Mortlock Island, off the coast of Bougainville have Polynesian similarity which is not unexpected given that there are groups in the Solomon Islands of Polynesian background. It does support the view that the Polynesians and Melanesians were linked in terms of their sea-faring capacities and relationships and their on-going traversal back and forth across the Pacific. 3 4

See examples on Manus in Fisher (2010), Owens et al. (2018a, 2018b). There are a couple of languages such as Fore that have 9 as needing one more to make 10.

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3.5 Displacement and Migration Although there is not a lot known about small groups being displaced, there is evidence from oral history of displacements. The Aitape people, in East Sepik Province, occupying a sandy land strip when the tsunami hit in 1998 were descended from a group who had migrated from West Papua. Daino (personal communication, 2014) talks of a group in the 1940s moving into her valley near Chuave in Simbu Province with their own language but accepted by her language group to stay. In Mid-Wahgi, the community accepted a displaced group and they now have their own role to play within the clan groupings (Muke, personal communication, 2014; Muke, 2000). One language that was recorded before it died out by Holzknecht (2001) was also indicative of the impact of families having the remaining children taken in by neighbours. There is evidence of people on Manus island migrating there a couple of centuries ago and around Madang in the last century. Displacements may account for different pockets of related languages.

3.6 Proto Languages The extensive work of Pawley, Ross and their colleagues (e.g., Bowden, Himmelmann, & Ross, 2010; Pawley & Green, 1985; Pawley & Ross, 2006; Pawley, Attenborough, Golson, & Hide, 2005; Ross, 2005) has added over many decades to the extraordinary knowledge of these languages and relationships between languages. It has been their work on Proto Oceanic and on Proto Trans New Guinea Phylum that has been one of the threads used by Lean and myself and colleagues to discuss the longevity of the counting systems (Owens & Paraide, 2019). The linguistic evidence establishes Proto Trans New Guinea Phylum (TNGP) for the hundreds of languages on the mainland through associated archaeological evidence of people residing in places still using languages of this phylum. Languages of older phyla are on the verges of the TNGP (see Fig. 2). A Proto language must have existed at the start of the phylum. That is 10 000 to 20 000 or more years ago in the case of this phylum based on archaeological evidence and language comparison and classification. By contrast, Proto Oceanic languages are believed to have begun in the Bismarck Archipelago (near Willaumez Peninsula, New Britain, a north east New Guinea island) about 4 200 years ago and spread north west to New Ireland and Manus, west along the mainland coast, east to Island Melanesia, and south around the southern coast. There is evidence of movement through both the pottery known as Lapita and obsidian rock from New Britain, albeit in different waves. Figure 1b illustrates the spread. The phonology of Proto Oceanic is fairly well determined (Pawley & Ross, 2006; Ross, 1989, 2010; Ross, Pawley, & Osmond, 2003). In general, the Oceanic languages are subject-verb-object languages but the subject-object-verb languages may have been in the original Proto Oceanic language or it has been a Non-Austronesian (Papuan) influence as it tends to only occur in New Guinea and

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(Source. Lean, 1992, available from Owens, et al., 2018, p. 12) Fig. 2 Language phyla in New Guinea

in some Solomon Island languages. However, the Polynesian languages have verbsubject-object order as do a few of the Oceanic languages in New Caledonia. This variation is not yet fully explained in terms of archaeological linguistics but it does suggest that this very basic structure of a language can be influenced by surrounding languages over time and through relationships. While there are some similarities of language across the wide ocean expanses for Austronesian Oceanic languages, there is surprising diversity among the nonAustronesian languages including within some phyla. However, there is some evidence regarding the similarities and the differences of neighbouring languages and why they might have occurred through trade, relationship building and warfare (Holzknecht, 1989; Owens, Lean, with Paraide et al., 2018a, 2018b; Pawley et al., 2005). One important point to raise is that the commonalities between the languages was not as a result of a lingua franca for trade purposes as some have suggested but given the sense of identity associated with the cultures, the Proto languages and the other associated evidence provided below, a result of intergenerational relationships being passed on to future generations for centuries (Terrell et al., 1997). These continue to this day with a strong sense of identity with the family, their customs and beliefs, and their language(s).

3.7 Social Values, Sociopolitics, and Language Each of the group of villages is autonomous and may be situated within a larger language group with considerable ties. There were, however, many times when food was needed either due to the seasonal weather, climatic changes, catastrophic changes

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like volcanoes, warfare or poor land quality or lack of area. Since each village and each language remained autonomous, allegiances were often formed through gifts and trade. These were especially important for warfare against other tribes or even within tribes. Shortages of food and goods together with negotiations about land and women were often sources of travel and trade. Trade routes occurred via the ocean and from valley to valley and from the highlands to the coast. Western Highland green stone and coastal shells are found across the highlands. There has been trade of food, pottery, wood, canoes, other implements, signs of importance, culture, designs, and practical subsistence knowledge such as growing food, making food, and building shelters. There was trade between the older non-Austronesian languages and local Austronesian languages and vice versa. Further details of these languages in terms of impact of one on another and resultant changes in counting systems are given in Owens and Lean with Paraide and Muke (2018). Foley (2010) noted, supported by my personal communications, that some languages have tended to reduce the use of other languages. This occurs in the Sepik and Madang coastlines in particular but also in the Eastern Highlands with the Usarufa men speaking the Kamano-Kanite and Fore languages (Kravia, personal communication, 2015) but not vice-versa while the Siane men and women also speak Chuave (Daino, personal communication, 2014). The Bel language tended to dominate the Madang and Rai Coast area of Madang Province due to long-term trading and a large number of languages, mainly Austronesian but also Non-Austronesia (Sondo, personal communication, 2018 and Leo, 2014). In Papua, Magi, a nonAustronesian language, has influenced the decline in use of Magori, an Austronesian language although there is also widespread use of Motu and English (Onagi, personal communication, 2014). Furthermore, with the numerous exogamous marriages, women were bilingual for their home and partners’ languages but men who traded or travelled were also likely to be multilingual: The effects of this intense language contact over many millennia have been profound: the languages show borrowing and diffusion of traits at all levels: lexical items, phonological patterns, bound morphology, word order, syntactic constructions, discourse styles, genres. (Foley, 2010: 797)

As a result, the counting systems, along with other aspects of culture and language, did change and it is necessary to decipher what might be the proto language counting system and what may have been borrowed from a neighbouring language. A counting system has a structure that is at a deep level in a language as it involves not only words or morphemes (frame words) but also how these morphemes follow an operative pattern. For example, in a base 10 system in English, forty five indicates the number is four tens plus five so the operative pattern is 1 to 9 × 10 + 1 to 9. One way of establishing knowledge about the counting systems and their interrelatedness is to analyse and classify them. Using frame words, and operative patterns for combining these words, e.g., it is possible to establish cycles that form the basis of the counting system. For example, a (5, 20) cycle system uses frame words 1 to 5 then 20 by which all other numbers are formed for example 6 to 9 are 5 + 1 to 5 + 4 or n + 1 to n + 4

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where n is another morpheme. Only when the cycle system is repeated in a regular way with a new frame word with a new power (e.g., 10 × 10 = 102 = 100 like the Hindu-Arabic base 10 system) is the word ‘base’ used to describe the system. Clearly time has been a factor in the development of these cultures but the environment has also been significant. House styles, garden foods, gathered foods, and the sources of protein vary from the coast to the ridges of the coastal provinces and to the highland ridges and valleys. The interweaving of time, place, culture and language created the mathematical tapestry.

4 The Tapestry Section on Number Systems—A Window into Diversity Culture impacts on when people count, how much they value counting, how much they value difference between languages, and how much counting is a part of relationship building within the language group and beyond. Our counting (Lean, 1992; Owens et al., 2018a, 2018b) and measurement data (Owens, 2015; Owens & Kaleva, 2008a, 2008b) were derived from first contact records with people who were first colonised in the late 1800s and early 1900s, linguistic records, from people who knew their languages and mathematical practices, through field trips, interviews, University of Goroka mathematics students’ reports, focus groups, anthropological studies, and both counting and measurement questionnaires completed by university students, staff and teachers. Examples of these systems and mathematical ways of thinking are provided and variation is explained by considering time and place, culture and language (Owens, Lean, with Paraide et al., 2018a, 2018b). Among the systems are those of different composite units, both of number cycles and for measurement. For example, while most Austronesian languages have a base 10 or decimal counting system, most of the non-Austronesian languages do not. Owens et al. (2018a, 2018b) provides many examples of counting systems with some analysis. A few examples are provided in Tables 1 and 3. Thus a language with an operative pattern of 6 = 5 + 1, 7 = 5 + 2 etc., indicates that the person has a group of 5 and 1 then a group of 5 and 2 units more. This is typical in a digit tally system where 5 is often the word for hand. This is a fairly entrenched system and common across many of the Trans New Guinea Phylum languages. However, this is not the only system within the phylum. Some also have groupings of 4. In order to explore the systems and possible longevity of systems and thus a genealogy, Lean collected about two-thirds of the languages of Papua New Guinea, and West Papua and most of the systems of the rest of Oceania. This diversity of personal oral knowledge and written records provide further colour hues for our tapestry. The ways of analysing or sorting the languages by their cycles, operative pattern and frame words become another important section of the tapestry.

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Furthermore, the grouping of the various systems tells of the diversity and that these languages do have significant differences but it is the comparison of neighbouring counting systems that begins to explain some of the ways in which number systems developed. First there have been spontaneous development of counting systems, mostly influenced by the use of hand gestures to tally but also from counting specific kinds of objects in the society. In fact, numerous systems, mostly Oceanic but not all, have classifiers attached to counting morphemes in order to express the counting words. The classifiers range from very few to over 100. There are pockets of systems using classifiers in Non-Austronesian languages that neighbour the Oceanic languages suggesting some kind of influence. The 4-cycle and 6-cycle systems also tend to be in specific areas (examples in Tables 1 and 2). The body-part tally systems are generally across the Sepik, Western, Hela and Southern Highland areas. It should be noted that the early investigators from groups such as the Cambridge Anthropological Expedition to the Torres Strait or more recent philosophical anthropological understandings may indicate quite controversial suggestions, especially the studies of the late 1800s and early 1900s. Owens et al. (2018a, 2018b) also provided evidence of longevity of counting systems in some cases for tens of thousands of years and of the way systems might develop spontaneously since body digits and body parts were useful for tallying, and pairs and other groupings were significant or naturally practiced in various societies. In some cases, system structures were borrowed from neighbours as well as modifications from the proto language’s system with evidence from both proto Trans New Guinea Phylum and Austronesian Oceanic (see Malcolm Ross and colleagues work) (e.g., Ross, Pawley, & Osmond, 2003). There are exceptions for both the AN and NAN groups, usually associated with neighbouring languages and trade. Some languages have systems with only 1 and 2 combined for other numbers. There are systems that have numerals for 1–5 but others combine 1 and 2 for 3, 2 and 2 for 4 with a numeral for five (with various variations on this) (Owens et al. 2018a, 2018b). These have cycles of 5 or (2, 5) respectively. Many of these then continue on to have combinations such as 5 + 1, 5 + 2, up to two 5s then +1, +2 up to three 5s or more likely these are called two hands and then two hands and a foot, the latter called digit-tally systems. Then there are variations of the base 10 systems. Some have pairs for numbers between 6 and 10 while of the base 10 systems have classifiers so that a different morpheme is used for different categories that are counted. Some of these also have a numeral classifier to distinguish hundreds and thousands. There are also systems with composite groups of 4 or 6. Some of the languages with groups of 4 also have composite groups of 8. The final kind of system, considered an early type of system (Lean, 1992; Owens et al., 2018a, 2018b) are the body-part tally systems in which not only the fingers (hands) and feet are used but others have the number to make 10 so 7 needs 3 more to complete the group of 10. Some other body parts usually up the arm from the small finger, across the head to the other side and down again. There are some languages that have more than one system of counting. Table 2 summarises the pattern of diversity of the counting systems for the NAN languages and the vitality of this section on counting within the tapestry of mathematics.

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Table 1 Examples of different counting systems Language system

Iqwaye (2, 5, 20) cycles

Fasu or Namo Me Body part tally

Hagen Enga–Mai dialect (4, 8)-cycles plus (4, 60) cycles 10 cycle

1

ungwonangi (1)

m¯eno (little finger)

tendta,tikpa (1)

me(n)dai

2

huwlaqu (2)

tetá (ring finger)

ralg (2)

lapo

3

huwlaqungwa or huwlaqanga (2 1)

isiá (middle finger)

raltika (3)

tepo

4

hyaqu-hyaqu (2 2)

kitafá (index finger)

timbikak (4)

kitome(n)de

5

hwolyempu (hand)

kakórea (thumb)

timbikak pumb ti pip (4 to next 1)

6

hwolye indeumoni ungwonangi (hand to the next 1)

namá (palm of hand)

timbikak pumb ralg pip (4 to next 2)

7

hwolye indeumoni huwlaqu (hand to the next 2)

yatipinu (inside wrist)

timbikak gul kalage ragltiki (4 to next 3)

8

hwolye indeumoni huwlaqungwa (hand to the next 3)

k¯ari (forearm)

engag or ki tendta (hands complete)

tukulapo (two arrows)

9

hwolye indeumoni hyaqu-hyaqu (hand to the next 4)

t¯okona (inside elbow)

engag pumb to gugl (hands to next 1)

tukutepon(ya) me(n) dai (1 of three arrows)

10

hwolye kaplaqu (two hands)

kaeyako (upper arm)

engag pumb ralg pip (hands to next 2)

tukutepon(ya) lapo (2 of three arrows)

11

hyule yengwonye ungwonangi (down-to leg 1 (two hands implied))

kinu (shoulder)

engag pumb ralg pip to pentipa (hands = 8 to next 2 to next 1)

tukutepon(ya) tepo (3 of 3 arrows group)

12

hyule yengwonye huwlaqu (down-to leg two)

kenó (collar bone)

engag pumb ralg pip gulg ragl (hands and 2 to next 2)

tukutepon (three arrows)

15

hyule umance hyelaq (leg half that-all)

pari (cheek bone)

engag pumb ralg pip timbikak pukit pumb ti or pumb rakltiki (hands 2 3)

mapun(ya) tepo (3 of sweet potato group)

20

hwolye kaplaqu hyule kaplaqu or amnye ungwonangi (two hands and two feet or one person)

t¯aku hi (other eye)

engag pump ragl pip wote engag pump ragl pip or ki ralg timbikak pukit (8 + 2 8 + 2 or 8 × 2 + 4)

yupun(ya) gato (ground earth group complete)

yungi (yugi) (time)

(continued)

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Table 1 (continued) Language system

Iqwaye (2, 5, 20) cycles

Fasu or Namo Me Body part tally

Hagen Enga–Mai dialect (4, 8)-cycles plus (4, 60) cycles 10 cycle

30

anmye ungwonangi amnye ungwoli …. amnye indeumoni hwolye kaplaqu (one, one person to the next two hands)

t¯aku nama (other palm of hand)

ki raltika or yanapun(ya) lapo (2 engag pump ragl of dog) pip engag pumb ragl pip wote engag gul ragl or ki ragltiki wote timikak pukit gul ragl (three hands or 8 + 2and 8 + 2 and 8 + 2 or 3 hands with 4 fingers with 4 fingers and 2 included)

40

[amnye hyule hwolye hyepu], amnye huwlaqu (two persons)

ki timbikak kujupun(ya) gato (I pumb tip pip or cut complete) engag pumb ragl pip engag pumb ragl pip engag pumb ragl pip engag pumb ragl pip (8x(4 + 1) or (8 + 2) + (8 + 2) + (8 + 2) + (8 + 2)

100

amnye hwolyempu kokoleoule hwolye hyelaqapu (five persons)

kolg mong kiki ki tenta or ki engag pumb rag pip gul ragl wote timbikak pukit or engag pumb ragl pip engag (8 × 8 8 × 4 2)

Note The Fasu system has 35 body points with the central one (18) being the nose ridge. Not all parts (neck and ear parts) are given here for brevity. Hagen has many variations so only one source is provided here. The unusual groups of four for the Mai dialect have a pattern shown in bold and applied to all groups of four, so 30 is part of the group, in English 32, in Mai dialect ‘dog’ and 40 completes the group (I cut complete) Source Owens, Lean, with Paraide and Muke (2018) based on work by Mimica (1988), May and Loeweke (1981), Mark (2003), and Lean (1992) respectively

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Table 2 Showing the distribution of counting system and tally types among the Phyla of the non-Austronesian languages Types Cycles

West Papuan

East Papuan

Torricelli

Sepik-Ramu

Trans N. G

Minor Phyla

Total

(2)

0

0

0

3

39

0

42

(2, 5)

0

1

16

5

86

1

109

(2’, 5)

0

1

3

5

17

1

27

(2”, 5)

0

0

5

3

31

1

40

(5, 20)

0

1

2

17

52

7

79

(4)

0

0

0

1

6

2

9

(6)

0

0

0

0

5

0

5

Body-parts

0

0

0

8

58

4?

70?

(5, 10)

2

12

0

3

4

0

22

(5, 10, 20)

5

0

0

0

4

3

13

(10)

1

8

0

1

2

0

13

(10, 20)

2

0

0

0

1

0

3

Note Trans N.G. = Trans New Guinea Phylum. These are numbers from collected data, which are most languages but not all languages in Papua New Guinea and Oceania. They exclude 11 West Papuan languages in North Halmahera. From data, it was difficult to classify some languages so the symbol ‘?’ is used to identify that this number may not be correct at the time that the data was collected in the 1970s and 1980s (today, too, more languages have been identified). 2’ are 2-cycle systems with numerals for 1, 2, and 4, all NAN and usually occurring with body-tally systems; 2” are 2-cycle systems with numerals for 1, 2, and 3 with many found also in AN languages—Buang family in Morobe and in Milne Bay usually as (2”, 5, 20) cycle systems Source Owens et al. (2018a, 2018b: 196)

From Table 2, the rich tapestry of diverse “colours” of patterns in the systems are evident. While the various hues of the (2, 5) systems dominate together with the (5, 20) systems, there is a surprisingly large number of body-part tally systems. These systems fall mainly on the western side of the mainland and are associated with the TNGP but have similarities to the Sepik-Ramu Engan languages or these are a subgroup of TNGP (Pawley, 2012; Ross, 2005). Interestingly these may have very old origins and may be used more for ceremonial and cultural purposes for specific groups (Owens, 2001). This type of system seems to be found only in PNG and Australia. Lean (1992) had provided a number of systems from East Sepik, Sandaun, Enga, Southern Highlands, Hela and Western Provinces, there was a possible system with unlikely features far to the east in Madang Province known only by an older man (Wassmann & Dasen, 1994) but not supported by my oral communications with younger men of the village area interested in their ethnomathematics. Dwyer and Minnegal (2016) have shown that one of the languages that Lean had considered was indeed several different languages and they provided even more data on these systems. These body-tally systems often occurred with 2-cycle systems. Many are truncated for various reasons such as introduced modern currency or similar tonal words being unmentionable in public (Owens et al., 2018a, 2018b; Saxe, 2012, nd).

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Details about the 6-cycle systems (see Table 3) have been confirmed to relate to trade from the Ndom Island to the mainland (Hammarström, 2009). Interestingly, in Kanum, it seems that there was a reasonably well developed base system but that different orders of words, and to some extent different morphemes (Donahue, 2008) depended on the relative size of the numbers. Donahue classified these as simple, moderate, and complex number systems. However, given the intricacies of counting in PNG there may be cultural reasons for these variations. It is also worth noting that in other language groups, I found people within the one language group would use the morphemes in different orders and ways to represent numbers suggesting a much more fluid approach to number than a western base 10 system might indicate (Owens et al., 2018a, 2018b). Without the detailed work of linguists like Hammarström, Donahue, Dwyer and Minnegal, it might not be possible to “see” the beauty and hues of these systems in our tapestry. In the examples, the numbers from 7 to 11 follow the pattern of 6 or another word +1, 2, 3, 4, and 5 (cf . Table 3). Various ways of showing 12 include double six and six plus group (probably hand). The second power of 6 has a new frame word5 like hundred is a new frame word for the second power of ten. The words in bold help to focus on the 6-cycle. The Kanum data differ in spelling from earlier data but are otherwise similar in pattern. Table 3 Three examples of 6-cycle numeral systems of the non-Austronesian Papuan languages Kimaghama

Ndom

Riantana

Kanum

1

növere, nubella

sas

mebö

aempy

2

kave

thef

enava

ynaoaempy

3

pendji

ithin

pendö

ylla

4

jando

thonith

wendö

eser

5

mado

meregh

mata

tamp

6

turo, ibolo-nubella

mer

törwa

ptae

7

iburo-növere

mer abo sas

mebö-me

aempy ptae 1 + 6

8

iburo-kave

mer abo thef

enava-me

ynaoaempy ptae 2 + 6

9

iburo-pendji

mer abo ithin

pendö-me

ylla ptae 3 + 6

10

iburo-jando

mer abo thonith

wendö-me

eser ptae 4 + 6

11

iburo-mado

mer abo meregh

mata-me

tamp ptae 5 + 6

12

-

mer an thef

törwa-me

tarwmpao 12

18

mer an ithin

nimpe

36

(ntaop) ptae (big) 6

216

tarwmpao 216

Note This data was sourced from Galis (1960, p. 148) and Drabbe (1926, pp. 6–7) with agreement from Boelaars (1950, p. 34) for Kimaghama, Donahue (2008) for Kanum complex numbers Source Owens et al. (2018a, 2018b: 120, 122)

5

Frame words are the numeral words on which other words are built; e.g. 1, 2, 3, 4, 5, 6, 36, 216, etc.

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The 6- and 4-cycle counting systems are not common but they are noticeable colours on the tapestry. One reason for this is the way in which the systems have higher powers. There are higher power, larger numbers and a sense of infinity in other systems including the digit-tally systems and those with powers of 10. An interesting confusion arose with the recording of numbers where reduplication of the morpheme was used for doubling. Some Gahuku speakers recorded the numbers which had logosi (‘five and five’) repeated for addition as logosi2 and when repeated 4 times (five, five, five, five) as logosi4 (Owens et al., 2018a, 2018b). Although this might suggest poorly understood school concepts inadvertently used to express traditional counting systems, it also reflects transcultural humour used for reduplication of words like kaukau as kau2 or kaukau paua (‘sweet potato power’). In summary, the tapestry threads of location of counting systems and the classification of counting systems are intertwined to present a larger part of the picture. That is, counting systems can develop spontaneously, be taken on by another group through relationships such as trade or reciprocity in intergenerational relationships or for privilege. The system may be highly valued such as non-Austronesian (NAN) Ekagi in West Papua with a base 10 (Lean, 1992) or the Iqwaye with a digit tally system of powers of 20 linked to mythology (Mimica, 1988) and oneness (counting to 20 completes the one, the whole, oneness) (Pickles, 2009). Counting systems may be valued for specific cultural activities. Counting systems of a particular category tend to be found in specific phyla for NAN languages and of clusters of Oceanic languages. The interpretation of the closeness of counting with body parts or objects does not reduce the notion of mathematical reasoning as the early writers suggested, nor can the wholeness and oneness as found in most of the digit tally systems (Iqwaye being one of them), be interpreted by or superimposed on western mathematical ways of thinking about number (Pickle, 2009).

5 Mathematics in Cultural Activities Mathematics in culture requires reading the knowledges critically in order to establish the impact of culture and the ways in which it has created the mathematics of the culture. In the cases of PNG, the tapestry of mathematics is influenced by several key aspects of the culture. These are: • • • • •

trade, group decision making and displays, valuing culture in mathematical language, intergenerational relationships, and a need for large numbers for cultural reasons.

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5.1 Trade In the discussion of the archaeological sites, mention has been made of trade and of the impact on languages through loan words but also major changes in the languages. Besides Boiken, Adzera is another language impacted by NonAustronesian languages in terms of the counting systems. They adapted a 2-cycle system rather than continue with the 10-cycle system as they were building relationships with a number of non-Austronesian language groups on both sides of the Markham valley. Trading routes from the highlands to the coast and between coastal and island communities and between seaboard and inland coastal villages were very common. To carry out trade, relationships between neighbouring villages occurred and a person able to speak multiple languages was highly regarded. However, much of the trade was more like reciprocity exchanges and the quality of the goods in a basket as well as the number of baskets or containers were taken into consideration. Interestingly, the trade of shells, pottery, obsidian, and designs in tortoise shell attached to hard shell symbolising leadership and male–female relationships together with the trade routes for the ocean-going canoes seem to have encouraged the Oceanic languages to spread widely. They carried some similarities in the languages that developed from Proto Oceanic. By comparison, for the Non-Austronesian languages, it seems that identity was closely linked with the individual language or even differences in dialects or villages. To maintain independent identity meant that the languages developed and remained quite different.

5.2 Group Decision Making and Displays Most decisions are made by the group whether it be for marriage and bride price or for comparing piles of food items. In counting systems with a cycle of 5, there was ready access to hands and “borrowing” hands of another indicated usually by a nod to the other person and then the next person in the group and so on. This occurs with the Yabiyufa6 system. Thus the counting in cycles of 5 would reach quite large numbers. This is extended to the digit-tally (5, 20) systems where another person represents each group of 20. It should be noted that group displays often accompanied counting or replaced the need to count as in Dobu-speaking groups on the Papuan islands. It also led to multiple counting systems as in Hagen where the counting system is basically a (4, 8) cycle system but when men counted the line of pigs, one counted up to ten and another keeping track of the tens. Ten involved bending down the four fingers of one hand, then the other (as they would for the (4, 8) cycle system) and then counting the thumbs and marking the full ten; for example, in Gawigl which is a dialect or similar to Hagen, Western Highlands, with the hands together and the 6

Many languages in the Eastern Highlands Province seem to have a 5-cycle system that may have been recently modified to incorporate a 10 or 20 cycle but Tok Pisin is so common that these languages are becoming endangered.

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thumbs were wiped down the lips. In recent times with large compensation claims, 100s and 1000s are marked by objects such as a coil of rope. In general Hagen has a (4, 8) cycle system with multiples of 8 used in counting as shown in Table 1 (Mark, 2003; Strathern, 1977; Vicedom & Tischner, 1943). Both systems have their places and are well recognised in cultural contexts.

5.3 Valuing Culture in the Mathematical Language Specific ways of counting occurred for different food items. For example, in many coastal areas, coconuts were tied together in pairs and a child might carry one pair in each hand, then as they got older they could carry two pairs and then three pairs in each hand. This is a total of 12 and so counting in 12s occurred in the Tolai or Tinatatuna language of East New Britain (Paraide, 2018). It seems that some of the 6-cycle systems were associated with tying taros in threes and then carrying three in each hand, a total of six. These were placed down in a display pattern for six groups of six. In the Sepik, balls of sago were often sold as a group of 10 and in Motu small fish were strung together in tens. Sometimes this led to the group being given different names for the different items as occurs in Motu, the significance is the recognised group of a specific item. Motu also has a counting system affected by pairs. Roro, the language to the west in the same Family, clearly has 6 = 2 × 3, 7 = 2 × 3 + 1, 8 = 2 × 4, and 9 = 2 × 4 + 1 while Motu has a specific numeral for 7. Some, but not all, Polynesian languages have cyclic patterns at 2 × 102 and 2 × 103 (not a base system at 202 and 203 ). These counting systems developed through cultural practices of counting in pairs, often to large numbers (Bender & Beller, 2006; Best, 1907). When counting in pairs using the words used for counting single items, it was understood that the counting of those objects was of pairs. In Yu Wooi or Mid-Wahgi, Jiwaka Province counting of the fingers in pairs was common without actually using ordered number words. As the fingers were bent two at a time, they counted two, two, two, two, two (i.e. hands).

5.4 Intergenerational Relationships Paraide (2018), a Tolai, provides details of how she learnt her Tinatatuna counting systems and arithmetic from collecting coconuts and taros with her mother with groups of 2, 4 and 12 as well as 3, 6, and 12 respectively, and 120 (ten groups of 12). Muke shares how his father would remember large numbers of pigs or exchange items using his body parts (Owens, Lean, & Muke, 2018a, 2018b). The Kaveve village reports, field trips and personal communications and interviews, and student projects in our study were often a result of how knowledge was part of village living and sharing (Owens et al., 2018a, 2018b).

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When certain practices were no longer carried out as a family group then these needed to be revived. However, families like Sondo’s from Malalamai, Madang Province, were deliberately involved in everyday and specific cultural activities by which the children learnt their families’ (mother’s and father’s) mathematics, science and language. Children eagerly watch as Elders display their knowledge as in Muke’s village on our field trip.

6 The Need for Large Numbers for Cultural Reasons “Counting does not exist in isolation. It quantifies and qualifies relations between people, objects and other entities” (Bowers & Lepi, 1975, p. 322) for many of the foundational societies of PNG. Goods were for prestige, power, and privilege so the display was important whether or not the goods were counted precisely as some valued counting whereas for others the display itself with approximate amounts was sufficient. For the Tolai counting fathoms of shell money, the generously looped length between the arms was then displayed in groups of a hundred and when there was 500 or 1000 they were bundled and tied into an annulus (Paraide, 2018). The Tolai and other Austronesian Oceanic groups use shells as money, usually sewn into a strip. There have even been banking systems using shell money and there was a resurgence in using during the COVID-19 crisis. Thus certain items are known to have certain values in terms of the shell money. The impact on counting and measuring as a result is significant. First there can be more than one composite unit or cycle for counting. Thus the composite units of 10, 12, 100, 1000 are used together with lengths such as fathom and half fathom. Second, large numbers are distributed; and the need for counting but also for display is reinforced. Bundles of 10 and 100 fathoms are made so there can be 300 displayed as three bundles. Furthermore, the money is distributed in small lengths providing a sense of division and parts of a whole. This places the counting numbers, whole numbers, into the continuous number system (Paraide, 2018). Muke (2000; Owens, Lean, & Muke, 2018a, 2018b) and Strathern (1971, 1977) and Mimica (1988) have all shown how highlands communities used large numbers and displays for cultural reasons with interesting and varied complexity.

7 The Tapestry Section on Time and Work Patterns Having provided the extraordinary section on counting systems as a window into diversity and a review of the cultural aspects that impact on the mathematical tapestry, we take a look at other sections of the tapestry. In Owens (2015), various cultural practices such as making houses and canoes were described and analysed in terms of the profound use of visuospatial reasoning such as having memory for shapes and amounts for different situations, trialling the impact of different heights of the stays

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for the cross-bars to the outrigger of a canoe and assessing balance, and the embodied memory in which time could be assessed whether asleep, sailing or walking. Each of these had an ecocultural aspect such as the location of the village and the need for food gardens or fish and travel. Movement patterns between places occurred on a regular basis. This section exemplifies further understanding of patterns when thinking differently from western modern cultures in terms of time and activity within the cultural context. Time was particularly related to everyday activity of the society in Kragur village and across the Kairiru language group (a Western Oceanic, Shouten Cluster (SIL, nd) spoken on Kairiru, Karesau and You Islands off the north coast of PNG, 10 km north west of Wewak (Smith, 1994)). Kragur is on the rough and higher northern end of the island. Patterns of behaviour are influenced by the worldview of cause and effect linked to the supernatural and the people of the present connecting with the supernatural. All accidents and illness are considered as a result of a failed relationship with someone. Certain people have the knowledge of magic to assist with gardens and fishing. People are expected to meet together to discuss and plan for a fishing trip for the konan7 fish to bring the fish together for catching. If someone is not present then the fish will not come together. The meeting times and lengths of time seem dependent on who comes, what is happening at sea, suggestions made by the group, and many other things. Groups meet according to relationships between individuals and across the group. Features of the group are identified. Groups vary. Family groups may be identified for the purpose of gathering together sufficient for a gift of, for example, taro. Points for discussion now arise around the gift, its nature and quality, who contributes, and whether some items should now be commodified and paid for. The latter is more and more expected today but that brings some conflict in establishing relationships upon which you can trust in times of difficulty. However, it is in the comparison of how time and work are understood in the subsistence community and in western workforces that the mathematical patterns emerge more clearly. Previously we have noted that time was embodied so people could arise in the middle of the night to participate in activities before the dawn or they knew how far they had walked or travelled on a canoe taking account of terrain and swells and tides (Owens, 2015). Smith (1994) provides several examples of time and work. In one, he goes with a villager to cut down a tree for a house that the community are helping to build. Through the day, they need to wait for various people to join them and although the man complains about the delay, it is expected. There are several side trips to do other things but they also work together to turn the log into a rectangular beam using adze, axes and knives. The time and work interplay with one informing the other in terms of meaning. Furthermore, the people talk about the moon or sun moving fast when they are busy in their subsistence life style which is always active, always out and about compared to doing a task for which they 7

Konan is the Leing Tau (Kairiru’s language) for these long-nosed small fish. They come in schools during the rainy season in the middle of the year close to the beach when the water is calm but now trees have been cut down and the water is too rough so they do not come often so they are hard to catch. When they grow big, the people call them Wurmak or smol tuna (Tok Pisin) (personal communication, Monika Sikas, 2021).

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are paid like being a driver. Some of the paid work was not regarded as work and bored them. They craved the active life of the village. Some village work was sitting around talking, it was still seen as valuable work unlike some paid work where you sit. Some paid work was even considered as being in a prison cell especially during the Japanese occupation where they were not allowed to move around at all but had to work for the Japanese. Time passed too slowly or rather the sun did not move quickly. Some even talked of making magic when working on a plantation to make the sun set more quickly. The sun can go rapidly or slowly. However, their contact with the western use of time such as school, church or employment meant they now had a seven-day week. Thus, as Hallowell (1955) suggested, time is different depending on the experience. There is a different temporal orientation reflected in the language of time for the Kragur. A year is denoted by the passing of Pleiades called abil and the moon cycle as kareo. The fact that 13 moon cycles matches 12 calendar day months is explained by the days not ending when the moon dies—quoting a villager, Smith reports “the sky goes fast but the days go slowly”. The task is valued over the passage of time. Time is not commodified as in the western view of time.

8 The Tapestry of Transactions Foundational (traditional) mathematics, from generations past to present, is associated with trade of objects and some of the trade routes that help to identify the longevity of cultures (Paraide et al., forthcoming 2022) and interactions of languages and counting systems (Owens et al., 2018a, 2018b). The transactions that occur in the Was valley among the Wola, not far from Mendi and Lake Kutubu in the Southern Highlands are discussed by Sillitoe (2010). The language is a variant of Angal or Mendi, of the Angal-Kewa Family, probably of the Engan Stock, of the Trans New Guinea Phylum (SIL, nd). Sillitoe (2010) provides a comparative analysis of the “economy” of the valley indicating the role land and food play in this subsistent community as well as the kind of transactions and goods. He also argues against the idea of scarcity of time and food as universal driving forces in transactions and against the notion of work for increasing wealth. Rather he considers the sociopolitical aspects of the societies for driving the economy. The establishment and maintenance of relationships is the main force in these societies. Politically it is related to status, not necessarily power to control or demand. Status is related to wealth and to actions involving wealth. Time such as five days earlier or seven days later can be named together with the moon cycles and two seasons but there is no sense of a period of time being a composite of another unit of time or that using extra time to work will create excess products or wealth. Their efforts contribute to their well-being but it is not work, there is no word for work per se. Transactions and effort for these transactions maintain social order. Thus the economy is for sociopolitical reasons and subsistence. However, there are patterns and Sillitoe was able to classify transactions according to their features and thus an ethnomathematical way of thinking.

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Sillitoe divides objects of transactions into categories in order to explain the transactions that take place and the importance of sociopolitical roles in transactions in various ways. Local consumables used in transactions in general include pigs, but taro, pandanus, marsupials, and cassowary are used in a few restricted transactions. Other objects obtained from elsewhere include cosmetic oil and salt in general transactions or cassowary in restricted transactions. More durable local materials for general transactions include feather headdresses (also from elsewhere), possum-teeth beard pins and cassowary eggshell headbands while products originating elsewhere that are durable for all general transactions include pearl shells, cowrie shell necklaces, nassa shell headbands, stone axes, black palm bows, steel tools and cash. Equivalence of items is not really an issue here. These are transactions for sociopolitical reasons rather than for amassing wealth. Some are enjoyed, others are practical, some are adornment, some commodities like the shells have lost their value since they are no longer scarce. While ‘men of the clearing’ (a meeting place) discuss and make decisions for the community or for exchanges and involve others in projects, there is not a stratified society based on wealth, family, land, or control of other people’s time and wealth. These men are leaders and in sociopolitical terms show their wealth and influence in exchanges. While exchange of time and effort does occur, usually at different times and not directly (e.g., helping to make a garden might be repaid by similar action sometime later), and food payments are made together with other small exchanges for time and effort, these transactions are for sociopolitical or subsistence purposes rather than for accumulating wealth. However, there may be economic transactions particularly for outsiders to pay or for a wide range of purchases such as a pig or to receive payment for assistance with a task, or a crop in the garden. These are specific transactions rather than lumped together as sales and commodities. Imported items may be sold on further perhaps in smaller quantities like the cosmetic oil and so some wealth might be made. Products were made like the possum teeth combs which were made from fortuitously capturing a possum and saving up or trading to get enough teeth to make the comb but without specific intention to gain wealth. Men will expect to “receive exchangeables in transactions, not work to produce them” (Sillitoe, 2010: 433) so they do not work or produce to make wealth. There are spheres of transactions—the sociopolitical sphere and that of household subsistence. Sharing taro, preparing a pig kill, or displaying shell money are all ways of noting one’s worth as part of the society and perhaps as a man of the clearing. Wealth needs to become part of the exchange system to be used sociopolitically and vice versa and until recently there was little possibility for these to occur due to a lack of cashpaying work. Durable exchange items tend to circulate and be collective property and other items like pigs and cosmetic oil are often divided up among many and passed on. Transactions support people’s social standing rather than their social standing being dependent on accumulation of wealth. The individual maintains personal rights within community. The valuing of these items is fuzzy, as Sillitoe (2010) says. There is considerable tacit knowledge, or knowledge as practice, and a general embodiment of values rather than precise values given to choices about land, food items, pigs, shell, cosmetic oil,

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wood or other items. People look after themselves and their immediate family so they are individuals but there are community expectations including assisting with tasks that may take more than one person to do, sharing land or assisting with exchanges. Reciprocal relations are important. Most garden work is done alone (roughly six times more likely to be done alone although one other person is also common) despite the hard work involved with clearing and fencing. Fencing is carried out to stop pigs from damaging the ground before or after planting. Steel axes probably halve the time taken compared to a stone axe but this does not seem to have resulted in more gardens being made, at least while there is no access to a market for selling produce. The shape of the garden depends on the landscape and it is tacit knowledge again that is used to decide if it is large enough for the family’s needs. Making the fences involves appropriate tools such as axes and wedges to split the logs. It also requires some mathematical tacit skills like use of tools, binding, spacing, sharpening both ends of the posts (to reverse in future when one rots in the ground), or making use of ditches. The ownership of land with rights to others’ land assists families when there is a need. The land holdings and gardens tend to be widely spread across the landscape with cultivable land available in pockets throughout so people can expand their holdings if they wished (to gain wealth) without exhausting their resources but they tend not to do this (Sillitoe, 2010). There is a well-accepted situation in terms of inherited land and land use that is flexible, taking into account family situations and needs. Men may make gardens on their inherited land, their wife’s land, his relatives’ land and occasionally on his wife’s family’s land. Efforts are made to keep land used by others recognised as one’s inherited land and sometimes there are disputes that need to be resolved. However, land is not sold or used to build up wealth. Except for disasters such as frosts and fighting, the Wola are able to plant sufficient sweet potato to feed families using available plots. The plots used will be negotiated. Most gardens are within an hour walk of the settlement. Some gardens for mixed vegetables are usually closer while taro gardens are often further afield and somewhat hidden—taros are distributed as part of the sociopolitical status. The work of the garden is divided by gender although occasionally this is not followed. Men usually start the garden cutting down rainforest, trees, and kunai grass if planting on an earlier garden that has been left fallow. Women keep the garden clear of weeds and harvest. They are likely to spend four to six hours a day in these activities. However, like the neighbouring Kailila and most other PNG groups, time is not considered as needing to be used today so that more time on work will gain more garden or wealth. Gardens are prepared for a purpose such as feeding the family or preparing for a payment or gift to another family. Sweet potato can last in the ground for some time but taro once picked needs to be eaten fairly quickly. The corms are given to relatives to maintain relationships. Size is important to impress in giving. The family themselves might end up with small corms (Sillitoe, 2010; personal observations). It is possible to consider the various variables that impact on the various transactions but it is in a fuzzy tacit agreement rather than in numerical values. Exact equivalence of goods is not possible. Goods may remain in one category or move between categories as discussed above. Importantly, the mathematical ways of thinking are

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different to western views of equivalence by using numbers. Patterns of exchange are reciprocal but not necessarily linear or direct. Sociopolitical reasons influence decisions on transactions which are discussed and open to others to see as a way of maintaining order and memory.

9 The Tapestry of Mathematics in Art Ethnomathematical studies often refer to art such as the sand drawings of the Pacific and Africa (Ascher, 1994) or wall art with each block having interesting symmetries and transformations (Gerdes, 1998). In PNG, mathematics occurs in the art of many cultural groups including the kapkap leadership designs of New Ireland and other Austronesian groups (Owens, 2015). The various designs themselves were traded. In the art of Sepik communities like the Abelam, the line is significant in each panel of artwork and connected through panels (Owens, 2016). The art on shields of the MidWahgi of Jiwaka Province (Muke, 1993; Paraide et al., forthcoming, 2022) involves three zones representative of head, torso and legs with differing colours and shapes depicted on them but connecting to spiritual supports in the battle. These areas of art indicate that shapes, lines and position were indicative of relationships and more than just a shape. Vandendriessche (2014, 2015 discussed the mathematics, particularly the operations, subprocedures and procedures of the string art commonly played in the Trobriands (as elsewhere but with local meaning and story, see also Haddon (1930, reprinted in 1979) and discussions in Owens 2015). The art of the Trobriands in particular in the Kula trade also display a section of the mathematical tapestry created by patterns that are followed in travel and in design. Kula was the trading between various groups in the islands of Milne Bay Province. Over considerable distance, the sailing canoes travelled to exchange shellbased objects of importance. There is considerable meaning behind the art of the bow (seen by all and resisting splash) and stern boards (a neat trim for the line of the boat and its movement). Both generally occurred at each end because the canoes can in fact go in both directions. The side splashboards were also decorated. The art of the boards forms part of the Massim style of these Milne Bay islands (or Louisiade Islands). It has been highly regarded and sort after since the mid-1800s and pieces can be found in museums. Importantly, symbols or procedures have cultural meanings. They can be positioned, connected and related. A study of the symbols used in the Kula art is productive in terms of patterns and mathematics. (Fig. 3a, b is a canoe on Kiriwina, Trobriands Islands). This analysis is based on the exceptional, detailed work of Campbell (2002) who initially developed it from museum pieces before an extensive period on Kiriwina to explore the meaning of shape and position, particularly of the Vakuan style (her book shows other canoes). Local people can distinguish between different island art and more local variations and that of the master artisan in the area. Euclidian shapes, especially those with straight sides, are limiting in terms of understanding shape from many Indigenous cultures whose emphases are on curves

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a. Canoe prow, 1973

d. End of bowl similar to prow head and body, C curve, tight S with curved end

b. Trobriands Canoe, 1973

e. C curve, horizontal opening downwards, repeated

g. Egret, prow-like carving, flower, circles, modified C curve (above brow of head)

h. Wave curve on head

c. Curve with tight single end and curved long section (like apostrophe; wave, snake, filled in apostrophe markings

f. Lower body section, single end of S with straight section and S-shapes flowing together and intertwined

i. Leader symbol on head similar to some prows at top – this is the head of small ebony statue

Fig. 3 Trobriand art on different objects (Source Owens collection)

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(Owens, 2015). In the case of Kula art, Campbell has classified a range of curves. Different places and carvers created their own styles. Figure 3 also shows carving undertaken by a follower but not the main empowered carver on wooden artefacts sold to tourists in 1973 (Owens collection) when their art was highly regarded. Tourist art was allowed and not seen as usurping the main carver’s role or power. However, you can see a number of the common curves (described in detail below) and that curves did not intertwine (compare with, for example, Celtic knots that intertwine). The boards themselves have different sections given anthropomorphic names, and like the art of the Mid-Wahgi, specific shapes are likely to be kept within them. Kula art also impressed to engage the spiritual for the journey and trade. Certain parts of the stern board are called head and nose, body or chest, throat, and tail and for the prow board, they are head, body or chest, wing or arm and sometimes a tail to one side. Certain curved shapes represent so-called animals (not necessarily real animals). Position, shape and orientation were important. The orientation often resulted from the position on the board (Campbell, 2002). Often to support the carved outline, further lines are carved parallel or concentric or in rotated position. Other carving is to fill in the spaces although if painted these would all be of the one colour. This is noticeable on the snake in Fig. 3c. Figure 3g is the representation of a bird, probably an egret and artistic license is taken to curve the beak. Campbell’s classification of curves had several subcategories for each category. One of these I will call the S curve. The curve may bend at both ends like an S but they can vary in their tightness, the equality or not of the ends, and the amount of curve with one variation being made of two Ss, one of which is in the rotated position so forming an enclosed twist (Fig. 3f). Another subcategory has one end curved and a straight section (more like an open P) (Fig. 3f middle) and another subcategory has one curve but a very slightly curved section more like an apostrophe (Fig. 3c, g). The curve with a straight section is often duplicated with slide or reflected symmetry and joined or unjoined ends for the curve. An important category features as the main curve on the head and body section of the prow board. It has the curve forming a C shape with more of a curve on both ends than a C, usually horizontally positioned with the opening up or down (Fig. 3d, g) or obliquely. Another category of curves is like a U with ends curved outwardly or repeated like a wave and either reflected or repeated beneath (Fig. 3c, h). Another category of shapes include the circles (several in Fig. 3), the parabola shapes with horizontal axes (Fig. 3f), and various enclosed shapes with concave curved ends and slightly curved sides that may be duplicated (see Fig. 4 where similar shapes are used on tapa cloth). These shapes often had another common feature of carvings which was to repeat the end of shapes for emphasis (below the circular hole on Fig. 3c). Some enclosed shapes resemble stylised whales or fish, bats, or other simple beautiful shapes with 3 or 4 curved sides. Each classification, especially the first group have multiple variations of subcategories. This is similar to classifying a variety of triangles in Indo-European school geometry. The categories could be seen as similar to classifying polygons, each type of polygon (named by the number of sides and angles) having its own properties. However, each curve and shape is used in specific sections of the boards.

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Fig. 4 Tapa designs

While the classifications and positions were recorded by Campbell from artefacts, the follow-up discussions in the villages gave meaning to the shapes and arrangements and confirmed the classification groupings that were made. The boards in Fig. 3a illustrate, however, the classificatory nature of the curves, and the impact of their position. The basic S curve is repeated around the key C curve on the prow board as a chain of basic Ss. However, when different sized curves in the S are used in the central position of the board, they are at the end of the chain. Wavy curves are used across the top for emphasis of the stylised figures in the centre top of the boards. Pigments are also used with different meanings and linked to their position in a carving. White is placed in grooves and across the surface of the carved wood with red used in oblique edges that drop to a deeper level of carving marked in black. From this discussion, the tapestry of patterns is evident by the categorising of shapes and by the associated positions in which they are placed on the board for different meanings and effects. Intergenerational knowledge is passed on to other carvers but only one person will be the new major carver of the group, an inherited position. For the other carvers, careful variations are made as in the tourist pieces, building on some basic curves, procedures and relationships between curves. Interestingly, paintings on tapa cloths from Oro Province next to Milne Bay Province have similar but unique non-intertwining curves and lines (Fig. 4).

10 Discussion Fuzzy mathematics is an important aspect of learning mathematics for students (Owens, 2009). It is a way of visualising mathematical concepts. It is often a result of embodied learning as, for example, a child placing counters, stones or other objects in a group for different numbers and noting the name and perhaps symbol associated with the number. This action leads to a sense of size for each of the numbers and their associated numerals. Children learn that 8 and 9 are bigger than 7, for example. This

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embodied learning and reckoning is an important way of thinking mathematically. In the examples given above, mathematical thinking is associated with doing, with actions. Silllitoe (2010) points out the Wola speakers have no word for work but there are various words for different “doings” for garden making, negotiating, and exchanging. From the cultural mathematics discussed here there are several important principles of mathematics evolving. These include the notion of patterns of activities including everyday village life. There is considerable mathematical knowledge as part of the tacit knowledge of the environment upon which decisions are made for gardening, fishing, or other activities. There is a memory of the past and these memories are used for comparison and evaluation of current practices albeit gardening or carving or negotiating. Sizes are tacitly and visually known and compared. Discussion is frequently a key aspect of making decisions and the group is important for knowledge to be accepted and valued in various discussions. In using known shapes or curves in art, these classifications and their use and permitted use, their position and variation is known from observation, practice and intergenerational knowledge sharing. None of the activities or symbols of communication are without cultural import. Counting as illustrated in Tables 1, 2, and 3 might seem clear cut. The system of counting for any language can be classified in terms of its cycles or tally system. However, in practice, the sense of size might not readily be explained in terms of the counting system. For example, there is no measuring of lengths or ways of assessing area of gardens for the Wola. They just look and decide on whether the garden will be large enough based on their previous experiences of the ground and soil, position in the valley, previous use etc. Similar approaches are found across PNG from my conversations, observations, interviews and students’ reports. Our classifying of systems is also based on extrapolating from data that may not be quite so straight forward as different groups within the same language group may have recorded different words. These systems may be relatively flexible, especially the 2-cycle system and as we found in different villages and with different informants of the Gahuku-Asaro Alekano system. Systems might occur together with another system or ways of making assessment of size usually in displays or gestures. Displays and gestures with counting in themselves are rituals of cultural significance as in the case of the Iqwaye (Mimica, 1988) and Tolai (Paraide, 2018). Finally counting is generally part of an open discussion in which various other aspects of the situation are also considered especially “oneness” or being one (Pickles, 2009). Thus counting is part of the mathematical tapestry created and used for decision-making.

11 Conclusion The rapidly changing cultures of today beg for knowledge of their past for the purposes of identity and understanding of their cultural ways of thinking that have mathematical connotations (Owens, 2015). Importantly they are recognising the

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patterns of actions, curves and other relations, classifications, and measures of some kind. Measures might relate to time or effort or value for transactions or sociopolitical position. However, there is a need also to see if there is a link with school mathematical connotations. In school, numbers might predominate but they are best understood in terms of relations between numbers if the whole of counting is to be applicable to other facets of life or mathematics. Establishing the idea of relationships is a significant way of understanding the essence of mathematics. Detail of relationships such as in art, transactions, and understanding of time are negotiated mathematically. How might there be trajectories in these mathematical understandings as we establish their incorporation into school mathematics? The learning of cultural knowledge is usually over time and with tacit intention. It may be however important to identify these learnings to recognise and strengthen the cultural mathematical ways of thinking and extending these to new areas of reasoning mathematically. Will these be linked then to school mathematics that has entered from a western world? Real life problems are fuzzy. Sometimes there is information that is not presented but either known from previous experience or needing to be sought. Looking for similarities in earlier problems may help but sometimes it is a matter of looking for all the key issues involved in the situation. Having the disposition to think of alternative approaches when faced with a problem is developed by students who have not learnt that mathematics is a set of procedures learnt from the teacher. Rather they have been presented with problems and situations throughout their life and schooling that encourages them to think of alternative approaches to problems. They have learnt to assess other students’ ideas on solving the problem. They do not have a mindset to following set procedures in problem solving. They know useful processes and strategies, and they know how to inquire. One of the important parts of our learning new concepts revolves around getting started. These preliminary ideas begin with “the stored memories and information processing strategies of the brain interact(ing) with the sensory information received from the environment to actively select and attend to the information and to actively construct meaning” (Osborne & Wittrock, 1983, p. 4). The beginnings of conceptualising are “primitive knowing, making and having images” (Pirie & Kieren, 1994). From these properties are noticed assisting students to learn new concepts and structures (Towers & Martin, 2014). Students will develop concepts that take them beyond the physical object that they are seeing to understanding the concepts embodied in the object or action on the object. Students begin to form relationships between ideas and the concepts develop. At this time, a word from the teacher or fellow student will help students to place the work in the schema stored in the mind, to tag the ideas and images, and to focus attention on the features of the object or experience. Fuzzy beginnings in students constructing their own concepts rather than following a set of procedures results in strong personal and flexible ownership of the concepts. That means they will be able to modify, extend, develop these ideas themselves and relate to other ideas in a mathematically thinking way. Furthermore this thinking often involves patterns and relationships and is the beginning of mathematical modelling so ethnomathematics is a beginning of this whole area of mathematics.

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The PNG student familiar with their cultural ways of reasoning mathematically has a head start on fuzzy mathematical beginnings that will permit stronger relationships between their cultural knowledge and mathematical modelling presented from a western or hybrid perspective. They have the capacity to tackle the new mathematical concepts. With creativity and emphasis on inquiry where details are established, sorted and related, and then applied to further areas of learning (Murdoch, 2019; Owens et al., 2015), mathematics may be established to link and grow school mathematics in ways that are meaningful to the society and those living within the society in its subsistence and new commodified arena. Furthermore, Indigenous knowledges are significant for the whole community and indeed world especially in changing times. Issues of climate change in particular highlight the significance of Indigenous knowledges relevant to science and mathematics that links to the sciences related to the use of land, sea, and air resources. Alternative worldviews are particularly important in respecting and relating to these spaces. Ways of thinking mathematically that are apparent in the Indigenous cultures of Papua New Guinea indicate that there are alternatives in terms of data, patterns and relations and decision-making that may create a stronger world in which to live, one that respects the Indigenous mathematical knowledges and provides for creative mathematical solutions. This chapter analyzes some of the mathematics of Indigenous peoples of PNG and provides a means by which formal schooling can bring about a balanced reconciliation of different epistemologies and worldviews in comparison to the dominant educational model of assimilation. The chapter highlights the importance of political, cultural and educational rights of Indigenous peoples worldwide.

12 PostScript PNG cultures do not weave cloth from fibrous string. They use the inner bark of trees to make the string but they also beat the bark flat to make tapa cloth for coverings. The string is used to make continuous string bags called bilums using figure-8 loops but can also be made into coverings for body parts by using tight figure-8 loops and variants. Other coverings consist of a group of large leaves strapped over private parts of the body, “grass skirts” made from shredded leaves or grasses, and some use penis goulds. Thus there was no need to weave cloth. However, the various bamboos, pitpits, grasses, pandanus and coconut leaves were used for weaving baskets of various kinds such as fish traps, large nut holders, or food-carrying baskets as well as mats, wall coverings and decorations such as armulets (see Owens, 2015 for illustrations of these artefacts). However, one art centre in PNG is Kainantu Pottery in the Eastern Highlands. It introduced weaving because there was a good source of carded wool from an agricultural farm in the same province (sheep would not survive in most areas and this farm does not appear to have continued for some years although the rugs are still made, perhaps from imported wool). Its tapestry weaving was that of pictures and abstract

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design rather than regular “zig-zag” or “diamond” patterns so often found in bilums, woven walls, and baskets (Owens, 2015). It is this kind of tapestry that provides the metaphor (Lakoff, 1987) for the fuzzy mathematics that visually provides fluid mathematical systems and ways of reasoning in PNG without strict single-answer representations. Nevertheless, all tapestries have threads and materials that weave together to form the creative art. Ethnomathematics, language studies, cultural practices, mathematical language, archaeological evidence, oral histories and stories, activities in practice, and field study observations have all joined together to lead myself and others to notice, represent, analyse and understand PNG mathematics. PNG mathematics has the usual mathematical features of representation, analysis, patterns, and relations but they are embedded in PNG cultures. Using language, creating with language, cultural and spiritual practices and associated reasoning, knowledge of the past, of ancestors and of places, continuing oral stories and discussions, practical mathematics in many cultural activities, and observations in the place of belonging have all been part of the crafting of mathematical reasoning in Papua New Guinean societies.

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Owens, K. (2016). The line and the number are not naked in Papua New Guinea. International Journal for Research in Mathematics Education. Special issue: Ethnomathematics: Walking the mystical path with practical feet, 6(1), 244–260. Owens, K., Cherinda, M., & Jawahir, R. (2015). The importance of an ecocultural perspective for Indigenous and transcultural education. In K. Owens (Ed.), Visuospatial reasoning: An ecocultural perspective for space, geometry and measurement education (pp. 245–273). Springer. Owens, K., & Kaleva, W. (2008a). Case studies of mathematical thinking about area in Papua New Guinea. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Annual Conference of the International Group for the Psychology of Mathematics Education (PME) and North America chapter of PME, PME32 - PMENAXXX (Vol. 4, pp. 73–80). Morelia, Mexico: PME. Owens, K., & Kaleva, W. (2008b). Indigenous Papua New Guinea knowledges related to volume and mass. In Paper Presented at the International Congress on Mathematics Education ICME 11, Discussion Group 11 on The Role of Ethnomathematics in Mathematics Education. Monterray, Mexico. Retrieved 2020 https://researchoutput.csu.edu.au/en/publications/ind igenous-papua-new-guinea-knowledges-related-to-volume-and-mass Owens, K., Lean, G., & Muke, C. (2018a). Chapter 3: 2-cycle systems including some digit-tally systems. In K. Owens, G. Lean, with P. Paraide, & C. Muke (Eds.), History of number: Evidence from Papua New Guinea and Oceania. Springer. Owens, K., Lean, G., Paraide, P., & Muke, C. (2018b). The history of number: Evidence from Papua New Guinea and Oceania. Springer. Owens, K., & Paraide, P. (2019). The jigsaw for rewriting the history of number from the Indigenous knowledges of the Pacific. Open Panel 106 Indigenous mathematical knowledge and practices: (crossed-) perspectives from anthropology and ethnomathematics. In A. Arante et al. (Eds.), Annals of 18th World Congress of United Anthropological and Ethnological Societies IUAES18 Worlds (of) encounters: The past, present and future of anthropological knowledge, Anais 18 Congresso Mundial de Antropologia (pp. 3468–3487). Florianópolis, Brazil: IUAES18. Paraide, P. (2018). Chapter 11: Indigenous and western knowledge. In K. Owens, G. Lean, with, P. Paraide, & C. Muke (Eds.), History of number: Evidence from Papua New Guinea and Oceania. Springer. Paraide, P., Owens, K., Clarkson, P., Muke, C., & Owens, C. (2022). Mathematics education in a neo-colonial country: The case of Papua New Guinea. Springer. Pawley, A. (2012). How reconstructable is Trans New Guinea: Problems, progress, prospects. Language and Linguistics in Melanesia, 12, 88–169. Pawley, A., Attenborough, R., Golson, J., & Hide, R. (Eds.). (2005). Papuan pasts: Cultural, linguistic and biological histories of Papuan-speaking peoples. Pacific Linguistics, Australian National University. Pawley, A., & Green, R. (1985). The Proto-Oceanic language community. In R. Kirk & E. Szathmary (Eds.), Out of Asia: Peopling the Americas and the Pacific (pp. 161–184). Journal of Pacific History, Australian National University. Pawley, A., & Ross, M. (2006). The prehistory of Oceanic languages: Current view. In P. Bellwood, J. Fox, & D. Tryon (Eds.), The Austronesians: Historic and cultural perspectives. Pacific Linguistics, Australian National University. Pickles, A. (2009). Part and whole numbers: An “enumerative” reinterpretation of the Cambridge Anthropological Expedition to Torres Straits and its subjects. Oceania, 79(3), 293–315. Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2 and 3), 165–190. Report from a seminar in Kárášjohka Norway. (2008). Ethics in Sámi and Indigenous research: Report of Kárášjohka, 23–24 October, 2006. Bredbuktnesv, Norway: Author. Ross, M. (1988). Proto Oceanic and the Austronesian languages of western Melanesia. Pacific Linguistics, C-98. Ross, M. (1989). Early Oceanic linguistic prehistory. A Reassessment. Journal of Pacific History, 24(2), 135–149.

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Indigenous Mathematics in the Amazon: Kinship as Algebra and Geometry Among the Cashinahua Mauro W. B. Almeida

Abstract The mathematical competence of non-literate cultures expressed in the design of complex sociological structures has been recognized since a path-breaking Appendix by the mathematician André Weil to Lévi-Strauss’s treatise on kinship structures. The import of Weil’s contribution was to highlight the role of symmetries underlying kinship structures and the algebraic concept of a group which can be seen as a general theory of symmetry. The kinship structure of the Cashinahua people who inhabit the south-western Brazilian Amazon is a unique example of symmetry in social organization. This point is illustrated here by means of a correspondence between the group of actions of Cashinahua kinship terms on Cashinahua namesake classes, and of symmetries connecting graphic patterns, showing an underlying non-trivial structure known as a dihedral group. Keywords Structuralism · Kinship · Marriage · Klein group · Dihedral group · Cashinahua

1 Introduction Anthropologists, science historians and mathematicians have argued that indigenous peoples have mathematical capabilities, embodied in social and technical activities. I support this position with the example of the Cashinahua culture, showing how Cashinahua kinship terms are combined so as to produce kinship terms. Numbers are combined to produce numbers, thus expressing the fact that kinship relations have a mathematical structure. Mathematics of the forest is modern mathematics. There is an extensive literature on the presence of “mathematical ideas” in nonliterate cultures across the world (Ascher, 1991). One important branch of this literature deals with numbers and counting, as well as with geometric patterns in textiles, baskets and pottery (on numbers, Zalavsky, 1973; Crump, T. 1990; Urton & Nina M. W. B. Almeida (B) Universidade Estadual de Campinas, Campinas, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_8

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Llanos, 1997; Verran, 2001; Pica & Lecomte, 2008; Ferreira, 1997, 2015); Everett, D. L. 2005; Everett, C. 2017; Dehaene 2009, 2009; Gilsdorf, 2012; Vilaça, 2018; Almeida 2015); on geometric patterns, Dawson, 1975; Washburn & Crowe, 1988, 2004; Ascher, 1991, 2002; Roe, 2004; Guss 1989). While numbers and geometric patterns are natural loci where to look for mathematical ideas among across cultures, since Tylor’s chapter on “The Art of Counting” (1874: 240–265), kinship and social relations are not so obviously related to mathematical reasoning. Nevertheless, the subject became one major area in the branch of cultural anthropology which deals with mathematical reasoning (Almeida, 2019). The connection between kinship and mathematics is not obvious, as one can see from the answer Lévi-Strauss obtained from Hadamard whom he asked for help with a complicated kinship system: “mathematicians know only the four operations and marriage cannot be assimilated to none of them”. After that failed attempt, Lévi-Strauss received a different answer from André Weil, who told him that “it is not necessary to define marriage in mathematical terms, for only the relations between marriages are of interest” (Lévi-Strauss and Eribon 1988: 79). Hadamard’s and Weil’s answers express different paradigms: Hadamard represented mathematics as a science of objects (e.g. prime numbers) while André Weil was an avatar of Bourbaki mathematicians, set upon expressing all mathematics based on axiomatic structures—a program in which only relations matter, objects being defined by the relations to which they belong. As an example, number 1 is characterized by how it behaves when combined with any number by an operation – that is to say, in a given structure. In “computer arithmetic” 1 + 1 = 0. In “military time” arithmetic, 11 + 1 = 0. In Peano arithmetic, addition of 1 to a number is defined by Peano axioms as the “successor” of a number—a definition that assumes as postulates that (1) 0 is a number, (2) to each number n there is a unique number defined as n + 1 (the successor of n); (3) no successor is 0. The resulting structure is the infinite set of natural numbers starting with 0 together with addition. The notable fact is that Russell and Whitehead introduced this structure—a linearly ordered system—using the language of the peerage system (Almeida, 2019: 93; Whitehead & Russell, 1910: 570). This was not an accident. Anthropology has rediscovered mathematical structuralism in many disguises. Marilyn Strathern’s characterization of Melanesian theories of persons as packets of relations implies that a person is a bundle of relations—not a set of intrinsic qualities (Strathern, 1988). Amerindians’ perspectivistic epistemology according to Viveiros de Castro can be expressed as the affirmation that amerindian ontologies belong to a group of transformations of subject’s points of view—relating human and jaguars’ points of view about their respective others, in the same sense as Lorentz transformations relate different coordinate systems (Viveiros de Castro 2015). Marshall Sahlins on “what kinship is, and what it is not” grounded kinship ontology on “commonality”, in a generalized sense that amounts to a group of transformations that conserve something in common—a mystical idea, property rights, or any other invariant property conserved along kinship transformations (Sahlins, 2013). Manuela Carneiro da Cunha argued for a historical-dynamic version of the structural view (Carneiro da Cunha, 1973).

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The domain of anthropology of kinship structures originated from Lewis Morgan’s treatise on kinship terminologies of indigenous America, India and Asia on the one hand, and Europe and Eastern Europe on the other hand—a comparison from which Morgan drew his widely misunderstood contrast between “classificatory” and “descriptive” terminologies (Morgan, 1963, 1997), which amounts to taking the “bifurcate merging rule” feature as having precedence over all other features for comparison. This is the contrast between “relationship systems” that group together collateral relatives and those that distinguish them—sociologically, logically and ontologically. Morgan’s work originated diverging paradigms. A first paradigm followed Morgan’s sociological association of his two major domains of kinship terminologies to distinct marriage patterns and property regimes.1 A second paradigm focused on the syntactical rules governing the combination of kinship terms to produce kinship terms.2 A third paradigm shifted the interest to the metaphysical ideas connected to kinship relationships.3 The present analysis belongs to the second paradigm: it focuses on kinship terminologies as logical systems. This approach was first given a clear formulation by Floyd Lounsbury (1956, 1969). While the sociological paradigm generated an extensive literature on lineages and alliance expressed in kinship language, Lounsbury’s approach inspired an equally huge literature on rules for combining kinship terms. Both approaches may be seen as attempting to represent respectively marriage and descent as demographic phenomena—as statistical structures—and, on the other hand, as algebraic patterns holding for filiation and affinity kinship expressions.4 These different approaches may be seen as evidence of the same underlying phenomenon. While keeping Lounsbury’s focus on the grammar of kinship terminologies, a more recent research program (Read, 1984; Read et al., 2012; Leaf, 1971, 2006) proposed to obtain terminological rules directly from native terms, independently from Western kinship vocabulary. In a parallel move, Thomas Trautmann combined the sociological and syntactical approaches, devising a formal language which expresses social rules of siblingship, generational difference and affinality as mathematical rules, applied to the case of Dravidian cognatic kinship rules.5 Trautmann’s formulation is the basis for my own formulation, which is a generalization of Trautmann’s approach (Trautmann, 1981; Almeida, 2010a, 2010b) corroborated by the impressive ethnography by Vaz (2010, 2014; Almeida, 2014). 1

This is tradition of functionalist theories of kinship, represented by Malinowski, Leach and Fortes. In Leach´s terms, “kinship is a language for expressing property rights” (Leach, 1961). 2 This program began with André Weil´s Appendix to Lévi-Strauss’ Elementary Structures of Kinship (1967), followed by an extensive literature (Courrège, 1971; Gregory, 1986, 2015; Lorrain, 1975; Samuel, 1967; Weil, 1967; White, 1963; Tjon Sie Fat, 1990), to which a further flood of papers was added since Louis Dumont applied Lévi-Strauss´ marriage theory as exchange to socalled cognatic societies of South India, where no descent categories are named (Dumont, 1953; Trautmann, 1981; Overing (Kaplan) 1975; Viveiros de Castro, 1998). 3 McCallum (2001), Sahlins (2013), Viveiros de Castro (2015). 4 Ballonoff tried to connect the two domains in a series of papers, unfortunately written in mathematical language inaccessible to cultural anthropologists (Ballonoff, 2017).

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The mathematical concept of structure is used here to represent indigenous mathematics. But there is more to it than that, because under the mathematical concept of structure, indigenous mathematics is just mathematics (Almeida, 2014, 2019; Vaz, 2010, 2014). This is a consequence of the view of mathematics as a science of structures, sets of relations subject to rules, not as a science of objects (Almeida, 1990). Accordingly, my goal is to show how Cashinahua kinship terms and their combinatorial rules are an instance of a mathematical structure: a group acting on sociologicalcosmological classes. As an aside: there is no claim here for substituting mathematical representation for ontological interpretation. On the contrary, the same mathematical structure permits multiple ontological interpretations. As a matter of fact, mathematical structures are paralleled by anthropological structures—for example, in Marilyn Strathern’s characterization of Melanesian persons as packets of relations—‘multidividuals’ in her formulation, which implies that a person is constituted by the bundle of relations that relate her to all other persons (Strathern, 1988). And by Marshall Sahlins’ view of kinship as “commonality”, which amounts to characterize kinship relations as a set of equivalence relationships—that is to say, connecting beings that share a common substance (Sahlins, 2013).

2 Note on the Cashinahua The Cashinahua inhabit the Southwestern Amazonia in villages strewn along the Purus and Jurua rivers and their tributaries (Lagrou, 2020). The first reports on them date from the twentieth century (Parissier, 1914), in the aggressive onset of the rubber boom economy. My fieldwork experience (starting in 1982) placed me in contact with second-generation survivors of this tragic encounter—caboclos, the local term for mixed-blood descendants of indigenous Panoan women and immigrant rubber tappers (Almeida, 1993). On the other hand, the Cashinahua retained their ancient social organization and cultural patterns in surviving villages the Jurua river’s headwaters, particularly along the Tarauacá river and its affluent, the Jordão river (Almeida, 1993; Aquino, 1977; Iglesias, 2010). The Cashinahua belong to the Panoan linguistic family, being distributed along the Ucayali river banks, the Javari river at the Brazil-Peru frontier and along the upper Purus river and the Juruá headwaters—the last Amazonian areas to be settled by the converging Peruvian and Brazilian expanding extractive frontiers at the end of the nineteenth century and the early twentieth century. The late contact with the extractive frontier accounts for the absence of historical records on the Cashinahua and other Panoan peoples prior to 1890, in contrast with the abundance of twentieth century records on Panoan groups and particularly on the Cashinahua by missionaries (Tastevin, 2009 [1914]), historians (Capistrano de Abreu, 1941; Iglesias, 2010), linguists (Rivet & Tastevin, 1929) and anthropologists (Dwyer, 1975; Chernela 1985; Kensinger, 1995; D’Ans 1973, 1990; Aquino, 1977; McCallum, 2001; Deshayes, 2003; Lagrou, 2007, 2009, 2020; Deshayes & Keifenheim, 2003; Martini and Sian Dua 2021). The kinship terminologies of Panoan groups

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have been object of special attention, being described as “Kariera” and “Two Section” (Melatti, 1977; Hornborg 1993; Read, 2018: 106 for an alternative interpretation), while standing out as a distinct pattern in contrast with Amazonian kinship systems described as “Dravidian” and “Crow-Omaha”. The Cashinahua are also known for the geometric pattens produced by women in looms and on human bodies. Among Panoan groups, Cashinahua designs are equalled in variety and geometric rigour only by the Shipibo, also a Panoan group, at the Ucayali banks. When we leave the linguistic boundary, one finds a similar geometric rigour in the dazzling Yekuana weaving patterns applied to bodies and textiles (Guss, 1989; van Velthem, 2014).

3 Cashinahua Name-Sake Classes Cashinahua persons are classified in classes each of which shares its own set of names, transmitted along male and female descent lines from a generation to the second generation after it. They are called “alternate generation name-sake groups” in the literature (Kensinger, 1995; McCallum, 2001). There are eight name-sake classes, divided in two moieties: INUBAKE and DUABAKE (children of Jaguar and children of Puma respectively).5 Within each moiety there are four namesake classes distinguished by two alternating generations and by gender. In the INU moiety, alternating generations are AWABAKE (Tapir’s children) and KANABAKE (Lightning’s children). In the DUA moiety, adjacent generations are YAWABAKE (Peccary’s children) and DUNUBAKE (Snake’s children). This is not all, because within each moiety and each generation persons are distinguished by gender: within the INU moiety a person is either INU (male) or INANI female), and in the DUA moiety a person is either DUA (male) or BANU (female). In sum, there are three binary oppositions of exogamous moieties, alternate generations and gender duality, that produce eight classes of persons, connected by kinship terms. In the next section I intend to show that the set of Cashinahua classes are connected by kinship terms that satisfy the axioms of a mathematical group. This amount to say that Cashinahua kinship terms express the symmetries apparent in Figs. 1 and 2. This will be done in two steps. In the first step, starting with a definition of a mathematical group, I show how kinship terms connecting the four Cashinahua male classes, and the four kinship terms connecting the female Cashinahua classes, constitute each of them instances of the abstract mathematical structure called the Klein group. The second step consists in exhibiting the full structure of eight Cashinahua classes as an instance of the so-called dihedral group of eight elements.

5

Duabake is translated by McCallum as “children of the splendor”; see also Montag (2008). I follow Camargo (1995, 2014) and her explanation for the association of ‘splendor” with the Puma yellow collor, in contrast with the spotted Jaguar skin.

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Fig. 1 Cashinahua eight name-same classes, divided by moiety generation and sex. Source McCallum (2001: 25). Moieties: left and right divisions. Generations: upper and lower lines. Gender: | and ~ columns

Fig. 2 A Cashinahua representation of the eight name-sake classes. Left: a plan of the four parties (Portuguese partidos) drawn by Alexandre Quinet under the guidance of shamans Agostinho and Tadeu Manduca Mateus (Ika Muru et al., 2014: 248) Right: Kensinger (1995: 152)

4 The Group Structure of Same-Sex Cashinahua Kinship Terms 4.1 Definition of a Group A group is a set S of actions together with an operation that associates to every pair a and b of actions in S an action also in S. The result of an operation on two actions a and b is expressed as ab. Actions in a group are also called permutations, transformations, operators or mappings according to the case. We may think of an operation as a machine that receives actions a and b as inputs in that order and produces action c as an output.

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In order that an operation on a set S of actions satisfies the structure of a group, the following axioms must be satisfied. G1. The existence of an identity action. There is unique action in S, symbolized by e, such that, for every action x in S, ex = x and xe = e. The identity action, combined with any other action, leaves it unchanged. The effect of the identity action is exemplified by adding 0 in addition, and multiplying by 1 in multiplication. In classificatory kinship terminologies, the role of identity term is represented by the same-sex sibling relationship.6 In Cashinahua kinship terminology, |betsa| and ~betsa~ (”|a brother”| and ~”a sister”~ respectively) perform the role of an identity action. G2. The existence of an inverse for every action. To each action x in S there is an action also in S, symbolized as x −1 , that inverts the effect of x. This is expressed algebraically as: x x −1 = e, x −1 x = e. In Cashinahua kinship terminology, the term |pui~ (relating a male to his sister) has ~pui| (relating a female to her brother) as its inverse, since |pui~pui| = |betsa| (”sister’s brother” = “brother” for a man), and ~pui|pui~ = ~betsa~ (“brother’s sister” = ”sister” for a woman). This axiom says that every kinship term has a reciprocal. Example. Sian Dua’s grandson and Sian’s grandfather are both Sian Dua, while two of Sian’s sons are Bane Dua, just Sian’s father (Martini and Sian Dua 2021). Sian once explained this point to a class at the State University of Campinas, as an unexpected comment a lecture on the Kariera section system7 : We have the same thing. This is why we respect our son: for our son is our father.

Sian’s statement is a literally true sentence in Cashinahua Rãtxa Hui-kuin kinship terminology: he says that his epan must be respected, while epan applies both to his “father” and to his “son” (in English and in Portuguese). This justifies also Sian’s referring to his father’s father (epan) as his (“older”) huchi, an address term for an older brother. By the same alternate-generation logic, his maternal grandfather was his chai velho (his “old” brother-in-law), as his daughter’s son was a his chaizim (little chai). These consequences were corroborated by Sian’s father Sueiro (personal communication) marriage with a second-generation-removed xanu, his chai’s granddaughter (personal communication in 1985). G3. Associativity. The result of first entering ab and then c in the group machine and of entering a and then bc yields the same output: (ab)c = a(bc). This apparently innocuous axiom is better clarified by an example. A Cashinahua example, which relies on the identity between |epa| as “son” and (|pui~eva~) as (“brother’s mother” = “mother”), and of |epa|pui~” (a son’s sister = a daughter) and ~eva~ as “mother”: 6

This is not a trivial condition. It is not valid in the Western kinship terminology. Sian was at the State University of Campinas under a scholarship for cinema studies, with no commitment to any research on kinship.

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Fig. 3 Name-sake classes and kinship words acting on them. Left: male moieties Jaguar and Puma (INU and DUA). Right: female moieties (INANU and BANU). Thick arrows: same-sex genitorchild terms (|epa|, ~eva~). Hollow arrows: same-sex in-law relationships. (|chai|, ~tsabe~). Dotted arrows: parent-in-law/children-in-law relationship (|kuka|, ~yaya)

(|epa|)(|pui~eva~) = (|epa|pui~)~eva~ a son’s (sister´s mother)” is a (daughter’s) mother As the example suggests, associativity, in the context of Cashinahua kinship structure, implies that half-siblings are not distinguished, in contrast with English kinship terminology where “a son’s mother” may be different from a “daughter’s mother” if son and daughter originate from different marriages.8 This is all that there is to a mathematical group.

4.2 The Klein Group of Cashinahua Alliance: Same-Gender Kinship Terms The following two graphs represent the structure of Cashinahua kinship terms acting on name-sake classes restricted each to a single gender. The same structure is shared by geometrical transformations acting on graphic patterns. It is easy to verify that in Fig. 3 the combination of kinship terms associated to arrows, and in Fig. 4 the combination of geometrical transformations associated to the same arrows, satisfy axioms G1, G2 and G3 in the definition of a mathematical group. In Fig. 3, the identity element is |betsa| (male diagram) and ~betsa~ (female diagram), thus verifying Axiom G1; to each kinship term there is a reciprocal kinship 8

Among the Kayapo (Mebengekrore) people of Central Brazil, there are “triadic” terms translated, in the context of a woman addressing her husband, as “your daughter”, meaning “your daughter who is also my daughter” – carrying a different connotation from “my daughter” (Lea, 2004).

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Fig. 4 The same structure as an algebraic group and as group of geometrical transformations. The actions e, f, a and fa are the elements (or actions) of the group. These elements are combined as follows: ee = e, ff = e, aa = e, fa = af. The same structural rules apply to corresponding kinship terms in Fig. 3. This amounts to say that the kinship structure acting on name-sake classes is isomorphic to the structure of geometric transformations acting on graphic patterns. The transformation f is geometrically an up-down reflection, and a is a right-to-left reflection. These two transformations generate the group

term that inverses its action, according to Axiom G2; and the combination of kinship terms is associative as demanded by Axiom G3. For example, the action chai can be performed along the path epa *(chai*epa) = epa*kuka or, equivalently, along the path (epa*chai)*epa = kuka*epa.9 In Fig. 4, the identity element is represented by the curled arrow associated with the symbol e; to every arrow corresponds an opposite arrow that inverts it; and associativity is clearly satisfied. The common structure underlying the graphs in Figs. 3 and 4 is the Klein Group. This is a commutative group, which says that the order of two actions f and a does not alter the result: fa = af. I argue that Cashinahua kinship terms, taken together both male and female terms, act on the full set of name-sake classes as a group (Almeida 2010a, 2010b, 2019). The first question is whether there is an operation on Cashinahua kinship terms in the mathematical sense described under the definition of a group given above. To answer this question in the affirmative it is necessary to show that every combination of kinship terms in a set K of kinship terms (where K stands for Kaxinawa) connecting male and female name-sake classes produces a kinship term in the set K. This is not an obvious assertion. I intend now to support this assertion by means of a set of graphs that show that all possible paths of kinship terms in the set K produces a kinship term. Graphs will represent simultaneously: kinship terms acting on name-sake classes, and geometrical transformations acting on figures – both associated to arrows. By so doing, I 9

The symbol * stands for the group operation of composing two kinship terms. Gender signs were omitted with loss, since only male relations are included in Fig. 3.

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show that there is an isomorphism between the structure of the full set of Cashinahua kinship terms K and the mathematical structure known as a dihedral group with eight elements.10

5 Cashinahua Structure of Eight Name-Sake Classes I now show how the Cashinahua mathematical expertise connect two same-sex structures into a single structure of eight name-sex sections related to each other by sex, generation and affinity differences.

5.1 The Cashinahua Eight-Sections Group The true name comes from a limited stock of names that are passed down through alternating generations from people in the category of maternal grandmother, in the case of girls, and paternal grandfather, in the case of boys (McCallum, 2001: 21).

A noteworthy fact is that each one of group axioms G1, G2 and G3 has a version as rules of kinship terminologies defined by Lewis Morgan as “classificatory”, their diagnostic feature being the terminological identification of “father” and “father’s brother” as “father” and of “mother” and “mother’s sister” as “mother” (Morgan 1977: 476). Using the notation of Fig. 3, Morgan’s diagnostic feature is expressed as follows: |fe| = |f|, ~fe~ = ~e~, and more succinctly, fe = e, ef = f . This means the existence of a merging term in classificatory terminologies amounts to the existence of an identity term in algebraic structures. In the multiplicative structure of integer, this says that, for every number n, the following holds: n = n1 = 1n. I argue that Cashinahua kinship terminology in its full extension has an algebraic structure. The first question is whether there is an operation on Cashinahua kinship terms in the mathematical sense. For while in the previous sections this assertion was justified by Figs. 3 and 4, it is not obvious that it holds for the whole set of Cashinahua kinship terms connecting the full set of eight Cashinahua namesake classes. I intend now to support this assertion, by showing an isomorphism between the structure of the full set of Cashinahua kinship terms and a mathematical structure, based on the ethnographic and linguistic evidence (Kensinger, 1995: 110; McCallum, 2001; 29; Camargo, 1995, 2014), and on my personal acquaintance with Sian Dua and his father Sueiro (Bane) and his son Bane. This is no trivial step. Indeed, the inclusion of sex difference in a calculus of kinship relations of the “Dravidian” type posed a challenge to theorists like Dumont (1953) and Viveiros de Castro (1998), while Trautmann’s solution to the problem has been largely ignored, with the exception of an important contribution by Tjon 10

For an analysis of a kinship structure based on dihedral group, see Ascher 1991: 74–77, based on Laughren´s data and analysis (1982). It suffices to say that a dihedral group is a non-commutative group generated by two generators. See below (Figs. 9, 10, 11 and 12).

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betsa: same-sex siblings (♂brother♂, ♀sister♀) pui: opposite-sex siblings (♂sister♀, ♀brother♂) epa : same-sex genitor/child, (♂father♂, ♂son♂) eva:same-sex genitor/child, (♀mother♀,♀daughter♀) ♂chai♂, ♀tsabe♀: in-law. (♂brother-in-law♂, ♀sister-in-law♀). ♂epa♂pui♀eva♀pui♂ = ♂chai♂ ♂FZDB♂ = ♂FZS♂ = ♂in-law♂ ♂ZMBS♂ = ♂MBS♂ = ♂in-law♂ Dotted arrows (pui): diagonal reflection. Thick arrows (epa, eva): vertical and

horizontal reflections.

Fig. 5 Name-sake classes with alliance relations betsa, pui, epa, eva: as in Figure 4. ♂chanu♀ = ♂epa♂pui♀eva♀ ♂W♀ = ♂FZD♀ ♂chanu♀=♂pui♀eva♀pui♂epa♂pui♀ ♂W♀ = ♂MBD♀ The two equations correspond to the two paths the lead to the chanu arrow. ♀chaita♂ = ♀eva♀pui♂epa♂ ♀H♂ = ♀MBS♂ ♀chaita♂ = ♀pui♂epa♂pui♀eva♀pui♂ ♀H♂=♀FZS♂ This is the representation of marriage relations between name-sake classes. “Chanu” is a man´s potential wife, and “Chaitan” is a woman´s potential husband.

Fig. 6 Name-sake classes and marriage relations

Sie Fat (Trautmann, 1981, Tjon Sie Fat 1990, Almeida, 2010a, 2010b). The underlying problem is as follows: to integrate a sex operator, a descent operator and an affinity operator in a single algebraic calculus.11 The Cashinahua terminology 11

This is in a nutshell the difficulty faced by Dumont´s interpretation of “Dravidian” cognatic kinship terminology in terms of the opposition between “consanguines” and “affines”: a father and a mother are consanguineous (“cognatic” assumption), as a “brother” and “sister” are. Now, a “father´s sister” is a composition of a “consanguineous with a consanguineous”, but it results in an

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M. W. B. Almeida betsa not represented. pui, epa, eva: as in Figure 4. Here the combination of the generators produces two cycles. Gray arrows: (♂epa♂pui♀)(♀eva♀)

Fig. 7 Name-sake classes and cross-generation affinity

betsa, pui, epa, eva: as in Figure 4. ♂chanu♀ = ♂epa♂pui♀eva♀ ♂W♀ = ♂FZD♀ ♂chanu♀=♂pui♀eva♀pui ♂epa♂pui♀ ♂W♀ = ♂MBD♀ The two equations correspond to the two paths leading to the chanu arrow. ♀chaita♂ = ♀eva♀pui♂epa♂ ♀H♂ = ♀MBS♂ ♀chaita♂ = ♀pui♂epa♂pui♀eva♀pui♂ ♀H♂=♀FZS♂

Fig. 8 Avuncular relations

“affine” (non-consanguineous) relationship, and a “mother´s brother” relationship is a composition of “consanguineous relationships with a consanguineous relationship”, but it produces an “affine relationship” as well. This could suggest an algebraic rule as “consanguineous plus consanguineous = “affine”. But this is inconsistent with “a brother´s brother” is a brother, and “a sister´s sister” is a sister (Almeida, 1990). Thus, an algebra of consanguinity and affinity is not trivial.

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illustrates a solution to the problem. The following graphs show how the eight namesake Cashinahua classes are connected by kinship terms generated by two kinship relations: the same-sex, alternating generation change (|epa|, ~eva~) and the opposed-sex, same generation alternance (|pui~, ~pui|), assuming the existence of a kinship term representing the identity of same-sex siblings (|betsa|, ~betsa~).12 The graphs will also show how the kinship actions are isomorphic to a group of geometrical transformations.13 In this representation there is no privileged Ego (defined by sex, generation of moiety), so that all rules will be expressed in a form that is independent from a choice of the speaker’s origin (or of the section chosen as the origin of coordinates). This assumption amounts to proposing that all points of view are equivalent for the expression of kinship rules (Figs. 5, 6, 7, 8 and 9).

5.2 Different Representations of the Dihedral Group Structure When commenting the implications of associativity, I have suggested that different ways of parsing a kinship sequence may convey different meanings, while having the same denotatum. I pursue this vein now, showing how the Cashinahua structure of kinship relations can be represented in alternative ways, thus opening the possibility of different readings. This point is illustrated in Figs. 10 and 11, represented in abstract form in Fig. 12 (as left and right graphs), and also in Figs. 13 and 14, where name-sake classes (as in Figs. 1 and 2) are connected as opposite parties along moiety, generation and gender oppositions by the group of kinship actions. Figure 9 shows the group of kinship terms acting on name-sake classes, generated by a rotation and a reflection. The rotation is the result of iterating the kinship action |achi~kuka| in the identity action: (|achi~kuka|)(|achi~kuka|) = |betsa|, (~kuka||achi~)(|kuka|achi~) = ~betsa~ In standard English notation, this is translated as: (|FZ~MB|)(|FZ~MB|) = |B|, (~MB| |FZ~)( ~MB|)(|MB|) = ~Z~. This cyclical structure is represented as two cycles in Fig. 9, namely the outer cycle and the inner cycle. These cycles are connected by the pui kinship term, represented by symmetrical dotted arrows. There is another representation of the Cashinahua dihedral group, represented in Fig. 11.

12

In Fig. 10, I will present a different set of generators: a reflection and a rotation. As it happens, this is the group of symmetries of a square. There are eight symmetries – transformations that leave the square invariant: two reflections (up-down, left–right), two diagonal reflections (along the south-west/north-east axis, and along the north-west/south-east axis); and four 90o

13

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M. W. B. Almeida Actions f and s are group generators, subject to the following constraints: ss = e, ff = e, fs ≠ sf and fsfsfsfs = (fs)4 = e The gendered structures of kinship terms above are related to this abstract structure by the following map: ♂epa♂ → f ♀eva♀ → f ♂pui♀ → s ♀pui♀ → s ♂betsa♂ → e ♀betsa♀ → e This map preserves all the rules. In mathematical jargon, it is a homomorphism. This is one representation of the dihedral group with eight elements.

Fig. 9 The dihedral group of eight elements. First representation Inner and outer cyclical groups (black arrows and inverse gray arrows) generated by ♂achi♀ = ♂epa♂pui♀ (♂FZ♀, ♂MBW♀) and ♀kuka♂ = ♀eva♀pui♂ (♀MB♂, ♀FZH♂) These generators run a four-step cycle (rotations by 90o of figures): ♂achi♀kuka♂achi♀kuka♂ =♂betsa♂. This path can be read either as ♂FZS♂FZS♂ = ♂B♂ or as ♂MBS♂MBS = ♂B♂ And this is the rule of bilateral crosscousin marriage. The kinship term pui is a reciprocal term connecting brothers and sisters (♂Z♀, ♀B♂). The gender sign was omitted. The identity term betsa was also omitted.

Fig. 10 Representation of the Cashinahua kinship structure as product of reflections and rotations

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Figure 11 combines the two separate single-gender alliance structures exhibited in Figures 3 and 4 into a single structure, in which the male and female structure are joined together. Hollow arrows (♂chai♂, ♀tsabe♀) represent the in-law relation between men (as brothers-in-law) and between women (as sisters-in-law) respectively. Black arrows (♂epa♂) represent the father-son recyprocal relationship, which transmit the alliance pattern along alternate generations. Gray, dotted arrows (♀yaya♀) represent the paternal aunt (♀FZ♀) action and its inverse and its inverse (♀BD♀) action, which transmit the alliance pattern along women.

Fig. 11 The dihedral group as a combination of two alliance structures

Fig. 12 Abstract representations of the Cashinahua dihedral group

Figure 11 represents a male structure (outer square) and an inner female structure (inner square) as alliance structures, connected by marriage. It also highlights the fact that men inherit their in-laws (|chai|) from their fathers along a male line. This is expressed as |chai| = |epa|chai|epa|. In English anthropological jargon, rotations, totalling eight symmetries. These symmetries are represented as the eight transformations of a thorn design in the following graphs.

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Fig. 13 Moieties, generation and gender connected by the group of kinship terms

with the added term A = |WB| = |ZY|, this means: |A| = |FAS|, or |A| = |FAS|). The inner square obeys the same rules (with ~tsabe~ substituted for |chai|, and ~yaya| substituted for ~epa|. Figure 12 shows the abstract representation of the kinship graphs of Figs. 10 and 11. All the above representations of the structure of Cashinahua kinship terms acting on name-sake classes share a common consequence: the partition of name-sake classes in three opposed pairs, opposed by generation, gender and moiety. Figure 14 represents kinship actions opposing name-sale classes: along generations (left thick same-sex filiation arrows), across gender (middle, marriage dotted arrows) and connecting alliance partners (right, hollow arrows). In sum: opposing generations (¼), gender opposition (middle) and moiety division (right). These opposed groups are related respectively by reciprocal genitor actions (thick

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Fig. 14 Moieties, Generation and gender connected by the group of kinship terms

vertical and diagonal arrows), opposite-sex sibling actions (middle, dotted lines) and exogamous moieties (right, double arrows).

6 Conclusions This chapter was an attempt at a visual demonstration of the structural similarity between the combinatorial rules underlying Cashinahua kinship terminology and the axioms of the mathematical structure of the dihedral group. This similarity, which in mathematical jargon is a homomorphism—since different Cashinahua words may convey the same relationship while preserving the structure—strongly suggests that the Cashinahua are acute applied mathematicians, or, more exactly: mathematical sociologists.14 This conclusion can be extended to a large set of cultures where the concept and practice of symmetries plays a central role in social organization, in graphical arts, in architecture and in cosmologies (Almeida, 2019; Ascher, 1991, 2002).

14 An isomorphism of groups is a one-to-one correspondence T between two sets A and B with operations * and ˆ respectively, such that T(a*b) = T(a) ˆ T(b). An example of isomorphism is the correspondence between multiplication of real numbers and sum of their logarithms: in this case, log(ab) = log(a) + log(b). A numerical example with base-2 logarithms: log 2 (4 × 16) = log 2 (64) = 6, and log 2 (4) + log 2 (16) = 2 + 4 = 6. An homomorphism is a many-to-one correspondence that preserves the structure. An example is the correspondence S between the set of integers (0, 1, 2, …) with addition + and the set {0,1} with “computer” addition +’. Here the correspondence maps even numbers to 0 and odd numbers to 1. Under this transformation, S(n + m) = S(n) +’ S(m). Numerical example: S (2 + 3) = S (5) = 1, S (2) +’ S (3) = 0 + 1 = 1. An homorphism is a transformation that collapses distinctions while preserving structure. In this text, it collapses gender distinctions while preserving the structure of kinship operations. This is not a universal feature of kinship terminologies.

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Another result is the use of representation theory of groups as indication of multiple possible interpretations of the same basic structure. Thus, the use of gender and generation as generators in the main part of the text (Figs. 5, 6, 7 and 8) does imply a bias, for analogous graphs could follow the representations generated by cycles of avuncularity and gender (Fig. 10) and by reflections, and by alliance, filiation and marriage (Fig. 11). Instead of mathematical language and semi-formal proofs, the language of graphs was used to convey isomorphisms (as the one-to-one structural correspondences between male and female kinship structures) and also homomorphisms (the many-toone structural correspondence between gendered kinship terms and algebraic structures where gender was omitted, thus collapsing two gendered structures into a single algebraical structure. The use of a graphic pattern extracted from Cashinahua women’s textile designs was not a casual choice either. The two-colour pattern in kinship diagrams above, a “thorn”, is a generative cell present in a rich set of textile designs produced by Cashinahua women weavers, which obey group-theoretical principles of symmetry (Almeida, 1990; Speiser, 1937). Although this aspect of Cashinahua mathematical competence could not be exhibited in this text for reasons of space, the use of the “thorn” motif and its geometrical transformations retained, as a hint to Cashinahua’s unity of sociology, cosmology and aesthetics. Acknowledgements This work is indebted to Viveiros de Castro’s exhaustive review of Dravidian Systems literature (1998) and to Thomas Trautmann’s monumental “Dravidian Kinship” (1981), as well as to Tjon Sie Fat’s complete synthesis and expansion of the mathematical theory of marriage systems (1990). I am also indebted to Dwight Read’s critiques and to Paul Ballonoff’s reception to my early paper on this theme in the Journal of Mathematical Anthropology and Cultural Theory. Tom Trautmann’s work and personal encouragement was fundamental, as well as the intellectual inspiration and friendly support of Marshall Sahlins and of my wife Manuela Carneiro da Cunha.

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The Western Mathematic and the Ontological Turn: Ethnomathematics and Cosmotechnics for the Pluriverse Michael Baker

Abstract This chapter calls for critical reflection on the onto-political relations between mathematics, technics, and cultures in the context of the ecological and technological crises of modern Western civilization. The relations between a culture’s mathematic, technical practices, and metaphysical worldview are intertwined elements within different civilizational complexes, comprising the geocosmo-politics of the modern world order. Reflected in the posthuman and postanthropocentric turns in the sciences and humanities, a variety of alternative ontologies of existence are emerging. As an intellectual field, the posthuman ontological turn is a bifurcated shift in the fundamental concepts and assumptions underlying the modern knowledge disciplines. The ontological turn has unsettled Western metaphysical concepts of nature and culture, calling the relations between mathematics and nature into question within an emerging historical problematic identified as the technoecological condition. I am exploring ways of overcoming the singular world ontology of modernity with a pluriversal ontological politics of knowledge and education. Ontological diversity is dependent upon technodiversity, while technodiversity is dependent upon renewed cosmological relations between technics and cultures. Mathematical knowledge practices are centrally about intervening in environments as well as providing metaphysical grounds for establishing truth. The technoecological condition involves the replacement of subject centered sense making with the environmentalization of sense, i.e., smart cities, ubiquitous computing and global media. Grounded in the mathematic of Western modernity, the process of cybernetization has concretized in world-wide systems of technological mediation that operate within sensory and intelligent environments. Biodiversity and technodiversity are being lost in an expanding mono-cultural technological civilizational complex. Underlying the production and consumption of cybernetic technologies are particular kinds of mathematics (i.e., symbolic, experimental, technological, computational), participating in multiple fields of knowledge, embedded in cultural-historical events,

M. Baker (B) Rochester, New York, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4_9

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processes, concepts and presuppositions about the world. This modern metaphysical paradigm shift involves a change in the ontological relations between mathematics and technology, reflected in the hybrid reformations of computational mathematics and STEM education, i.e., biotechnology, artificial intelligence. An historical ontology of cultures and mathematics brings matter and meaning together, focusing on the material/ideological relations between mathematics, technics, cultures, and environments. Cosmotechnics is proposed to the knowledge disciplines to balance the cosmological relations between cultures, technics, and natures. We can learn to live within the limits of the earth system. From this recognition of the cosmoontological relations between the modern Western mathematic and modern technology, does ethnomathematics contribute to a project of producing technodiversity in a world propelled towards technological singularity? A multiple ontologies perspective, associated with the ontological turns in anthropology, sociology, geography, decoloniality, and philosophy of technology and media, contributes towards a pluriversal project of overcoming technological modernity that is neither fascist, nationalist, nor technocapitalist. Today, the human future is thrown into question by our technological capacities. The convergence of new technologies (for example, biotechnology, robotics, informatics, and nanotechnology) in projects of controlling life has radically reconfigured our sense of the human condition, both through technological capacities already at our disposal and through emergent imaginations of what human futures are possible, desirable, and good. We have begun to achieve unprecedented capacities to manipulate not only our external environment but the internal environments of our bodies as well. In light of these emerging and anticipated capacities, questions about human progress, redemption, or demise are increasingly asked in relation to imagined technological futures (Tirosh-Samuel & Hurlbut, 2016: 5). Our time is thus one in which it is urgent that the West—or what remains of it—analyze its own becoming, turn back to examine its provenance and its trajectory, and question itself concerning the process of decomposition of sense to which it has given rise (Nancy, 2008: 30). The question that remains to be answered is what will be the fate of these indigenous ontologies and practices when confronting modern technology, which is the realization of naturalism? Or are these “practices” able to transform modern technology so that the latter acquires a new direction of development, a new mode of existence? This is one of the most crucial questions, since it is also about how to escape both colonialism and ethnocentrism (Hui, 2017b: 7).

Keywords Comparative metaphysics · Technoecological condition · Cybernetics · Transhumanism · Technodiversity

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1 Introduction What are the relations between mathematics and technics, and cultures and technics (Hansson, 2020; Skosvmose, 1988, 2016)? To what extent is technical diversity interrelated with cultural diversity, understood as diversity of ways of knowing and being human? To what extent are different cultural traditions of mathematics interrelated with different cultural forms of technics? What are the cultural relations between mathematics, technics and the making of particular kinds of worlds? How are different cultural-civilizational traditions of mathematics and technics related to different ways of knowing and being involved in the ordering of worlds (Restivo, 1992; Verran, 2007)? How has the modern Western mathematic participated in the metaphysical framework responsible for producing instrumental knowledge of nature and enabling technical ways of making the world modern? What are the metaphysical/cosmological relations between the computational mathematics and digital technology (Fisher, 2017)? How does “pure” mathematical knowledge participate in the production of scientific concepts and objects, e.g., “artificial intelligence”? How are the underlying metaphysics of the Western mathematic changing in the emerging computational age of the digital and the anthropocene? These kinds of ontological and cosmological questions point towards trajectories of thought capable of understanding and critiquing to the deceptively reductive and unsustainable technomathematically made worlds we are living in the twenty-first century. Situating the universalizing metaphysical traditions of modern reason in the philosophy of mathematics and technology within the modern/colonial world system, this chapter points towards the philosophical-cosmological renewal of local knowledge practices and cultural techniques in response to the emergent technoecological condition (Manzini, 2015; Mignolo, 2000). This chapter questions the historical-ontological relations between technology, mathematics, and cultures in the context of the monocultural crisis of Western modernity. Understanding the historical-ontological relations between mathematics, technology, and culture connects cultural diversity of mathematical knowledge with cultural diversity of technical knowledge. This chapter is an inquiry into the onto-political relations between the Western mathematic and technology at the end and transformation of the metaphysical framework that made the singular world ontology of modernity/coloniality (Mignolo & Walsh, 2018). The self-understanding of the modern age and the hegemony of the West (cultural/material/ontological/cosmological) were grounded in a dualistic metaphysical framework that separated nature from culture and technics from nature. In determining what entities are along with the totality of entities in the universe, metaphysics grounds an age (Heidegger, 1977a). “By giving shape to our historical understanding of “what is,” metaphysics determines the most basic presuppositions of what anything is, including ourselves” (Thomson, 2000: 298). Epochal narrations, forms of knowledge, and material and technical practices are expressions of metaphysical philosophies of existence that emerge and change together over time. Identified by naturalism and rationalism, modern metaphysics provided the basic

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conceptual parameters, assumptions, and standards for a unified world sensing intelligibility that reduced “nature” to the calculative (Descola, 2013; Thomson, 2005; Winograd & Flores, 1986). If naturalism has succeeded in dominating modern thought, it is because such a peculiar cosmological imagination is compatible with its techno-logical development: nature should be mastered for the good of man, and it can indeed be mastered according to the laws of nature. Or put another way: nature is regarded as the source of contingency due to its weakness of concept, and therefore it has to be overcome by logic (Hui, 2017a: 6).

Associated with the anthropocene, the digital, and de-westernization, the crisis/transformation of the modern metaphysical worldview and project can be characterized as: ecological, technological, geopolitical, and cultural-cosmologicalontological. Geopolitically, this epochal metaphysical transformation of Western hegemony in the modern world order is marked by historical processes of dewesternization, associated with capitalist development and deracination in China, along with varied local and regional movements for cultural autonomy – “cosmopolitan localism” (Falk, 2004; Mignolo, 2011). The monocentric world order is today exploding. Out of the explosion and the debris of the crashing monocentric world order, two trajectories emerge in coexistence: one is being called a multipolar world order, and the second is pluriversality as a universal project. The first is a state-led project of dewesternization. The second is the project of the emerging political society of decoloniality (Mignolo, 2018a: 92).

A multi-polar world order continues the global capitalist geopolitics of the modern European international system of states competing for political-economic-military power, determined now by technological advancements associated with “machine learning” (Kissinger, 2018). It is rather obvious that every nation-state is going to have its own Ministry of Accelerationism (e.g., Dubai appointed its Minister of Artificial Intelligence in 2017), and it is hard to imagine that this will be an emancipatory politics and not one that only further strengthens the synchronization of the global axis of time (Hui, 2019a: 264–265)

The decolonial, de-westernization project orienting this chapter aims to overcome the singular world ontology of modernity with a pluriversal ontological politics of knowledge and education (Berque, 2002; Mignolo, 2011). The metaphysical relations between the modern Western mathematic and modern technology are interpreted as geo-political-ontological forces involved in making the modern world order, in terminal crisis (Dussel, 1993). The onto-political relations between mathematics, cultures, and technics are changing in the metaphysical transformations related to the computational and ontological turns (Holbraad & Pedersen, 2017; Hui, 2010). Over the past few decades, the subject-object, form-matter metaphysics of modernity has been supplanted by competing post-dualist and posthuman metaphysical conditions and philosophies (Hayles, 1999; Hui, 2019a, b). In the modern dualist metaphysics, mind and matter, body and soul, life and nonlife were separated in a dead cosmos (Koyré, 1957). Reflected in the ontological turn occurring across the knowledge disciplines, the cosmos is returning to life, while matter is re-imagined to

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be more relationally and recursively dynamic in humans, organisms, and machines, i.e., neovitalism, animism, autopoiesis, symbiogenesis, Amerindian perspectivism, organology, cybernetics (Haraway, 2016; Ingold, 2000; Melitopoulos & Lazzarato, 2010; Rose, 2013; Vivieros de Castro, 2014). The “ontological turn” involves a contradictory convergence occurring across knowledge disciplines, bringing matter and thought (materiality of knowledge) together in varying ways (Pellizzoni, 2015). What entities are, along with the totality of entities (an epoch’s foundational metaphysics) are changing once again, along with the epistemic sensibilities of modern existence (Barad, 2007; Thomson, 2005). This turn overall involves a bifurcated shift in the metaphysical concepts and assumptions underlying the modern knowledge disciplines. The contrasting variety of ontological speculation includes reductive, totalizing tendencies of algorithmic computation in cybernetic systems of control, as well as, renewed relations with technics and nature in the recognition of multiple ontologies or ways of knowing and being (Pellizzoni, 2015). Among a few of the many currents, the ontological turn involves a resurgence of questions concerning technology and understanding the synchronization occurring under the ontology of the digital. The computational turn is part of a broader ontological shift in understanding the nature of nature and human beings. Computational physics and mathematics illustrate the culmination and transformation of modern metaphysics, described as calculative reasoning. Computational cybernetic systems are associated with the creation of new conditions and experiences of existence, raising ontological questions about the modes of existence of technical objects (Simondon, 2016). What is a digital object? (Hui, 2016a). The technical object no longer features as a meaningless tool, or as an instrument that is a mere means to achieve the ends of an already constituted and meaning-giving subject. It is no longer a separate, minoritized object situated at the abyss of non-sense; no longer the accursed share, or the impossible outside of meaning. This inferior object that was always considered a mere thing in the work of interiority and the theatre of intentionality now appears at the very heart of the culture of sense, opening up a new stage and a new environment of sense. Whereas sense used to come about through a meaning-making act, it now becomes a transcategorial notion, an assemblage emerging from the non-signifying collaborative practices of humans, objects, and machines (Horl, 2013b: 12–13).

Existence is increasingly determined by technical systems in a new kind of industrialization based upon digital technologies (e.g., AI, machine learning, surveillance) and transhumanist ideologies (Hui, 2019a). Humans, machines and nature are joined together in cybernetic environments that are both technological and metaphysical. The mechanical and organicist philosophies of nature, life, and biology have merged in the technosciences and posthuman humanities. At the end or culmination of modern metaphysics, contending ontologies of existence have emerged within a posthuman politics of existence in which the ontological becomes political, i.e., ontological politics, political ontology (Blaser, 2009, 2013; Joronen & Hakli, 2017; Pellizzoni, 2015). At stake is the ontological freedom of earthly existence. The ontological turn has unsettled the Western dualistic metaphysical concepts of nature and culture, calling the relations between mathematics and nature into question

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within an emerging historical philosophical problematic, identified as the technoecological condition. Awareness of an emergent technoecological (or eco-technological) condition has grown since the end of the twentieth century, that portends the beginnings of an anxious and uncertain epoch, characterized as a transition from a technical to a technological world (Horl, 2015). “Modernity ends with the rise of a technological consciousness, meaning both the consciousness of the power of technology and the consciousness of the technological condition of the human” (Hui, 2016a: 42). Adapting to this eco-technological condition calls for a shift in cosmology and a new ontology (Hamilton, 2020; Horl, 2013a). We need (or more accurately, will have thrust upon us) a new way of understanding what it is to live on the Earth beyond the four ontologies described by Descola, one that emerges from a sense of the distinctiveness of the Earth in the new epoch and the emerging relationship of humans to it, one of intense engagement but with no possibility of mastery (Hamilton, 2020: 116).

Largely overlooked in the two thousand year history of Western philosophy since Aristotle’s distinction between techne and nature, questions of technology and ontological relations between technics and nature are now central to understanding the posthuman metaphysical worlds modern humans inhabit and enact (Braidotti & Hlavajova, 2018). Since the nineteenth century, the evolution of modern technics has accelerated massively while modern philosophy has, until recently, inadequately understood the ontological significance of technology (Stiegler, 2018). Exemplified in the technosphere, biotechnology, and technosciences overall, the enlightenment distinction between techne and nature “has given way to a generalized techno-nature”, or “mechano-organicism” (Haff, 2013; Hui, 2020a: 55; Lindberg, 2018: 95). The reversal of Platonism—spoken of from Nietzsche to Heidegger to Deleuze as the task of philosophy and as a new beginning for thought—is also (perhaps even mainly) a machinebased event, implementing an utterly unPlatonic “mathematics as instrument of fiction”. Nowhere is this so evident as in the computer-supported transition from a classical to a transclassical knowledge culture, one that shifts the sciences into close proximity to the arts. Under the aegis of the technosciences, the age-old tension between episteme and techne loses its force, and the order of knowledge is no longer characterized by procedures of truth and the production of certainty, but by the signatures of nonlinear mathematics and non-trivial machines (Horl, 2016: 95).

The modern humanist paradigm of technics as an extension of instrumental reason has been succeeded since the mid-twentieth century by an evolving variety of posthuman philosophies of the relations between nature and technics, developing around the present form of “technoecological rationality” (Horl, 2018). A general ecological mode of thought is being proposed as an opening towards understanding and critiquing the techno-environmentalization of thought occurring across the technosciences, education, and everyday life (Horl, 2017). In the twentyfirst century, the environmental becomes technological, while the technological becomes environmental, i.e., smart cities, ubiquitous computing. Ecology becomes a concept of cybernetics, and not only pre-synthetic biology (Hui, 2020a). Environmentality is to be situated in the moment of our entering into techno- ecological conditions, into their manifest infrastructurality and processuality, which encompasses

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radical relational spatializations and temporalizations on multiple scales.... Environmentality is thus not just an ontological but an ontogenetic key category that, ultimately, can only be grasped speculatively (and thus no longer phenomenologically), beyond the established semantics of “environment.” ....Environmentalization, in other words, also has a TechnologyForm and a Media- Form, whose sites remain to be detailed. What can already be said, though, is that the twenty- first century’s “environmental media” (Hanson, 2015: 37), which together stand for the “becoming- environmental of computation” (Gabrys, 2016: 4), turn out to be central agents in this profound historical transformation. .... Becoming- environmental, in a word, is the cipher of our age, and to decipher it via formal analysis is a, if not the, central task of critique today (Horl, 2018: 156, 157).

The technoecological condition is marked by the autonomization of contemporary technology and the emergence of the Anthropocene and the technosphere in which technology participates in altering the conditions of the earth system along with the biosphere, lithosphere, hydrosphere, and atmosphere (Haff, 2013). What we are witnessing today is a shift from the organized inorganic to the organizing inorganic, meaning that machines are no longer simply tools or instruments but rather gigantic organisms in which we live. In the time of Schelling, and later in Hutton and Lovelock, nature is considered as a gigantic organism, of which we are a part. However, this term general organism, which was attributed to nature considered as a source of contingency, seems more appropriately to designate the technological system that we are now inhabiting—for example, smart homes, smart cities, and the Anthropocene. Instead, we are observing the becoming of an “artificial earth,” and we are living within a gigantic cybernetic system in the process of forming: This comprises our contemporary condition of philosophizing (Hui, 2019a: 28).

Within this context of a disruptive eco-technological transformation of the modern metaphysical worldview and project, what are the relations between cultural diversity of mathematical knowledge and cultural diversity of technics? This chapter questions computational mathematics within the computational turn and the fourth industrial revolution (Hui, 2010; Schwab, 2017). Exploring sources in anthropological ontology and new materialist philosophy of technology, I will interpret technics as ontogenetic, as world-making forces—the geo-cosmopolitics of technics (Lindberg, 2018). “Insistence on the ontological force of technological apparatuses transverses received philosophical and ontological divides and revitalizes the notions of ‘nature’ and ‘the human,’ which are now understood as coevolving with technology” (Hoel & van der Turin, 2013: 187). The ontological force of technics, grounded in the modern Western metaphysics of mathematically conceived mechanisms, have altered the planetary conditions of existence in negatively inhuman ways that diminish and threaten the futures of existence, i.e., from Holocence to Anthropocene, from mechanical and organic nature to cybernetic techno-natures, from planetary space to interplanetary spacelessness (Turnbull, 2006). Human and non-human world sustaining earthly environments are being completely disrupted and destroyed. The worldwide problems brought about by technological modernity are leading to the exhaustion of resources and life on earth while space exploration becomes a commercial fantasy for leading tech moguls in the United States and England. “This liberation from the earth directly confronts humankind with the infinite universe and prepares for a cosmic nihilism” (Hui, 2020a: 57).

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The unilateral globalization that has come to an end is being succeeded by the competition of technological acceleration and the allures of war, technological singularity, and transhumanist (pipe) dreams. The Anthropocene is a global axis of time and synchronization that is sustained by this view of technological progress towards the singularity. To reopen the question of technology is to refuse this homogeneous technological future that is presented to us as the only option (Hui, 2017a: 9).

To what extent and how are the modern knowledge disciplines and education into them participating in making unsustainable mono-technological cultures associated with eco-modernity and transhumanism? Could pluriversal disciplinary knowledges and a multiple ontologies curriculum contribute towards the renewal of diversity in technical relations between the moral and cosmic ordering of worlds? I am questioning the sources of the technoecological crisis as a destructive imbalance in the material/ideological conditions of modern synchronized life (Rosa, 2013). “The contemporary crisis is the result of deeply entrenched ways of being, knowing, and doing” rooted in an uprooting metaphysics of modern Western rationalism/naturalism (Escobar, 2017: 19). Understood as uprooting by “removing from a native environment or culture”, deracination is a worldwide experience of the metaphysics of colonization, modernization, globalization, and cybernetization (Hui, 2016b; Merriam-Webster, 2003). Modern ways of relating with the world, by separating knowledge from the world, and culture from nature, have produced destructive cultural pathologies and profound forms of alienation, insufficiently understood in the human sciences. Interpreting the French mechanologist, Gilbert Simondon, “the fundamental alienation of industrial societies resides in the misunderstanding and the ignorance of technology” (Hui & Morelle, 2017: 508). This separation between morality and technics and nature is related to the detachment of technics from nature and technics from culture (Hui, 2016b). The modern world order is characterized by an unstable imbalance between moral relations and technical practices as the metaphysical/ontological conditions of life are being transformed in the knowledge disciplines and technical existence itself. Morality is a marginal afterthought in the metaphysical-industrial-geopolitical acceleration of technological progress, i.e., telecommunications advertisements such as, “5G” is coming; buy a new cell phone. In sum, the recent turns towards ontological questions in the sciences and humanities involve a renewed questioning of the natures of existence and an uncertain overcoming of the metaphysical worldview of Western modernity. The phrase metaphysical worldview expresses how I am exploring the renewal of the concept of cosmology, a term usually interpreted as a subfield within physics, i.e., astrophysics. Broadly, ontology can refer to the objects and conditions of existence and experience (material, life-worlds themselves) as well as the metaphysical traditions of interpretation of existence and experience (philosophical-hermeneutic ways of knowing and being). “Ontologies are modes of being giving rise to ‘understandings’ of the nature of reality and one’s place in it” (Hamilton, 2020: 111). In the Western tradition, ontology is part of metaphysics as interrelated forms of inquiry into the nature of nature and existence overall. “By ontology we mean the study of what exists (or what is real), including our own conditions of being” (Blaney & Tickner, 2017: 296). In the modern philosophical tradition, the ontological realm

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has been determined and suppressed within a dualistic epistemology of the real, i.e., subject-object, mind-matter. Both subjects and objects participate in making the world what it is, such that the world itself and cultural ways of producing knowledge are relationally constitutive of historical cosmologies. Understood generally as ways of knowing and being, ontology is inseparable from epistemology and cosmology. “Cosmology entails a series of assumptions about the origins and the evolution of the cosmos. In this sense, ontologies and cosmologies can be said to co-constitute each other” (Blaney & Tickner, 2017: 296). Metaphysics, ontology, epistemology, axiology, and cosmology all together comprise the “formative parameters of civilizations” (Davutoglu, 2014b). From a comparative civilizational perspective, the concept of cosmo-ontology refers to the co-constitutive relations between a civilization’s ways of knowing and being (ontologies) and worlds (cosmologies) (Arnason, 2003; Hamilton, 2017, 2020). This chapter is divided into five parts that together propose a critique and alternative to the transforming hegemony of the Western mathematic in the totalizing reductions of realities to abstract symbols of calculation and manipulation (Weisse et al., 2021). Part one distinguishes technology and technics and introduces the concepts of multiple ontologies, multiple technics, and multiple natures in a critique of modern technology. Part two connects technology and mathematics within an emerging background horizon described as the technoecological condition and the end or culmination of the modern metaphysical tradition. Part three introduces the ontological turn and the philosophical/political project of technodiversity. Part four problematizes mathematics in the computational turn, contrasting technological singularity with pluriversality. Part five asks, what is the role of ethnomathematics in the technoecological condition and the project of technodiversity?

2 Multiple Technics, Multiple Natures, and Multiple Ontologies The philosophical concept of technics is part of the history of philosophy from the Hellenic period in ancient Greece, and should constitute “one of philosophy’s core inquiries” (Hui, 2016b, p. 10; Stiegler, 1998). Technics is more broadly defined than technology to refer to embodied-cultural-techniques involved in the production of worlds (Brüning & Knobloch, 2005; Leroi-Gourhan, 1943, 1993; Macho, 2013). Technics and culture are profoundly related in processes in which human beings and civilizations evolve (Leroi-Gourhan, 1993). Since the end of the eighteenth century technics and culture have been antagonistically related in the mono-culture of technological modernity (Simondon, 2016). Technology is the modern Western expression of human technics. Technology defines the epoch of modernity as ruled by instrumental-calculative thinking. Heidegger proposed that there is a rupture between what the ancient Greeks called techn¯e and what he referred to as modern technology, for they differ in their essences. Techn¯e has

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its essence in poiesis, i.e., bringing-forth, while modern technology or enframing [Gestell], sees everything as standing reserve or as resources to be exploited. We should ask ourselves, however, where the position of, say, ancient Indian technology, Chinese technology or Amazonian technology is in Heidegger’s analysis (Hui, 2020b: 1).

According to Chinese philosophical engineer Yuk Hui, technics refers to “the general category of all forms of making and practice” (Hui, 2016b: 4). Technics are not only anthropologically universal or transcultural, but also profoundly cultural and indeed ontological in the sense of world-making. The contemporary global technological system illustrates the world making ontological powers of technics (Bratton, 2015; Ellul, 1964; Gabrys, 2016). Situated within its own historical metaphysical and cosmological system, modern technology brings forth a technological world in which all existence becomes “standing reserve” (Heidegger, 1977b). Modern technology constitutes European and North American forms of knowledge and rationality that emerged in the late Renaissance and early modern transition, founded upon a mathematic of nature and a geometry of space (de Beistegui, 2003; Elden, 2006). As such, the skills and artificial products of every culture are not captured by the modern term “technology”. The Greek and English words techne, technics, techniques, and technology do not necessarily translate into other languages and civilizations with their own conceptual, metaphysical and cosmological systems. “Instead of taking for granted an anthropologically universal concept of technics, one should conceive a multiplicity of technics, characterized by different dynamics between the cosmic, the moral and the technical” (Hui, 2020a: 54). The same can be said of the modern concept of a singular nature and the relations between cultures and nature. Non-European cultures have different relations with and maybe no equivalent concepts for the Western metaphysical concept of “nature”, i.e., Pachamama. In different ontologies or collective ways of knowing and being, what may be considered “nature” has different roles and relations with different groups of people, comprising different cosmologies (Descola, 2014). The metaphysics of modern naturalism is characterized by an ontological opposition between nature and culture, in which a singular nature is conceived as a universal (factual) world upon which there are multiple cultural perspectives. According to aboriginal anthropologist Mario Blaser, this modernist assumption of one world with multiple perspectives is constitutive of a particular kind of politics, characterized as “reasonable” (Blaser, 2016). Universal science plays the primary role in “reasonable politics” as the arbiter of the factuality of different perspectives (Blaser, 2016: 550). According to Blaser, there is “an implicit equivalence between technological prowess and apprehending reality as it is” (Blaser, 2016: 550). “Reasonable politics operates on the basis of turning differences into perspectives on the world. Differences made into perspectives are amenable to be ranked according to putative degrees of equivalence between perspectival representations of the world and the factual world itself” (Blaser, 2016: 549). Modern naturalism, mathematical logic, and technological development go hand in hand. From the ontological turns in anthropology and science and technology studies (STS), this mono-naturalist framework of a singular nature and multiple cultural perspectives is now confronted with an alternative framework of multiple ontologies

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with multiple natures and multiple technics. Multinaturalism replaces a singular nature multiculturalism as different cultures have different conceptions and relations with their environments that do not separate the world itself from representations of the world. Similarly, multiple cosmotechnics replaces universal technology as technics is no longer separate from cultures. Nature is no longer opposed or inferior to culture, while technics is no longer opposed to natures, but part of evolving relations between cultures, natures, and technics. According to Hui, technics is an ontological category, which I associate with world-making. The anthropologically universal and world-making aspects of technics consists in the extension of somatic functions and the externalization of memory (Hui, 2016b; Leroi-Gourhan, 1993; Simondon, 2016; Stiegler, 1998, 2018). Interrelated with this universally human (anthropogenic) aspect, the cultural (cosmogenic) aspect of technics is located in the cosmological understandings and cultural techniques that comprise different cultures and civilizations in different historical periods and epochs. If technics is world making, it “must be interrogated in relation to a larger configuration, a ‘cosmology’ proper to the culture from which it emerged” (Hui, 2016b: 10), including mathematical knowledges.

3 Mathematics, Technology and the Technoecological Condition The relations between mathematics and technology have not yet received much scholarly attention despite their reciprocal relations and practical importance (Hansson, 2020). This lack of scholarly interest in understanding the relations between mathematics and technology is due in part to the ways both concepts have been understood. Mathematics has been conceived as completely independent of material reality, while technology is commonly understood as a material object only. In addition, both Western mathematics and modern technology have been understood as universal and hence separate from the historical-cultural contexts within which they are enacted as ontological (world-making) practices/techniques. There are some important countervailing perspectives however in the philosophy of mathematics and technology that reject these dualistic universal presuppositions. Oswald Spengler (1880–1936) for example, categorized mathematical knowledge as rigorous science, a true art, and a metaphysics of the “highest rank” (Spengler, 1961: 43). The highest ranking status of the modern Western mathematic is related to its world building (ontological) functions. In its complex relations with science, technology, engineering, and design, modern Western mathematics can be understood as participating in the metaphysical/ontological production of the material and ideological worlds comprising the modern worldview and project (Escobar, 2017). Philosophers in the Aristotelian tradition have always put the truth value of predictive mathematical descriptions on a different plane from the description of essential qualities of beings found in natural philosophy. Mathematics is powerful because it projects a vision of reality

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upon the world that allows for predictive control. But it can stand in the way of accessing the layers of reality that reveal themselves through engagement with the world (Scherz, 2019: 970).

According to Spengler, mathematics in the plural refers to mathematical practices from multiple cultures or civilizations. “Every philosophy has hitherto grown up in conjunction with a mathematic belonging to it” (Spengler, 1961: 43). Spengler interpreted the meanings and components of world views according to styles and stages in culture, number and mind (Restivo & Collins, 1982). “Thus, number thought is not merely a matter of knowledge and experience, it is a ‘view of the universe’” (Restivo & Collins, 1982: 279). For Spengler, there is no ontological divide between mathematical knowledge and the physical, material world or the “essential qualities of beings found in natural philosophy” (de Freitas et al., 2016). Mathematical practices and material practices are intertwined in the cultural metaphysics of particular worldviews and ways of life. For Spengler, each civilization and each epoch have their own particular mathematic. From this comparative civilizational perspective, the “modern Western mathematic” can be understood in the singular. Each civilization and each epoch also have their own technics interrelated with their own historical cosmological relations with the universe. Different civilizations have distinctive traditions of thought or ways of knowing and being that should be allowed to flourish in their own autonomy and historicity. It seems every form or type of modern technology, and maybe ethnotechnics overall, have their own underlying mathematical knowledges (understood broadly as in “multimathemacy’), that to a large extent make different material/cultural forms of technical existence possible (Pinxten, 2016). One mundane example from a recent textbook, Mathematics for Machine Learning divides “mathematical foundations” into (1) linear algebra; (2) analytic geometry; (3) matrix decompositions; (4) vector calculus; (5) probability and distributions; and (6) continuous optimization (Deisenroth et al., 2020). In the Foreword, the authors identify a moral dilemma with cosmological significance regarding who will control the recursively automated technology they are teaching how to make mathematically (Winner, 1977). Machine learning is the latest in a long line of attempts to distill human knowledge and reasoning into a form that is suitable for constructing machines and engineering automated systems. As machine learning becomes more ubiquitous and its software packages become easier to use, it is natural and desirable that the low-level technical details are abstracted away and hidden from the practitioner. However, this brings with it the danger that a practitioner becomes unaware of the design decisions and, hence, the limits of machine learning algorithms (Deisenroth et al., 2020: 1).

In a 2014 British newspaper article, The Independent, Stephen Hawking and three other science professors distinguished between short term impacts of AI in terms of “who controls it”, and the long term impact of “whether it can be controlled at all” (Hawking et al., 2014). One can imagine such technology outsmarting financial markets, out-inventing human researchers, out-manipulating human leaders, and developing weapons we cannot even understand. Whereas the short-term impact of AI depends on who controls it, the long-term impact depends on whether it can be controlled at all (Hawking et al., 2014).

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In addition to these profound cultural, cosmological, and eschatological questions, the mathematical foundations of Artificial Intelligence have their own culturalrhetorical relations in “modes of exposition, goals, kinds of argumentation, criteria of validity, and clientele” (Davis, 2008: 23). Modern technology and mathematics also share a history in the industrial “control revolution” from the late-nineteenth century to the present, from which the universal metaphysics of cybernetics emerged (Beniger, 1989; Goffey, 2014; Horl, 2013a). Cybernetics belongs to organicism, a larger paradigm originating in the eighteenth century critique of mechanism. Distinguished from vitalism and its immaterial foundations, organicism is founded on mathematics (Hui, 2020a: 55). As one form of organicism, cybernetics has been interpreted as a unified conceptual schema for the representation of reality (Hui, 2019a; Jonas, 1966). Joining mechanism with organicism, modern machines are cybernetic machines in their assimilation of recursive behaviors of organisms. Today cybernetics becomes the modus operandi in machines ranging from smartphones to robotics and spacecraft. The rise of cybernetics was one of the major events in the twentieth century. Different from mechanism, which is based on linear causality, e.g., A–B–C, it rests on a circular causality, e.g., A–B–C–A, meaning that it is reflective in the basic sense that it is able to determine itself in the form of a recursive structure. By recursion we mean a non-linear reflective movement which progressively moves towards its telos, be it predefined or auto-posited (Hui, 2020a: 55).

Mobilizing the concepts of feedback and information, cybernetics analyzes all beings – animate and inanimate, nature and society (Hui, 2020a: 55). Cybernetics endeavors to eliminate dualisms by connecting different orders of magnitude. As a universal mode of thinking, cybernetics aims to be a universal discipline, able to replace philosophical thought and unite all knowledge disciplines (Hui, 2020a: 58). Cybernetic technosciences participate in the making of a kind of second nature, as the concept nature is replaced by an ecology of technical systems, monitored and governed in algorithmic rules and calculations (Hui, 2020a; Pellizzoni, 2015). Becoming ecological involves “the mechanistic-technological triumph of modernity over nature” (Hui, 2020a: 59). Techno-nature in the techno-capitalist sciences involves the planetary domestication of life in which humans become livestock, i.e., biotechnology (Heidegger, 1977a; Hui, 2019a: 185–186; Sloterdijk, 2017). The technology of livestock domestication has gradually merged with the self-domestication of the human being, which may be understood in terms of what Foucault calls governmentality. Human beings’ intervention into the environment constitutes a specific kind of governmentality which Foucault calls environmentality (Hui, 2020a: 60).

In the face of emerging techno-capitalist, eco-modernist, transhumanist eschatologies, (grounded in an unstable detachment between technics, natures, and cultures), ontological diversity across the knowledge disciplines is now a necessary condition for ecological sustainability and cultural autonomy in the brave new environments enacted through cybernetic techno-sciences (Burdett, 2015; Hottois, 2021). Drawing on Erich Horl and Yuk Hui, the technoecological condition refers to the present conjuncture of the Anthropocene and the digital, the ecological crisis

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and the computerization of everyday life, The Sixth Extinction (Kolbert, 2014) and The Fourth Industrial Revolution (Schwab, 2017). The simultaneous emergence of the Anthropocene and digital capitalism appear to name two of the most far reaching consequences of colonial, industrial, technological modernity and the disruptive ontological-ecological changes witnessed around the world today, i.e., global warming, mass migration, fascism, stateless wars and ongoing militarization, COVID-19, alternative-right movements and fascist political leaders, unstable corporate-state politics of social media, surveillance capitalism, and the geopolitics of computational technology (Sassen, 2014; Zuboff, 2015). The material and ideological conditions of existence are changing, rapidly and disruptively, involving all dimensions of life in a new planetary historicity (joining geological with human time) and algorithmic modes of self-governance (Bratton, 2015; Chakrabarty, 2020; Gabrys, 2016; Hamilton, 2017; Nancy & Barrau, 2015; Reed, 2019). “The Anthropocene is at once the crisis of naturalism and the crisis of modernity. It is under such a crisis that modernity is again called into question” (Hui, 2017b: 6). The various intertwined networks of modern technics constitute a milieu that conditions human action (Hui, 2016c). Remaking the objects of nature into digital entities, a new cybernetic state of nature has emerged, supplanting earlier states of nature based on the figures of Man and Nature from sixteenth century natural law theory (Horl, 2016; Jahn, 1999, 2000; Kuhn, 2017). We are currently going through a fundamental ecologization of the image of thought and the image of being. The concept of a general ecology, as I am developing it, has a double meaning: on the one hand, it refers to a fundamental change in experience, and a fundamentally new position, that characterizes being and thinking under the conditions of a cyberneticized “state of nature”; on the other, it refers to the new description that this transformation demands, and the new philosophico-conceptual politics that it entails (Horl, 2013b: 21).

Drawing from the cultural-historical turn in the modern knowledge disciplines, mathematics and technics are viewed as cultural forms of rationality and epistemology that participate in making particular environments, modes of existence, worldviews, and cosmologies (Aerts et al., 2011; Restivo, 1985). All forms of knowledge and cultural techniques reflect imaginary/material expressions of the world and human existence (Santos, 2014). All forms of knowledge and cultural techniques are derived from and express historical cosmo-ontologies, reflected in particular ways of knowing and being. From a decolonial, multiple ontologies perspective, ontology is understood broadly as ways of knowing and being, intertwined with the cultural techniques enacting unique worlds. Particular ways of knowing and being comprise particular historical cosmologies. Anthropologically, cosmologies relate to how people participate, know, and experience the worlds they are living (Abramson & Holbraad, 2016). Etymologically related to the ordering of worlds, cosmologies are comprised of different worlds, viewed and lived through different ways of knowing and being that participate in the ordering and disordering of worlds. Embedded in historical-cultural relations, mathematical and technical knowledges are cosmo-ontological in the ways they participate in making and unmaking worlds.

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4 The Ontological Turn and Technodiversity The ontological turn involves a diverse array of currents from across the knowledge disciplines, all suspending, altering, or replacing the modern metaphysics of nature and subjectivity. There is no broad disciplinary or cross-disciplinary consensus on the meaning of ontology, and even less agreement on its emancipatory possibilities (Pellizzoni, 2015). Although there are many conflicting views on what ontology may mean, there is wide agreement that the uses of ontology and related concepts are proliferating within and across the sciences and humanities that question and revise inherited Western ways of understanding ourselves, others, and material and social worlds (Jackson, 2018). I interpret the ontological turn broadly as a crisis related, bifurcated and uncertain transformation of the modern cosmology. The ontological turn reflects a cosmoontological transformation of the modern worldview and project. From the sixteenth century onwards, the modern world order was grounded in anthropocentric and humanist rationalism. The ontological turn is a contradictory posthuman and postanthropocentric transformation of the modern humanist, dualistic metaphysical tradition. The emergence of this turn toward ontological questions over the past two decades across the knowledge disciplines is interrelated with technological and ecological crisis of the twenty-first century. Occurring across the modern knowledge disciplines, the recent shift towards ontology reflects an epochal transformation of the ontological and epistemological presuppositions of the nature of nature and human beings (Ferrando, 2019). The renewal of ontological questioning involves a contradictory overcoming of the modern metaphysical (ontological, epistemological, ethical) paradigm of truth and certainty of knowledge about the existence and essence of the world. Understood as an epistemic paradigm shift, the ontological turn can be interpreted as an uneven and bifurcated (ontological opening and closing) transformation in the Western metaphysical tradition (Hui, 2016b; Pellizzoni, 2015). The recent turn to ontology is being taken in ways that further expand the reach of the modern powers of mathematical physics, science, and technology within the modern/colonial history of control (Beniger, 1989; Mignolo, 2000). This turn to ontology includes a geopolitics of cloning environments in a techno-capitalist system of computational algorithms, undertaken within a capitalist civilizational ethos of technoscientific control and regulation for the “survival”, “benefit”, and “progress” of human existence, i.e., biotechnology, biontology, algorithmic governance, surveillance capitalism (Pellizzoni, 2015; Rifkin, 1998; Stingl, 2016). Mathematization took its most materialized form in algorithms. The algorithm understood as a process or an operation only expresses abstract thinking, and gains a quasi-autonomy when it is realized in machines. This is one of the modes of existence of the algorithm in its most “objectified” form. An algorithm is fully expressed in its functionalities in operation. How different is the algorithmic contingency and the contingency of the laws of nature? (Hui, 2015: 131).

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The bifurcation in the ontological turn is reflected in both the reductionist and nonreductionist images of cybernetics with divergent socio-economic-political implications. In the reductionist image all organisms are recognized as “feedback systems”, “imposing a predictive determinism within an economy of finality” (Hui, 2019a, p. 274). A non-reductionist image calls for understanding both the possibilities and the limits of cybernetics. Both philosophical and mathematical/technical understandings of technics are necessary in order to situate and assess the consequences of rapidly evolving cybernetic systems (Beniger, 1989; Hui, 2019a). In addition to a techno-capitalist narrowing and digital enclosures of existence, the ontological turn also involves an opening within the knowledge disciplines towards more resonant and humanly possible relations with the world (Jackson, 2018; Rosa, 2019). For the modern knowledge disciplines and education into them, the ontological turn appears to include an opening towards multiple worlds of processual relations of existence, occluded within the singular world, static entity metaphysics of colonial-industrial modernity (Savransky, 2017). Reflected in the proliferating and contending states of nature and technological futures, we are living through a fragmentation and uncertain transformation of the modern civilizational cosmology (Burdett, 2015). How are and how should the knowledge and education disciplines be(ing) transformed (O’Sullivan, 2001)? The metaphysical transformation that comprises the ontological turn in mathematics is evident in fields comprising the socio-political-cultural turns in mathematics (Boylan & Coles, 2017; Kollosche, 2016; Guittterrez, 2017; Pinxten et al., 1983; Raju, 2018; Restivo, 1985, 1992; Skovsmose & Greer, 2012). There are recent openings in the philosophy of mathematics towards the world producing dimensions of mathematics, “connecting humans to equations” in the cultural historical conditions of existence (material/discursive/technical), i.e., “formatting power” (Eglash, 1997; Pinxten, 2001; Ravn & Skovsmose, 2019; Restivo, 1992; Restivo & Bauchspies, 2006; Skovsmose, 1994). As we understand it, a philosophy of mathematics must be able to capture the profound contingency and historicity of mathematics. We need to address the social formatting of mathematics, as well as the mathematical formatting of humans and the social. This brings us far beyond the classical topics of a philosophy of mathematics as well its regular methodological approaches (Ravn & Skovsmose, 2019: 171).

As an intellectual movement, the ontological turn is most visible in mathematics literature under “new materialism” and in the critical theory of neoliberalism (Chorney, 2017; de Freitas & Sinclair, 2013; de Freitas et al., 2016; Mikulan & Sinclair, 2019; Wolfmeyer et al., 2017). Computation and the body are among the key concepts, along with questions related to the kinds of environments produced in particular mathematical knowledge practices, i.e., teaching mathematics in classrooms. A shift towards ontology in rethinking mathematics and culture is illustrated in the special journal issue title “Alternative Ontologies of Number: Rethinking the Quantitative in Computational Culture” (de Freitas et al., 2016). In an era of digital technologies, exponential rates of data production, and market-based technocratic governance, diverse practices of numeracy, quantitative inquiry, and software

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analytics become ever more ubiquitous. Yet, the dominant epistemological and ontological assumptions about number continue to rest on outdated philosophical frameworks. What is more, the data sciences continue to operate according to an ontology in which the natural world is assumed to be underwritten by mathematical laws (Manovich, 2013; Ruppert et al., 2013). Not only have these assumptions been deconstructed by posthumanist scholars, but recent work in philosophy begins to point toward alternative ontologies of number (Wade, 2008; Chatelet, 2000; Deleuze, 1994; Meillessoux, 2008; Rotman, 2000) and new ways of theorizing measure and quantification (Barad, 2007; Kirby, 2011; Parisi, 2013) (de Freitas et al., 2016: 431).

The ontological divides between mathematical knowledges and the material world, reflected in the persistently Platonic philosophies of mathematics, are being bridged. Understood as an overcoming of the nature/culture divide in historical metaphysical conditions of existence, the ontological turn is emerging from a variety of twentieth-century mathematical knowledge practices and posthuman philosophies of mathematics related to the computational turn, involving the mathematization of “mechanical, biological, social, economic, and organizational” mechanisms (Hui, 2015, p. 131; Kuhn, 2017; Buhlmann, 2020). First order cybernetics for example, from the 1950s, emerged in the mathematization of information and communication in technical systems of control and regulation (Horl, 2015: 2; Shannon & Weaver, 1964).

5 Computational Mathematics: Singularity Versus Pluriversality From this broad historical-philosophical-cosmological interpretation of the present, the metaphysical relations between computational mathematics and digital technology are problematized. From Galileo, Newton and Descartes onwards, the modern Western mathematic becomes the ground for the metaphysics of a singular world horizon, separating humans from nature and making the rational subject the master of a calculable nature. The mathematical metaphysics of modernity is now undergoing a cosmo-ontological transformation, visible in the popular imagination (or selfunderstanding) of Silicon Valley as “technological singularity” (Kurzweil, 2005). Singularity is another “end of history” narrative within a global neoliberal capitalist cosmotechnics (Kurzweil, 2005; Shanahan, 2015). Singularitarianism pushes instrumentalism to its limit, and in a very peculiar way. The Singularity marks the point of convergence between technological evolution and the spiritual ascension of a chosen group of humans. Technology becomes the vehicle for spiritual transcendence, the instrument for the fulfilment of human nature. The logical end point of the autonomous evolution of technology coincides with the consummation of the cosmic destiny of humanity. Both moments are one and the same. In this way, technology not only mediates but substantializes the union of humanity and the universe in a sort of Holy Trinity (human– machine–cosmos) (Vaccari, 2020: 44).

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Computational mathematics and digital capitalism illustrate how the modern mathematic and the modern cosmology are interrelated in the cosmopolitical/technological shift associated with the confusing array of posthumanisms (Betancourt, 2016; Bostrom, 2008; Braidotti, 2019; Ferrando, 2019; Simon, 2019b). Biotechnology and cybernetics radically destabilize the ontological foundations of what counts as human. Imagining technological futures has become a particularly consequential locus for both religious and secular engagements with notions of the human person. The theological reflections on the human person are increasingly undertaken in response to—and in terms of—the possibilities of human biotechnology (Smith, 2010), even as efforts to constitute appropriately secular public deliberation render the moral questions subsidiary to scientific and technological accounts (Hurlbut, 2015b) (Tirosh-Samuelson & Hurlbut, 2016: 6).

While the planetary ecological crisis remains inadequately addressed, the universe becomes a digital reality in an onto-epistemology of natural numbers. Many mathematicians, physicists and computer scientists view reality as ultimately a reflection of the postulated digital nature of the universe. Penrose presents the example of electric charge that supports the notion that discrete number entities exist in the physical universe. This is reinforced by the quantum-mechanical view that at the deepest level, time and space are discrete, losing their appearance of continuity. Discreteness implies digital properties, which can be traced back through the rational, real and complex number systems to the natural numbers. If we fully understand the ontology and epistemology of the natural numbers, it will assist in understanding the reality of the cosmos. These ideas are hypotheses in need of further observations, insights and revelations to prove their value in the ongoing debate on the nature of reality. Until then, nature and reality in the final analysis remain a mystery to us (Kuhn, 2017: 6).

While Western humanist and anthropocentric metaphysics continue to orient everyday modern life, conflicting posthuman ontologies are transforming the knowledge disciplines with new concepts and technocapitalist ventures, e.g., postanthropocentrism, transhumanism, eco-modernism, accelerationism (Asafy-Adjaye et al., 2015; Bessent, 2018; Bostrom, 2008; Burdett, 2015; Kurzweil, 2005; Mackay & Avanessian, 2014; Landgraf et al., 2018; Manzocco, 2019; More & Vita-More, 2013). As an alternative to contending transhumanist technological agendas, a multiple ontologies perspective proposes the preservation and proliferation of technical diversity in the pluriverse. As a philosophical project, a multiple ontologies, pluriversal perspective questions the presuppositions and methods of all forms of inquiry into the conditions of existence and experience—“natural” or “social”. Overcoming modernity involves “undoing and redoing” the translations of fundamental concepts (e.g., techn¯e, physis, metaphysics, mathematics) organized within historical systems of thought, providing the metaphysical/ontological grounds for making “true” knowledge and ordering “real” worlds (Hui, 2017a). Historical ontology becomes a central task of a post-Eurocentric philosophy oriented towards overcoming the self-destructive consequences and tendencies of technological acceleration. A brief description of the uses of indigeneity in relation to ontology will further introduce the distinctions between technological singularity and ontological pluralism (Blaser, 2012; Hunt, 2014). Indigenous is a politically charged concept,

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central to the ontological turn in at least three ways: (1) as comparative metaphysics, (2) as philosophical anthropology, and (3) as social movements for cultural survival and autonomy (Escobar, 2017). First, indigenous philosophies of existence are being compared with a variety of new materialist and other relational ways of understanding knowing and being in the Anthropocene (Chandler & Reid, 2019; Ingold, 2000). “Indigeneity is increasingly becoming a crucial marker for imagining new modes of living and governing in our contemporary condition of climate crises and economic uncertainty” (Chandler & Reid, 2019: 1). At the center of this new onto-political imaginary is the concept of relationality, and contending uses across the knowledge disciplines reflect the broader bifurcation. Assuming that existence is constituted in the myriad of relations that precede and produce any individual objects, things or entities, relationality is one of the most consequential onto-political concepts (Barad, 2007; Debaise, 2012). New forms of animism for example are being described (similar to earlier anthropological descriptions from Levy-Bruhl and others) as a central phenomenon of participation in new digital cultures (Horl, 2016). “For the primitive mentality to be is to participate. [Pour la mentalité primitive être c’est participer.] ….If this participation was not given, was not already real, individuals would not exist.” (Lévy-Bruhl, 2019: 22–23, original emphasis) (Quoted in Horl, 2016: 98). The relational cosmologies of indigenous peoples are being recognized and compared with the ways the nature/culture divide and other dualistic metaphysical categories are being rethought in the modern sciences and humanities. Inanimate matter, human and non-human, are reanimated and interrelated as ontological forces of existence. In a period of global warming, pandemics, and quasi-autonomous algorithms, the world is acting on its own in ways that are more unpredictable and catastrophic (Abramson & Holbraad, 2016; Hui, 2019a). Contingencies and catastrophes now function recursively in the internal self-regulating processes of cybernetic systems (Hui, 2015). Second, in anthropology, indigenous peoples are no longer only conceived as non-modern others of the modern self, but ontologically distinct in their own ways of knowing and being (Descola, 2014, p. 236; Pandian & Parman, 2004). This shift in understanding diversity of humanity was central to the cultural turn and the emergence of ethnomathematics in the late twentieth-century critiques of Eurocentric modernity. According to Hui, “the ontological turn in anthropology tackles the problem of modernity by proposing an ontological pluralism” (Hui, 2017a: 8). One task of the new ontological ethnographer is to suspend their own scientific-cosmological concepts and ontological presuppositions about the world in an effort to begin understanding other ways of knowing and being in their own terms and processes. The ontological ethnographer also learns to recognize the intercultural incommensurability between different worlds and concepts (Aman, 2015; Holbraad & Pedersen, 2017). Like learning a second language, this comparative ontological opening suggests a cosmopolitical sensibility of multiple cosmologies and their mutual historical

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interrelations within the world order. The recovery of the cosmological sensibilities of different cultural relations with the world is not a direct return to preexisting indigenous cosmologies, but the renewal of local cosmologies in relation to a pluriversal cosmology appropriate for the planetary technoecological and geophysical conditions of life on earth in the twenty-first century. There is an important sense in which a concern with cosmology is more apposite to the contemporary world than it ever was to putatively primitive ones. In close correlation with political and economic shifts that have taken place since the 1970s—broadly speaking, from what we, following James Scott (2014), call ‘high modernity’ to neo-liberalism—a new cosmological sensibility has begun to emerge, and this new orientation toward the cosmos is generating novel ways of being concerned with it (Abramson & Holbraad, 2016: 41).

Anthropology and other knowledge disciplines are redirected towards understanding and contributing to the existence of non-modern ways of knowing and being, becoming part of the permanent decolonization of the singular world horizon of Eurocentric modernity (Savransky, 2017; Viveiros de Castro, 2014; Mignolo & Walsh, 2018). The ontological turn in anthropology and cognate disciplines involves an opening in the modern disciplines of knowledge towards a pluriversal understanding of the universe—the many worlds within the world (de la Cadena & Blaser, 2018). And third, indigenous movements around the world are involved in ontological struggles over the survival of their relational cosmotechnical ways of knowing and being, threatened by further encroachments of a mono-techno-culture and the loss of sustainable habitats. The technoecological condition is the result of the technocapitalist form of power in which existence has been reduced to mercyless market forces imposed as the best or only alternatives for survival and success (Horl, 2013a, 2015; Peters, 2017). “There is a strong empirical and conceptual link between the emergent ontologies in the biophysical sciences and neoliberal rationality of government” (Pellizzoni, 2015: 66). Within these new technoscientific cultures and neoliberal rationalities, all relations are reduced to “calculable, rationalizable, exploitable ratios, in the form forcefully wielded by the mathematics of power” (Horl, 2017: 8). “Capitalism… redefines human and nonhuman worlds in a way that unravels relationships of interdependence and institutes the most inextricable network possible of chains of dependence” (Stengers, 2002: 2). A pluriversal, multiple ontologies perspective, comprising a few of the many conflicting currents in the ontological turn towards the relations of existence, provides an alternative opening towards “overcoming modernity”, that is neither fascist, nationalist, nor technocapitalist (Hui, 2016b). Every culture must reflect on the question of cosmotechnics for a new cosmopolitics to come, since I believe that to overcome modernity without falling back into war and fascism, it is necessary to reappropriate modern technology through the renewed framework of a cosmotechnics consisting of different epistemologies and epistemes (Hui, 2017a: 9).

Hui suggests ontological diversity may be impossible without technical diversity, and technical diversity now involves rethinking the relations between technics

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with cultures (epistemologies, ontologies and cosmologies). In the present technoecological condition, ontological diversity may only be possible under conditions of renewed technical diversity, a cosmopolitics of cosmotechnics. “Ontological pluralism can only be realized by reflecting on the question of technology and a politics of technology” (Hui, 2017a: 6). The ongoing loss of bio, ethno, and techno diversity calls for philosophical renewal of the (unconscious and alienated) relations between technics and natures and technics and cultures. “Modernity functions according to a technological unconsciousness, which consists of a forgetting of one’s own limits” (Hui, 2016a: 42). Illustrated in the present loss of biodiversity, ontological differences are being eclipsed in the geo-politics of technology driving the politics of the global geoculture (Davis, 2002; Kissinger, 2018; Kolbert, 2014; Shiva, 1993). “Biodiversity is the correlate of technodiversity, since without technodiversity, we will only witness the disappearing of species by a homogenous rationality” (Hui, 2020a: 63). The elimination of technodiversity entails the further elimination of cosmo-ontological diversity and further deracination. Critical reflection on technodiversity promises alternatives to the singular mono-technological world ontology becoming entirely global in the twenty-first century, i.e., “the global technical system” (Clayton & Archie, 2021; Hui, 2019a; Vaccari, 2020). The possibilities of technodiversity in other words, are dependent upon the question of the historical-philosophical renewal of local and regional concepts, knowledges, and techniques. The inquiry into the relation between machine and ecology is less about how to design more intelligent machines, but rather requires first of all a discovery of cosmotechnical diversity, while such diversity has to be thought through by going back to the question of locality, therefore re-articulating the concept of technics by resituating it within the geographical milieu, culture and thinking. The task that is left to all of us is the effort to rediscover these cosmotechnics in order to reframe modern technologies, namely, by reframing the enframing [Gestell]; only through such a reframing can we imagine a “new earth and people that do not yet exist” (Deleuze and Guattari 108) (Hui, 2020a: 65).

6 Ethnomathematics and Cosmotechnics The various critiques of the historical-social-cultural-ethical dimensions of the Western mathematic since the 1990s share a common background in the philosophical questioning of the self-destructive consequences of modernity, nascent across the disciplines in the 1960s. The modern Western world order is now recognized as one among multiple ways of ordering existence within the co-existence of multiple cultural modes of knowing and being. The modern Western world order is also increasingly recognized as self-destructive and unsustainable in its historical trajectories. There are historical and more recently changing world making metaphysical relations between modern technology and the modern Western mathematic. Particular forms of mathematics undergird particular technologies, particular epochs, and the present technoecological condition. From this perspective, cosmoontological diversity of mathematical knowledge and education contributes to the

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diversity of technics and the kinds of worlds produced in particular technical practices. From this historical ontology of the relations between mathematics, technology, and cosmology, could cultural diversity of mathematical knowledge (ethnomathematics) contribute to a comparative metaphysical philosophy of technodiversity in a world propelled towards technological singularity (Kurzweil, 2005; Shanahan, 2015; Vaccari, 2020)? The emergence of ethnomathematics in the closing decades of the twentieth century involves a decolonial-ontological critique of Eurocentric modernity in the practices of modern Western mathematics (D’Ambrosio, 2003). Scholars in mathematics, education, anthropology and other cognate fields, positioned somewhat outside Occidentalism, developed a cultural-practice critique of the Eurocentric ontoepistemology of the Western mathematic (D’Ambrosio, 1985, 2003; Verran, 2000). Ethnomathematics began imagining, researching, teaching, and learning Western and non-Western mathematics in the historical–geographical environments of local and regional ways of knowing and being (culture and history), i.e., “multimathemacy”, colonial education, indigenous metaphysics (Pinxten, 2016). Adding the prefix ethno to cultural-historical forms of knowledge production involves a critique of the ontological separation of modern scientific knowledge from its geo-politicalcultural-historical locations (Harding, 1989; Mignolo, 2000). Recognizing the ethnos of all knowledge practices involves a rethinking of the modern separation of nature from culture, and knowledge from world (Descola, 2013; Santos, 2018). Mathematical knowledge practices participate in the making of environments or milieus. Biophysical and cultural-technical relations are inseparable in making lived worlds (Pinxten & Francois, 2011; Pirani, 2005). From a general ecology of thought (Horl, 2017), the material/technical world is recognized as an ontological force in making cultural-natural worlds. Natures, cultures, and technics are inseparable in the ways they participate in the production of particular forms of existence. In these processes of adaptation and adoption, we see that there is a reciprocity between the living being and its environment, which we can also call its organicity, namely, the fact that they do not only exchange information, energy and matter but also constitute a community. A human community is far beyond the sum of the human actors that constitute it, it also includes their environment and other non-human beings (Hui, 2020a: 57).

In the twenty-first century, it appears ethnomathematics has a leading role in the ontological renewal of the relations between technics and cosmologies towards advancing cosmotechnical diversity. The role of ethnomathematics can be situated in the politico-ontological project of cosmotechnics responding to the present crisis/transformation of technological modernity. A pluriversal cosmopolitics has emerged as an alternative to neo-Kantian cosmopolitics and the universalized cosmology of capitalist civilization (Fry & Tlostanova, 2021; Santos, 2018; Moore, 2015). This interpretation of the technoecological present (e.g., smart cities, media communications, computational mathematics, Green Revolution), calls for reorienting the knowledge disciplines and education towards recognizing ontological differences in a pluriversal cosmopolitics (Blaser, 2016).

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Cosmotechnical pluralism contributes to comparative inquiry that includes ethnomathematic forms of inquiry, i.e., relations between mathematical knowledges, environments, and cultural techniques. Parallel and joining currents associated with cosmotechnics, the comparative study of cultures and mathematics questions the ways different mathematical knowledges and technical activities interrelate within different environments and cultural practices. Being comparative means to be always aware of the cosmotechnical conditions of the culture to whom the one performing the inquiry belongs, and to relate these to the cosmotechnics proper to the inquired culture. This would lead our attention to the technical activities shaping these cultures’ worldviews and to a comparison between their cognitive apparatuses. .... A cosmotechnical pluralism is an inquiry in the understanding of technics in different cultures and an analysis of the ways these understandings shape the experience as well as the becoming of the technical systems of these cultures (Pavanini, 2020: 35).

According to Simondon (2016), the universalized technical practices making the modern world are out of balance in the sense that modern technology became the ground of all existence, instead of one of many figures. As the “universalizing ground of everything”, technology is detached from the broader realities of the world that enable and constrain it. I gave a preliminary definition of cosmotechnics as unification between the cosmic order and the moral order through technical activities, in order to suggest that technology should be re-situated in a broader reality, which enables it and also constrains it. The detachment of technology from such a reality has resulted from the desire to be universalizing and to become the ground of everything. Such a desire is made possible by the history of colonization, modernization and globalization, which, being accompanied by its history of economic growth and military expansion, has given rise to a mono-technological culture in which modern technology becomes the principle productive force and largely determines the relation between human and non-human beings, human and cosmos, and nature and culture (Hui, 2020b: 2).

From an image of flourishing cultural techniques within a plurality of local knowledge traditions, cosmotechnics aims to re-attach modern technology with the world or cosmos (Santos, 2014). Cosmotechnical pluralism is intended to: open the future toward a plurality of heterogeneous technological trajectories, each driven by a different cosmotechnical imaginary. As such it projects a profound “reframing” of the current planetary enframing so as to “recosmicize the Earth” (Augustin Berque) and doing so in a variety of locally specific ways (Lemmens, 2020a: 6).

Modern Western technology is involved in an historical episteme that is fundamentally cosmo-ontological (world-making) and only one of many diverse forms of cosmotechnics (Hui, 2016b). From a pluriversal curriculum, the modern knowledge disciplines and education into them can learn to understand the modern Western cosmotechnics in relation to non-Western cosmotechnics. Non-European cultures can consider their own cosmotechnics in relation to the history of their own philosophical/cultural/material/technical ways of knowing and being and their historical relations with modern Western technologies. Hui mentions an “indigenous technology methodology” (Hui, 2020d). These kinds of cosmotechnical reflections point

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towards non-universalized ways of thinking about technics, modern technology and education. No one culture’s technics are better than another, except in relation to each culture’s own historical cosmo-ontology. As the ontological turn in anthropology involves a reconciliation between culture and nature, cosmotechnics involves a reconciliation between technics and nature. Cosmotechnics involves a renewal of intuitive and perceptual sensibilities, artistic and craft creation, and a resurgence of local and collaborative ways of knowing and being. Reopening concept of technics as multiple cosmotechnics and the future of technological imaginations – necessitates the rediscovery of the nonmodern epistemologies and the reinvention of epistemes through the regime of aesthetics as responses to the current crisis from the point of view of localities, or as what Augustin Berque call recosmosizing [recosmiser] (Hui, 2019a: 277).

Cultural-politics is intimately connected with ethno-technics (Bardin, 2018, 2019; Simondon, 2016). “The intervention of human beings in the environment defines the process of hominization, the evolutionary and historical becoming human and its politics” (Hui, 2020a: 57). Cosmotechnics includes the question of technical innovations that contribute towards a shift from technologies of control towards harmonious and reciprocal relations between technics and nature, as well as new forms of technically mediated collaboration and education, i.e., biomimicry, noncapitalist social network infrastructures (Blok & Gremmen, 2016). A variety of knowledge discourses on the multiplicities, pluralities, and ontological politics of worlds (human and non-human) have emerged at the end of modern Western universalism and its anthropocentric, humanist cosmology (Jullien, 2014; Mignolo, 2000; Wallerstein, 2006). From a multiple ontologies perspective, the end of universal Eurocentrism is not the end of universalism and the beginning of relativism, but the recognition of pluralism as universal conditions of planetary existence, i.e., biodiversity and sustainability, technodiversity and the ethnosphere (Davis, 2002). The new universal expresses a new political imaginary outside the ideological strictures of the modern nation-state. It is the condition of possibility for a planetary ideal of a new humanity – the non-human basis and destiny of every human – that brings together the planet’s cultural and ecological elements in a singular cosmological embrace (suggesting that both natural and cultural life are holistically related as vibrant multiplicities) (Turnbull, 2006: 136).

Bio, ethno, and techno diversity are ontologically interrelated and become more noticeably important for life to evolve in an era moving abruptly towards mono-agrocultural technological singularity and ecological collapse (Kurzweil, 2005; Shiva, 1993). How can we liberate our cosmo-ontological relationships with modern technology and encourage the flourishing of technodiversity? How could these changes be part of a pluriversal ethnomathematics curriculum, informed by a “political ecology of machines” and a philosophical/political project of cosmotechnics (Dunker, 2020)? In contrast to a modern singular universal world orientation, a pluriversal world orientation is characterized by diversity in the objects, thoughts and practices of technics. One civilization’s formatting power over what becomes “real” for everyone

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may be losing its functionality in the fragmentation of social/technical imaginaries (Sneath & Pedersen, 2009; Taylor, 2004; Vazquez, 2017). The pluriverse is a planetary cosmology in which the plurality of ways of knowing and being and their life-sustaining conditions flourish in their own unique co-existence with the earth (Braidotti, 2016, 2019; Mignolo, 2011; Simon, 2019b). Although the future may appear more dystopian for the new ontological realist, there are cosmoontological openings reorienting the ethnomathematical, ethnoscientific, ethnotechnical knowledges, and the critical posthuman humanities, towards a cosmotechnics within a decolonial pluriversal imaginary (Braidotti, 2019; Mignolo, 2018b; Mignolo &Walsh, 2018).

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Conclusions. Some Possible Lines for Further Research in Ethnomathematics Rik Pinxten and Eric Vandendriessche

In the concluding chapter we, the editors of this volume, want to highlight some challenges and opportunities for future research. By definition such a chapter is speculative rather than a summary or mere comment on the contributions in this book. Still, some of the research related in the present volume is included here, since it points to a specific ongoing work that is highly innovative and complements in part, the broad landscape we outlined in these conclusions. We sincerely think that the globalization of the world invites such speculative suggestions. Globalization in a colonial perspective was clearly meant to trigger a uniform westernized (and probably Christian and free market) future for humanity. However, in the present predicament, globalization tendencies show a much more diversified and indeed “poly-form” world in emergence. Yes, education through schooling is prevalent worldwide, and mathematical skills matter more and in a more universal way than ever before. But on the other hand, the colonial uniformization of minds looks less successful and possibly less likely, let alone desirable to many, than in times past. A mixture of goals, values and dreams seems to take the stage today. Cultural habits and beliefs change rather rapidly, but they do not disappear in favor of one uniform western perspective on the world. Moreover, in various indigenous societies worldwide the decolonization process (d’Ambrosio, o.c.) entails claims to take local cultural concepts and learning strategies more into account in the educational system. Also, success or failure of educational programs gradually teaches the old dominant conquerors to show more cautiousness and more modesty with regard to their own value system and knowledge as well as to those of other traditions. We think that it is at this point—which may well be a turning point in history—that the relevance of anthropological and ethnographic studies on indigenous mathematics and of ethnomathematics in educational contexts may be reconsidered. Taking this critical perspective one is immediately confronted with unforeseen and promising new horizons. In these conclusions, we outline a few ideas in this respect.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. Vandendriessche and R. Pinxten (eds.), Indigenous Knowledge and Ethnomathematics, https://doi.org/10.1007/978-3-030-97482-4

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More Ethnography of Diverse, Local Mathematical Knowledge For one thing, the “old” idea in anthropology that many oral cultures are likely to disappear will continue to motivate researchers to document more of them. In that sense, a first line of thinking is, obviously, that of further empirical studies of cognitive procedures and products of what is generally recognized as mathematics. One line of differentiation in the material found was introduced by Bishop (1988): he distinguished between Mathematics (with the capital M) and mathematics. The former category holds all those elements of knowledge that are understood as part of the modern scientific discipline which goes under the same name. This category was developed in academia in western countries primarily, notwithstanding some historical inputs from other origins (such as India, Islamic countries, etc.). For a long time, mathematics was generally presented—at least in Europe and North America— as a disciplinary field that emerged in ancient Greece, and would have achieved a mature form—in relation to systematic uses of hypothetico-deductive methods— through the works of Euclid, Archimedes and Apollonius in particular (Brunschvicg, 1912; Boyer 1968; Bourbaki, 1984). In contrast, other mathematical traditions, from different parts of the world (from Asia in particular), were most often considered as basic/elementary mathematics, lacking in abstraction and rigor (Granet, 1934; Kline, 1972; Rey, 1937). In the last decade though, fundamental research in the history of ancient mathematics has brought to light sophisticated mathematics, based on algorithmic practices, developed in the ancient worlds (Mesopotamia, China, India in particular), in various historical, cultural and social contexts (Keller, 2015; Proust, 2019; Yiwen, 2016). De facto, these new outcomes in the history of mathematics have significantly yielded epistemological gains regarding our understanding of mathematics (Chemla, 2012), and Science at large (Chemla & Fox Keller, 2017), leading in return to new historical insights on mathematical traditions from the Hellenistic period until the nineteenth century (Acerbi & Vitrac, 2014; Netz, 2009; Smadja, 2015). At the same time, other projects in the history of mathematics have concentrated on mathematical practices carried out in non-scholarly communities from Europe and the United States (Durand-Richard, 2016; Morel, 2017; Tournès, 2018). In the old differentiation, Bishop (1988) referred to counting and measuring formats and such. We think it is very important to point to several other domains of thinking possessing mathematical aspects: rituals, artworks, cosmologies, even farming and hunting or fishing, and other domains can be salient constituent parts of cultural traditions with a more or less rich mathematical knowledge (Pinxten, 2016). Since such domains, with their typical activities and products, may be more relevant to particular traditions than the decontextualized western mathematics, it is important that the research on this reservoir of situated or contextualized knowledge should be studied by anthropologists and social scientists and educationalists in the

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most open way possible. For these reasons we will not continue to use Bishop’s distinction between a capital and a lowercase m in mathematics. The relevance of this open-ended research is enhanced by the fact that children grow up, until now, during early childhood in family or peer group contexts, developing a worldview and cognitive and linguistic anchors in the pre-school world they inhabit. When they make the step to primary school, it is obvious that the gradual transition towards or integration into the implicit worldview of that school system should at the very least, be guided pedagogically: it should not be left exclusively to the pupil to get a grip on likenesses and differences between the local or oral and the school worldview. In the present volume, Petit (chapter “Re/Creating “Evocative Images” (sunannguanik iqqaigutinik): Procedural Knowledge and the Art of Memory in the Inuit Practice of String Figure-Making”) shows how string figure-making is very popular with the Inuit. This traditional game holds great opportunities for teachers: the implicit mathematics in the games can be made explicit and used for pedagogical purposes. The same can be said about Tiennot’s elaborate analysis of the solo game in a completely different part of the world (chapter “Modeling of Implied Strategies of Solo Expert Players”). It is particularly challenging to try and use this preschool knowledge and develop sophisticated mathematical exercises from there. On the other hand, the profound description by this author suggests a useful and maybe pathbreaking link between so-called traditional or ethnomathematical knowledge (Bishop’s small m) and the algorithm cybernetic knowledge outlines in the Baker’s meta-perspective in the present volume (chapter “The Western Mathematic and the Ontological Turn: Ethnomathematics and Cosmotechnics for the Pluriverse”). On another level, it is good to know that some cultures learn and think in socalled verb languages (with practically no nouns, and hence no basic grammatical structure of “noun phrase” and “verb phrase” in Chomskian terms; Navajo and other Athapascan languages are examples of verb languages, Pinxten et al., 1983). Hence, reasoning with sets and objects is different, if not almost inconceivable in the pre-school mental setup of the child raised in such a cultural-linguistic context. Less dramatically, the Turkish language does not grammatically distinguish between singular and plural objects in nouns. When Turkish immigrants have to depend on the presumed “naturalness” of such grammatical signifiers (singular versus plural form) in the European language of their textbook, they are lost, except when the teacher is aware of this difference and can guide the child (as was shown in lengthy fieldwork with Turkish immigrant children, Center of Excellence, Ghent University: Huvenne, 1994). Similar examples exist. It follows from the few examples mentioned and from the many ethnographic data we have by now that it is important to gain more knowledge of the very diversified cognitive categories, learning strategies and cultural contexts in the world. The ethnographic reports we have so far are not really representative of the vast and varied array existing in the world.

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More Anthropological Research Cultures, ways of thinking, styles and learning procedures differ around the world. In the short history of anthropology, we have learned that ethnographic descriptions offer the basic material concerning human ways. They help us to come to grips with the tremendous cognitive diversity which is so typical of our species. At the same time, they reveal two profound shortcomings of empirical work as such. On the one hand, the researcher in human affairs is a human being. We have learned from the so-called self-reflective trend in anthropology of the past decades that systematic conscientious analysis of the researcher’s cognitive categories, attitudes and values constitute a subconscious frame in the researcher’s mind. This frame will allow a partial, often distorted or truncated and one-sided view, resulting in research questions that cannot really be called “objective” (see e.g. Fabian, 1984, on the way that for centuries, temporality notions biased research on time in other cultures). Hence, the researcher should be aware that his or her own biases might play an important role in the actual research processes with subjects from other cultures. On the other hand, and in an even more general sense, human beings enter into processes of ethnographic research in a deeply insecure way, which results in the utter impossibility of the famed scientific rule of “reproducibility of any particular research”. Humans can misunderstand, lie, withhold data when there is a lack of trust or understanding between researcher and subject, and so on. In a more general and intrinsic way: human beings, more than members of any other species and in contrast with “inanimate” material (of the natural sciences) learn from each contact with others. Hence, a second run in order to check on the data of the first moments of research is not possible in the sense in which it happens in inanimate phenomena: the subject is always changing in and by the process of research. In a second round the subject will recognize or not, refuse to collaborate on the basis of former experiences, and so forth: hence, genuine corroboration of results is only partially achieved. In light of all of this, the invitation to collect more ethnographic data stands, but it also suggests a critical and cautious handling of empirical data. Although a few ethnomathematical studies have been carried out in that perspective in the last decades (Chemillier et al., 2007; Pinxten et al. 1983; Vandendriessche, 2015), there is a lack of in-depth ethnographical/ethnomathematical studies, aiming to analyze the form of rationalities/reasonings that underlie these practices possessing a mathematical character, carried out worldwide in various societies. Progress on that issue would allow us, on the one hand, to better understand the kind of mathematics students could learn while practicing activities such as string figure-making, basketweaving, sand drawings, “as such”, in and of themselves (see Vandendriessche, chapter “Sand Drawing Versus String Figure-Making: Geometric and Algorithmic Practices in Northern Ambrym, Vanuatu” of this book) in the math classroom, and, on the other hand, to contribute to an epistemological reflection on the nature of mathematics.

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One way of deepening our knowledge in social sciences in general and in anthropology in particular is by striving for comparative studies. Comparison will allow for model and theory building. The listing of mathematically relevant (or salient) activities may be a starting point for comparative research: a vast amount of ethnographic data can at least be categorized in this first scheme of up to twelve different types of activities with mathematical relevance (Pinxten, 2016). At the same time, the comparison of the combinatorial/geometrical/arithmetical properties at work in these activities (when practiced in a similar way in different societies) should contribute to an improved understanding of the nature of mathematical knowledge and practices involved in these various activities (cf. Chapter “Sand Drawing Versus String Figure-Making”). Such a formal comparison of cultural and cognitive aspects of different practices comprising a mathematical character should further contribute to developing a new epistemological framework for the study of such practices. On the basis of such categorization, it will then be possible to go to the next step, i.e. using particular categories with specific groups of cultural subjects in the process of sophisticating formal thinking in ever more varied and/or abstract contexts towards mathematical knowledge in a school setting. This is the subject of the last paragraph of these conclusions.

Mathematics Education and Ethnomathematics Our general point is that children don’t enter primary education with a “clean slate”, but with a worldview and perspective on natural and social reality, with experiences and learning procedures that have been formed in the linguistic and cultural world they learned to acquire and share by growing up. We take the stand that it is a good educational choice to take this pre-school background into account when introducing children to a corpus of knowledge and learning procedures such as Mathematics education. The tremendous, but selective dropout rate due to math classes we have witnessed so far is, we think, largely the result of disregarding the children’s preschool mental context. Dropout data are high in particular immigrant groups in Europe. For example, any city in Belgium today shows that 20–25% of the pupils leave school after many years without any diploma (PISA, 2010). The dropouts, who enter society without sufficient qualifications to compete for the better jobs in the adult world, are identified primarily as children of immigrant populations. At the same time, we now know that certain groups (notably Japanese and Chinese populations) have high performance rates. Our claim here is that the type of studies ethnomathematics is engaging in may shed more light on the cultural aspects (e.g. religious, cognitive and moral traditions, but also learning strategies) and contextual constraints that are likely to play an important role in the pupil’s transition from childin-a-particular-culture to successful learner in the presumably universal mathematical knowledge.

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In some research the relevance of the pre-school worldview for mathematics education has been a focus. For example, Bishop et al. (2015), Barton (2006), but also older work (Pinxten et al., 1987) investigate cognitive differences next to contextual aspects, gender differences, linguistic structural diversity, and so on. This is not to say that all or any of these dimensions will always and exclusively determine whether performances in mathematics classes will be good or bad. But it certainly looks as if the set of these dimensions offers a panorama of children’s mental setups from which educational programs can then selectively and particularly work to reach out in a differential way to each child or group of children (as cultural subjects). An obvious reason why these “setups” are important is that they allow for an approach of education that respects and enhances the child’s motivation and preferences, rather than negating, or worse killing them. A very important—and so far, hardly recognized issue—is the education and preparation of teachers instructing in other-cultural or multicultural settings. The thorough report on a Brazilian policy is just one example of the way such teacher training can be offered in a more respectful and possibly rewarding way (Oliveira et al., chapter “Indigenous School Education: Brazilian Policies and the Implementation in Teacher Education”). But obviously much more research of this kind can and should be done. Starting with more knowledge and a better understanding of these aspects of the children’s worldview makes it possible to look at the entries for mathematics education that seem most appropriate in any particular case. We clarify this statement with an example. Designing is one of the mathematically salient activities we pay attention to. In the cultural tradition of Pacific seafarers, the initial “catamaran” was developed as an alternative to simple canoes. Ages ago, by means of (ethno-) mathematical thinking these seafaring cultures developed a canoe with a stabilizer, carved from simple trees found on the island they inhabited. Over the years this “catamaran” (as it became known after westerners discovered this strange and ingenious boat in the nineteenth century) was used rather successfully to navigate on the ocean and make trips of up to hundreds of miles between the islands. They dug out a boat from a palm tree, in one piece. Then a stabilizer is cut and attached at the appropriate distance from the main hull, and attached by means of small beams. Undoubtedly many generations of experience go into this boat making. But moreover, islanders developed a star chart made of twigs on the basis of which they could orient themselves on the open sea. Finally, knowledge of sea currents, of the relative warmth of the sea and so on complete this kit of procedures and cognitive data for the “illiterate” Micronesians (Gladwin, 1973). With the Second World War, the motor boat was introduced in the area. However, today the cost of oil is too high for the islanders and depending on the motor boats would starve them. Hence, the old knowledge and technology of boat building and seafaring is picked up again and promoted in schools, this time with the addition of solar energy GPS devices. It is obvious that children know about this, since they are raised in this context. Our suggestion is that such cultural knowledge should be the basis of at least some of the mathematics classes, with full recognition of and respect for the indigenous tradition (amongst others: Rubinstein, 2004).

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Examples like this abound, from string games (this volume: Petit, Vandendriessche) to house building, music and rituals, and so on. The importance of the particular context and its specificities for the educational process has obvious relevance for the cognitive development of children, in the West as elsewhere. This may sound like a sensible focus for (so-called) nonwestern cultures, but Chronaki & Lazaridou’s forceful analysis (chapter “Subverting Epistemicide Through ‘the Commons’: Mathematics as Re/Making Space and Time for Learning”) demonstrates how this point is as relevant for children in the (partially) westernized context of northern Greece. In an unexpected way the analysis of the nature and the role of indigenous mathematical knowledge in Papua New Guinea by Owens (chapter “The Tapestry of Mathematics—Connecting Threads: A Case Study Incorporating Ecologies, Languages and Mathematical Systems of Papua New Guinea”) makes a similar point. Quite obviously, this opens a line of research in ethnomathematics that may revolutionize mathematics education in diverse parts of the world. In yet another focus in a still very different part of the world (i.e., Amazon cultures) Almeida (chapter “Indigenous mathematics in the Amazon: kinship as algebra and geometry among the Cashinahua”) shows how kinship systems, which are the social foundation of most education around the world, use mathematics. Thus, socialization processes, rituals and even ethical training in ever so many different cultural settings provide insights and procedures that will enable a child’s entry into the world of mathematics. Educationalists cannot ignore this rich source of preschool knowledge. Obviously, studying these issues and developing educational programs for ethnomathematics from them, will only be possible by using interdisciplinary approaches, involving anthropologists, psychologists, mathematicians, and (local) educators. The products of such a collaboration may yield surprising multicultural material and teaching procedures. Our suggestion is to promote such interdisciplinary and multicultural research in the context of particular groups and graft any basic course (e.g. on the level of primary education) in mathematics teaching on them. In this way, we want to promote the “reality-based” approach of Freudenthal (e.g. 1985) and diversify it in the context of the many cultural traditions all over the world, in order to allow for a better entry in the field of mathematical thinking by working with the children’s insights (Bishop et al., 2015). Of course, this will entail that a uniform pedagogy, let alone unique and uniform teaching curricula and materials (like the New Math approach) should be dropped. This may come as a shock to mathematicians of the rationalistic school, but it meets the criticisms voiced by those who want to develop links between the social sciences and mathematics education. Chances are that along the way children may indeed come to love mathematics in a variety of its manifestations instead of hating it as an ill understood discipline that first and foremost has you fail in school (Hersh & John-Steiner, 2011).

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