Ubiratan D’Ambrosio and Mathematics Education: Trajectory, Legacy and Future (Advances in Mathematics Education) 3031312929, 9783031312922

Table of contents :
Preface
Encantamento
Preface
Enchantment
Acknowledgments
Contents
Past and Future: Ubi’s Way of Seeing Education in the Present
1 Ethnomathematics in the Slums of Brazil
1.1 Calculators, Ethnomathematics, Guatemala and México
1.2 Digital Technology, Ethnomathematics, Orality and Video Mathematics
2 Final Remarks
References
Part I: Roots of Ethnomathematics
The Presence of Professor Ubiratan D’Ambrosio in the Development of Graduate Mathematics Education in Brazil
1 Introducing the Theme of the Chapter
2 The Contribution of Professor Ubiratan D’Ambrosio to the Creation of the Graduate Program in Mathematics Education at UNESP-Rio Claro
3 Characteristics That Pose Obstacles to the Becoming (Devir) of the Program as a Result of Being in the Mathematics Department: Professor Ubiratan D’Ambrosio’s Contribution
4 Exposing Understandings
References
Ubiratan D’Ambrosio and the Development of Researchers in (Mathematics) Education
1 Introduction
2 Encouraging Curiosity, Creative Daring, and Epistemological Humbleness: Solidarity, Respect, and Cooperation
3 Respect, Solidarity, Cooperation, and the Ethics of Diversity
4 Final Considerations
References
D’Ambrosio’s Legacy in Teacher Ethnopedagogical Space for Glocalization
1 Introduction
1.1 Background
1.2 Rationale and Relevance
2 Theoretical Foundations
2.1 Problematizing Cultural Signs from an Ethnomathematics Perspective for Pedagogical Action
2.2 The Emic Approach to Cultural Signs Based on the Study of Regional Ethnomathematics
2.3 Glocalization of the Ethnopedagogical Space as a Result of Teacher Enculturation
3 Empirical Foundations: Model Design Process
3.1 Phase 1: Design and Validation of Instruments
3.2 Phase 2: Course Design and Organization
3.3 Phase 3: Course Implementation
3.4 Phase 4: Systematization of Information
4 Description of the Teacher Education Model
5 Final Considerations
References
ISGEm and NASGEm: Two Elements of the D’Ambrosio Intellectual Legacy
1 Introduction
References
Ubiratan D’Ambrosio as Historian of Mathematics and Science
1 My First Personal Encounters with Ubi
2 The Kenneth O. May Award Medalist
3 The International Visibilization of the History of Latin American Science
4 The Latin American Management of Institutional Spaces in History of Mathematics and Science
References
The APUA – Ubiratan D’Ambrosio Personal Archive and the Research on the Production of New Knowledge: History of Mathematics, Ethnomathematics and Mathematics Education
1 Initial Considerations
2 About Personal Archives
3 Non-fiction Biography: Another Writing of Ubiratan D’Ambrosio’s Trajectory
4 Knowledge, Scientific Disciplines, Disciplinary Fields...
5 The GHEMAT Documentation Center: Collections and Examples of the History of the Production of New Knowledge
6 The APUA – Ubiratan D’Ambrosio Personal Archive
7 Final Considerations
References
Ubiratan D’Ambrosio and His Contribution to the History of Science and Mathematics
1 Our Academic and Personal Relationship
Remembering Ubiratan D’Ambrosio (1932–2021)
1 Action in the International Community
2 Action in the Luso-Brazilian Community
3 Other Participations in Portugal
Part II: Ethnomathematics in Action
“Ethnomathematics Has Worked, and VEm Brasil Is Proof of That”
1 Ubiratan D’Ambrosio and VEm Brasil: Scenes, Scenarios, and Backstage
2 Enchantments, Tensions, and Expectations on the Path to VEm Brasil
3 The VEm Brasil as a D’Ambrosio’s Ethnomathematics Manifestation
4 Inspiration for the Future
References
Influences and Contributions of Ubiratan D’Ambrosio in the Development of Ethnomodelling as a Research Concept Related to Ethnomathematics and Modelling
1 Initial Considerations
2 My Personal Relations with Ubiratan D’Ambrosio: A Lasting Friendship
3 Personal, Professional, and Academic Life of Ubiratan D’Ambrosio
4 D’Ambrosio’s Role for the Development of Ethnomathematics
5 D’Ambrosio’s Influences and Contributions to the Development of Ethnomodelling
6 Final Considerations
References
The Importance of Ubiratan D’Ambrosio in Latin America
1 Presentation
2 What Is Latin America?
3 Who Was Ubiratan D’Ambrosio?
4 The Development of Ethnomathematics in Latin America
5 What Did Ubiratan D’Ambrosio Mean for Latin America?
6 D’Ambrosio and Its Importance in Latin America
References
Ethnomathematics and Complexity: A Study of the Process of Elaboration of a Peruvian Andean Textile
1 Introduction
2 Diversity and Interculturality in Peru
3 A Multidisciplinary Research from Complexity
4 A Rapprochement Between Ethnomathematics and Complexity Theory
4.1 Colonialism and Epistemicide
4.2 The Quipu: Inca Writing System
4.3 An Inca Mathematical Tool: The Yupana
4.4 Epistemologies of the South and Epistemic Rights
4.5 The Science of Weaving in Peru
4.6 Ethnography and Participant Observation in the Province of Melgar in Puno
5 Construction of the Base of the Loom Structure
5.1 Construction of Rectangles When Making the Loom Base
5.2 Construction of the Rectangle from the Width
5.3 Construction of the Base of the Loom from the Measurement of the Length
5.4 Rectangle Construction from Diagonals
5.5 Construction of the Rectangle by Matching the Measure of the Diagonals
6 Definitions and Mathematical Properties Related to the Rectangle
7 Identification of Mathematical Definitions and Properties in the Construction Processes
8 Impact of Emerging Objects in Pedagogical Practice
References
Part III: Trends in Ethnomathematics
The Political Dimension of Ubi D’Ambrosio’s Theorizations of Ethnomathematics: Criticalethnomathematics
1 Introduction
2 Paulo Freire’s Epistemology
3 Ethnomathematical Knowledge
4 Re-conceiving Mathematical Knowledge: Ethnomathematics and Freire’s Epistemology
4.1 Reconsidering What Counts as Mathematical Knowledge
4.2 Considering Interactions Between Culture and Mathematical Knowledge
4.3 Uncovering Distorted and Hidden History of Mathematical Knowledge
5 Conclusion: Implications for Further Ethnomathematical Research
Appendix: Criticalmathematics Educator: A Definition
Postscript (2023)
References
References for Postscript
Voyaging Beyond the Horizon: An Ethnomathematics Legacy in Hawai‘i and the Pacific
1 Background
1.1 Overview of Ethnomathematics in Hawai‘i and the Pacific
2 University of Hawai‘i Ethnomathematics Program
2.1 Ethnomathematics Examples
3 Embracing Changing Winds and the Moananuiākea Voyage
References
Ethnomathematics in Nepal: Research and Future Prospects
1 Few Words for Ubiratan D’Ambrosio
2 Two Mathematics: Small m and Capital M
3 Impact of Ethnomathematical Movement in Nepal
4 Ethnomathematics Research Practice in Nepal
4.1 Ethnomathematical Practices in Indigenous Communities
4.2 Ethnomathematics and School Pedagogy
4.3 Cultural Artifacts and Ethnomathematics
4.4 Ethnomathematics and Cultural Activities
4.5 Ethnomathematics in Cultural Games
4.6 Ethnomathematics and Cultural Metaphors
4.7 Ethnomathematics in Teacher Education
5 Future Prospects of Ethnomathematics Research in Nepal
6 Concluding Remarks
References
Ubiratan D’Ambrosio: Alchemist of the Mathematics Universe
1 Introduction
2 Part I: Anthropology, Education, and Mathematics in Everyday Life
3 Part II: Ethnosciences – Proportions, Ratios, and Percentages
4 Part III: In the Spirit of Gift Exchange – Ubiratan’s Legacy of Solidarity and Generosity
References
Ethno-biomathematics: A Decolonial Approach to Mathematics at the Intersection of Human and Nonhuman Design
1 Why Does Ethnomath Tend to Miss Out on the Biological Dimensions of Indigenous Mathematics?
2 Prior Work Toward Ethno-biomathematics
3 Biomorphic Form in Cultural Perspective
4 Contemporary Applications
5 Conclusion
References
Ubiratan D’Ambrosio, Curriculum, and Humanistic Mathematics: A Journey of Contrasts from the Modernist Rails to the Postmodernist Awareness
1 Introduction
2 An Analysis of Contrasts
2.1 First, the Background
3 The Modern Curriculum Tradition: Control by Depersonalization
4 The Tradition of the Perception of Mathematical Knowledge: Absolute Objectivity Through Depersonalization
5 The Humanistic Philosophy of Mathematics: Culture as the First Contrast
6 From Description to Seductive Strangeness: The Last Contrast Brought About by Postmodernity
7 Concluding Remarks
References
Final Summary

Citation preview

Advances in Mathematics Education

Marcelo C. Borba Daniel C. Orey   Editors

Ubiratan D’Ambrosio and Mathematics Education Trajectory, Legacy and Future

Advances in Mathematics Education Series Editors Gabriele Kaiser, University of Hamburg, Hamburg, Germany Bharath Sriraman, University of Montana, Missoula, MT, USA Editorial Board Members Marcelo C. Borba, São Paulo State University (UNESP), São Paulo, Brazil Jinfa Cai, Newark, NJ, USA Christine Knipping, Bremen, Germany Oh Nam Kwon, Seoul, Korea (Republic of) Alan Schoenfeld, University of California, Berkeley, CA, USA

Advances in Mathematics Education is a forward looking monograph series originating from the journal ZDM – Mathematics Education. The book series is the first to synthesize important research advances in mathematics education, and welcomes proposals from the community on topics of interest to the field. Each book contains original work complemented by commentary papers. Researchers interested in guest editing a monograph should contact the series editors Gabriele Kaiser ([email protected]) and Bharath Sriraman ([email protected]) for details on how to submit a proposal.

Marcelo C. Borba  •  Daniel C. Orey Editors

Ubiratan D’Ambrosio and Mathematics Education Trajectory, Legacy and Future

Editors Marcelo C. Borba São Paulo State University (UNESP) São Paulo, Brazil

Daniel C. Orey Universidade Federal de Ouro Preto Ouro Preto, Minas Gerais, Brazil

ISSN 1869-4918     ISSN 1869-4926 (electronic) Advances in Mathematics Education ISBN 978-3-031-31292-2    ISBN 978-3-031-31293-9 (eBook) https://doi.org/10.1007/978-3-031-31293-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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

Preface

Encantamento (…) Just as someday, if you have something to offer, someone will learn something from you. It’s a beautiful reciprocal arrangement. And it isn’t education. It’s history. It’s poetry. (...) Assim como um dia, se você tem algo a oferecer, alguém aprenderá com você. É um belo arranjo recíproco. E isso não é educação. É história. É poesia. J.D. Salinger, The Catcher in the Rye

Quando comecei a escrever este prefácio, veio à lembrança o Apanhador no Campo de Centeio (The Catcher in the Rye), clássico de J.D.  Salinger, um dos autores favoritos do meu pai. O protagonista Holden Caulfield descobre seu lugar no mundo através da “compaixão” e do olhar para o “outro”. Em sua jornada, revela um profundo sentimento de amor e responsabilidade em relação às pessoas. Ao mesmo tempo, enquanto desperta para a vida adulta, Holden se angustia com a perda da pureza e do encantamento, próprios da infância: seu amadurecimento se dá através de descobertas e experiências, e sua indignação cresce ao longo da história. A angústia de Holden culmina na imagem do “apanhador” que dá título ao livro, tentando desesperadamente salvar as crianças que se lançam no abismo da vida adulta. Meu pai gostava muito desse livro e uma vez me explicou que, nessa metáfora, enxergava seu próprio papel como educador. Nesse sentido, faço a conexão do livro de Salinger com três das melhores lembranças que guardo de meu pai: (i) sua busca contínua por aprender, (ii) sua capacidade de ouvir o outro, e (iii) sua capacidade de se encantar. Desde minha primeira infância, lembro de quando ele retornava de viagens e congressos e partilhava suas descobertas com a família, de forma entusiasmada. Lembro de sua empolgação ao falar sobre ideias novas, livros e teses que estava lendo, pessoas que tinha conhecido e trabalhos que estava escrevendo. Sem falar de sua fascinante biblioteca, construída no decorrer de sua vida, livro-a-livro. Ele tinha uma mala especial só para carregar livros. Ao retornar de viagem, ele abria essa

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mala e retirava os livros novos, um por um, explicando do que se tratava cada livro e como essa nova “descoberta” impactava suas ideias. Sua biblioteca versava sobre todo tipo de assunto. Desde história, filosofia, literatura, arte, cinema, música, religião, esoterismo, antropologia, sociologia, astronomia, medicina... até, evidentemente, matemática! Era impressionante sua capacidade de associar todos esses campos de conhecimento e incorporá-los às suas ideias e trabalhos, sempre empolgado por descobrir novas abordagens. Gosto de comparar esse entusiasmo de meu pai ao encantamento presente em minhas filhas – hoje com 13 e 8 anos. Crianças sempre se encantam com a descoberta do novo: a primeira leitura; a primeira vez dormindo fora de casa; o primeiro dia de aula; o primeiro contato com o mar; o encontro com outras crianças... Esse tipo de encantamento é comum a crianças do mundo inteiro. Da mesma forma, em qualquer parte do mundo é normal perdermos a capacidade de nos encantarmos, na medida em que envelhecemos e nos tornamos adultos. Lembro vivamente de nossa viagem à África, quando ficamos juntos, em família, durante quase dois meses em Bamako, no Mali. O aprendizado dele durante esse período foi visível até para nós, crianças, pois ele nos mostrava tudo com o encantamento de criança. E, com essa experiência, ele nos ensinou a questionar alguns paradigmas de nossa própria forma de vida, mostrando que havia outras religiões, outras formas de organização social, outras estruturas familiares. Tudo isso nos ajudou, desde crianças, a valorizar a diversidade e a respeitar outras formas de pensar. Assim como essa experiência ajudou a moldar nossa própria consciência, também moldou a visão de mundo do meu pai e o ajudou a descobrir, na prática, o sentido daquilo que viria a ser chamado de “etnomatemática”. Quando começou a pandemia, ele começou a fazer conferências diárias por vídeo e ficou empolgado com as imensas possibilidades que essa integração traria para a educação, aproximando pessoas e culturas no mundo inteiro. Por videoconferência, ele fez inúmeras palestras em vários lugares do Brasil e do mundo. Ao falar com ele no dia seguinte, contava como tinha acabado de “voltar” de uma viagem a esses lugares. Tendo passado a vida viajando pelo mundo, os últimos meses de sua vida foram um alento por retomar o ritmo de palestras e viagens que experimentou desde a juventude. Meu pai sempre foi apaixonado pelo que fazia e, até os últimos dias de sua vida, continuava a demonstrar entusiasmo por novas ideias e novas descobertas. Ele conseguia transmitir para nós, em casa, o entusiasmo pela busca, pela descoberta, pelo aprendizado. E, na medida em que ele conseguia nos contagiar com esse amor pela aprendizagem, sei que ele fazia o mesmo com colegas e alunos. Voltando à epígrafe de Salinger, posso afirmar que meu pai viveu sua vida de forma generosa, oferecendo seu tempo, suas ideias, sua atenção, a quem precisasse. Ele sempre estava disponível para colegas, alunos e familiares e sempre interessado em ouvir. Mas, mesmo para pessoas fora de seu círculo, meu pai valorizava suas ideias. Numa ocasião, eu e meu pai caminhávamos juntos pelo centro da cidade, eu ainda adolescente, quando fomos abordados por um morador de rua. Não pediu dinheiro, nem comida, apenas queria falar. Meu pai parou para conversar com ele e

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permaneceu ouvindo, com profunda atenção, por vários minutos. Depois, o homem se despediu e retomou seu caminho. Nesse dia ele me explicou que o verdadeiro respeito pelas pessoas é enxergá-las, ouvir suas ideias e se interessar por elas. Eu testemunhei inúmeras situações semelhantes, quando meu pai generosamente se dedicava a ouvir as pessoas, independente de quem fossem e, com essas trocas, ele sempre aprendia alguma coisa. E, como disse Salinger, essa troca nem sempre é “educação”. É história. É poesia. Durante toda a vida, tive o privilégio de tê-lo como guia e mentor, sobre qualquer assunto. Embora eu nem sempre concordasse com todas suas opiniões, ele sempre sugeria uma perspectiva diferente, um novo caminho, um exemplo histórico para me ajudar a tomar minhas próprias decisões. Ele me ensinou a duvidar, a questionar, a ser curioso, buscar o aprendizado constante e, sempre, a olhar também sob a perspectiva dos outros, com a humildade de mudar de opinião. E, acima de tudo, com ele aprendi a manter viva a capacidade de me encantar. Em nome de nossa família, agradeço de coração pelo enorme carinho de todos pelas homenagens prestadas à memória de meu pai. Sei que ele está acompanhando – e se encantando – com cada artigo, cada debate, e sobretudo com a riqueza das ideias apresentadas neste livro. São Paulo, Brazil Janeiro de 2023

Alexandre Silva D’Ambrosio

Preface

Enchantment (…) Just as someday, if you have something to offer, someone will learn something from you. It’s a beautiful reciprocal arrangement. And it isn’t education. It’s history. It’s poetry. (...) Assim como um dia, se você tem algo a oferecer, alguém aprenderá com você. É um belo arranjo recíproco. E isso não é educação. É história. É poesia. J.D. Salinger, The Catcher in the Rye

As I began to write this foreword, one of my father’s favorite authors J.D. Salinger’s classic The Catcher in the Rye came to mind. In the book, the protagonist Holden Caulfield discovers his place in the world through “compassion” and looking at the “other.” On his journey, he reveals a deep sense of love and responsibility towards people. At the same time, while awakening to adult life, Holden is anguished by his own loss of purity and enchantment. Typical of childhood, his maturation takes place through discoveries and experiences and his indignation grows throughout time. Holden’s anguish culminates in the image of the “catcher” that lends the book its title, desperately trying to save children who plunge into the abyss of adult life. My father was very fond of this book and once told me that, in this metaphor, he saw his own role as an educator. In this sense, I would like to connect Salinger’s book with three of the best memories I have of my father: (i) his continuous quest to learn, (ii) his ability to listen to others, and (iii) his ability to be enchanted. Since my early childhood, I recall when my father returned from trips and congresses and shared his discoveries with the family, enthusiastically. I remember his excitement as he talked about new ideas, books, and theses he was reading, people he had met, and papers he was writing. And there was his fascinating library, built up over the course of his life, book by book. He had a special suitcase just for carrying books. Upon returning from a trip, he would open this suitcase and remove the new books, one by one, explaining what each book was about and how this new “discovery” impacted his ideas.

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His library included all sorts of subjects. From history, philosophy, literature, art, cinema, music, religion, esotericism, anthropology, sociology, astronomy, medicine... to, of course, mathematics! His ability to associate all these fields of knowledge and incorporate them into his ideas and work was impressive, and he was always excited to discover new approaches. I like to compare my father’s enthusiasm to the enchantment present in my daughters, now aged 13 and 8. Children are always enchanted by discovering something new: the first reading; the first time sleeping away from home; the first day of school; the first contact with the sea; meeting other children. This type of enchantment is common to children all over the world. Likewise, in any part of the world, it’s normal for us to lose the ability to be enchanted as we age and become adults. I vividly remember our trip to Africa, when we stayed together as a family for almost two months in Bamako, Mali. His learning during that period was visible even to us children, as he showed us everything with the enchantment of a child. And, with this experience, he taught us to question some paradigms of our own way of life, showing us that there were other religions, other forms of social organization, and other family structures. All this has helped us, since childhood, to value diversity and respect other ways of thinking. Just as this experience helped shape our own consciousness, it also shaped my father’s worldview and helped him discover, in practice, the meaning of what would come to be called “ethnomathematics.” When the pandemic began, he started doing daily video conferences and was excited about the immense possibilities that this integration would bring to education, bringing people and cultures together around the world. Through video conferences, he gave numerous lectures in various places in Brazil and around the world. When talking to him the next day, he told us how he had just “returned” from a trip to these places. Having spent his life traveling the world, the last few months of his life were an encouragement to pick up the pace of lectures and travel that he had experienced since his youth. My father was always passionate about what he did and, until the last days of his life, he continued to show enthusiasm for new ideas and new discoveries. He managed to convey to us at home his enthusiasm for seeking, discovering, and learning. And to the extent that he managed to infect us with this love of learning, I know he did the same with colleagues and students. Going back to Salinger’s epigraph, I can say that my father lived his life generously, offering his time, his ideas, and his attention, to anyone who needed it. He was always available for colleagues, students, and family and always interested in listening. But even for people outside his circle, my father valued their ideas. On one occasion, my father and I were walking together downtown, I was still a teenager, when we were approached by a homeless man. He didn’t ask for money or food, he just wanted to talk. My father stopped to talk to him and listened with deep attention for several minutes. Afterwards, the man said goodbye and resumed his path. That day, he explained to me that true respect for people is seeing them, listening to their ideas, and being interested in them. I have witnessed countless similar

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situations where my father has generously dedicated himself to listening to people, no matter who they were, and from these exchanges he has always learned something. And, as Salinger said, this exchange is not always “education.” It’s history. It’s poetry. All my life I have had the privilege of having him as a guide and mentor, on any subject. While I didn’t always agree with all of his opinions, he always suggested a different perspective, a new path, or a historical example to help me make my own decisions. He taught me to doubt, to question, to be curious, to seek constant learning and, always, to also look from the perspective of others, with the humility to change my mind. And, above all, with him, I learned to keep alive the ability to be enchanted. On behalf of our family, I sincerely thank everyone for their enormous affection and for the tributes paid to my father’s memory. I know that he is following, and enchanted, with each article, each debate, and above all with the wealth of ideas presented in this book. São Paulo, Brazil Janeiro de 2023

Alexandre Silva D’Ambrosio

Acknowledgments

As co-editors, we wish to thank Beatriz Kajikawa Delgado, a mathematics undergraduate UNESP student in Brazil, who throughout this project has made heroic efforts and worked tirelessly to keep all authors and the editors on task. Her keen eye and patience with us all is extraordinary. Without her, this project would have never happened! Thank you! As well, we can do no better than thank, from the bottom of our hearts, both Professors Gabriele Kaiser and Bharath Sriraman for their very kind invitation to organize this memorial. Organizing this book for us has been a distinct honor and during the loss of losing our mentor helped us greatly. Thank you! We are more than grateful to the D’Ambrosio family, especially Alexandre D’Ambrosio for the preface and Maria José D’Ambrosio for their suggestions, love, and encouragement. It was an honor for us to interact with Ubiratan’s son and wife in the organization of this book. Obrigado! And it goes without saying how the work, as presented here, with the tremendous diversity, experiences, and perspectives of the authors, stands in memorium to the memory of Ubiratan D. Ambrosio. His inspiration, example, and work, as reflected upon here, stand as the greatest example of his legacy and for the work in classrooms in dozens of countries. As the readers will see, his influence as felt by thousands of teachers, educators, researchers, and students worldwide is partially reflected in this book. And most importantly, during the many difficulties found in many countries that Ubiratan talked about – pandemics, economies, injustices, prejudices, extremes of wealth and poverty – Ubiratan taught each of us to look outside our individual “gaiolas epistemologicas” (epistemological cages) and share our hope and ideas peacefully together. This book offers that hope. Marcelo C. Borba Daniel C. Orey

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Contents

 Past and Future: Ubi’s Way of Seeing Education in the Present����������������    1 Marcelo C. Borba and Daniel C. Orey Part I Roots of Ethnomathematics The Presence of Professor Ubiratan D’Ambrosio in the Development of Graduate Mathematics Education in Brazil ����������   17 Maria Aparecida Viggiani Bicudo Ubiratan D’Ambrosio and the Development of Researchers in (Mathematics) Education����������������������������������������������������������������������������   29 Maria da Conceição Ferreira Reis Fonseca and Cristiane Coppe-Oliveira D’Ambrosio’s Legacy in Teacher Ethnopedagogical Space for Glocalization������������������������������������������������������������������������������������   41 María Elena Gavarrete Villaverde, Margot Martínez Rodríguez, and Marcela García Borbón ISGEm and NASGEm: Two Elements of the D’Ambrosio Intellectual Legacy ������������������������������������������������������������������������������������������   63 Tod L. Shockey, Patrick (Rick) Scott, and Frederick (Rick) Silverman  Ubiratan D’Ambrosio as Historian of Mathematics and Science���������������   71 Luis Carlos Arboleda The APUA – Ubiratan D’Ambrosio Personal Archive and the Research on the Production of New Knowledge: History of Mathematics, Ethnomathematics and Mathematics Education ��������������������������������������������������������������������������   83 Wagner Rodrigues Valente and Luciane de Fatima Bertini

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Ubiratan D’Ambrosio and His Contribution to the History of Science and Mathematics����������������������������������������������������������������������������  101 Sergio Nobre  Remembering Ubiratan D’Ambrosio (1932–2021) ��������������������������������������  109 Luís Saraiva Part II Ethnomathematics in Action  “Ethnomathematics Has Worked, and VEm Brasil Is Proof of That”��������  121 Olenêva Sanches Sousa Influences and Contributions of Ubiratan D’Ambrosio in the Development of Ethnomodelling as a Research Concept Related to Ethnomathematics and Modelling ����������������������������������������������  145 Milton Rosa  The Importance of Ubiratan D’Ambrosio in Latin America����������������������  163 Armando Aroca and Maria Cecilia Fantinato Ethnomathematics and Complexity: A Study of the Process of Elaboration of a Peruvian Andean Textile������������������������������������������������  179 María del Carmen Bonilla-Tumialán Part III Trends in Ethnomathematics The Political Dimension of Ubi D’Ambrosio’s Theorizations of Ethnomathematics: Criticalethnomathematics����������������������������������������  203 Marilyn Frankenstein and Arthur B. Powell Voyaging Beyond the Horizon: An Ethnomathematics Legacy in Hawai‘i and the Pacific ������������������������������������������������������������������������������  241 Linda H. L. Furuto and Antonina Monkoski-Takamure  Ethnomathematics in Nepal: Research and Future Prospects��������������������  253 Jaya Bishnu Pradhan  Ubiratan D’Ambrosio: Alchemist of the Mathematics Universe����������������  275 Mariana K. Leal Ferreira Ethno-biomathematics: A Decolonial Approach to Mathematics at the Intersection of Human and Nonhuman Design����������������������������������  289 Ron Eglash Ubiratan D’Ambrosio, Curriculum, and Humanistic Mathematics: A Journey of Contrasts from the Modernist Rails to the Postmodernist Awareness����������������������������������������������������������������������  305 Carlos Mathias Final Summary������������������������������������������������������������������������������������������������  323

Past and Future: Ubi’s Way of Seeing Education in the Present Marcelo C. Borba

and Daniel C. Orey

As mathematicians and math educators we have a responsibility towards issues of sustainability, climate change and pandemics, which are urgent. Ubiratan D’Ambrosio (Como matemáticos e educadores matemáticos devemos nossa responsabilidade perante questões de sustentabilidade, de alterações climáticas e de pandemias, que são urgentes. D’Ambrosio) (2018, p. 197)

Abstract  In this chapter, the authors wish to share two stories that show how they came to know Ubiratan D’Ambrosio and how his philosophy changed their lives and work. The authors, as young scholars, first encountered Ubiratan from two distinctive but similar perspectives. At the time, vital issues related to the mix of mathematics education, with new technologies, issues of diversity and culture, and social class were interacting together. For each of us, it began to make sense when, separately, we had the bounty to spend quality time with a man who became our mentor and teacher and remained so for many years. As Ubiratan reminded us, Janus had an eye in the future and one in the past. D’Ambrosio had an eye on the mathematics of the ancients, of the poor, and the mathematics found outside of formal Western academia contexts. As well, he was always concerned with mathematics-with-­ technology. Janus is present in this chapter, and we believe in the whole book. Like Janus, D’Ambrosio clearly had an eye in the past and another in the future, as his way of seeing mathematics education in the present. Keywords  Janus · Ethnomathematics · Humans-with-media · Pandemic

M. C. Borba (*) São Paulo State University (UNESP) Rio Claro, São Paulo, Brazil e-mail: [email protected] D. C. Orey Universidade Federal de Ouro Preto, Ouro Preto, Minas Gerais, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_1

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The quote above comes from a paper, published in Portuguese, where D’Ambrosio foresaw the current pandemic. In the paper, he used more than once, the word pandemic in the plural “pandemics.” He talked about how we are coming out of one pandemic that has come after many before, and many more to come. Ubi foresaw how the COVID pandemic would so drastically and rapidly change the world we now live in. And like the Roman god Janus, we can use this opportunity to look before us and into the future, at the same time. Ubi often used the example of the Roman god Janus, in his work. Janus, who the Romans named the first month of the year in his honor, could look back and forth at the same time. Ubi, as one of the renown international specialists in the history of mathematics and who developed the program of ethnomathematics, taught us to look backwards in history and then to the future at the same time. From both his global and historical perspective, he described a certain tension between the past and future of humanity to address mathematics education in its current form. Technologies of communication and information have been particularly influential in new directions of society, in particular of education. The transition from orality to writing marked a new role for the teacher. From the sole repository of accumulated knowledge, the teacher became a guide and interpreter of registered knowledge. The emergence of hardware, in the form of documents and books, initiated a companionship between teacher and hardware. It is also remarkable how the emergence of writing strengthened individual memory, contrary to the concerns of Thamus when Theuth explained to him the discovery of writing. The conservative king was afraid that the new invention would implant forgetfulness in the souls of men. (D’Ambrosio, 2005, p. XIV, preface in Borba & Villarreal 2005)

The above quote illustrates how he analyzed different media in different moments of history, and to support our use of digital technology in mathematics education in the present. D’Ambrosio (2005), in this preface which may not have gained the importance it should, develops the notion that education is a result of a tension between past and future. Extensions of memory, to use Levy’s (1993) terminology could affect the memory, as feared by the King in the old days and by many who feared the implementation of digital technologies in (mathematics) education. Mathematics Education, once such a dichotomy between humans and hardware (to use D’Ambrosio’s way to refer to books and to computer hardware) should incorporate new technologies, should assist us to look into the future. From this dialogue he invented the ethnomathematics trivium (Rosa & Orey, 2015). The Trivium model as outlined by Ubiratan is an ethnomathematics-based program used by educators to identify pedagogical actions in the form of teaching– learning practices. The curriculum proposal is based on D’Ambrosio’s Trivium, composed of literacy, matheracy, and technoracy. The Trivium supports the development of school activities based on a foundation of ethnomathematics and modeling (Rosa & Orey, 2015). In Ubiratan’s three-point curriculum model, literacy is the capacity students have to process information present in their daily lives; matheracy is the capacity students have to interpret and analyze signs and codes in order to propose models and to find solutions for problems faced daily; and technoracy is the capacity students have to both use and combine different instruments to solve

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increasingly complex problems. And of course, numeracy plays an important role in this curriculum model. Such a look into a Janus-Trivium of the future does not discard any view of education that values the past, it merely asks us to be more cognizant, aware, mindful, and flexible as new situations arrive from the future. His unique way of using the context of and the use of the history of mathematics formed his constant call for a new epistemology and became what he called “A new Trivium,” in his approach to history of mathematics. D’Ambrosio (2018) recognized that he built on the work of many scholars such as Ascher and Ascher (1981) who wrote an ethnomathematical inventory of the mathematics of traditional Andean people who developed the Quipu. Ascher and Ascher presented strong evidence that the base 10 system was present in what is called Latin America today, hundreds of years ago. If mathematics history, such as this is used in curriculum, one could think of integrated curricula of history and mathematics in a way that would empower students from this area and at the same time could have important context for learning the base 10 system, the all-important concept for Western-Academic mathematics, and a very important concept for school mathematics universally. Considering all of this, we would like to explore, using a Janus-Trivium perspective, this tension between the traditional and the future. We will present a few well-­ known examples from the slums in Brazil, which was the first master’s research developed based on the notion of ethnomathematics. We will also present how the history and colonial architecture, as that found in Ouro Preto, provides us with a profound interdisciplinary source for curriculum. And looking into the future we will discuss the whole of digital videos in mathematics education and how Ubi’s Program Ethnomathematics will include forms of digital video culture.

1 Ethnomathematics in the Slums of Brazil As D’Ambrosio was preparing his memorable talk at ICME-V in Adelaide, Australia, in July 1984 he was also teaching in the first and oldest graduate program in mathematics education in Brazil. The program had its first classes held in March of the same year at Universidade Estadual Paulista (UNESP), Rio Claro (see Bicudo’s chapter in this book). In preparation for his trip to Australia, Ubi gave us a “pre-session” of his talk in which he summarized his understanding of ethnomathematics. In Brazil at that time, except for the book by Carraher et al. (1995), street mathematics and school mathematics (Na Vida Dez, na Escola Zero), there were few books in libraries that could be considered in the domain of ethno/mathematics education. To hear what became one of the most remarkable addresses at an ICME conference was something amazing for the first cohort of graduate students in the Mathematics Education Graduate Program at UNESP, Rio Claro. Among the students, was a very young Marcelo, the first author of this chapter!

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At a coffee break, students were completely amazed with the number of new ideas and how profound they sounded. The notion of ethnomathematics was known only by two or three students at the time. The notion that mathematics was culturally shaped and that every cultural group may have a unique way of expressing mathematical thinking was astonishing, especially for Marcelo who had just left undergraduate studies with emphasis on academic mathematics. Other courses with intense face-to-face interactions between graduate students, some fresh from undergraduate studies, mixed with others who were experienced didactical book authors, made the notion of ethnomathematics at the center or core of the debate. The influence of the rehearsal of Ubi’s talk was monumental for all involved. One witnessed a truly innovative moment. Some like Marcelo changed the focus of his research, and combined Ubi’s ideas with Paulo Freire’s, a Brazilian educator who have worked with the notion that culture was fundamental for literacy, for adults who had not had access to formal literacy or academic study as young people. After many informal and formal talks between Eduardo Sebastiani and Maria Viggiani Bicudo – professors of the graduate program at the Universidade Estadual de Campinas (UNICAMP) – Ubi designed a study in which included long-­ term field work, combined with participatory research, which was going to be developed in the favelas (slums, or poor communities) of Campinas, in the state of São Paulo. The main idea was to study ethnomathematics found in the communities, composed of migrants, from different parts of Brazil, and who were leaving in precarious conditions. Most adults were bricklayers, cleaning persons, or unemployed living in shacks, without adequate plumbing or resources. Many parents were illiterate, where traditional forms of literacy were not available to them. The project showed researchers that they, in accordance with Freire (and soon Ubiratan), were indeed “literate,” just not in traditional forms. Children from this community would go to school and “were ignored.” In conjunction with this project, the city government decided to create an “informal school” with two lay teachers. A team from UNICAMP, where the graduate program in mathematics education was located – became involved in helping the project in the “Favela de Vila Nogueira  – São Quirino.” At the time, there were no mathematics educators on the team. Ubi and Sebastiani, who were professors at UNICAMP, and taught in the graduate program at UNESP, made the connection and Marcelo became part of the team. The leaders of the project were students of Paulo Freire’s. With the end of the military dictatorship in Brazil, Freire had returned from exile. As part of this project, it is thought that the kinds of mathematics for the adults and children were documented for the first time. Such a study would serve in future work in pedagogical projects combining modeling, incorporating mathematics education, ethnomathematics, and Freire’s ideas (Borba, 1987, 1990 FLM). Both work in Mathematical and Ethnomodeling Modeling (Rosa & Orey, 2015, 2016) and Freire emphasize the idea that students may have participation in designing curricula, by having their voices heard in the choice of themes to start pedagogical actions, whether the goal was to teach Portuguese or to teach mathematics. D’Ambrosio dialogued with such ideas, not only because he, with Bassanezi, helped

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to develop modeling in Brazil, but because he was also profoundly interested in social justice, a major point for Freire and those living in the Campinas favelas at the time. In his field notebook, Marcelo collected themes of interest for children between 8 and 13 years old, which included the mathematics coming from bricklayers, pipe fixers, and other jobs employing their parents and members of their community. The pedagogical proposal developed was in line with modeling as seen by researchers such as Prof. Joni Meyer, a longtime collaborator with Ubiratan D’Ambrosio. Also D’Ambrosio - in the introduction to Bassanezi’s important book on mathematical modelling – presented an early use of the term “ethno-modelling”. (Bassanezi, 2002). The very idea that poor people, many of them with various levels of literacy and matheracy, knew that mathematics expressed in a unique or particular way was powerful. Ubi’s rehearsal of his keynote address for ICME-Adelaide was for the graduate students in the UNESP course a starting point, a Janus moment for ethnomathematics. And the students in the graduate program of UNESP, Rio Claro, were honored to witness it. By then, it developed into a formal Master’s program and since 1993 has added a doctorate program. The master’s thesis by Marcelo, the second one defended in the program, in 1987, with Ubi as a member of the evaluating committee, reported on the ethnomathematics of students regarding a vegetable garden and included the planning for it. As well, the geometry of a “soccer field” the community built was studied. Besides the dissertation, a book published by Paulo Freire and two of his doctoral students, in which Marcelo participated, documented this work. The autograph, the cover, and the summary of it can be seen in the pictures below. Ubi was happy to see this connection with Paulo Freire happening.

1.1 Calculators, Ethnomathematics, Guatemala and México About the same time, 1986, Daniel had returned to the United States from Guatemala to begin master’s and doctoral work in New Mexico. Both the state of New Mexico and Guatemala are still involved in important dialogue between past and present, indigenous and Western technologies, both with a rich ancient history, where many of the languages and customs are still intact. On a day in the early 1980s Daniel found himself bargaining for a certain object in a Highland Maya market. The vendor did not speak Spanish, but when he asked the price, she took a calculator out of her huipil (blouse), typed the number in, and handed it back to Daniel. He looked at it, shook his head, typed in a new price, and handed it back. This went on for a few minutes until both the buyer and seller were satisfied. It was so automatic and natural that it wasn’t until he was on the bus back to Guatemala City, when he asked himself “What had just happened?” Calculators at the time were new, and many mathematics educators refused their use, yet there in rural highland Guatemala he witnessed a new use for it. This was the moment he began to watch how people of diverse groups used different forms of technologies, geometry, count, order, pattern,

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model, etc. Some of which is directly connected to ancient architectures, sciences, and astronomies. Master’s work invited him to a project that allowed him to travel around New Mexico, Arizona, Colorado, and West Texas, giving people their first access to technology. At the time there was no internet, and the project used Texas Instruments TI computers in a van on donated hospital gurneys (the type used in ambulances). It was research impossible to replicate now, with the worldwide diffusion of smart phones and Wi-Fi. But then as project leaders used LOGO and worked with children, teachers, and families, he saw how all people, no matter their age, culture, gender, social class – including groups of Catholic nuns, or indigenous community elders – took control of the turtle and quickly saw the potential for the computer. In the midst of a civil war, he took this model back to Guatemala and worked in two schools there, where like in the States, after a brief introduction he turned on the computer, introduced participants to LOGO and watched. His doctoral work took him to a school in Puebla, México, that Apple had endowed with a huge number of computers (Apple IIe’s) and watched; again, how the teachers and students implemented mathematics using LOGO. As well the school was a test site for Apple de México and was debugging the very first Spanish language keyboard. The Apple representatives listened intently to the children as they discussed command letter keys for different accents, etc. (Fig. 1). At the same time, his doctoral advisor, Patrick Scott, then at the University of New Mexico (UNM), was translating Ubiratan’s first book into English. Daniel remembers looking over Prof. Scott’s drafts and discussing them with a group of

Fig. 1  Patzún, Chimaltenango, Guatemala 1982. (Source: Photo by Daniel Orey)

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graduate students. It was a moment not unlike Borba’s experience above when Ubiratan shared his talk in Australia. After defending his Ph.D. work, Daniel took a faculty position at California State University Sacramento (CSUS) and participated in the annual meeting of Northern California Math Council at Asilomar, where Ubiratan was invited to give the opening lecture. After Ubi’s talk he introduced himself (as Professor Scott’s former doctoral student) and was able to be 1:1 with Ubi for 48 h. Together they walked the grounds of Asilomar, the beaches, took a drive along the coast, where Daniel asked 1000s of questions, and in the end was invited to apply for a Fulbright to work with Ubiratan and Rodney Bassanezi in the Mathematical Modeling and Ethnomathematics specialization at the Pontificia Universidade Católica de Campinas (PUCC). After a rigorous process, he was offered a semester at PUCC as a visiting scholar, thus being one of the first Fulbright Scholars in Ethnomathematics. It was during this time that Daniel began looking at how the voice of the people under study needed to be honored in the context of modeling. That is, a movement away from the researcher being the outsider going into an exotic location to study how “they” do it, towards “participants telling us how they do it.” Something he began to see willingly in Guatemala and New Mexico, but the PUCC specialization allowed him to learn and to focus on using the tools of mathematical modeling to uncover ethnomathematics in a group of people. Learning to step away and let “they who do the work” tell us how they do it eventually led towards they themselves publishing and assuming positions of power and authority over their own work. The time at PUCC gave him the opportunity to practice, learn, and refine tools to do this work, and was where he first met Marcelo Borba and Milton Rosa. In New Mexico, and now in Guatemala, a very large and powerful group of indigenous voices has immerged – indigenous people who have higher education degrees, master’s, and doctorates; indeed, they are hundreds of MDs, PhDs, lawyers, engineers, archeologists, professors in universities. A more than worthy goal for us here in Brasil that we need to work towards the same. Something happens when, as in Daniel’s case, his mathematics or anthropology professor talked about his own culture and offered observations about Daniel. He was often humbled; it is still a powerful life lesson. A few years later, Daniel was invited to spend a year at the Universidade Federal de Ouro Preto (UFOP) as a CNPq visiting scholar, in the Department of Mathematics. A colleague in the United States, Kay Toliver, was involved in creating math trails in Harlem, New York, so he began to play with this idea with math specialization students at UFOP and at a public elementary school in Ouro Preto.1 The result has been a math trail, numerous graduate research projects, and numerous visits from projects participants outside of Ouro Preto to the trail. Participants are currently documenting their own models and stations developed in the historic center. Daniel and many others clearly see what happens when the voice of the insider is honored  See: A Trilha das Matemáticas de Ouro Preto/The Ouro Preto Math Trail (https://sites.google. com/site/trilhadeouorpreto/). As well as various masters’ projects and research archived at: https:// ppgedmat.ufop.br/disserta%C3%A7%C3%B5es 1

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and given respect. Each tour or visit creates new problems; the latest was related to how many discarded masks did we encounter as we walked along the streets. Questions arose for instance “Was there a difference to location and or side of the street?” At one point the children began inventing surprisingly complex, but to them very serious, math questions. At first, the university students were flabbergasted with, then excited to solve and work with, the 9- and 10-year-old students. The best, to my mind as the then outsider, now as a UFOP professor insider were: • How many marbles can fit inside of the giant pillar at the top of Alvarenga street? • How many fuscas (the nickname for Volkswagen bugs in Brazil) are there in Ouro Preto? In the State of Minas Gerais? In Brazil? • How many “paralelepípedos” (cobble stones) are on the Alvarenga street? • Why is the first house on Rua Alvarenga Number 7? And not 1 or 0? Why is the second one 12? Then 32? What is happening? • Are the curves along the wall of the Colégio Arquidiocesano de Ouro Preto mathematical curves? Both university and elementary learners became active inquirers and not passive recipients of math as Seymour Papert2 the inventor of LOGO would say. All these questions are grounded, in what the kids and university students began to see as we walked along the street. All of these led to profound forms of mathematics; the activity created a synergy and excitement. And, to date, not one participant in the Math Trail to Daniel’s knowledge leaves with the question that learners in traditional math classes in Guatemala, California, Nepal, and Brasil ask, “when will we ever use this?”

1.2 Digital Technology, Ethnomathematics, Orality and Video Mathematics What has brought the two authors together over time has been our mutual experience, indeed our Janus moments in regard to technology, including research and interest in culture, technology, and mathematics education. Ubiratan said: Like Janus, the ancient Roman god whose double-faced head signified his knowledge of the present and the future, education has always been a two-faced enterprise. The past establishes goals and methods of education, and the other face tries to capture the future and suggests and proposes new directions of thought and new styles of action for the next generation who will in a few years, take over routines and societal innovation in the many diverse contexts worldwide. History tells us that this face of education has always been sensitive to emerging technologies. (D’Ambrosio, 2005, preface in Borba & Villarreal, 2005)

 Orey had the honor of spending two summer schools with MIT Professor Seymour Papert and his LOGO research team. 2

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Very few people in mathematics education will connect the work of Professor Borba to ethnomathematics, but it influenced him from the very beginning. Most people will connect his work to digital technology. What most people will be surprised is the connection of both to his work. His early work in the slums of Campinas, reported earlier, had the recognition of researchers who built upon this early work, where new developments were made. The publications between 1989 and 2005 of his work demonstrate the influence of Ubiratan and how ethnomathematics taught Marcelo how mathematics can change with different modes of expression, as he first observed as he saw in the Brazilian favela. It was this idea that inspired him to think of computers changing the mode of mathematics being expressed, as it can be shown in his doctoral dissertation (Borba, 1993). It was open terrain for the notion of humans-with-media as units who produce knowledge. The construct humans-with-media is mostly used to emphasize the role of digital technology in changing the way mathematics is expressed or in the very nature of what mathematics may be (Engelbrecht et al., 2023). Villarreal and Borba (2010) illustrate how different media – compass, paper and pencil and informatics – has shaped mathematics throughout history. Such a discussion, during the pandemic, has been expanded, during the pandemic to other “artifacts.” Borba (2021a, b) shows how homes are important in collectives of humans-with-media-homes during the pandemic. The environment, the size of home, the culture of knowledge of the home, and social class, all influenced the learning and teaching in the model of education that prevailed for the most part of the pandemic. This construct was also used to empower the ethnomathematics expressed by writing. Collectives of humans-with-orality generate mathematics in Brazilian favelas, as mentioned before, in parallel with academic mathematics that is produced by collectives of humans-with-paper-and-pencil. Artifacts and orality are used to demonstrate truth in mathematics produced by a brick layer and paper and pencil produced demonstrations. In the last ten years, a movement regarding use of videos in mathematics education in Brazil allowed us to remember the Janus’ metaphor brought by Ubi almost 20 years ago to refer to digital technology. Video is on the one hand the future, the present, as it is an important, if not the most important, means of communication of this century. Ubiratan in his Trivium model laments how traditional education ignores the power of the diverse forms of visual and social media that we all find ourselves using. On the other hand, initiatives such as the Digital Videos and Mathematics Education Festival (https://www.festivalvideomat.com/) show that we can document old traditions or different cultural perspectives. For example, a video from the second festival, in which a father teaches how to count in his native indigenous language, is an example of this.3 Different videos were placed in the festival, from

 See: https://www.youtube.com/watch?v=ATGgiPTwIuw&list=PLiBUAR5Cdi61ylU9-09NUE wqdqtWnsFry 3

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ones that are close to video-classes, those embedded in criticism of social reality, to ones that explore different cultural aspects. In 2022, Marcelo was invited to give a keynote address at the VI Brazilian Congress of Ethnomathematics that took place in the Amazon, in the state of Tocantins. The title of the talk was “Mathematical knowledge and practices produced in different societies and their technological innovations from the perspective of Ethnomathematics.” It seemed that the title of the talk, chosen by the organizers, is influenced by Janus’ metaphor. In this talk, Marcelo connected his work in collective of humans-with-media-with-orality with the one of humans-with-media-withdigital videos. Several Brazilians did not know that Paulo Freire had edited a book (Freire et al., 1998), in which the work of Borba and ethnomathematics in the favelas was part of. The autograph by Paulo Freire on an education book, with a chapter on ethnomathematics, two co-authored chapters with Freire, when Marcelo was doing his doctorate in digital technology in mathematics in the United States, was again a sign of the presence of Janus, not noticed at first but in reflection became obvious. The idea of popular education, coined by Brandão (1999), a colleague of Paulo Freire and of Ubiratan, was present, making the connection between “popular knowledge,” education, and academic knowledge, as one may see in the summary of the book, in Portuguese, below (Figs. 2 and 3). Presenting the look at the past that we want to overcome, the valuable knowledge that is produced even in such adverse conditions was inspiring for many educators in the Janus moment. At the same time, showing videos produced by indigenous parents teaching their children how to count in their own language was the best connection of past and future: Janus! Digital technology, produced in their own voice,

Fig. 2  A book by Freire with an autograph to the first author of this chapter

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Fig. 3  A summary of the book by Freire involving chapters on ethnomathematics

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and from their own perspective, gives extraordinary agency in preserving traditional culture and language for a new generation of indigenous Brazilians. In that congress, some of the first indigenous Brazilian master students in mathematics education presented their own research, being maybe the true embodiment of Janus! We are now in the process of having a special category for indigenous people and their work in the Digital Video and Mathematics Education Festival.

2 Final Remarks Janus is a perfect metaphor for how both collaborators in the chapter and this book made vital changes in their lives and awakened to ethnomathematics and began to use it in their own teaching and research. As well, this is a common theme of the authors found in this book as well. The passing of our dear professor and mentor was a shock. Here, in this book, and after some time, time we all needed to reflect and begin to move forward after his passing is one more Janus moment for those interested in ethnomathematics, culture in mathematics and technology, and who are interested in the many faces of social class, ethnicity, history, gender, and sexual orientation and how they influence our work as presented here in the following chapters. The Program Ethnomathematics as outlined by Ubiratan and encouraged by him to diversify along the many unique lines and cultural-national contexts offers us a glimpse here as well (e.g., Borba e Souza, 2021). It goes without saying that Ubi had an enormous influence on thousands of students, educators, researchers, and colleagues throughout the planet, not only on the two authors of this chapter but on all the authors of this book, as well as in many other mathematics educators, including the ones that could not, for several reasons, write for this book. As co-editors of the book, it has been a tremendous honor to work with the authors who present their perspectives and work here in this book. Together, in solidarity, we share our mutual love and respect for Ubiratan’s ideas and work. To see it presented here in the following chapters has been for us one more Janus moment (Fig. 4). In solidarity, and what brings all the authors in this book together, Ubi was our mentor. He shared how he would take all the time and opportunity to patiently work with both author/editors at any time. As we worked on this book, we found it was a common thread for the authors found in this book. Ubiratan saw ethnomathematics as a look to the past and a look to the future! Like any mentor/teacher, he looked at knowledge (past) to be taught, and thinking of this knowledge, to develop a sense or curiosity (future). As educators we are called to think of connecting our mutual pasts and futures in creating new Janus moments for our students! (Fig. 5). Ubiratan Forever!

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Fig. 4  Ubi and Marcelo in the ICMI, 100-year celebration in Rome, Italy, 2008

Fig. 5  Daniel Orey and Milton with Ubi in São Paulo at a surprise birthday celebration

References Ascher, M., & Ascher, R. (1981). Mathematics of the Incas: Code of the Quipu. Dover Publications. Bassanezi, R. C. (2002). Ensino-aprendizagem com modelagem matemática. Editora Contexto. Borba, M. C. (1987). Etnomatemática: uma Proposta pedagógica para a favela da Vila Nogueira São Quirino. Anais do I ENEM Encontro Nacional de Educação Matemática, PUC  – São Paulo, Brasil (resumo), fev. Borba, M.  C. (1990). Ethnomathematics and education. For the learning of mathematics. International Journal of Mathematics Education, 10(1), 39–43.

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Borba, M.  C. (1993). Students understanding of transformations of functions using multi-­ representational software. Tese – Cornell University. Disponível em: https://igce.rc.unesp.br/#!/ pesquisa/gpimem%2D%2D-­pesq-­em-­informatica-­outras-­midias-­e-­educacao-­matematica/ material-­gpimem/teste/ Borba, M.  C. (2021a). The future of mathematics education since COVID-19: Humans-with-­ media or humans-with-non-living-things. Educational Studies in Mathematics. Accepted: 16 February 2021 – Published online: 27 April 2021. https://doi.org/10.1007/s10649-­021-­10043-­2 Borba, M. C. (2021b). El futuro de la educación matemática a partir del COVID 19: humanos-­ com-­medios ou humanos-com-coisas-viventes. Revista de Educación Matemática (RevEM), 36(3), 5–27. Editora Universidade Nacional de Córdoga. Borba, M. C., & Villarreal, M. E. (2005). Humans-with-media and the reorganization of mathematical thinking: Information and communication technologies, modeling, experimentation and visualization (Vol. 39). Springer. Borba e Souza. (2021). En memoria del pensador y educador matemático brasileño Ubiratan D’Ambrosio: creador del Programa Etnomatemática. Revista de Educación Matemática (RevEM), 36(2), 89–94. Editora Universidade Nacional de Córdoga. Brandão, C. R. (1999). Pesquisa Participante. Editora Brasilense. Carraher, T., Carraher, D., Schliemann, A. (1995). Na Vida Dez, na Escola Zero [Mathematics in the streets and in schools] 10. ed. São Paulo: Cortez. D’Ambrosio, U. (2018). Etnomatemática, justiça social e sustentabilidade. Estudos Avançados, [S. l.], 32(94), 189–204. https://doi.org/10.1590/s0103-­40142018.3294.0014. Disponível em: https://www.revistas.usp.br/eav/article/view/152689 Engelbrecht, J., Borba, M. C., & Kaiser, G. (2023). Will we ever teach mathematics again in the way we used to before the pandemic? ZDM – Mathematics Education, 55(1), 1–16. Freire, P., Nogueira, A., & Mazza, D. (Org.). (1998). Na escola que fazemos: Uma reflexão interdisciplinar em educação popular (1st ed.). Editora Vozes. Lévy, P. (1993). As tecnologias da inteligência: o futuro do pensamento na era da informática. Editora 34 Rosa, M., & Orey, D. C. (2015). A trivium curriculum for mathematics based on literacy, matheracy, and technoracy: An ethnomathematics perspective. ZDM Mathematics Education, 47, 587–598. https://doi.org/10.1007/s11858-­015-­0688-­1 Rosa, M., & Orey, D. C. (2016). Humanizing mathematics through ethnomodelling. Journal of Humanistic Mathematics, 6(2), 3–22. https://doi.org/10.5642/jhummath.201602.03 Villarreal, M., & Borba, M. C. (2010). Collectives of humans-with-media in mathematics education: Notebooks, blackboards, calculators, computers and … notebooks throughout 100 years of ICMI. ZDM, 42, 49–62. https://doi.org/10.1007/s11858-­009-­0207-­3 Marcelo C.  Borba is a professor of the graduate program in Mathematics education of the Mathematics Department of Universidade Estadual Paulista (UNESP), campus of Rio Claro, Brazil. His work is archived at: https://igce.rc.unesp.br/#!/gpimem.  

Daniel C. Orey is a professor in the Departamento de Educação Matemática in the Instituto de Ciências Exatas e Biológicas at the Universidade Federal de Ouro Preto. He is Professor Emeritus from California State University, Sacramento and is a Senior Fulbright Specialist – Nepal/Brasil. His work is archived at: https://www.oreydc.com.  

Part I

Roots of Ethnomathematics

The Presence of Professor Ubiratan D’Ambrosio in the Development of Graduate Mathematics Education in Brazil Maria Aparecida Viggiani Bicudo

Abstract  This chapter focuses on Professor Ubiratan D’Ambrosio’s presence in the Graduate Program in Mathematics Education at Universidade Estadual Paulista-­ Campus Rio Claro, São Paulo, Brazil (UNESP-Rio Claro). This was the first graduate program in Mathematics Education in Brazil, and it was designed with professors of the Mathematics Department in the Institute of Geosciences and Exact Sciences. Professor D’Ambrosio, who at the time was a professor at the State Universidade Estadual de Campinas (UNICAMP), was invited and accepted to be part of the faculty of this program, contributing in an outstanding manner to the proposal of innovative disciplines in the curricula of the science areas and respective teaching and research procedures. It is claimed that his presence has been remarkable throughout the history of this program, from its inception to the present day. His way of being and his conceptions about education and science have radiated light and encouraged debates and research among the participants of this program. Keywords  Stricto sensu graduate course in mathematics education · Mathematics teaching · Mathematics education · Cultural-historical approach · Mathematics education in devenir

1 Introducing the Theme of the Chapter The objective of this chapter is to present the stricto sensu graduate studies in mathematics education in Brazil, which started with the creation of the Graduate Program in Mathematics Education at Universidade Estadual Paulista – Campus Rio Claro,

M. A. V. Bicudo (*) Program in Mathematics Education, UNESP – RC, São Paulo, Brazil; http://lattes.cnpq. br/143272807891052 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_2

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São Paulo, Brazil (UNESP-Rio Claro). I was one of the organizers of this project1 and managed it from the first negotiations within the mathematics department of Instituto de Geociências e Ciências Exatas (Institute of Geosciences and Exact Sciences) UNESP- Rio Claro, until its approval and inception through the publication of Resolution UNESP No. 29/83. That resolution created the Graduate Program in Mathematics; Field of Study: Teaching Mathematics and Fundamentals of Mathematics. The ideas for the program began in 1982, with meetings between professors in the mathematics department, and included educators from several other departments at UNESP-Rio Claro and two other universities in the State of São Paulo: Universidade Estadual de Campinas (UNICAMP) and Pontifícia Universidade Católica de São Paulo (PUCSP). At that time, it was common in Brazil to view mathematics teaching as a small branch of mathematics. This was the first stricto sensu graduate program in the area of education at UNESP-Rio Claro and the first in mathematics education at any university in Brazil. And according to Professor Ubiratan D’Ambrosio, as he stated in several lectures, it was the first in Latin America. Unlike other education programs on mathematics and didactic of mathematics in operation in other countries, with which we collaborated, such as Bielefeld, Germany, and France, our program was proposed and, to this day, operates within the mathematics department of Instituto de Geociências e Ciências Exatas at UNESP-Rio Claro. This is an important characteristic that distinguishes and strengthens it, but also which has brought and still brings obstacles. I will comment on the specificities that are revealed in the beginning of the program, highlighting the presence of Professor D’Ambrosio.

2 The Contribution of Professor Ubiratan D’Ambrosio to the Creation of the Graduate Program in Mathematics Education at UNESP-Rio Claro Having been created in the mathematics department, this constituted a strongly favorable factor for the program. This is because, since the very creation of the department in 1959, it encompassed a core of ideas and attitudes that show its lecturers’ concern with the education of mathematicians and teachers who teach mathematics. The program has produced excellent mathematicians who went on to work at many renowned Brazilian universities, as early as the 1960s and 1970s, such as in the Engineering Department of USP-São Carlos and the Mathematics Institute

 In this text, this word refers to the idea presented in Being and Time, by Heidegger, (1962) of launching ideas conceived to the forefront to happen. The ideas can be actualized in the becoming (devir) of individuals. It is not being taken as synonymous with a logical program that, a priori, models, within the scope of its proposals, what could be. The project, in its movement of actualization, welcomes programs, which bring ideas and values that take on characteristics of the historical-­ social temporality in which it is contextualized. 1

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(IME), at UNICAMP, as well as many educators and researchers dedicated to teaching mathematics. It is important to emphasize the participation of Professor Ubiratan D’Ambrosio in the developmental phase of our program, who, with Mario Tourasse Teixeira and Lourdes de La Rosa Onuchic, constituted the faculty of the department. They were familiar with issues in mathematics and mathematics education and the development of professionals who would later become mathematics teachers. In the years that followed, Professor D’Ambrosio did not remain in the department. However, the spirit that animated the activities conducted by that group of educators endured. His ideas continued through subsequent initiatives implemented through the 1960s, 1970s, and the beginning of the 1980s, which aimed at educating mathematics teachers, through the implementation of undergraduate courses in mathematics, extension courses, development and specialization courses in mathematics education, and scientific- academic events with teachers who were involved with practice and discussions regarding teaching and learning mathematics. Such actions show the opportunities which had been cultivated, since 1959, and would welcome the future project in graduate studies program that started at UNESP-Rio Claro in the 1980s. Stricto sensu graduate programs in Brazil were established as an organized system in the mid-1960s. Before that, only a few institutions and universities offered doctorates along the French model. The stricto sensu graduate program at UNESP-­ Rio Claro, created in 1965 (Bicudo, 2014), followed the North American format. In terms of Brazilian Federal legislation, that year was key for graduate studies in Brazil, as the Minister of Education under President Castelo Branco, Flávio Suplicy de Lacerda, summoned the Higher Education Council to define requirements for postgraduate courses in Brazilian universities (FAPESP, 2002). The proposals for this level of education had three objectives: • To develop competent teachers to meet the demand for an expansion of higher education • To stimulate scientific research through the instruction of researchers • To ensure state-of-the-art education for technicians and intellectual workers. The graduate program at UNESP-Rio Claro was organized at two levels: master’s and doctorate Even though it had been instituted in 1965, the program was consolidated and developed due to the combination of two factors: the 1968 Teaching Reform (BRASIL, Diário Oficial da União, p.  10369) and the creation of a network of research-funding agencies, of which graduate studies became the largest beneficiary. As well, resources were allocated for a continuous institutional evaluation system of programs. The first nationwide legislation regarding the graduate system in Brazil was marked by a centralization of the requirements that governed different institutions. UNESP-Rio Claro, which, from its inception, was a multicampus institution, incorporated the new national paradigm of graduate studies. In the 1970s, to avoid program duplicity in the 15 existing campuses, it was determined that, in each area of

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knowledge, such as mathematics, there would be only one program, with different fields of study. Even though mathematics courses and respective departments were available in two campuses, in the cities of Rio Claro and São José do Rio Preto, both in the state of São Paulo, there could only be one graduate program in mathematics, with specific fields of study. Therefore, in view of the pertinent legislation, the feasible way for creating the graduate program at UNESP, in the early 1980s, was to create a mathematics program, in Rio Claro, with two fields of study: Fundaments of Mathematics and Mathematics Education. In 1984, both fields of study started their programs at the master’s level. It must be noted that, at that moment in Brazilian history, a master’s program was highly valued. A master’s was a 4-year program and the requirements for quality research were strict. In the country, there were very few master’s programs, in all areas, and the authorization for its establishment required a strong proposal with a body of renowned researchers. In addition, UNESP did not have stricto sensu graduate programs in education or mathematics, which might contribute to the parameters and the development of research. Therefore, engendering, conceiving, and proposing this project was a bold and innovative move within the scope of UNESP and Brazil as a whole. Ubiratan’s involvement at this stage was paramount. The innovation of this program resulted both from its proposal and structure, as well as the philosophy that supported the articulation of the disciplines and research. The debate regarding mathematics education, characterized by interdisciplinary aspects, required teaching not only pertaining to the hard sciences, specifically to mathematics, but to be treated in an interdisciplinary manner, intertwining humanities, education, and mathematics. This was at the time, a political-ideological struggle, spearheaded by Professor D’Ambrosio and involved the community of mathematics teachers throughout the country. The stricto sensu graduate program in mathematics education, at UNESP, Rio Claro, was established within the historical and political context and, as previously mentioned, conceived within the mathematics department. It included highly qualified and respected scholars among Brazilian mathematicians, such as Mario Tourasse Teixeira and Irineu Bicudo who worked with foundations of mathematics, as well as mathematical logic; Luiz Roberto Dante who worked with teaching and learning mathematics; Maria Aparecida Viggiani Bicudo in the field of education; and Maria Lúcia Wodwotski, in statistics. Such educators led the movement that articulated meetings with other departments, emphasizing ideas about mathematics, philosophy of mathematics, cognition, and active teaching practices, as well as the overall philosophy of education. The discussions soon revealed a detachment from current positivist conceptions of science and a willingness toward thinking more broadly. The discussions involved observing significant points, among such were the understanding of the production of mathematical knowledge, considered within historical and scientific contexts; the importance attributed to the study of the fundaments of mathematics, with emphasis on the roots of mathematical science in the Western world; learning issues, criticism regarding behavioral and Piagetian concepts, turning into aspects of humanistic psychology, with studies based on phenomenology; and a constant quest for ways of teaching which viewed teachers and

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students as being together, in a dialogical stance, while conducting teaching and learning activities, viewing education, as an action not derived from scientific theories such as psychology or sociology, for instance, but dedicated to contemplating the professional development within social, historical, political, and cultural contexts. When the guidelines of the proposal for the graduate project were clearly defined within the mathematics department, we sought to collaborate with academicians from other universities, aiming to constitute a robust faculty to contemplate mathematics education. Teachers whose conceptions and values met the ideas that were in motion were invited from UNICAMP, located 90 kilometers from Rio Claro, Professors Ubiratan D’Ambrosio, Eduardo Sebastiani Ferreira, and Rodney Bassanezi, were invited. Together they conducted groundbreaking work regarding the history of mathematics, focused on education, ethnomathematics, and mathematical modeling. As well, they focused on teaching. Professor Joel Martins was invited from PUCSP, located 170 kilometers from Rio Claro. At the time he was working with phenomenology and seeking ways to conduct qualitative research, within the perspective of that philosophic school. This group was joined by two scholars in the area of education who worked at the Rio Claro campus, professors Maria Cecília de Oliveira Micotti and Lucila Maciel dos Santos, also renowned in the community of researchers in education. This assemblage represented a positive beginning for the soon to be established field of mathematics education in Brazil. Its influence has been felt in the dissemination of graduate programs in mathematics education in Brazil, ever since. As well as the hiring of research associates, holders of master’s and doctorate degrees granted by the Rio Claro program are teaching and researching in many institutions throughout the country. That early group of scholars contributed experience that has influenced higher education, teaching, and research in Brazil. Their visions of science and education opened a promising horizon for change in concepts and generated an impetus that supported the creation of specific lines of research. These include ethnomathematics, mathematical modeling, teacher education in mathematics, history of mathematics education, philosophy of mathematics education, and mathematical education and highlight the very specificity of mathematics. At the same time, it instilled a spirit into the program which endured, from its beginning, and was always open to what is innovative. With this vision, as time went by, other lines of inquiry were created, such as technology in mathematics education. Philosophical and scientific rigor was a requirement of the program; however, it was not viewed as based on a rigid methodology that imposed investigation methods that may not necessarily lead to true knowledge. The spirit and values of the program remained vigorous and were in line with the ideas initially set out; the influence of this program has extended into the present moment. It goes without saying that the participation of Professor Ubiratan D’Ambrosio was vital for the inception, that is, the genesis, origin, and manifestation, of the idea and project of the program that, in its historicity, has effected the phýsis of the program, which was determined as follows:

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M. A. V. Bicudo everything that is born is destined to be what it should be, not anything else. This destined birth is phýsis, by which what is born is subjected to a process of realization, and, as such, archê. (...) neither phýsis or archê are expressions of anarchy (...), nor occasionality... Together these terms designate what always occurs frequently or ordinarily (...), but with such effectiveness that it always fires (as if it were a biological trigger) whichever is best among all possibilities. (SPINELLI, Miguel, 2006, pp. 36–37)

As phýsis, the program kept on this movement of becoming or evolving and changing, while persistently producing scientific and philosophical knowledge regarding mathematics education for many decades in Brazil. The initial proposal was advanced, and strongly directed by the guidelines, and has endured into the twenty-first century. As stated earlier, the program was established in mathematics with two fields of study: mathematics teaching and fundamentals of mathematics. Regarding teaching, which later became the graduate program in mathematics education, it became clear, in the first year of operation, that it could not be confined by boundaries of what would be construed as a branch of mathematics. It was necessary to launch it as a separate project to be updated through decisions made by participant faculty members. While the program revealed a disparity between with its denomination, that is, mathematics teaching, the ideas discussed and the actions triggered Brazilian federal legislation. The graduate system was then modified at the national level, thus enabling the creation of more than one graduate program in the same field of knowledge at UNESP. After that, discussions about restructuring the program were conducted, and it became necessary to choose a name that would encompass the conceptions assumed by the faculty and students that animated it. The name agreed upon was Graduate Program in Mathematics Education and its philosophic and scientific foundations, which was in line with mathematics education. What did that mean? It was clear that it was not mathematics teaching, as the goal was not study of mathematical science per se, under the point of view of its progress. The goal was education, conceived as formation – Bildung – of mathematicians who understood the science, its modes of production, history, and ways of being of students, who learn it and live in specific historical and social contexts. A decade was spent amid debates regarding this issue and with numerous clashes with mathematicians who were dedicated solely to (pure) mathematics, seen as a science constituted and produced in the Western world. This issue will be explored below, as Professor D’Ambrosio’s influence was fundamental. The designation philosophical foundations was intended to encompass education, psychology, didactics, sociology, and other fields that deal with aspects of humanistic disciplines, as well as mathematics. Likewise, scientific foundations, meant to be understood as aspects of the procedures and tools used by different sciences, intertwined with the treatment of essential ideas underlying the thought process of such sciences. Such ideas supported and directed the project. They needed to be broken down into activities: from different disciplines and investigations, to be presented in a master’s thesis and, beginning in the 1990s, in doctoral dissertations. That was a substantive challenge. How would research be presented rigorously and scientifically, as most of it did not follow the positivist model? A new model

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was generated. The disciplines needed to be distributed in such a way so as not to create a single view of science and education. The products, dissertations and theses, should be scientific, however, not confined by a single model, the positivist model. As the area emerged, and the country was changing, the demands we and our students had to tackle were significant. We knew that it was important to have a vehicle for communication and dissemination of scientific and academic work. Thus, in 1985, the Boletim de Educação Matemática  – BOLEMA (Mathematics Education Bulletin) was created. Also, there were few to any individuals qualified in mathematics education, so, we invested in the development of educators in the field. In those two instances, the leadership and contribution of Professor D’Ambrosio were vital. To achieve the goal of developing professionals in the field of mathematics education, he shared his vast network of contacts with scholars and researchers working in renowned international teaching and research centers, so that the first candidates would go to the United States, Germany, and England, for instance, to study with the aim of obtaining doctorates and returning to Brazil with new ideas. Two students from the two initial classes of the program, which held master’s degrees, Marcelo de Carvalho Borba and Sérgio Roberto Nobre obtained doctorates. The former went to the United States, and the latter to Germany. Both were subsequently hired by the mathematics department where they remain to this day. Nowadays, they lecture and research, have significant achievements in their teaching careers at UNESP, and are recognized both nationally and internationally. In addition, at ICME, which took place in Hungary, in 1988, Professor D’Ambrosio included BOLEMA in the list of mathematics education journals of ZDM (International Journal of Mathematics Education), in Germany. Professor D’Ambrosio was responsible for the international recognition of the newly created program, promoting contacts that ensured visits and exchanges from educators and researchers from all over the world, such as the United States, Germany, Cuba, Sweden, France, Portugal, Mexico, as well as other Latin American countries, to Rio Claro, thus nurturing a community of teachers and students with ideas and debates from other academic centers. UNESP-Rio Claro became and to this day remains a hub for new ideas and practical proposals regarding mathematics education. The program remains in motion to this day, and Professor D’Ambrosio’s presence and strength were vital due to the clarity it brought to all involved. In the 1980s, he was involved in the study of history of mathematics and ethnomathematics. His presence among students and teachers from the very first year of the program was memorable. Every week, he travelled to Rio Claro to teach, advise students, lecture, and participate in scientific meetings, departmental meetings, and festive gatherings. His presence created a strong bond and was strengthened, at the emotional and friendship level, which influenced academic-scientific and political articulations. His voice showed the graduate program abroad, as it became more and more renowned internationally, as he developed his theory of ethnomathematics and

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the ethnosciences, in conjunction with colleagues, researchers, and students at UNESP. It was necessary to ensure the rigor of the investigations presented as dissertations with the objective of obtaining a master’s degree through the program. However, given its groundbreaking characteristics in the context of Brazilian education, there were no parameters for comparison. Nonetheless, it was clear that both the community of researchers in the field of mathematics and education took notice of what was being done, at UNESP-Rio Claro. Initially, we faced prejudgment and disparaging remarks, based on the biased notion, devoid of analytical and reflective foundations, that those who taught mathematics did so because they did not comprehend education or mathematics. Professor D’Ambrosio raised his voice against this pre-conception throughout Brazil. Within the program, we were aware that it was necessary to face such bias and show the rigor of our work, even when conducting qualitative research. So, how does one conduct research in areas such as ethnomathematics, history of mathematics, or philosophy of mathematics education? The question arises in investigations conducted within the scope of philosophy of education, focused on mathematics education, which came to be known as the philosophy of mathematics education. The resulting discussions regarding qualitative research, and possibilities of assuming it in the hermeneutic-phenomenological line, showed ways of rigorously investigating, without becoming hostage to the “scientific method,” of a positivist nature. As a result of those discussions conducted in the 1980s, the first book with the expression Mathematics Education (Educação Matemática) was published in Brazil, in 1987, (Bicudo, s/d), as well as a qualitative research book entitled Qualitative Research in Psychology: Fundamentals and Basic Resources (Martins & Bicudo, 1989). The latter contains a critique of the positivist paradigm and deals with qualitative research in psychology and ways of conducting it. Given the amplitude of the debate, it opened the possibility for working with the procedures and visions indicated in education and mathematics education. The first book contained five chapters, namely, For a Mathematics Education as Intersubjectivity, by Cleide Faria de Medeiros; The Mathematics Teacher in Elementary and High School, by Maria Aparecida Viggiani Bicudo; Only Multiplication Tables, by Maria Cecilia de Oliveira Micotti; Critical Mathematics Education: An Application of Paulo Freire’s Epistemology, by Marilyn Frankenstein; and Pedagogical Action and Ethnomathematics as Conceptual Milestones for Teaching Mathematics, by Professor Ubiratan D’Ambrosio. The chapter by Professor D’Ambrosio set a solid foundation that sustained the development of ethnomathematics in Brazil and the program that he built in the following decades, which he called An Education for Peace. It is important to say that the program also welcomed positivist research. Statistics was part of the assumed curriculum from its inception, as well as investigations conducted under that conception. Professor D’Ambrosio’s strong presence is still felt today in the spirit that animates the program UNESP-Rio Claro.

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3 Characteristics That Pose Obstacles to the Becoming (Devir) of the Program as a Result of Being in the Mathematics Department: Professor Ubiratan D’Ambrosio’s Contribution Being in a mathematics department, which has a tradition of forming good mathematicians, also generates conflicts and often creates animated ideas, debates, and production by teachers who work in mathematics education, and following mathematical theories of traditional mathematical disciplines, such as algebra, analysis, calculus, and geometry. The former already assumed a sense of interdisciplinarity as a characteristic of mathematics education and the latter worked disciplinarily. The views of science in both categories of teachers were diverse and focused on dealing with teaching content and evaluation of student learning. The views of mathematics teachers are in line with those assumed, proclaimed, and defended by Sociedade Brasileira de Matemática – SBM (Brazilian Mathematical Society), for whom it influenced how to teach in Brazil, and for those working with mathematics education, when teaching mathematics, the focus was on student learning. The discrepancy among those views transcended the walls of the mathematics department of UNESP-Rio Claro and was present in the heated clash between SBM and the nascent Sociedade Brasileira de Educação Matemática – SBEM (Brazilian Society of Mathematics Education), which was created in 1987. The voice of Professor D’Ambrosio was paramount in that debate and ongoing dialogue. His leadership has mobilized many mathematics teachers all over of Brazil. And, all members of the Rio Claro graduate program, along with Professor D’Ambrosio, fought, by publishing articles and exposing ways to understand and differentiate mathematics teaching from mathematics education. This debate culminated in publishing the first BOLEMA (Journal of Mathematics Education) in 1985, and the first book on mathematics education in Brazil, participating in meetings and organizing events, such as I Encontro Nacional de Educação Matemática – ENEM (The First National Meeting of Mathematics Education), held in São Paulo, in 1986, chaired by Professor Tania Campos, from PUCSP, and the II ENEM, in 1987, which took place in Maringá, Paraná, when SBEM was created. Today, two graduate programs coexist in the department at UNESP-Rio Claro. One is the Stricto Sensu Mathematics Education Program, described within the scope of current Brazilian legislation as academic, and a Mathematics Teaching Program, with nationwide penetration, proposed and coordinated by SBM, described as professional. As for the research conducted, the difference in the conception of science often generates discordant judgments between the two groups, but we have learned to live with our differences. As for this author, I believe that, although coexistence is peaceful today, it has much to be desired. There is no deeper articulation that, not unlike an archeological excavation, pursues the roots of mathematical science production and its different disciplines, as understood under the dimension of Western thought, offering these ideas to the understanding of students and teachers so that, in an

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analytical and reflective way, they can critique this science in terms of its proclaimed and assumed hegemony in the world of science and technology. Such hegemony transcends its region of inquiry, with values of certainty and accuracy, in accordance with its manner of constituting and producing knowledge and extends to other social practices, such as politics and economics, and human ways of comprehending the world, such as religion and sociocultural interactions. In a naturalized way, the values of certainty and accuracy, which speak of mathematical truths and are appropriated to mathematical science in the context of their theories and applications, are transposed to other realities and ideologically imposed as being superior to other types of knowledge. I see educators of the self-proclaimed field of pure mathematics acting a little less rigidly while conducting their work. I also see educators in the field of mathematics education venturing into investigations conducted within history of mathematics, oral and documentary; ethnomathematics; and critical mathematics education, for instance, without bringing mathematical thought into the core of their research.

4 Exposing Understandings Ubiratan D’Ambrosio’s spirit endures in the acceptance of differences, in the acknowledgement of mathematical practices in other cultures, effected by other peoples, and the non-predominance of any single culture, people, individuals, or knowledge produced by them. I believe in, in addition to this teaching and the values embedded in it, which must be impressed upon our way of being close to the other, respecting them as equal and respecting differences. Mathematics education is responsible for fostering the understanding of essential ideas in the mathematical sciences and includes its diverse modes of production. We should contemplate, critically and reflectively, the reason that knowledge is taken as the basis of scientific research in all areas, worldwide, both in the West and the East, as well as the support of computer science and its apparatus supporting cyber-reality. Mathematics education should emphasize and open possibilities for students and future teachers, researchers, and lecturers who educate teachers and researchers, in the field of mathematics education to understand and work with the specificities of science historically produced in Western civilization, and do mathematics, expressed in different languages and practiced in different cultural instances of communities within the same civilization and other civilizations. I believe it is necessary to be alert and practice Hegelian dialectic, of becoming (devenir), in which equality and difference are presented in an articulated manner, transmuted into languages and practices. Choosing a single way of criticizing mathematical science, emphasizing other ways of effecting it, would be the core of a dictatorship, based on hegemony. Certainly, this is not the spirit that animated and still animates the discourse of Professor Ubiratan D’Ambrosio, present in his texts and media messages. The objective of the program presented herein was to always welcome different conceptions and stances, channeling strength to sustain scientific-academic debate.

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References Bicudo, M. A.V. (2014). A Pós-Graduação em Educação Matemática de Rio Claro: historiando sua trajetória. In R. NARDI (org.), Pós-Graduação em Ensino de Ciências e Matemática no Brasil: memórias, programas e consolidação de pesquisa na área. Livraria da Física. Bicudo, M. A.V. (organizadora). Educação Matemática. Editora Moraes Ltd., s/d. Brasil. Legislação Informatizada – LEI N° 5.540, DE 28 DE NOVEMBRO DE 1968 – Publicação Original. Diário Oficial da União - Seção 1 – 29/11/1968, Página 10369 (Publicação Original). Educação Superior: Graduação e Pós-Graduação. Coordenadora: Maria Aparecida Viggiani Bicudo (Unesp). Pesquisadores: Leonor Maria Tanuri (Unesp) e Helena Sampaio (USP). FAPESP (Francisco Romeu Landi – coordenação geral). (2002). Indicadores de ciência, tecnologia e inovação em São Paulo 2001. Fapesp. Heidegger, M. (1962). Being and time. Harper & Row Publishers. Martins, J., & Bicudo, M. A. V. (1989). A Pesquisa Qualitativa em Psicologia. Fundamentos e Recursos Básicos. EDUC-Editora da PUCSP e Editora Moraes Ltda. 1ª edição. Spinelli, M. (2006). Questões Fundamentais da Filosofia Grega. Loyola.

Ubiratan D’Ambrosio and the Development of Researchers in (Mathematics) Education Maria da Conceição Ferreira Reis Fonseca and Cristiane Coppe-Oliveira

Abstract  In this chapter, we revisit Professor Ubiratan D’Ambrosio’s practices in the arts of teaching, advising, and researching, sharing our approaches to these practices as his former graduate students. We do so through a dialogue between the content of his letters addressed to us and other students, texts published by him and about him, and reflections on our formation as researchers. Inspired by the solidarity, respect, and cooperation that simultaneously produce and are produced in the Ethics of Diversity that guides Professor D’Ambrosio’s educational practices, we reflect on his legacy for research and researchers, for knowledge and for those who know and recognize themselves by knowing, for the field of Mathematics Education, as well as for those who form it and form themselves within it. Keywords  Ubiratan D’Ambrosio · Arts of teaching and advising · Constitution of researchers · Mathematics education

1 Introduction It has always been a great challenge to talk about Professor Ubiratan D’Ambrosio while observing the various movements he has created throughout his trajectory as an educator. We highlight here his action and posture as a postgraduate professor and, as such, a researcher, trainer, and advisor of researchers and teachers of higher education in mathematics education. As students, and later colleagues, we had the privilege of sharing the practices of this educator in postgraduate classrooms, research projects, and research groups, experiencing what we called here the D’Ambrosian art of researching, teaching, and advising. M. d. C. F. R. Fonseca (*) Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil C. Coppe-Oliveira Universidade Federal de Uberlândia, Ituiutaba, Minas Gerais, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_3

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When we were faced with the challenge of writing about the legacy of Professor Ubiratan D’Ambrosio as a postgraduate professor, we feared encapsulating his legacy into adjectives that did not do justice to the fertility, generosity, and dynamic character of his contributions. Because of this, in our reflection here, although we resort to some of what was said or written about him, we tried to focus on the professor himself in action, based on two of his writings addressed to his advisees and other graduate students, which thematize research, teaching, and guidance practices. We also outlined a text, written by him together with Marcelo Borba discussing the motivations and trends of research in mathematics education in Brazil, which also tells us about their understanding of the field and, thus, how he and his students composed it. The exercise of memory and attentive and affective (re)reading of these texts, after 8, 12, and 35  years since their writing and our first reading  – years during which we were establishing ourselves as researchers, teachers, and advisors in (mathematics) education – also inserts us into the historical movement of graduate studies in Brazil. During this period, several programs or research lines in mathematics education were created in the country, following the first, which was created in 1984, within the Department of Mathematics in the Institute of Geosciences and Exact Sciences of the UNESP/State University Julio de Mesquita Filho (UNESP-­ Rio Claro), in Rio Claro, São Paulo. Ubiratan was an inspiration and collaborator for its creation until the last month of his life, ending his trajectory with his participation in a doctoral thesis committee1 and the advisory of two other doctoral students who defended their thesis after the supervisor’s death.2 The three texts we put into dialogue were motivated by different intents: in D’Ambrosio (2014), the advisor Ubiratan systematizes some clarifications, advice, and tips that he considered important to his advisees at the beginning of the mentoring relationship; D’Ambrosio (1987) is a letter he wrote as the teacher of the subject “Trends in Mathematics Education” on the first Graduate Program in Mathematics Education in Brazil, to give a student feedback on the reports and research project  Souza, Valdirene Rosa. (2021) Presença africana na arquitetura e na educação brasileira: uma perspectiva decolonial sob a égide da Etnomatemática (African presence in Brazilian architecture and education: a decolonial perspective under the aegis of Ethnomathematics) [Doctoral thesis, Universidade Estadual Paulista]. Rio Claro-SP. https://repositorio.unesp.br/ handle/11449/204702 2  Ribeiro, Renato Douglas Gomes Lorenzetto (2021) Aspectos Socioculturais e Políticos na Especialização do Conhecimento do Professor de Matemática: Interfaces entre o Programa Etnomatemática e o modelo do Conhecimento Especializado do Professor de Matemática (MTSK) (Sociocultural and political aspects in the specialization of Mathematics teacher knowledge: interfaces between the Ethnomathematics Program and the Mathematics Teacher Specialized Knowledge model (MTSK)) [Doctoral thesis, Universidade Estadual Paulista]. Rio Claro-SP. https:// repositorio.unesp.br/handle/11449/216014 Leão, Marcílio. (2021). Educação Matemática, sociedade e meio ambiente: reflexões sobre violência social e ambiental. Um estudo transdisciplinar e crítico em uma pesquisa Etnomatemática. (Mathematics Education, Society and Environment: reflections on social and environmental violence. A transdisciplinary and critical study in an Ethnomathematics) [Doctoral thesis, Universidade Estadual Paulista]. Rio Claro-SP. https://repositorio.unesp.br/handle/11449/216157 1

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she had presented for her final evaluation; the third text, D’Ambrosio and Borba (2010) is an article published in a prestigious international journal, in a special issue on research in mathematics education in Brazil, in which the researcher Ubiratan narrates his version of the history of the field, intertwined with the country’s history, heir to the economic and social relations resulting from its colonization, which, in turn, define the thematic, theoretical, and methodological interrelationships established in this field in the national context. My dear advisees, This is a welcome letter to advisees, with many important instructions and suggestions. It’s long but important to read. [...] I will make some preliminary considerations, about bureaucratic/administrative care, my style of guiding, and my research concept. I end with some important tips about bibliography. (D’Ambrosio, 2014, p. 1, author highlights) Dear Conceição: I hope that your trips to Rio Claro have been rewarding and that you have enjoyed the course well. It was exciting to have you in my class. About your reports: … (D’Ambrosio, 1987, p. 1) In this issue of ZDM, we present some facets of mathematics education in Brazil and illustrate how they are intertwined. We will focus on six major trends and their specific connections: modeling, use of technology, ethnomathematics, philosophical aspects, historical perspectives, and political dimensions of mathematics education. Different authors were invited to address each of these trends and asked to illustrate the influence of at least one of the other trends listed above. […] The unusual option of not providing references in this paper is due to the fact that much of this narrative comes from personal memories of the authors, but mainly because the articles of this special issue provide the most relevant references. (D’Ambrosio and Borba, 2010, p. 271 and p. 279)

Such as the intention of these texts, the breadth, longevity, and interaction conditions of the author with their recipients are also very different. For several years, the first text (D’Ambrosio, 2014) was given at the beginning of teaching activities to the group of his new advisees, who, in the following years, would have a relationship of proximity and partnership with him. The second (D’Ambrosio, 1987) is addressed to a single graduate student who had taken his course taught in a given year, thus ending a weekly coexistence that he asks to continue through correspondence exchange. The third (D’Ambrosio & Borba, 2010) would reach a broad set of readers of a mathematics education international journal, who have already accessed or will access his article – among these readers, many knew D’Ambrosio personally, others have read the article but never met the author, and many others will be able to access the article in the future and will no longer have the chance of this meeting. However, all readers – of the collective letter, of the individual letter, of the article published in the journal – will be able to see him in countless videos of his multiple participations in different events, in which his interventions were always long-­ awaited moments, very engaging, and much commented.

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M. d. C. F. R. Fonseca and C. Coppe-Oliveira My advising style is summarized in actions that complement each other. We must always be in touch. Face-to-face is often difficult. The best way is the e- mail [email protected]. I usually reply as soon as I get a message. But sometimes I don’t respond right away (for various reasons) and then I forget (reason: age?). So, after a reasonable time without an answer, complain. (D’Ambrosio, 2014, p. 1, author highlights) When you feel like it, write. […] Happy holidays, happy new year, and happy readings! Hugs (D’Ambrosio, 1987, p. 2) This issue of ZDM will focus on these different trends, i.e., philosophical issues, political and historical matters, ethnomathematics, modeling, and technology. As mentioned above, these trends will be connected to each other and to other trends, as well. We hope that the reader will be able to feel the flavor of these various aspects of mathematics education in Brazil, which are enriched by some examples from Argentina. (D’Ambrosio & Borba, 2010, p. 272)

Despite the different production and reading conditions of these texts, we could identify some very typical traits of Ubiratan’s actions as a teacher, an advisor, and a researcher, which we want to discuss in this text and make up the D’Ambrosian art of researching, teaching to research, and guiding researchers.

2 Encouraging Curiosity, Creative Daring, and Epistemological Humbleness: Solidarity, Respect, and Cooperation The production of knowledge is often motivated by curiosity about the unknown. In the research field, especially that carried out within a graduate program, several factors act as inhibitors of curiosity and daring that should stimulate researchers. There are limitations of financial resources for the feasibility of carrying out the research, pressures linked to requirements for completing the course, and, still and above all, a certain embarrassment or fear of transgressing rules dictated by formalized research traditions, which circumscribe themes, methodologies, references, and analysis directions to what is already known and accepted. However, all of us who had the privilege of enjoying the contributions of Ubiratan D’Ambrosio as an advisor, a teacher, or research reference had our investigative curiosity, and our thematic, methodological, or analytical boldness sharpened, as we were favored by his example as a researcher and analyst with a broad, respectful, and non-trivial perspective; his cooperative dedication as a teacher, who wished to share his knowledge – and his lack of knowledge – provoking reflections and stimulating and offering subsidies to the theoretical, empirical, and analytical adventures of his students; and his provocation and complicity as a demanding but sympathetic advisor, who would stimulate, challenge, but also welcome and support his advisees’ attitudes and investigative paths. Indeed, when talking about the tapestry that constitutes research in mathematics education in Brazil, D’Ambrosio and Borba (2010), in a way, attribute a collective character to the movement that was effectively described and driven by the

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Ubiratan’s trajectory as a researcher, with its critical and rebellious stance toward theoretical buildings, thematic choices, and methodological procedures: We recognize that mathematics education has developed as a result of trends, which has led to intense theory building in recent years in diverse areas of mathematics education. Theories in general tend to treat students as, ideally, rational agents. From this perspective, mathematics education is seen as structured and organized into different fields of ­specialties. However, among most Brazilian mathematics educators, there is a feeling that this sort of reductionist approach is nothing more than a theoretical idealization, and that it fails to help us understand the cognitive successes and failures of our students. We believe that the complexity of Brazilian society, where pockets of wealth coexist with the most shocking poverty, has contributed to the adoption and generation of different strands in mathematics education, crossing the boundaries between trends. This is a predominant characteristic of mathematics education in Brazil. (p. 277)

The interest and the search for connections between different fields and his laborious exercise of weaving relationships between contributions from a wide and varied set of studies made him an erudite teacher who yearned to share, compassionately, with new generations what he gathered with diligence and creativity. In the letter written to a student, when commenting on her observations about the dynamics of his course, Professor Ubiratan refers, in several passages, to his position as a teacher: [...] at the time of class I get ready to give myself, and I try to give myself completely, without much concern if I’m well-seasoned or if I’m being indigestible. […] I’m eager to get out what’s going on in my head. [...] But we, as teachers, want to give ourselves with love, and this makes us not worry too much about others being this or that way, learning this or that. I think Saint Augustine said, “To love is to let the other be” and he, himself, said that “the act of educating is the act of love par excellence.” (D’Ambrosio, 1987, p. 1)

On the other hand, on three occasions in this same letter, he confesses some lack of knowledge, while expressing his curiosity and cooperative disposition to seek something new: I liked the style of references and the idea of suggesting a soundtrack for each reflection. Many of them I didn’t know and couldn’t find them. How about a cassette with the collection? [...] I don’t know the SEE-JUDGE-ACT3 the scheme you mention as publicized by ecclesial movements. But the similarity [to my proposal] is not surprising. [...] I really liked the 9/16/87 report. I know something about the Boffs,4 but I think I’ll take a closer look. Your reports have strongly drawn my attention to their work. (D’Ambrosio, 1987, p. 1, our highlights)

 It refers to the dynamics adopted by the Basic Ecclesiastical Communities of Latin America to know the population’s problems, their analysis, and decision-making to face them. It is inspired by the methodology adopted in the encyclical Mater et Magistra of Pope John XXIII, published on May 15, 1961. 4  Ubiratan refers to theologians Leonardo and Clodovis Boff, exponents of Liberation Theology in Brazil. Years later, Leonardo Boff became internationally known for his defense of environmental causes and the rights of the poor and excluded populations. 3

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Conversely, the teacher also stimulated the interest of his students in broad and varied topics and, respectfully and generously, revealed unconventional possibilities for studies in mathematics education: Your vision of education as the rescue of the balance between the individual and the collective is interesting. It would be worth exploring the idea. [...] I prefer to give a panoramic view, and it cannot be more than superficial, and then, as interest is piqued, “wind up” the student. Proposing readings, seminars, works, and even courses on well-defined topics, to deepen. Even when approaching Calculus, I do it like this. Have you seen my book CALCULUS AND INTRODUCTION TO ANALYSIS? The Preface explains much of this approach. [...] The bibliography you handle is great. I am attaching something of mine that might be of interest. And I suggest you familiarize yourself with Bachelard. Do you know any mystics, like Meister Eckart? I think it’s inevitable that you will go deeper into the study of symbols and then you will have to look at mathematics linked to mysticism. (D’Ambrosio, 1987, p. 1, our highlights)

Addressing specifically to his advisees, Ubiratan D’Ambrosio (2014) explained his position as an advisor, who respectfully welcomed and encouraged the curiosity of his advisees and sought, together, to instill in them confidence or encourage them to live with the inevitable insecurities of those wanting to produce new understandings: As an advisor, I feel like a partner in my students’ research trajectory toward a Master’s or Doctor’s degree. The research is yours; the topic is your choice. I understand this partnership, in which I will try to follow your ideas, as a resource for your doubts and questions, uncertainties, which normally arise during the research. I will do my best to help you. My mission in this partnership is to accompany you and my great reward is to learn a lot from you. [...] I believe that research can only be done well if there is an authentic interest on the part of the researcher, in fact, a vibration, for the topic. I usually say: that the first step is for the researcher to fall in love with the topic. The advisor acts like Cupid! The advisor/advisee relationship should be a rewarding experience and an opportunity for academic and human growth for both. (D’Ambrosio, 2014, p. 1 and p. 3, author highlights)

However, although he offers himself as a support and reference for researchers in training, the advisor Ubiratan does not deceive them with the promise of a linear and pre- defined trajectory, from which he would be a guide to point out serene and accurate paths for the success of the investigation. On the contrary, he warns about a journey of constant methodological and analytical re-elaboration. The metaphors used refer rather to the role of the advisor as someone who encourages adventure into the unknown and is willing to cooperate in the search for new ways to proceed, rather than the teacher who recommends paths already taken: [...] as the research progresses, the methodology is reorganized. Research progress cannot be subordinated to a pre-established methodology. As Pierre Bourdieu says “Get rid of methodological watchdogs”. […]

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Even the final result, which is the thesis itself, may not be final, in the sense that the conclusion does not necessarily exhaust the topic, often opening more research topics than closing them. [...] Those who work in academia know that research is part of our daily lives. The validity of research, research methods, and lines of research are discussed. Metaphorically, I say that research is like flying through unknown spaces. I expect my students to fly and invite me to fly with them in a wide space, usually unknown to me. Together we can venture into the new. (D’Ambrosio, 2014, pp. 4–6, author highlights)

Nonetheless, in these and many other opportunities in which the researcher, professor, and advisor Ubiratan D’Ambrosio spoke about doing research, teaching in a graduate course, or guiding a graduate student – always highlighting genuine curiosity as the engine of investigative exercise and the respectful, supportive, and cooperative partnership between researchers, between teacher and students, and between advisor and advisees as a loving and fertile pact between companions in the adventure of producing knowledge – we also recognize another characteristic of his performance in these three roles. It is the epistemological humbleness that gives this curiosity and this partnership an ethical dimension. Indeed, abandoning the presumptuous attitude that establishes certain knowledge and ways of knowing as a model to think of all humanity and considers as lacking beings those who do not adopt it requires a posture of epistemological humbleness. This humbleness allows us to recognize the dignity and diversity of ways of knowing and the knowledge they produce. This recognition questions the universality or the superiority of specific ways of knowing and identifies the power relations that define hierarchies, naturalize impositions, and erase certain ways of knowing to the detriment of others. Epistemological humbleness is, strictly speaking, at the heart of the ethnomathematics perspective. Clarifying the scope of ethnomathematical studies, D’Ambrosio and Borba (2010) highlight exactly the cultural character of any knowledge and the recognition of its cultural roots as a condition to analyze the relationships that individuals and groups establish with knowledge and knowledge production: It is important to note that, indeed, ethnomathematics focuses on the mathematics of native populations and kids in slums, not only in Brazil but just about everywhere. In poor countries, as in more affluent societies, particularly in Europe and in the USA, ethnomathematics has been regarded as a response to demographic dynamics and social inequity. Socially, it is true that ethnomathematics aims at valuing and supporting knowledge production of those who are the “losers” in this long process of globalization. But ethnomathematics is not only this. For a long time, professional groups such as engineers, biologists, and physicians have employed their own ethnomathematics as the theoretical bases for their activities. The notion of ethnomathematics is sometimes erroneously seen as challenging academic mathematics, which is sometimes, again erroneously, regarded as free of cultural influence. Research in ethnomathematics reveals fundamental roots, sometimes unnoticed, of mathematics. (D’Ambrosio & Borba, 2010, pp. 278)

Thus, the hierarchy to which knowledge has historically been submitted is not based on a neutral consideration of the nature of knowledge or the effectiveness of ways of knowing, because the configuration of knowledge and the evaluation

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criteria of effectiveness are always established by the interests of those who want to know, the intentions of using knowledge, and the consequences of these uses for the groups that have such knowledge and that impose or withhold it from other groups. However, the acknowledgment that all knowledge is cultural and historical (and therefore committed to the interests and power relations of those invested in its production, use, dissemination, and evaluation) is also the acknowledgement of the inevitable incompleteness of knowledge and ways of knowing. In turn, the incompleteness of knowledge reflects the incompleteness of the being who knows – and who is only willing to know because he is aware of his incompleteness. We refer here to a reflection by Paulo Freire, of whom Ubiratan confessed to being a disciple.5 For Freire, it is the awareness of incompletion that inserts an ethical dimension into human life: In fact, the incompleteness of being or its inconclusiveness is characteristic of life experience. Where there is life, there is incompleteness. But only among women and men did incompleteness become conscious. [...] The support gradually became a world and life, existence, as the human body becomes a conscious body, capturing, apprehending, transforming, creating beauty, and not an empty “space” to be filled by contents. To repeat, the invention of existence involves, necessarily, language, culture, and communication at deeper and more complex levels than what happened and still happens in the domain of life. The “spiritualization” of the world, the possibility of beautifying as well as making the world ugly, all of this would inscribe women and men as ethical beings. (Freire, 1996, pp. 50–51, author highlights)

3 Respect, Solidarity, Cooperation, and the Ethics of Diversity Transdisciplinarity refuses the arrogance of certainty and proposes the humbleness of searching. Ubiratan D’Ambrosio (2001)

As we pointed out above, when trying to outline their vision of research in mathematics education in Brazil, D’Ambrosio and Borba (2010) attribute to the complexity of Brazilian society, marked by social inequality and structural racism, and its dramatic consequences for the majority of the country’s population the need and the willingness of Brazilian researchers – many of whom, supported by D’Ambrosio’s writings, classes, and guidance – to adopt or generate “different strands in mathematics education, crossing the boundaries between trends” (p. 277).

 In 1995, alongside Maria do Carmo Domite, Ubiratan D’Ambrosio interviewed Paulo Freire, beginning his intervention with the following words: “Oh, I must say that for me it is a privilege, a very rare thing, for me to be able to interview the master. I was never formally your student, but I think I belong to that army of educators from all over the world who consider themselves disciples of Paulo Freire. And to have this opportunity for a conversation is a great honor for me.” https:// www.youtube.com/watch?v=O_TC3nSz3MM 5

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For these authors, this overflowing, beyond the limits of a single trend, discipline, or theoretical-methodological framework, seems to be a need, as well as the wealth and the contribution of Brazilian research for mathematics education: Thus, we see relevant contributions by Brazilian mathematics educators to areas such as ethnomathematics, mathematical modeling, and digital educational technology. It is impossible to separate the various strands, which have been pursued by mathematics educators in Brazil. They grew together in a symbiotic process and have mutually benefited from each other. The organization of this special issue was strongly influenced by our perception of these characteristics and by the historical, cultural, social, and economic factors, which are interwoven as in a tapestry of ideas upon a wider canvas (the complex scenario of Brazilian society) and aiming at respect, solidarity, and cooperation among the individuals that compose the scenario. (D’Ambrosio e Borba 2010, pp. 277–278)

As a pioneer advocate of transdisciplinarity against the fragmentation of knowledge into disciplines, typical of the understanding of the world established in modernity and imposed on colonized peoples, Ubiratan D’Ambrosio, like Paulo Freire, attributes an ethical character to our dispositions to know and, thus, of doing research and of constituting ourselves as researchers. It is in this perspective that we understand D’Ambrosio and Borba’s mention of respect, solidarity, and cooperation between people as an objective and a condition to produce knowledge that can be considered relevant to face the challenges of life in society. This refers to the D’Ambrosian proposition of an Ethics of Diversity, which the teacher, advisor, and researcher recommends we prioritize: 1. Respect others with all their differences. 2. Solidarity with others in their fulfillment of the needs to survive and transcend. 3. Cooperation with each other in the preservation of common natural and cultural heritage (D’Ambrosio, 2001, p. 153, author’s highlights).

These ethical principles, which, as we have shown, permeate Ubiratan D’Ambrosio’s practices as a researcher, a graduate schoolteacher, and as an advisor of researchers, are highlighted in his book on transdisciplinarity (D’Ambrosio, 2001), in which he points out the radicality of its use. D’Ambrosio (2001) distinguishes respect from the mere acceptance of the other depending on one’s conversion. He points out that an acceptance that requires the transformation of the other and wants to produce the other is “the origin of the great violence of a religious and partisan nature” (p. 154). Bringing this discussion to the theme of this chapter – research and development of researchers – we have to admit that the warning that we must respect each other, “not because I ‘modeled’ the other to what I like, (…) not because he mirrors me, nor because I converted him” (p. 154, author’s highlight), challenges researchers, teachers, and advisors to strip away their certainties, which is difficult to assume in our actions. However, for Ubiratan, only this respect that welcomes differences can produce respect for us, which is indispensable for inner peace. Similarly, solidarity, which implies “giving the shoulder to the other to cry or laugh, and dance or sing along in emotional needs, (…) eat, but eat together, commune” (p. 154), requires the availability of time, attention, affection, interest, and memory to accommodate the demands of students and advisees, but also of other

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participants in research and graduate programs, to which professors and researchers must dedicate their empathy and guide their students to do the same. Thus, it is part of a researcher’s training to learn to be available, for example, to a student of Adult Basic Education, whose calculation practices in focus will only be identified after a long period in situations outside the classroom, after much conversation about topics other than their math skills, and when a relationship of trust and camaraderie is established between the researcher and that student. A teacher whose teaching practices I want to investigate may be looking to the researcher for help with the immediate challenges she is facing in class. The graduate program secretary, in turn, expects an attitude of empathy from professors and graduate students, as their job of organizing classes and schedules depends on the punctuality and accuracy of the information that professors and students must send them. The program coordinator needs the solidarity of professors and master’s and doctoral students because his/her success in managing the program requires that advisors clarify the organization of the course to their advisees. These individuals must also want to be included, seeking to understand the conditions of the program, the available resources, how to access them, using their rights, and assuming their responsibilities. These types of care, assumed by researchers, professors, and graduate students, even though they cause apparently restricted interference, are permeated by the same solidarity that drives much broader actions, indispensable and urgent in such an unequal world. Like big actions, these small concerns are also decisive for social peace. Finally, the cooperation proposed and lived by Ubiratan D’Ambrosio is, for him, after all, what makes life possible: “life is only possible because there is cooperation in the broadest sense” (D’Ambrosio, 2001, p. 154). As former students, advisees, and research colleagues of this researcher, professor, and advisor, we were all witnesses to how he assumed cooperation as the genuine way of researching, teaching, and learning, activities that only make sense and can be carried out because of our incompleteness. The humbleness that makes us recognize our incompleteness, the incompleteness of our knowledge, and the power and limitations of every way of knowing is what makes us serene or fearful, refusing certainties and seeking memories and hopes. It also demands and enables coexistence, and, in this sense, it is what promotes environmental peace.

4 Final Considerations Throughout history, humanity has been looking for explanations about who it is – and has believed itself to be God’s favorite – what it is – and has been believed to be a complex system of muscles, bones, nerves, and humor – what it is like – and has believed to be an anatomy with will – and above all how much it is – and has believed itself without limitations to its will and ambition.

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Distortions in the way man has judged have induced power, arrogance, greed, envy, avarice, arrogance, and indifference. Violations of Peace, in all its dimensions, are fundamentally the result of these distortions. Hence, the violations of human dignity and the elimination of the individual, the unfeasibility of an equitable society, and an immeasurable aggressiveness against nature. There has never been an attempt to face the core of the issue: the issue of knowledge itself, conveniently fragmented into disciplines to justify – discouraging criticism – our actions in each sector. (D’Ambrosio, 2001, 151–152)

Discussing the legacy of acting as a researcher, but mainly as a researcher trainer, necessarily leads us to discuss how this actor understands knowledge, its production, use, and dissemination. This is what we have tried to address in this chapter. We have aimed to show a little of a graduate school professor that understands each activity of researcher in development as an action inscribed in history, culturally constituted, and committed to a project of society. Directly connected to the processes of knowledge production, the researcher we sought to present here has no illusions about the value of knowledge disconnected from the interests that motivate it and the repercussions it triggers. As an advisor, he warns his advisees to assume the ethical dimension of research, teaching, and knowledge, as well as its production, its relationship with other types of knowledge, and ways of producing it. Therefore, the relationship between advisor and advisee must be, for both, an opportunity not only for academic growth but also for human growth. The experience of going through the supervision by Professor Ubiratan strongly influences the human dimension of this development, nourished by a relationship that, solidarily, respectfully, and cooperatively, gains affectionate paths in these bonds grounded on the Ethics of Diversity. Always solidary, respectful, and cooperative, the advisor, the teacher, and the researcher, who come together in the trajectory of Ubiratan, criticize the separation of knowledge, which acts in the unfair and unequal distribution of resources and rights, including the right to produce knowledge, to attribute value to the knowledge produced by people and social groups, to access the diversity of knowledge produced in different ways, and to analyze their production conditions for a better understanding of the intentions and scope of each knowledge. This researcher, advisor, and teacher bets on solidarity, respect, and cooperation, to establish an Ethics of Diversity, which he teaches us not only through his precious writings and well-attended presentations but in the legacy of his actions in these and many other roles, which we will keep in our affective memory and intellectual thinking. However, we will also find it in our attitudes reflecting the privilege we had of his presence in our professional development. We will also identify it when guiding the decisions of our students, thus inserting us into the complex chain of researcher development and the production of knowledge, which is also constituted by the principles and legacy of Ubiratan D’Ambrosio.

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References D’Ambrosio, U. (1987). Carta à aluna Conceição Fonseca, para envio de feedback sobre os trabalhos que ela elaborou ao longo da disciplina [Letter to the student Conceição Fonseca, sent as a feedback on her works during the subject]. Authors’ personal archive. https://sites.google. com/view/gen-­numeramento/arquivos D’Ambrosio, U. (2001). Transdisciplinaridade. Palas Athena. D’Ambrosio, U. (2014). Carta de Boas Vindas aos Orientandos [Welcome letter to advisees]. Authors’ personal archive. https://sites.google.com/view/gen-­numeramento/arquivos D’Ambrosio, U, & Borba, M. (2010). Dynamics of change of mathematics education in Brazil and a scenario of current research. ZDM Mathematics Education, 42, 271–279. Frere, P. (1996). Pedagogia da Autonomia: saberes necessários à prática educativa. Paz e Terra. 27ª. Ed.

D’Ambrosio’s Legacy in Teacher Ethnopedagogical Space for Glocalization María Elena Gavarrete Villaverde and Marcela García Borbón

, Margot Martínez Rodríguez

,

Abstract  This chapter shows the contribution of D’Ambrosio’s ideas to the development of ethnomathematical research in Costa Rica, by means of his contribution to pedagogical practices, specifically in relation to the teaching education of elementary teachers. It describes a model that proposes activities for continuous teacher education, beginning with glocalization and cultural signs into the mathematical enculturation process to generate the construction of contextualized didactic resources, through the appropriation of mathematical knowledge that comes from the teacher’s own context. Since 2002, D’Ambrosio has had a deep impact in the ethnomathematics development in Costa Rica, given that his ideas served as a guide for new researchers to construct ethical work with indigenous communities in this decade and has also shaped projects oriented to teacher training in diverse regions of this country. Inspired by these ideas, this research team has developed the current proposal, to work in particular with differentiated groups, such as coastal, rural, and marginal urban communities, among others. His legacy is in both areas, and this paper shows and explains one example of the impact of D’Ambrosio’s contribution to the evolution of ethnomathematics in Costa Rica. Keywords  Cultural context · Enculturation · Ethnomathematics · Teacher education · Glocalization · Cultural signs

1 Introduction This chapter describes a model of teacher education that is theoretically and empirically based on the International Ethnomathematics Program, created within the framework of Ubiratan D’Ambrosio program. According to D’Ambrosio (2006), M. E. G. Villaverde (*) · M. M. Rodríguez · M. G. Borbón Universidad Nacional Costa Rica, Heredia, Heredia Province, Costa Rica e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_4

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this program has a dynamic character, since it promotes new methodologies and approaches as well as new ways to see what is considered as mathematics. It is intended to show the progressive evolution of this model from theoretical, methodological, empirical, and reflective perspectives. The term “Ethnomathematics,” in singular and capitalized, will be used here to refer to the International Ethnomathematics Program and “ethnomathematics” in plural and lowercase will be used to refer to different ways of conceiving mathematics from the social and cultural contexts (ethno). The evolutionary process described here has tried to encourage teachers to reflect on the nature and origin of mathematics from a sociocultural perspective. The way to stimulate these reflections has been to animate a search for and recognition of local or regional ethnomathematics, based on examples developed during meetings with teachers, as will be shown later, in which an attempt has been made to generate retrospective discussions about the nature of local ethnomathematics in the teacher’s school environment and what the teacher’s conception of mathematics is. The activities that have been used to inspire the search for local ethnomathematics have enriched the group of teachers in terms of research capacities, epistemological reflection, and planning tools for participatory and active didactic activities that facilitate the construction and the sociocultural contextualization of knowledge (Albanese et al., 2014).

1.1 Background According to Shirley and Palhares (2016, p. 13), D’Ambrosio declares that “it often has been seen that ethnomathematical examples demonstrate new ways of looking at mathematics and lead to better understanding of the concepts, procedures, and uses of the curricular content.” He also states that ethnomathematical examples help to get a better understanding of procedures and concepts. In this sense, the Universidad Nacional (Costa Rica) created the Museum of History of Mathematics (MHM) project, over two decades ago, with the objective to incorporate the sociocultural and political aspects in the teacher education, as well as the development of philosophical reflection on mathematics as a human product and as a social construct (Gavarrete et  al., 2016). Based on experiences developed in this project and having studied the abovementioned D’Ambrosio ideas, a research assumption was proposed: expanding epistemologically conceptions of mathematics is important in the area of pedagogy, since it motivates the development of teacher creativity, both in terms of didactic action and in research. The tool chosen to epistemologically develop these conceptions was the study of regional ethnomathematics in various geographical areas of Costa Rica. Since 2015, the Teachers Education in the Sociocultural Vision of Mathematics project was proposed, with the expectation of guiding teachers in the use of methodologies to identify regional ethnomathematics in their environment, which could be included in their classes, as well as to promote the integration of elements of

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regional cultural identity and mathematical contents into curricular development, thus seeking a development of a diversity of contextualized didactic actions where the intention was the ethnomathematical study of cultural signs. Based on these expectations, it was determined that mathematical enculturation was the ideal mechanism for the study of regional ethnomathematics, since, as described in Gavarrete et  al. (2016), it is a mechanism that allows teachers to develop professionally as researchers while improving their professional practice and promoting meaningful learning with cultural relevance. The project development began in 2016, and one of its activities consisted of the first application of the Mathematical and Ethnomathematical Enculturation course, which was designed by considering the promotion and development of innovative methodologies, acquisition of professional scientific and research competencies, and the theoretical perspective of Ethnomathematics. This course is the second to be held under this initiative in Costa Rica, which aimed at teacher education. The first course, developed by Oliveras and Gavarrete (2012), had a similar approach regarding indigenous teachers. Many courses have been conducted within the framework of Ethnomathematics (Gerdes, 1998; Naresh, 2015; Oliveras, 1996; Presmeg, 1998), as well as in a culturally responsible education perspective (Parker et al., 2017). From the work of these authors, elements that are relevant to the proposal described in this chapter were considered. For example, universal mathematical activities are taken from Bishop (1995, 1998,1999), the idea of an ethnomathematical study of a cultural sign is taken from Oliveras (1995, 1996), while the importance of individual research work was drawn from Presmeg (1998), Gerdes (1998), and Aroca (2010). On the other hand, Shirley (1998, 2001) takes into consideration the different ethnomathematical visions, adjusted to the interests of teachers, and Gavarrete (2012) considers the observation of their own ethnomathematics and the development of individual microprojects. The Mathematical and Ethnomathematical Enculturation course has been implemented in several regions of the country, including coastal, agricultural, urban marginal, and indigenous areas, contributing to an emerging process of consolidation of the model that will be described. It considers five sequential stages that are addressed as processes: • • • • •

Adaptation of the course to a specific region. Implementation. Production, selection, and edition of teaching resources. Validation of teaching resources. Return of results.

The execution of these five stages in other educational regions favors the consolidation of a training model as described later, in Sect. 4 of this chapter, that makes it possible to assess the role of the sociocultural vision of mathematics in teacher training, that is, it follows the sense of social justice and sustainability that D’Ambrosio’s legacy has promoted, as supported in Shirley and Palhares (2016,

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p. 14), that the work with “students of another cultural group or living away from their home culture” has a big impact in the development of ethnomathematical experience. Those authors add that every teacher should consider the necessary changes in their methodology and practice to fit the content to students’ interests. On the other hand, the model proposed in this paper allows to offer a set of contextualized teaching units with a new epistemological perspective on mathematics.

1.2 Rationale and Relevance The formulation of this model takes into consideration both international and national factors. International trends and demands for mathematics education are considered to improve attention to sociocultural diversity and promote equity and social justice. For example, UNESCO (2012) establishes that quality of mathematics education must be accessible to all and favor full personal development. It should, thus, be designed in a manner that allows each student to express their knowledge and skills in the best possible way. In light of these disciplinary axes, four workshops were designed for the Mathematical and Ethnomathematical Enculturation course that addressed the mathematical areas of Numbers, Measurements, Geometry, and Statistics and Probability, which were specified in the program. The relevance of this proposal may be justified from different perspectives. Currently, Ethnomathematics follows approaches that embrace social justice, civil rights, indigenous education, professional contexts, recreational practices, and urban and rural contexts and that humanity evolves into a model of social sustainability, balance, and harmony; this is to promote taking a socially responsible attitude and leave a stable and sustainable world to the next generation (D’Ambrosio, 2018). On the other hand, as stated by García (2015), the exercise of teaching must be undertaken (as far as possible), considering a sociocultural approach to mathematics and mathematics education, but this epistemological approach enriches professional practice, since it can offer students situated learning, concrete examples of the social and cultural use of mathematics in their daily contexts, and. [...] to conceive of a class as a learning community, focused on the purpose of sharing and developing mathematical meaning and resignifying learning as the quality of the mathematical discourse with which the student participates in that community. (García, 2015, p. 32). Another factor that makes the adoption of the referred model, as described in Sect. 4, relevant is the urgent need in Costa Rican Education for interventions with curricular innovation that enriches methodologies proposed by the Mathematics Study Program of the Ministry of Public Education (MEP). For example, Chavarría states (2018):

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Despite the fact that the MEP Study Program (2012), through the specification of the curriculum, provides some proposals on how to approach mathematical knowledge through introductory problems that include aspects of history, contextualization and technology, a review of the proposed tasks has not been carried out to verify if they respond precisely to possible experiences in the life and culture of Costa Rican students and if they are consistent with the theoretical approach of active contextualization, explained in the very foundations of the curriculum. (p. 3). Chavarría and Albanese (2021) also state that even when the Ministry of Public Education theoretically promotes contextualization in Costa Rica, in a practical sense, it is revealed a total absence of rural and indigenous contexts, prevailing an artificial contextualization and delimited to the urban region of the country. Another relevant aspect in the formulation of this model is that the activities carried out by teachers participating in the Mathematical and Ethnomathematical Enculturation course favor the use of culturally relevant pedagogies, which according to Rosa et al. (2017) incorporate reference frameworks of students, communities, and schools into teaching, in accordance with the foundations of the Ethnomathematics Program instituted by Ubiratan. D’Ambrosio. Thus, a cultural sign becomes the protagonist of a contextualized proposal through a teaching unit that addresses the elements of the MEP Study Program. Participating teachers have made the connections between mathematics, culture, and teaching practice evident, confirming the ideas of Panes Chavarría et al. (2018) about the beliefs of teachers regarding mathematics and culture, as they establish that “they are the product of the experience that teachers develop in their learning communities” (p. 587), which can model the teaching practice, where beliefs about “the role of mathematics in the sociocultural environment are immersed in a dynamic context of social and cultural changes.”

2 Theoretical Foundations The theoretical foundations of this model consider different approaches from the International Ethnomathematics Program (D’Ambrosio, 2007, 2008) and Mathematical Enculturation (Bishop, 1988, 1999, 2001) as global organizations that have established the roots of the approach that disseminates Mathematics (plural) as a social product and which in turn, are interrelated and oriented to consider historical-philosophical and sociocultural perspectives on mathematics in teacher education. The loss of universality of mathematics and the growing consideration of the sociocultural context in its practices have given impetus to Ethnomathematics as an area of research. The precursor of this approach was D’Ambrosio (1985), who initially defined ethnomathematics as mathematics practiced by identifiable cultural groups. Further, D’Ambrosio (2008) described ethnomathematics as the ways, styles, arts, and techniques (Tics) to explain, learn, know, and relate to (Mathema)

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the natural, social, and cultural environment (Ethno) to incorporate a broader conception of knowledge that admits the existence of diverse ethnomathematics, all of it, considering that techniques are developed and help to promote of cognitive capacities in different sociocultural environments. The theoretical justification for the existence of many different forms of mathematics is based on the contributions of various disciplines, and at the same time, it counters the ethnocentric vision of Western mathematics, which Bishop (1988, p. 180) recognizes as “decontextualized and abstract mathematics, which, however, are those that have dominated and can be socially applied by anyone, which is why Western mathematics is clearly universal.” According to Bishop (1988), for many years mathematics was unrelated to its cultural environment. However, anthropological research and comparative studies of different cultures have shown that “Mathematics is a cultural fact and that other cultural groups have created ideas that are clearly other mathematics” (Bishop, 1988). However, to carry out curricular restructuring in the field of mathematics education, Bishop (1988) states that mathematics must be considered as a cultural fact and that we must first try to “culturalize” it. To arrive at this “culturalization,” Bishop (1988, 1999, 2001) specifies six criteria with which mathematical activities can be culturally characterized. These activities are counting, locating, measuring, designing, playing, and explaining, each of which develops important ideas for our mathematics and are necessary and sufficient for the development of mathematical knowledge. In this regard, Shirley and Palhares (2016, p. 16) state that Bishop introduces the concept of enculturation by arguing that students’ way of thinking in specific forms will only make them be “governed by systems of power and authorities.” The model that will be presented in this work is a proposal for curricular innovation based on ethnomathematics that can be considered as an application of culturally relevant pedagogy (Rosa & Orey, 2013). According to Gavarrete et al. (2017), this proposal includes theoretical, methodological, and empirical foundations. This model increases the possibility of “unfreezing mathematics” (Gerdes, 1985) since it allows teachers to provide and use meaningful learning materials and create environments that include different cultural traits, customs, and traditions. This approach helps to promote teacher creativity, with the purpose of including various activities in lessons that enable students to make meaningful connections between their lives and school-related experiences. As Rosa and Gavarrete (2016) state, by doing this, teachers show their respect to the background of students and are able to fulfill the cultural congruence in classrooms. Regarding the above, Neupane and Sharma (2016) state that the creation of alternative teaching and learning processes by using an ethnopedagogical approach enriches students’ cognitive and reflective capacities, which allows them to formulate better questions, collaborate, and reflect critically on their own identity, which supports a view of this model as an innovative proposal with collateral effects in the affective domain and beliefs (Gómez-Chacón, 2010).

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2.1 Problematizing Cultural Signs from an Ethnomathematics Perspective for Pedagogical Action The interest in guiding teacher training from the perspective of the Ethnomathematics. Program (D’Ambrosio, 2008) lies in the fact that it contributes to improve investigative skills that allow “unfreezing mathematics” (Gerdes, 1985) inserted in cultural objects and by doing so promoting the necessary creativity in teaching for developing a mathematics curriculum in connection with a sociocultural environment. To be able to present mathematics as a science that is available to all, it is important to open a discourse aimed at highlighting the connections between mathematics and other disciplines (Oliveras, 1996). In order to do this, it is essential to promote in teachers the development of competencies or skills that help them to analyze classroom problems and provide solutions that are open and consistent with the reality. The route for guiding these connections lies in the recognition and study of the mathematics implicit in a cultural sign, which is conceived as any feature or element of a culture, tangible or intangible, that has some mathematical potential that may be useful in school classrooms (Oliveras, 1996; Gavarrete, 2012; Oliveras & Gavarrete, 2012) through a sequence of activities constructed to introduce a mathematical concept (Gavarrete & Albanese, 2015). This process can also have an educational effect on teachers, when they assume the role of researchers (Oliveras, 2005) who ethnomathematically problematizes the elements in context that have potential for teaching mathematics in the classroom. The problematization of the cultural signs through Ethnomathematics involves different ways to mathematize reality in order to examine how mathematical ideas and practices are processed and used in daily activities. “Mathematization” is understood as the process in which the members of a differentiated group (cultural or regional) develop specific mathematical tools that can help them organize, analyze, understand, and solve specific problems located in a specific real-life context or situation (Rosa & Orey, 2010, cited in Rosa & Gavarrete, 2017). The treatment of a cultural sign from this perspective promotes the development of a culturally relevant pedagogy (Rosa et al., 2016) and teacher awareness toward mathematics as a cultural phenomenon that is socially shared (Bishop, 1999). In this context, ethnomathematical analysis of a cultural sign promotes reflection about the professional practice of teachers and on their own concepts of mathematics, which directly contributes to pedagogical innovation (Shirley, 2001). Culturally relevant pedagogy studies the consistency between the referential frameworks of students, communities, and schools and the fundamental principles of the Ethnomathematics Program (Rosa and Gavarrete, 2016), so that teachers contextualize the teaching of mathematics and learning, which relates to mathematical content to the sociocultural experiences of the students. To develop a culturally relevant pedagogy, the model described in this chapter is intended to assist teachers to improve competencies or skills to analyze problems in

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their classrooms and provide solutions that are open and consistent with the realities of the times and the environment. Attention is focused on the study of mathematics implicit in a cultural sign, where national or regional idiosyncrasies are alluded to through the study of cultural signs (Gavarrete and Albanese, 2015). For example, in the proposals developed by Chavarría et al. (2017) and Gavarrete and Albanese (2018), Costa Rican addresses and directions are approached through analytical geometry that shows the results of problematizing this valuable cultural sign in terms adapted to the Costa Rican reality to design culturally relevant pedagogical actions.

2.2 The Emic Approach to Cultural Signs Based on the Study of Regional Ethnomathematics The model that, as it was said, will be described in Sect. 4 of this chapter proposes an ideal use of cultural and mathematical knowledge in dialogues oriented around a cultural sign. In this regard, we consider the constructs proposed by Rosa and Orey (2012), who have drawn a distinction between the emic and etic approaches to characterize aspects of culture that are or are not related to an Ethnomathematical perspective. Table  1 shows a comparison between the two approaches from the perspective of these authors. From the perspective of Rosa and Orey (2012), the etic approach refers to an interpretation of aspects of another culture based on the categories of the researchers themselves, while the emic approach is one through which a certain culture is understood based on its own referents. The etic approach is thus considered as being based on the external vision of the observers, while the emic approach corresponds to an internal vision, that is, sociocultural aspects associated with knowledge of the cultural signs linked to the emic perspective, while the mathematical contents that are implicit in the sign correspond to the etic perspective. Concerning the active contextualization proposed for addressing cultural signs in this model, Albanese et al. (2017) state that the contextualization of tasks becomes meaningful from an ethnomathematical perspective when respect for culture is promoted from an emic perspective. Similarly, Rosa and Orey (2012) state that if problems created are consistent with situations familiar to the members of the Table 1  Differences between the emic and etic approaches Emic approach Perspective of members of the local community (internal) Local (internal) vision Prescriptive cultural translation Mental structures Cultural transcription Source: Rosa and Orey (2012, p. 3)

Etic approach Observer perspective (external) Overview (external) Descriptive analytical translation Behavioral structures Academic transcription

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differentiated group, then these situations can emerge in a real context, and mathematical concepts or procedures come into practice in the same way as found within a particular cultural group context. As Orey and Rosa (2015) point out, each person conceives the world according to their own cultural background and is deeply influenced by the paradigm in which they are immersed. When reflecting on mathematics or ethnomathematics, a difference is established from the point of view of mathematical knowledge, giving rise to dialogic reflections in which the etic (global) vision is as important as the emic (local) vision. These reflections are essential in stimulating a creative process that enriches the perception of local ethnomathematics and favors creativity in the design of contextualized resources. According to Rosa and Gavarrete (2016), the application of culturally relevant pedagogies and the ethnomathematical perspectives that accompany them in classrooms allow the validation and incorporation of cultural backgrounds. This results from the ethnic heritage of students, as well as their current interest in the teachers’ daily instructional practices, which produces students’ empowerment in intellectual, social, emotional, and political areas. This approach affects students’ reality and sociocultural and historical contexts. Thus, through the mathematization of reality, teachers can perceive the human part of mathematics and enrich students’ academic skills, while at the same time they encourage a change in their attitudes toward academic instruction.

2.3 Glocalization of the Ethnopedagogical Space as a Result of Teacher Enculturation The inclusion of cultural aspects in a mathematics program has long-term benefits for mathematical achievements of students, since these aspects enrich the perception that mathematics is part of students’ daily lives, which allows for a deepening understanding of its nature through improvement of students’ ability to make meaningful connections, favoring the formation of an ethno-pedagogical space, or educational environment, where culturally relevant pedagogy is implemented. This model proposes broadening the vision of mathematics teacher education curricula using ethnomathematics, which agrees with the ideas of Naresh and Kasmer (2018), who make a proposal based on their professional and empirical knowledge. In this proposal, the theoretical field of ethnomathematics was used to visualize a dynamic and equity-oriented teacher training curriculum that favors the creation of spaces for teacher reflection on their practices in teaching mathematics in a more just and democratic manner. The model described in this chapter seeks equity and justice based on a balance of knowledge that is achieved through “glocalization,” which is defined by Orey and Rosa (2015) as the relationship between local and global knowledge and is related to a dialogical approach to knowledge, in which the dialogue occurs between the emic and etic visions of the members of culturally differentiated groups.

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The process of “dialogue” between both visions integrates earlier knowledge from various epistemological positions about mathematics: on the one hand, Western mathematics, which D’Ambrosio (2008) has also called ethnomathematics, and, on the other hand, regional or local ethnomathematics, which are associated with the ancestral knowledge of a distinct group. Glocalization promotes the process of contextualization of mathematics (Albanese et al., 2017), allowing a better understanding of the relationships between local and global knowledge from the emic and etic perspectives, respectively. This provides a much broader vision of mathematics which includes ideas, notions, procedures, processes, methods, and cultural practices rooted in different environments. This in turn contributes to an increase in the evidence of cognitive processes, learning capacities, and attitudes that are fostered in the classroom and reflection on the social and political dimensions of ethnomathematics, enriching the development of innovative approaches for a dynamic and glocalized society (Orey & Rosa, 2015). Within the formative model addressed in this chapter, it is considered that the path to developing a dialogue between the emic and etic perspectives can be consolidated through “Mathematical Enculturation,” a theoretical and methodological mechanism that leads to an appropriation of mathematical knowledge of their own context. This allows teachers to develop professionally as researchers, improve their teaching practices, and promote meaningful learning with cultural relevance (Bishop, 1999). The enculturation process, as summarized by García (2015), is an “intentional, situated process, aimed at generating interaction and intersubjectivity between subjects who learn mathematics. It is essential that the mathematics teacher understands the individual and social nature of this process and the didactic ruptures that underlie it” (p. 31), and at the same time, it is an interpersonal process, that is, it arises from social interaction of individuals (Bishop, 1999), developed within a given knowledge framework, but with the objective of re-creating and redefining that framework. To integrate the glocalization process in the mathematical enculturation of teachers, it is necessary to provide a method for mathematizing knowledge, based on the fact that it must try to cultivate mathematics while considering it to be a cultural fact (Bishop, 1988). Thus, in this model, the culturalization of mathematics is founded on the six “universal mathematical activities” proposed by Bishop (1988, 1999, 2001). Enculturation of teachers in regional ethnomathematics favors the creation of contextualized didactic resources that consider mathematical practices developed in any culture (Bishop, 1988, 1999, 2001) to achieve a transversal view of mathematics education (D’Ambrosio, 2007, 2008) that strengthens the identity of the regional culture of teachers and improves the process of teaching and learning mathematics. Glocalized integration of the emic and etic perspectives in the framework of mathematical enculturation of teachers involves a creative and interactive process with participants that include those who live in the culture, those who are born within it, and those who want to identify the mathematical knowledge involved in the knowledge of the differentiated group, which results in ideas, norms, and values that are similar from one generation to the next, although it is inevitable that they

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differ in some respects due to the “re-creation” of these ideas, norms, and values by the next generation (D’Ambrosio, 2016). In this formative model, the glocalization of cultural signs that results from the interaction and enculturation process of teachers based on ethnomathematics implies a relativist epistemological view of mathematics, which takes into account its interdisciplinary role in terms of school content. Likewise, the methodology described below promotes the teacher’s role as a cultural activator, a partner in the discovery of one’s own ethnomathematics within an ethnopedagogical space.

3 Empirical Foundations: Model Design Process In this section, the four evolutionary phases of the training model design are presented, which emerged from the planning, implementation, and evaluation of the Mathematical and Ethnomathematical Enculturation course.

3.1 Phase 1: Design and Validation of Instruments In this phase, the instruments that made possible the detection of changes in the teachers’ perceptions before and after the training process were designed and validated. The indicators established for the impact measurement instruments, which recorded the initial situation and changes in the perceptions of teachers, were defined guided by the experience and theoretical foundation provided by the researchers, as well as from evaluations by international advisers. The choice of indicators considered the teachers’ knowledge of their own regional identity, the cultural traits they identified in the environment where they carried out their teaching, and the capacities that they display before and after the educational process.

3.2 Phase 2: Course Design and Organization This phase consisted of the design and organization of the course “Mathematical and Ethnomathematical Enculturation,” which had the purpose of promoting awareness of both the historical and philosophical dimensions of mathematics, as of the social and cultural vision of mathematics, as well as promoting training of primary school teachers as mathematical enculturators. It was proposed that teachers take advantage of their regional identity in the investigation of mathematics in their environment and that they use it for the design and construction of contextualized didactic resources based on their teaching environment.

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The duration of the course was 40  h, in a combined modality (classroom and non- classroom hours). The classroom sessions integrated discussion and reflection on the results of contextualized teaching material production workshops, and the non-classroom sessions were dedicated to ethnomathematics research. The classroom sessions were focused on training and awareness raising about the D’Ambrosio Ethnomathematics Program and the theory developed by Alan Bishop about the sociocultural vision of Mathematics and Mathematical Enculturation as a professional training process. The non-classroom sessions were a space that allowed the development teachers as researchers into their context and as professionals capable of reflecting on their own practice. Each teacher investigated a cultural sign that promoted mathematical activities through whose use mathematical content could be presented in the classroom. In this way, teachers not only carried out an investigative process but were also able to reflect about the classroom implementation of pedagogical strategies in a given context that was close to both the teachers and their students. Classroom sessions were alternated with non-classroom sessions to permit answering questions in the classroom arising from individual ethnomathematical research undertaken by teachers as part of course activities. In the three classroom sessions, theorizing and socialization of strategies were carried out to address the sociocultural vision of mathematics in the classroom. • The first session provided awareness raising on the sociocultural vision of Mathematics and Mathematical Enculturation as a teacher training process (Bishop, 1988, 1999, 2001), as well as a presentation of the main foundations and theoretical constructs of the Ethnomathematics Program (D’Ambrosio, 2007, 2008). This session also included a workshop on traditional measurement systems and their relationship with the International System of Units, as they related to the mathematical activities of counting, measuring, and explaining. This workshop was called “Measuring as my grandparents did.” • The second session was structured around two workshops entitled “Patterns with harmony” and “Where am I heading.” The first one focused on the analysis of traditional designs of typical Costa Rican carts and tessellations present in various cultural elements; the combination of these analyses involved the study of isometric transformations that are related to the mathematical activities of designing, playing, and explaining. The second workshop focused on Costa Rican addresses and directions, which led to a discussion on essential aspects of spatial location and the mathematical activities of locating, measuring, and designing. In addition to developing the workshops, this session reflected on the ethnomathematical studies that teachers carried out on regional cultural signs. • The third and last classroom session included the workshop “Winning or losing: what does it depend on?” whose central theme was the study of probabilities and which involved the mathematical activities of playing, explaining, and counting. In addition, in this session the teachers socialized results obtained in the non-­ classroom sessions, in which they carried out an ethnomathematical investiga-

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Fig. 1  Workshops, disciplinary areas, and mathematical activities involved. (Source: Prepared by the authors)

tion of a regional cultural sign which was used as an input for the construction of a contextualized teaching unit. Figure 1 shows the distribution of each workshop with respect to the disciplinary area it addresses and the related mathematical activities. This training process was accompanied by an evaluation of teachers that focused on identifying their research capacity, ability to think critically, creativity, effective oral and written communication, active participation, and healthy competitiveness. In addition, as part of the evaluation, each participating teacher developed a teaching unit focused on a cultural sign, as a product of their research and reflection process, as described previously.

3.3 Phase 3: Course Implementation The implementation of the training course began with the selection of target populations located in different geographical areas, taking into account regional cultural characteristics in the country. Using these criteria, rural, marginal urban, indigenous, and coastal practicing primary school teachers were selected. In each region, the instruments for detection of changes in perception were applied; the course was carried out, as well as a process of construction of teaching units, considering the particularities of each context.

3.4 Phase 4: Systematization of Information After the implementation phase, the information collected by the instrument use was systematized, as was the independent work carried out by the teachers. The final product of each implementation was an anthology that brings together the

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cultural signs studied by teachers in their ethnomathematical research, as well as teaching units for approaching different subjects of mathematics in primary school but considering the cultural sign as the central focus. Each teaching unit includes the following sections: 1. Description of the cultural sign. This is based on the investigative activity of teachers and allows readers to locate themselves in a Costa Rican context where part of the history of the cultural sign is explained, as well as the reason to be considered this way. The story may well have been told by an important person in the area, or it might have been found in sources including books, articles, and videos. 2. Bishop’s Universal Mathematical Activities (1988, 1999), allowing development through cultural signs. 3. Curricular aspects associated with the Mathematics Programs of the Ministry of Public Education of Costa Rica. 4. Sequence and didactic activities, consisting of the proposal of various exercises and contextualized problems, in which the cultural sign is used as a didactic resource. 5. Evaluation of the knowledge acquired using a rating scale. Once the course had been implemented in different regions of the country, and after the relevant results were analyzed, the conditions were satisfied for the development of a proposal for a teacher training model.

4 Description of the Teacher Education Model A teacher education model, as defined by García and Chavarría (2016), is understood to be “the compilation or synthesis of theories, methodological and evaluative strategies that guide teachers in the preparation and execution of course programs” (p. 43). In this way, as the authors add, the model becomes a guide for action by representing the design, structure, and components of the training process. The theories underlying this proposal have already been widely explained but are summarized below (Table 2). The stages of the consolidation process of the model for teacher training with respect to the sociocultural vision of mathematics are illustrated in Fig. 2. Stage I: Adaptation of the Mathematical and Ethnomathematical Enculturation Course, Considering the Sociocultural Context of the Population of Teachers This stage involves the theoretical foundation adaptation of the sociocultural vision of mathematics and ethnomathematics to the needs of the region context where the course will be offered. In addition, it includes a review and update of diagnostic and evaluation instruments, which allows the documentation of the initial and final teachers’ perceptions, about didactic resources related to local history and sociocultural vision of mathematics.

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Table 2  Theoretical contributions to the training model, by authors Author Allan Bishop (1999)

Contribution Characterization of universal mathematical activities (counting, locating, measuring, designing, playing, explaining) present in most cultures that serve as the basis for the design of teaching units María Luisa Ethnomathematical inquiry into a cultural sign, understood as the Oliveras (1995, investigation, through a holistic study, of the mathematics implicit in the 1996) cultural sign Norma Presmeg Proposal on an approach to ethnomathematics through individual (1998) investigations of students, who use their own diversity in the design of activities for learning mathematics under the assumption that mathematics is a cultural product Paulus Gerdes Suggestions for the development of awareness in teachers about the cultural (1998) and social bases of mathematics, so that their ability and openness to working in multicultural environments can be improved (D’Ambrosio Ethnomathematics has allowed educators to acknowledge and use cultural 2006) diversity within the classroom to highlight the amazing diversity of worldwide mathematical practices Armando Aroca Research experience, in which students develop a project that describes a (2010) local practice or knowledge, reflecting the ethnomathematics that people produce Lawrence Shirley Different ethnomathematical views, with a focus on the interests of teachers (1998, 2001) and their training. Also, the concept that teachers should have different teaching skills that fit the different backgrounds of their students, since these can determine their attitude toward mathematics and education María Elena Observation of personal ethnomathematics and development of individual Gavarrete (2012) microprojects in a training experience for indigenous teachers in Costa Rica Source: Prepared by the authors

Fig. 2  Stages of the formative model. (Source: Prepared by the authors)

It also includes a review and update of the design of the 40-h course, so that it will adjust to the context where it is being offered. This assures that the workshops that are carried out in classroom sessions can be modified to cover mathematical skills and content that the team of researchers consider to be pertinent or at the request of educational authorities of the region.

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Stage II: Implementation of the Mathematical and Ethnomathematical Enculturation Course with the Selected Population The implementation of the course is evaluated based on the indicators and instruments designed for this purpose. The selection of the population that will take the course must consider the characteristics of regional culture and socio-geographical elements of different areas, as well as the arrangement of entry into the region with the responsible educational authorities. The course schedule, the convenience of dates for classroom sessions, and the number of participants, among other administrative aspects, must be managed at this stage. This stage includes obtaining the informed consent of participating teachers and the execution of the 40-h course previously described. In addition, as a product of the course’s implementation, these teachers create contextualized didactic resources based on the sociocultural vision of mathematics and the process of Mathematical Enculturation, which involves the participation of teachers. The production of classroom activities involves glocalization, as defined above, in both its design (by the researchers) and in its execution (by the participating teachers). The global aspect of these activities corresponds to the contents of Western mathematics and the guidelines of the Mathematics Study Program of the Ministry of Public Education, while the local aspect corresponds to the contribution of teachers through their investigation of cultural signs of the region and the adaptation of activities to local contexts. Stage III: Selection and Editing of Teaching Resources Generated from Ethnomathematical Research in the Populations Served The team of researchers selects contextualized teaching units elaborated by teachers as a product of their learning process, which have been generated from the ethnomathematical investigation of regional cultural signs. The selection criteria are based both on the representativeness of the cultural sign chosen by the teacher and on its potential to generate classroom activities that demonstrate the use of mathematics at a socio-historical level. A detailed didactic resources edition is then made to facilitate it in return to the region where it was developed, expecting it will be productively used in the local context. This editing includes the unification of the format, a search for suitable images, identification of curricular aspects, revision and complement of the investigation of the cultural sign, construction of complementary didactic activities, the proposed evaluation, and the identification of universal mathematical activities. Stage IV: Validation of Contextualized Teaching Resources The validation of contextualized teaching units, which will be compiled in anthologies, is carried out in two ways. The first of these was performed by the researchers responsible for the project and guided by the advice of the international specialists who have collaborated with this team throughout the entire process that led to the production of the course, as well as by the authorities related to the mathematical discipline of each educational

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region where the course has been implemented. This first validation makes use of an instrument designed by the researchers specifically for this purpose. The second is the responsibility of primary school teachers from each of the educational regions that participated in the enculturation process through collaborative work for the presentation of teaching units. Workshops are organized to collect the observations of these professionals on the relevance, viability, and any other aspects of teaching units that are considered relevant to improvement of the units. Appropriate modifications are then made based on the results of this validation process. Stage V: Return of the Results in the Regions Served The final stage is concerned with returning the revised teaching units to each of the regions being served, including a Teacher’s Guide that brings together not only the units themselves but also the theoretical foundations of mathematics and ethnomathematics, a region infographic, and the teachers’ reflections on their learning over the training process. The devolution is carried out in workshops, where the teaching units are presented, but participating teachers will also be able to try them and possibly use them in the classroom.

5 Final Considerations One of the most poetic quotes of D’Ambrosio, for these authors, is “mathematics is powerful enough to help us build a civilization with dignity for all, in which iniquity, arrogance, violence and bigotry have no place, and in which threatening life, in any form, is rejected” (D’Ambrosio, 2004, p. IX). This philosophy inspired the work developed in Costa Rica. The proposed model stimulates reflection in teachers on the significant connections between mathematical content, context, culture, and society, to promote the development of empathy, consideration, and the skills necessary for the appreciation and education of all students, in particular those who are marginalized due to their social status and their cultural, economic, or political environments. This educational model allows teachers: 1. To develop professionally as researchers and mindfully improve their teaching practice. 2. To promote meaningful learning with attention to cultural relevance through a process of enculturation, which covers both theoretical and methodological aspects. 3. The approach for the study of mathematical knowledge in a regional context of the teachers involved in investigations of ethnomathematics in their own environments. Rosa and Orey (2021) state that the promotion of the understanding of diverse and alternative mathematical ideas and practices developed by our vernacular

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cultures teaches us that we need to empower not only our students, but our scholars as well, about the knowledge developed by their own culture. As the project matures and extends itself, this requires ongoing research and dissemination of findings about both historical aspects and current features of culture, so that teachers can use and adapt this information. Results of work carried out by the teachers or by their colleagues are also required to provide true contextualization of teaching. All this represents a real challenge to discover living ethnomathematics and attend to diversity from a global framework, as proposed by UNESCO (Unesco, 2012). The model, as described here, seeks to contribute to and enrich pedagogical activities in the Ethnomathematics Program, creating a training process that will allow teachers to build ethnopedagogical spaces in their classes. This will enable educators to meet a need evidenced by D’Ambrosio (2016) – that of teaching students about real-world problems and instilling in them the desire to seek and work toward this goal, since students who do not value or acknowledge their own cultural roots may easily assimilate the dominant culture without critically reflecting on its values. Ethnomathematics draws on the sociocultural experiences and practices of diverse groups of students, their communities and society in general, using them not only as vehicles to make mathematical learning more meaningful and useful but also to provide students with the perceptions that mathematical knowledge is embedded in diverse environments and is part of historical knowledge that has not been properly recognized or valued. That is, D’Ambrosio contributions to education through the Ethnomathematics Program had a high impact and have been enthusiastically accepted by the teachers who have met them. Furthermore, they have been means for people belonging to cultures that have been marginalized to feel proud of their legacy. Through the divulgation of D’Ambrosio’s legacy, this team is honored to be part of the hundreds of scholars that could get “in the paths of the search for peace and social justice through the appreciation and respect for mathematical knowledge developed locally by members of different cultures” (Rosa & Orey, 2021, p. 443) and bring his ideas to local teachers.

References Albanese, V., Santillán, A., & Oliveras, M. L. (2014). Etnomatemática y formación docente: el contexto argentino [Ethnomathematics and teacher training: The Argentine context]. Revista Latinoamericana de Etnomatemá tica, 7(1), 198–220. Albanese, V., Adamuz-Povedano, N., & Bracho-López, R. (2017). Development and contextualization of tasks from an ethnomathematical perspective. In A.  Chronaki (Ed.), Mathematics education and life at times of crisis (pp. 205–211). University of Thessaly Press. Aroca, A. (2010). Una experiencia de formación docente en Etnomatemáticas: estudiantes afrodescendientes del Puerto de Buenaventura, Colombia [An experience of teacher training in Ethnomathematics: Afro-descendant students from the Port of Buenaventura, Colombia]. Educaçao de Jóvens e Adultos, 28(1), 87–96.

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Bishop, A.  J. (1988). Mathematics education in its cultural context. Educational Studies in Mathematics, 19, 179–191. Bishop, A. J. (1995). Educando a los “culturizadores matemáticos” [Educating the “mathematical culturizers”]. UNO: Revista de Didá ctica de las Matemá ticas, 6, 7–12. Bishop, A. J. (1998). Equilibrando las necesidades matemáticas de la educación general con las de la instrucción matemática de los especialistas [Balancing the math needs of general education with those of specialist math instruction]. Suma: Revista sobre Enseñanza y Aprendizaje de las Matemá ticas, 27, 25–37. Bishop, A. J. (1999). Enculturación matemática, la educación matemática desde una perspectiva cultural [Mathematical culture, mathematics education from a cultural perspective]. Paidós. Bishop, A. J. (2001). Lo que una perspectiva cultural nos cuenta sobre la historia de las matemáticas [What a cultural perspective tells us about the history of mathematics]. UNO: Revista de Didáctica de las Matemáticas, 26(8), 61–72. Chavarria, G., & Albanese, V. (2021). Problemas matemáticos en el caso de un currículo: Análisis con base en el contexto y en la contextualización [Mathematical problems in the case of a curriculum: Analysis based on context and contextualization]. Avances de Investigación en Educación Matemática, 19, 39–54. Chavarría, J., Albanese, V., García, M., Gavarrete, M. E., & Martínez, M. (2017). Ubicación espacial y localización desde la perspectiva sociocultural: validación de una propuesta formativa para la enculturación docente a partir de Etnomatemáticas [Spatial location and location from the sociocultural perspective: validation of a formative proposal for teacher enculturation based on Ethnomathematics]. Revista Latinoamericana de Etnomatemática, 10(2), 26–38. D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the learning of Mathematics, 5(1), 44–48. D’Ambrosio, U. (2004). Preface. In F. Favilli (Ed.), Ethnomathematics and mathematics education (V–X). Tipografia Editrice Pisana. D’Ambrosio, U. (2006). Ethnomathematics: Link between traditions and modernity. Sense Publishers. D’Ambrosio, U. (2007). La matemática Como ciencia de la sociedad [Mathematics as a science of society]. In J. Giménez, J. Diez-Palomar, & M. Civil (Eds.), Educación Matemática y Exclusión (pp. 83–102). Graó. D’Ambrosio, U. (2008). Etnomatemática. Eslabón entre las tradiciones y la modernidad. Limusa. D’Ambrosio, U. (2016). An overview of history of ethnomathematics. In M. Rosa, U. D’Ambrosio, D. Orey, L. Shirley, W. Alangui, P. Palhares, & M. E. Gavarrete (Eds.), Current and future perspectives of Ethnomathematics as a program (pp. 5–10). Springer. D’Ambrosio, U. (2018). Etnomatemática, justiça social e sustentabilidade [Ethnomathematics, social justice and sustainability]. Estudos Avançados, 32(94), 189–204. García, B. (2015). Competencias matemáticas, expectativas de aprendizaje y enculturación matemática [Mathematical competences, learning expectations and mathematical enculturation]. Escenarios, 13(1), 22–33. García, M., & Chavarría, J. (2016). Propuesta de un Modelo Metodológico para el abordaje de las Didácticas específicas en la carrera Bachillerato y Licenciatura en la Enseñanza de la Matemática de la Universidad Nacional (Final report) [Proposal of a Methodological Model for the approach of the specific Didactics in the Baccalaureate and Bachelor’s degree in the Teaching of Mathematics of the National University (Final Report)]. Universidad Nacional. Gavarrete, M. E. (2012). Modelo de aplicación de etnomatemá ticas en la formación de profesores indígenas de Costa Rica [Ethnomathematics application model in the training of indigenous teachers in Costa Rica]. Unpublished doctoral thesis. Departamento de Didáctica de la Matemática, Universidad de Granada, Spain. Gavarrete, M. E., & Albanese, V. (2015). Etnomatemáticas de signos culturales y su incidencia en la formación de maestros [Ethnomathematics of cultural signs and its impact on teacher training]. Revista Latinoamericana de Etnomatemática, 8(2), 299–315. ISSN: 2011-5474.

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Gavarrete, M.  E., & Albanese, V. (2018). Abordar la ubicación espacial y el Plano cartesiano desde la Interculturalidad [Addressing the spatial location and the Cartesian plane from Interculturality]. UNO: Revista de Didáctica de las Matemáticas, 82, 23–30. Gavarrete, M.  E., Chavarría, J., & Martínez, M. (2016). Museo de historia y filosofía de las matemáticas: evolución y alcances Para la formación docente en Costa Rica [Museum of history and philosophy of mathematics: Evolution and scope for teacher training in Costa Rica]. In E. Mariscal (Ed.), Acta Latinoamericana de Matemática Educativa (pp. 1107–1113). Comité Latinoamericano de Matemática Educativa. Gavarrete, M.E., Albanese, V., Martínez, M., García, M., & Chavarría, J. (2017). Enculturación Matemática y Etnomatemática: fundamentos teóricos, metodológicos y empíricos de un proyecto de formación docente en Costa Rica [Mathematical and Ethnomathematical Enculturation: Theoretical, methodological, and empirical foundations of a teacher training project in Costa Rica]. In Federación Española de Sociedades de Profesores de Matemáticas (Eds.), Libro de Actas del VIII Congreso Iberoamericano de Educación Matemática (pp. 360–368). CIBEM. Gerdes, P. (1985). Conditions and strategies for emancipatory mathematics education in undeveloped countries. For the Learning of Mathematics, 5(1), 15–20. Gerdes, P. (1998). On culture and mathematics teacher education. Journal of Mathematics Teacher Education, 1(1), 33–53. Gómez-Chacón, I.  M. (2010). Tendencias actuales en investigación en matemáticas y afecto [Current trends in research in mathematics and affect]. In M. M. Moreno, A. Estrada, J. Carrillo, & T. A. Sierra (Eds.), Investigación en educación matemá tica XIV (pp. 121–140). SEIEM. Ministerio de Educación Pública. (2012). Programas de Estudio Matemáticas. Educación General Básica y Ciclo Diversificado [Mathematics study programs. Basic general education and diversified cycle]. Ministerio de Educación Pública de Costa Rica, Costa Rica: Author. Naresh, N. (2015). The role of a critical ethnomathematics curriculum in transforming and empowering learners. Revista Latinoamericana de Etnomatemática, 8(2), 450–471. Naresh, N., & Kasmer, L. (2018). Using Ethnomathematics perspective to widen the vision of mathematics teacher education curriculum. In T. Gau Bartell (Ed.), Toward equity and social justice in mathematics education (pp. 309–326). Springer. Neupane, R., & Sharma, T. (2016). Crafting cultural intelligence in school mathematics curricula: A paradigm shift in Nepali school education. RIPEM, 6(1), 285–308. Oliveras, M. L. (1995). Artesanía andaluza y matemáticas, un trabajo transversal con futuros profesores [Andalusian crafts and mathematics, a transversal work with future teachers]. UNO: Revista de Didáctica de las Matemáticas, 6, 73–84. Oliveras, M.  L. (1996). Etnomatemá ticas. Formación de profesores e innovación curricular [Ethnomathematics. Teacher training and curricular innovation]. Comares. Oliveras, M. L. (2005). Microproyectos para la educación intercultural en Europa [Microprojects for intercultural education in Europe]. UNO: Revista de Didáctica de las Matemáticas, 38, 70–81. Oliveras, M. L., & Gavarrete, M. E. (2012). Modelo de aplicación de etnomatemáticas en la formación de profesores para contextos indígenas en Costa Rica [Ethnomathematics application model in teacher training for indigenous contexts in Costa Rica]. Revista Latinoamericana de Investigación en Matemática Educativa, 15(3), 339–372. Orey, D., & Rosa, M. (2015). Three approaches in the research field of ethnomodeling: Emic (local), etic (global), and dialogical (glocal). Revista Latinoamericana de Etnomatemá tica, 8(2), 364–380. Panes Chavarría, R., Friz Carrillo, M., Lazzaro-Salazar, M., & Sanhueza, S. (2018). Matemática, cultura y práctica docente: Un análisis de creencias y elecciones socioculturales [Mathematics, culture and teaching practice: An analysis of sociocultural beliefs and choices]. BOLEMA, 32(61), 570–592. Parker, F., Bartell, T.  G., & Novak, J.  D. (2017). Developing culturally responsive mathematics teachers: Secondary teachers’ evolving conceptions of knowing students. Journal of Mathematics Teacher Education, 20(4), 385–407.

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Presmeg, N. (1998). Ethnomathematics in teacher education. Journal of Mathematics Teacher Education, 1(1), 317–339. Rosa, M., & Gavarrete, M.  E. (2016). Polysemic interactions between ethnomathematics and culturally relevant pedagogy. In M. Rosa, U. D’Ambrosio, D. Orey, L. Shirley, W. Alangui, P. Palhares, & M. E. Gavarrete (Eds.), Current and future perspectives of ethnomathematics as a program (pp. 23–30). Springer International Publishing. Rosa, M., & Gavarrete, M.  E. (2017). An Ethnomathematics overview: An introduction. In M.  Rosa, L.  Shirley, M.  E. Gavarrete, & W.  V. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education (pp. 3–19). Springer. Rosa, M., & Orey, D. (2012). O campo de pesquisa em etnomodelagem: As abordagens ê mica, ética e dialética [The field of research in ethnomodeling: Emic, etic and dialectic approaches]. Educação e Pesquisa, 38(4), 865–879. Rosa, M., & Orey, D. C. (2013). Ethnomathematics: Connecting cultural aspects of mathematics through culturally relevant pedagogy. Mathematics Education and Society, 8(3), 887–897. Rosa, M., & Orey, D. C. (2021). Ubiratan D’Ambrosio: Celebrating his life and legacy. Journal of Humanistic Mathematics, 11(2), 430–450. Rosa, M., D’Ambrosio, U., Orey, D. S. L., Alangui, W., Palhares, P., & Gavarrete, M. E. (2016). Current and future perspectives of ethnomathematics as a program. Springer International Publishing. Rosa, M., Orey, C. D., & Gavarrete, M. E. (2017). El Programa Etnomatemáticas: Perspectivas Actuales y Futuras [The ethnomathematics program: Current and future perspectives]. Revista Latinoamericana de Etnomatemática, 10(2), 69–87. Shirley, L. (1998). Ethnomathematics in teacher education. In M. L. Oliveras & J. Fuentes (Eds.), Ethnomathematics and mathematics education: Building an equitable future. Proceedings of First International Conference on Ethnomathematics (CD-ROM). Granada. Shirley, L. (2001). Ethnomathematics as a fundamental of instructional methodology. ZDM, 33(3), 85–87. Shirley, L., & Palhares, P. (2016). Ethnomathematics and its diverse pedagogical approaches. In M.  Rosa, U.  D’Ambrosio, D.  Orey, L.  Shirley, W.  Alangui, P.  Palhares, & M.  E. Gavarrete (Eds.), Current and future perspectives of ethnomathematics as a program (pp.  13–17). Springer. UNESCO. (2012). Challenges in basic mathematics education. Author.

ISGEm and NASGEm: Two Elements of the D’Ambrosio Intellectual Legacy Tod L. Shockey, Patrick (Rick) Scott, and Frederick (Rick) Silverman

Abstract  Fettweis (Rohrer A, Schubring G, For the Learning of Mathematics 31:35–39, 2011) is credited for coining the term “Ethnomathematics” in his research of the 1930s. During the 1980s, D’Ambrosio began an international conversation using the term, unknowing of Fettweis’ use earlier. Barton (For the Learning of Mathematics 19:32–35, 1999) acknowledges D’Ambrosio’s 1984 International Congress on Mathematics Education address as “establishing the field in its contemporary form” (p. 32). At the 1985 National Council of Teachers of Mathematics Annual Meeting held in San Antonio, Texas, Kilpatrick “emphasized the importance of Ethnomathematics” two nights after a talk given by Bishop on “The Social Dimensions of Mathematics Education in Research” (Scott P, International Study Group on Ethnomathematics Newsletter. 1.1. https://web.nmsu.edu/~pscott/isgem. htm, p.1, 1985a). According to the first newsletter of the International Study Group on Ethnomathematics (ISGEm), published in the fall of 1998 (Scott P, International Study Group on Ethnomathematics Newsletter. 1.2. https://web.nmsu.edu/~pscott/ isgem.htm, 1985b), D’Ambrosio was convinced of the importance of Ethnomathematics, which led to the formation of ISGEm, International Study Group on Ethnomathematics. Gloria Gilmer served as the first chair of ISGEm, and Rick Scott served as the ISGEm newsletter editor. As Ethnomathematics was gaining international momentum, particularly as a Working Group at the 2000 International Congress on Mathematics Education in Tokyo, the North American Study Group on Ethnomathematics was formed in 2002 with Lawrence Shirley serving as the first president. This chapter explores the historical records of ISGEm T. L. Shockey (*) University of Toledo, Toledo, OH, USA e-mail: [email protected] P. (Rick) Scott University of New Mexico, Albuquerque, NM, USA e-mail: [email protected] F. (Rick) Silverman University of Northern Colorado, Greeley, CO, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_5

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and NASGEm to illustrate the timelines of these global groups through the lens of D’Ambrosio’s intellectual legacy. Keywords  Ethnomathematics · International Study Group on Ethnomathematics History

1 Introduction The 1985 National Council of Teachers of Mathematics (NCTM) annual meeting in San Antonio, Texas, was the confluence of scholarship that led D’Ambrosio to share with Gloria Gilmer, Gilbert Cuevas, and Rick Scott “that the concept of Ethnomathematics had generated enough interest that it was time to form a study group” (Scott, 1985a, p. 1). D’Ambrosio had been discussing Ethnomathematics in various presentations according to Scott (1985a), but it was at the 1985 NCTM meeting where Jeremy Kilpatrick “emphasized the importance of Ethnomathematics” in his address and the keynote given by Alan Bishop in the research pre- session highlighted “The Social Dimensions of Mathematics Education in Research.” In an available room at the 1985 NCTM conference, D’Ambrosio, Gilmer, Cuevas, and Scott discussed and formed the International Study Group on Ethnomathematics (ISGEm). It is noteworthy that in 1985 D’Ambrosio published his seminal paper, “Ethnomathematics and Its Place in the History and Pedagogy of Mathematics” in For the Learning of Mathematics. In this meeting, Rick Scott agreed to serve as the newsletter editor for ISGEm. Scott’s important work as ISGEm editor began with the publication of the August 1985 ISGEm Newsletter, which serves as the foundation for chronicling the birth and growth of ISGEm. An outcome of this 1985 meeting was that each of these four pioneers reached out to colleagues who had a shared interest in Ethnomathematics, thus beginning a global network of scholars and teachers. In the first ISGEm newsletter, Scott had included an introductory bibliography. By newsletter two in the spring of 1986, Claudia Zaslavsky and Marcia Ascher were involved and had opened up the conversation pertaining to how Ethnomathematics might be defined. In her book Africa Counts, Zaslavsky had used the term “Sociomathematics” which she wrote to the founding ISGEm group stating “I assume that Ethnomathematics is synonymous with ‘Sociomathematics’” (Scott, 1986, p. 1). It was in this newsletter that Marcia Ascher revealed that she and her husband had written a paper titled “Ethnomathematics” unaware of D’Ambrosio’s use of the word. Ascher’s paper appeared in 1986 in History of Science (Ascher & Ascher, 1986, pp. 125–144) a journal published in England. This is included here to indicate that Ethnomathematics was very quickly becoming a field of interest and inquiry within an international audience. In the 1985 NCTM annual meeting, D’Ambrosio had proposed that Ethnomathematics be a special session at the Sixth Inter-American Conference on Mathematics Education held in Guadalajara, Mexico, as well as at the 1986 NCTM Annual Meeting.

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Momentum was gaining for Ethnomathematics, and through their contacts, D’Ambrosio, Gilmer, Cuevas, and Scott had organized an international network which began to reveal scholarship occurring globally. At the 1986 NCTM Annual Meeting, the International Study Group on Ethnomathematics had grown to 60 members from 11 countries (Scott, 1986). The autumn ISGEm Newsletter of 1986 included publications of research occurring globally through a newsletter section titled “Have You Seen.” Founding members, identified in the newsletter as the “ISGEm Advisory Board,” shared continuing plans for ethnomathematical presentations to occur at upcoming meetings. It was at this ISGEm business meeting (held during the spring 1986 NCTM annual meeting) that the conversation opened for ISGEm to begin the process of becoming an affiliate of both the International Congress on Mathematical Instruction (ICMI) as well as the NCTM according to Scott (1986). At the 1987 NCTM Annual Meeting, ISGEm held a “Research Section on Ethnomathematics.” In this session, chaired by Gloria Gilmer, D’Ambrosio spoke on “Socio-Cultural Bases of Mathematics Education: Research Status Worldwide,” and Marilyn Frankenstein gave a talk on “Teaching Mathematics in a More Useful Way to Public and Community Service Workers” (Scott, 1987a). The 1988 International Congress on Mathematics Education (ICME) held in Budapest included an ISGEm presence, particularly Theme 7 – Curriculum: Towards the Year 2000 co-chaired by John Malone and Cristine Keitel. According to Scott (1987a), “The mathematics curriculum in the Year 2000 and the changing character of social demands on, and need for, mathematics looks particularly relevant to Ethnomathematics,” and a call was announced in the newsletter for those interested to make contact and to send submissions for Theme 7. It was at this 1987 meeting that Claudia Zaslavsky, assisting with seeking funding to support ISGEm projects, and Elsa Bonilla of Mexico, translating ISGEm newsletters to Spanish, were announced as new members of the ISGEm Advisory Board. As an organization, ISGEm was establishing its presence in the global mathematics education community. It is also noteworthy that the growing bibliography published in the ISGEm Newsletter was very broad in what was included that may be said to also “look particularly relevant to Ethnomathematics.” One of the impacts ISGEm was having through the newsletter was sensitizing global scholars to existing literature that fit in the emerging theoretical framework of Ethnomathematics. Below is a list of the initial publications listed in the ISGEm bibliography through the September, 1987 ISGEm Newsletter (Scott, 1987b). Alice, M. (1986). Hypatia’s heritage. Beacon Books Ascher, M. (1981). Code of the Ouipu: A study in media, mathematics and culture. University of Michigan Press Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1982). Na vida dez, na escola, zero: Os contextos culturais de aprendizagem de mathematica. Cuadernos de Pespuisa, 42, 79–85 Carraher, T.  N., & Schliemann, A.  D. (1983). Escola: Algoritmos ensinados a estragias. Revista Brasileira de Estudos Pedagogicos, 64, 234–242

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Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in the schools. British Journal of Developmental Psychology, 3, 21–29 Carraher, T.  N., Carraher, D.  W., & Schliemann, A.  D. (1987). Written and oral mathematics. Journal for Research in Mathematics Education, 18(2), 83–97 Closs, M. (1986). Native American mathematics. University of Texas Press D’Ambrosio, U. (1985). A methodology for Ethnoscience: The need for alternative epistomologies. Theoria Segunda Epoca (San Sabastian), 1(3), 397–409 D’Ambrosio, U. (1986). Matematica per paesi ricchi e paesi poveri: anologie e differnze. L’Educazione Matematica, 1(2), 187–197 D’Ambrosio, U. (1986). Culture, cognition and science learning. In J. J. Gallager & G. Dawson (Eds.), Science education and cultural environment in the Americas (pp. 85–92). NSTA/NSF/OAS D’Ambrosio, U. (1986). Some reflections on the western mode of thought. In E. Hat-tori (Ed.), Science and the boundaries of knowledge: The prologue of our cultural past (Final Report of Venice Symposium). UNESCO Hemmings, R. (1980). Multi-ethnic mathematics. New Approaches in Multicultural Education, 8 Hunting, R.  P. (1985). Learning Aboriginal world view and ethnomathematics. Western Australia Institute of Technology Huygens, C. (1986). The pendulum clock or geometric demonstration concerning the motion of pendula as applied to clocks (J. Blackwell, Trans.). University of Iowa Press Moore, C. B. (1982). The Navajo culture and the learning of mathematics. NIE Moore, C.  B. (1985). Cat’s cradle, mathematics, minorities and Kurt Vonnegut. Northern Arizona University Scribner, S., & Cole, M. (1974). Culture and thought. Wiley and Sons Sjoo, M., & Mor, B. (1987). The great cosmic Mother. Harper & Row Wilson, B. (1984). Cultural contexts of science and mathematics education: A bibliographic guide. Center for Studies in Science Education Yanez Cossio, C., & Jerez, A. (1984). Elementos de analisis en matematicas quichua v castellano. Pontifico Universidad Catolica de Ecuador Zaslavsky, C. (1985). Bringing the world into the math class. Curriculum Review, 24, 62–65 The attentive reader may find errors in this bibliographic listing; every attempt was made to be as accurate as possible. Through the newsletter, ISGEm was seeking contributions from its growing global membership. By 1988, ISGEm’s promotion of Ethnomathematics was taking hold. At the 1988 NCTM annual meeting, the first panel discussion, David Davison, Gloria Gilmer, Rick Scott, and Claudia Zaslavsky, was held. In the same year, Professor D’Ambrosio was the chair for “What Can We Expect from Ethnomathematics?” which was part of the International Congress on Mathematics Education (ICME-6) agenda. It is reported that “Mexican mathematics teachers responded enthusiastically to what one of them called the ‘novedosa etnomatemáticas’,” a response to a plenary address at the 1987 National Association of Teachers of Mathematics in

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Mexico meeting held in Jalapa. In 1988, led by Gloria Gilmer, a panel discussion on “The Role of Ethnomathematics at the University Level” included Marcia Ascher, Arthur Powell, and Solomon Garfunkle. Finally, ethnomathematics was on the agenda for the Second Central American and Caribbean Meeting held at the University of San Carlos in Guatemala in the spring of 1988. By the fall of 1988, the ISGEm Newsletter, English version, was being sent to approximately 200 recipients, and the Spanish version to another 200 folks, mainly in Latin America. While no count was provided, D’Ambrosio was personally sending copies to folks throughout Brazil. In 3 years, the number of countries receiving the ISGEm Newsletter had grown from 11 to 45. It was in 1988 that ISGEm established a constitution, produced by Luis Ortiz-­ Franco, for tax purposes in the United States. One outcome of the constitution was the establishment of defined roles for individuals, and in 1989 Gloria Gilmer was recognized as the President of ISGEm. It was also in 1989, through a request on the membership form, that ISGEm was informed of the research of David Kufakwami Mtetwa who was doing his dissertation “students’ beliefs about mathematics in Zimbabwe.” David went on to state that he planned “to do research aimed at uncovering the Ethnomathematics of various groups in Zimbabwe and bringing it into the classroom. In particular I wish to study the Ethnomathematics of Zimbabwe traditional medicine practice.” This is noteworthy as it is the first mention of a dedicated research agenda, particularly for a new academic beginning his career as a scholar. As ISGEm was working toward an ethnomathematics definition, Scott (1990) reported the following from the spring 1990 ISGEm business meeting: Ubiratan D’Ambrosio discussed the meaning of Ethnomathematics. He pointed out that most Mathematics as taught in schools is eurocentric. There is usually the assumption that the mathematical practice of various cultures lacks a theoretical basis and is non-academic. ISGEm is trying to correct those assumptions. The “Ethno” refers to any identifiable cultural group, “math” is a way of understanding reality and “tics” means a technique. Therefore, Ethnomathematics is a technique for understanding reality used by a cultural group. Greek-based mathematics is but one strand.

Readers may be aware that there were many definitions for ethnomathematics emerging from scholars around the world. Contained in the spring 1990 newsletter were reports of Turner’s research in Bhutan, Gilmer’s work using ethnomathematics in curriculum development, and an introduction to Gerdes’ upcoming book Geometry of the African Sona. It seems at this time, through the ISGEm membership, ethnomathematical scholars were connecting, and there was a much broader body of work than may have been previously understood. ISGEm’s newsletters through ensuing several years begin to include short papers on presentations and research being conducted. While ethnomathematical scholarship was being published globally, there was no “central” outlet, a role that the newsletter was taking on. In the spring of 1998, ISGEm announced the first International Congress on Ethnomathematics, hosted by Maria Luisa Oliveras at Universidad de Granada. In this short span of 8 years, ethnomathematics had grown from a topic of conversation to a field of study and application to teaching that now

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included an international audience of scholars and teachers that were coming together to share their findings. According to Lawrence Shirley: In the 90s, and especially after the First International Conference on Ethnomathematics in Granada, Spain, in 1998, a growing number of non-US members were joining--and, significantly, starting their own national “branches” of ISGEm. We had a feeling we were too US-centered to act like we were the international organization. I don’t remember exactly when, but around 2000 or 2001, we renamed ourselves as NASGEm, an affiliate of ISGEm.

In 2003, the North American Study Group on Ethnomathematics, under the presidency of Lawrence Shirley, published their first newsletter. This publication included the minutes for the ISGEm business meeting, which was a regularly occurring event at the National Council of Teachers of Mathematics Annual Meetings. In this newsletter, following the lead of ISGEm, NASGEm was publishing work by ethnomathematical scholars. Shockey and Orey served as the editors, and it is worth sharing that the newsletter had the “appearance” of a journal as discussions were underway for a dedicated journal for ethnomathematical scholarship. By 2007, Wiseman and Engblom-Bradley had taken over the NASGEm newsletter, certainly elevating it to a new status. Since 1985 the International Study Group on Ethnomathematics, the North American Study Group on Ethnomathematics, and numerous other groups around the globe have been supporting the research program put forward by D’Ambrosio. Nearly 15 years ago, the Journal of Mathematics and Culture (JMC) was organized as an outlet for ethnomathematical scholars. The commitment to establish the JMC took place following a small gathering that included Tod Shockey, Jim Barta, Miriam Leiva, Rick Silverman, and Ron Eglash, if memory serves us correctly, at an NCTM annual meeting. The Journal has a dedicated editorial board that considers submissions in Arabic, English, Italian, Portuguese, and Spanish. The interested reader is directed to https://nasgem.wordpress.com/ to learn more about the history of ISGEm and NASGEm. In closing this chapter of the book, at least a few words about the character and humanity of the man, Ubiratan D’Ambrosio, Ubi, as we called him, are in order. He was in many ways a renaissance man, well versed in philosophy, for example, as his writings demonstrate. He was humble, consistently rejecting, when others said, that he was the founder of Ethnomathematics, rather indicating only that he was among those who brought attention to the presence of mathematics as an integral activity in the lives of people of diverse cultures, various professions whose members do not take direct notice of the mathematics that occurs naturally in what their practitioners do, and such. He cared deeply about children’s learning of mathematics and wrote with their teachers in mind, as well as with mathematics educators and mathematicians and their students in mind. He brought his advocacy for a broad view of mathematics to diverse audiences focused variously on mathematics itself, as well as to conferences that included teachers Pre-K through high school grades and also to those of other disciplines. One of his opinion pieces that appeared in Teaching Children Mathematics offered insights from ethnomathematics for relating mathematics and culture in the processes of learning and teaching of mathematics

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(D’Ambrosio, 2001). Ubi exhibited inspiration, and let us add, charisma, to those of us who had opportunities to be with him personally; those qualities were accompanied by his passion, so evident in his personal presentations. His powerful, humane, and intelligent influence on us and others strongly impelled us to follow his example with our students and colleagues. In 2022 the 7th International Congress on Ethnomathematics will bring together international scholars and teachers working in ethnomathematics, a legacy of honor, respect, and love for Professor Ubiratan D’Ambrosio.

References Ascher, M., & Ascher, R. (1986). Ethnomathematics. History of Science, 24(2), 125–144. Barton, B. (1999). Ethnomathematics: A political plaything. For the Learning of Mathematics, 19(1), 32–35. D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44–48. D’Ambrosio, U. (2001). In my opinion: What is ethnomathematics, and how can it help children in schools? Teaching Children Mathematics, 7(6), 308–310. Rohrer, A., & Schubring, G. (2011). Ethnomathematics in the 1930s – The contribution of Ewald Fettweis to the history of ethnomathematics. For the Learning of Mathematics, 31(1), 35–39. Scott, P. (1985a). International Study Group on Ethnomathematics Newsletter. 1.1. https://web. nmsu.edu/~pscott/isgem.htm Scott, P. (1985b). International Study Group on Ethnomathematics Newsletter. 1.2. https://web. nmsu.edu/~pscott/isgem.htm Scott, P. (1986). International Study Group on Ethnomathematics Newsletter. 2.1. https://web. nmsu.edu/~pscott/isgem.htm Scott, P. (1987a). International Study Group on Ethnomathematics Newsletter. 3.1. https://web. nmsu.edu/~pscott/isgem.htm Scott, P. (1987b). International Study Group on Ethnomathematics Newsletter. 3.2. https://web. nmsu.edu/~pscott/isgem.htm Scott, P. (1990). International Study Group on Ethnomathematics Newsletter. 6.1. https://web. nmsu.edu/~pscott/isgem.htm

Ubiratan D’Ambrosio as Historian of Mathematics and Science Luis Carlos Arboleda

Abstract  D’Ambrosio’s participation in a number of events in history, epistemology, and science education in the early 1980s will be discussed initially. This will illustrate his interests at the time and, in turn, highlight his contribution to the professionalization of these disciplines in Latin America, particularly in Colombia. The second part will examine three facets of Ubi’s activity as a historian of mathematics that earned him the Kenneth O. May Medal. These facets were his work for the renewal and expansion of the field by introducing the epistemological perspective of Ethnomathematics, his original point of view on the pedagogical appropriation of the history of mathematics based on Ethnomathematics, his contributions to the history of mathematics in Latin America and to the study of non-Western scientific cultures, and his criticism of Eurocentrism and intellectual and social colonialism. Finally, we will present the efforts made by Ubi, together with other presidents and founding members of the Latin American Society for the History of Science and Technology (SLHCT), to manage institutional spaces favorable to the development of the Latin American history of science and mathematics and also to contribute to its visibility at the international level. Keywords  Ubiratan D’Ambrosio · History of mathematics · History of science · Mathematics education · Ethnomathematics · Latin American science

1 My First Personal Encounters with Ubi If memory serves me well, my first meeting with Ubi was in Campinas, and not in Mexico City as it should have been. We both had been appointed members of the Latin American Council of the Latin American Society for the History of Science and Technology-SLHCT at its constitutive meeting in August 1982  in Puebla. L. C. Arboleda (*) Universidad del Valle, Cali, Colombia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_6

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However, Ubi could not attend this meeting, as I understand, because at that time he was leaving his position as Chief of the Unit of Curriculum of the Organization of American States and was returning to his country to assume the functions of ProRector for University Development at UNICAMP. We would have to wait until the following year to meet at the installation of the Advisory Council of the SLHCT, which took place in Campinas, Brazil, from February 21 to 25, 1983. Months later, he came to Bogota with several members of the SLHCT to the International Seminar on Methodology for the Social History of Science in Latin America (November 2–4, 1983). Ubi oriented his talk at the event to highlight the importance of taking into account in historical research, along with the specific aspects of science and mathematics, the social and cultural aspects derived from the process of colonization in our countries (D’Ambrosio, 1993) (Fig. 1). In this sense, Ubi contributed to the establishment in the region of new and enriching lines of critical thinking about “colonial science” and, in general, about scientific practice. Ubi articulated in his presentations both the sociological and historical views of his early work on goals and aims of mathematics education, with his recent experience of participation in the famous Pugwash Conferences, created by Einstein, Russell, and other scientists, philosophers, and humanists to analyze the relations between science and global issues. Underlying this critical approach was their concern about the deplorable state of the world, in which “the enormous progress of science, technology, and now, techno-science, all strongly depending on mathematics, do not represent progress in the global quality of life. What to do about it as mathematicians and mathematics educators? I believe the challenge is not only to advance more mathematics and to teach it better, but to restore human values and ethics to our practices as mathematicians and mathematics educators” (Magalhães Gomes, 2006; p. 2). Fig. 1  Visit of SLHCT directors to the historical site of the Astronomical Observatory of Santafé de Bogotá in the framework of the International Seminar on Methodology for the Social History of Science in Latin America. (November 2–4, 1983). With Jorge Arias de Greiff, director of the Observatory in the center: Ruy Gama, Juan José Saldaña, Ubi, and Luis Carlos Arboleda

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The aforementioned seminar in Bogota was decisive for the implementation of the Social History of Science Project in Colombia, which would become a milestone in the establishment of this new field of study in the country. Within the framework of this project supported by Colciencias, the state agency for the promotion of science and technology in Colombia, and the OAS, activities of various kinds were carried out in the Social History of Science, among which the most important result was the publication in 1993 of an interdisciplinary work that compiled in ten volumes the results of several years of research (Quevedo, 1993). The first of these volumes on Theoretical-Methodological Foundations contains the works of the seminar including Ubi’s work on Social History of the Sciences: Methodological Aspects (Quevedo, 1993; vol. 1, p.  179–183). With respect to these projects on national histories in S&T, it should be recalled that in those years the publication of the first collections of history of science and technology in the region (Mexico, Brazil, Colombia, etc.) was booming. These are still fundamental references in various academic and professional circles. Thus, the presence at the seminar of the international guests, the representatives of the SLHCT and the researchers of the Colciencias Project, is precisely associated with a fundamental turn in the institutionalization and professionalization of the history of science in Colombia. A year later Ubi returned to Bogota to participate in the International Conference on the Nature of Epistemological Inquiry (February 5–8, 1984), organized by the International Physics Center, the Colombian Society of Epistemology, and the National University of Colombia. The objectives of the Conference were to highlight the figure of Professor Carlo Federici and his work in logic and mathematics teaching and to promote in the country the development of epistemological reflection in itself and in its relations with education. Ubi presented a paper on The concept of time and its epistemological implications. After his first two visits to Bogota, Colombia, mentioned above, Ubi made a third visit to Cali to participate in the Latin American Seminar on Alternatives for the Teaching of the History of Science and Technology (Universidad del Valle, Cali, November 4–10, 1984). This seminar inaugurated the SLHCT section on this thematic area created under my responsibility at the previous year’s SLHCT Board meeting in Campinas. At that time another section was also formed, coordinated by Ubi and dedicated to the methodology of research in the history of science in the Latin American region. In the lively discussions of the Cali seminar, Ubi made valuable suggestions on the pedagogical uses of the history of science and mathematics based on his own experience on the practice of the mathematics educator. In this regard, he was named Fellow of the American Association for the Advancement of Science that same year, with the mention “For imaginative and effective leadership in Latin American Mathematics Education.” At the 1984 Cali Seminar, Ubi proposed a theoretical framework for inquiring into the past and present of scientific activity in our cultures, using an epistemology that was open, flexible, and less restricted than the epistemology of academic and institutionalized science. It was an alternative program of Ethnoscience, conceived to design and implement initiatives in History and Science Education in such a way as to be able to “understand at the same time both Western science and other forms

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of knowledge of a scientific nature, structured according to an ordering and logic substantially different from those of the former and which, therefore, allow us to consider them as ‘other sciences’” (D’Ambrosio, 1986). In the short term, the reflection on this approach contributed to feed our methodological elaborations on the cultural dynamics of knowledge both in the Seminar on History of Sciences that we taught with several colleagues at the Universidad del Valle and in the interdisciplinary and interinstitutional group of the project of Social History of Sciences in Colombia (Fig. 2). In the following months, Ubi’s ideas on the relationship of ethnomathematics with the history and pedagogy of mathematics began to spread in our university circles through some of his publications and writings (D’Ambrosio, 1985, 1992b). The exchanges of professors and students of the Institute of Education and Pedagogy of Cali with Ubi on these topics became more and more frequent. Gradually, the ethnomathematics approach was incorporated into the lines of research of our groups, both in the social history of mathematics and in mathematics education. Under his supervision and that of Paulus Gerdes, we were mainly interested in exploring, in the field of ethnomathematics, intercultural and interdisciplinary methodologies to analyze numerical calculation and information registers in the quipu of the Incas; geometrical figures in the sand representations of the Bushoong in Africa; logic in the kinship relationships of the Walpiri in Australia; probabilities in the games of the Maori in New Zealand; numerical systems in the Incas, Mayas,

Fig. 2  Latin American Seminar on Alternatives for the Teaching of the History of Science and Technology, Universidad del Valle, Cali, November 4–10, 1984. From left to right, Ubi D’Ambrosio, Clara Lucía Higuera, Jorge Valderrama, Ángel Zapata, Regino Martínez, Pedro Rovetto, LCA, Diana Obregón, Ernesto Rueda, Juan José Saldaña, Consuelo Mariño, Germán Cubillos, Jean-­ Claude Guédon, Blanca Inés Prada, José Luis Villaveces, Jorge Puerta, Rubén Darío Lozano, Margarita Posada (?), Simón Reif, Celina Lértora

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and Yorubas; and geometric figures in the tissues of the Arhuaca mochilas of the Sierra Nevada de Santa Marta in Colombia. In 1994 Ubi returned to the Universidad del Valle, Cali, to inaugurate a cycle of meetings of the Mathematics Education Group at the Institute of Education and Pedagogy. His talk on Mathematics and Citizenship was in the same line of thought as one given in 1990 at the Vancouver group on The role of Mathematics Education in the construction of a more just and democratic society (D’Ambrosio, 1990). Ubi repeatedly disseminated these ideas in several countries of the region, for example, at the first congress of REDUMATE, the Mathematics Education Network of Central America and the Caribbean, on a broader sense of mathematics for social justice (D’Ambrosio, 2014). In addition to his talk, at the 1994 conference, Ubi was positively involved in discussions about the creation of the master’s program in Mathematics Education at the Universidad del Valle. From then on, his role would be decisive in advising the first graduate theses in ethnomathematics and in the development of this area of study in other universities, mainly in the southwestern part of the country. He also stimulated the creation of the Latin American Network of Ethnomathematics and the virtual journal in ethnomathematics at the University of Nariño. Currently, several Colombian PhDs with training in ethnomathematics are working in universities in the country. Others are doing their doctoral studies in this field in Brazil and other countries. All of them recognize that, directly or indirectly, Ubi’s work or his teachings had a significant influence on their initial motivation and the orientation of their work (Blanco, 2006).

2 The Kenneth O. May Award Medalist To better understand Ubi’s contribution to the history of mathematics and science, it is important to refer to the Kenneth O. May Award, which was conferred to him in a solemn session during the 21st International Congress on the History of Science and Technology (Mexico, July 2001), the first to be held outside Europe and the first to be convened under the rubric of Science and Cultural Diversity. The International Commission for the History of Mathematics established this prize in 1977 to honor the memory of the mathematician and historian of mathematics Kenneth O. May and to recognize his distinguished services to the international community, in particular with the publication of the first World Directory of Historians of Mathematics and the creation of Historia Mathematica, one of the most important scientific journals in this field of study. Dirk J. Struik and Adolf P. Yushkevich were the first historians of mathematics to receive the prize at the 18th International Congress for the History of Science in Hamburg in 1989. From 1993 onward, a bronze award medal began to be given along with the prize. The next recipients of the award and medal were Christoph J. Scriba and Hans Wussing at the 19th International Congress of the History of Science in Zaragoza, René Taton at the 20th Congress in Liège in 1997, and Ubiratan D’Ambrosio and Lam Lay Yong at the 21st Congress in Mexico. In 2005 the Kenneth O. May prize

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and medal were awarded to Henk Bos at a special ceremony. The most recent Kenneth O.  May award medalists have so far been Ivor Grattan- Guinness and Rhada Charan Gupta who received it at the 23rd Congress in Budapest in 2009. From my personal point of view, the main issues of Ubi’s activity as a researcher, lecturer, and master’s and doctoral thesis advisor in the history of mathematics, which made him worthy of the highest distinction in this field, were the following: • First, the most obvious to the international community, his contribution to the renewal and expansion of the field of history of mathematics by introducing the epistemological perspective of Ethnomathematics (D’Ambrosio, 2002) • Then, the program of his creation and on which he has been working for more than 25 years, on the pedagogical appropriation of the history of mathematics based on the ethnomathematics approach (D’Ambrosio, 1985, 1992b, 1997) • Finally, his research on the history of non-Western scientific cultures and, in particular, on the history of Latin American science, based on his radical critique of Eurocentrism (D’Ambrosio, 1997, 2000, 2001) (Fig. 3) These three issues had been gaining increasing interest in the communities of historians at the international level and in Latin America throughout the 1990s. It is enough to refer to the Colloquium Science and Empires. A comparative History of Scientific Exchanges: European Expansion and Scientific Development in Asian, African, and Oceanian Countries was organized by the REHSEIS team (Research on Epistemology and History of Exact Sciences and Scientific Institutions) of the CNRS (National Center for Scientific Research) at UNESCO headquarters in Paris, from April 3 to 6, 1990. The SLHCT, then under the presidency of Ubi, played a prominent role in the colloquium. Ubi was in charge of the opening lecture, and Juan José Saldaña participated in the Final Round Table. Arboleda, Dantes, Vargas, Obregón, and Lopes, among other members, presented papers in sections of the event (Petitjean et al., 1992). It is worthwhile to review some of the central ideas of Ubi’s conference where an outline of the abovementioned history of science program can be seen.

Fig. 3 Ubiratan D’Ambrosio receiving the Kenneth O. May Award at the 21° International Congress on the History of Science and Technology in Mexico City, July 2001. (https://www.mathunion. org/ichm/prizes/ kenneth-­o-­may-­prize-­ history-­mathematics)

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Ubi criticizes Eurocentric approaches to history that account only for the winning ideas in the cultural exchanges of human history. He also objects the exclusive emphasis on studying the emission and reception of the mode of thought labeled as science, without concern for the cultural dynamics and cognitive processes of the development of ideas in the encounter between winners and losers. On the contrary, what our countries are concerned with is to extend historiographical interests to the study of the generation, transmission, institutionalization, and dissemination of knowledge. It cannot be ignored that throughout our history and even today, the vast majority of our population survives, produces, reproduces, and explains its reality with a different structure of knowledge (…) Their thinking has contributed to the development of our knowledge, and, for this reason, it should not be considered under the current prejudices of “ad- hoc knowledge,” “superstition,” or “folklore.” Hence, it is the responsibility of the History of Science to investigate alternative epistemologies of a broad nature, as Ethnomathematics has been doing (D’Ambrosio, 1992a).

3 The International Visibilization of the History of Latin American Science The founding document of the SLHCT, known as the Bucharest Declaration, was signed by a dozen Latin American historians who participated in a strictly individual way in the 16th International Congress on the History of Science in 1981. It originated as a concerted desire to articulate at the regional level our activities in the history of Latin American science and to incorporate it as a legitimate object of study in the universal field of the history of science (Bucharest Declaration, 1981). The dynamic deployed in the following 4 years by the SLHCT headed by Juan José Saldaña (whose death in October 2022 we deplore today), Ubi, and other colleagues of the Latin American Board allowed the region to have for the first time an organized presence in an international congress of HCT, in this case the 17th Congress in Berkeley, California. In particular, the papers presented at the Symposium Cross Cultural Transmission of Natural Knowledge and its Social Implications: Latin America were published in a book that had an excellent diffusion (Saldaña, 1988), mainly as a result of the interest aroused among historians of science by the chapter written by Ubi: Socio-cultural influences in the transmission of scientific knowledge and alternative methodologies (D’Ambrosio, 1988). With the choice of Mexico as the venue for the 21st International Congress, the History of Science Division of the International Union for the History and Philosophy of Science (UIHFC) made a significant move that transformed the custom of bringing together researchers from the most varied backgrounds and cultures to meet in congresses organized exclusively in countries of the northern hemisphere. Such a decision was, of course, a recognition of the maturity that Latin American studies on science had already reached, and of their undeniable impact at the

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international level. This was mainly achieved thanks to the systematic articulation of regional activities through the SLHCT; its international dissemination through Quipu, Latin American Journal of HCT (Quipu, 2011); and the presence of Latin American personalities in international leadership positions. The most notable case is Juan José Saldaña who at the time of the congress was serving as the Secretary of the Executive Committee of the UIHFC. I will refer later on to the outstanding positions held by Ubi. The motto Science and Cultural Diversity for the Mexico congress was the most appropriate to highlight the distinctive feature of the Kenneth O. May award that was conferred at the same time to Ubi and Lam Lay Yong. It sanctioned the international community’s recognition of the legitimacy of new objects of historical studies in relation to the conventional subjects of European and Anglo-Saxon academic science (ICHM, 2001). Referring in her notice of the award to Ubi’s contribution to the expansion of new fields in the history of mathematics, Kirsti Andersen, president of the International Commission for the History of Mathematics, states the following: “from now on no serious historian of mathematics could write a general book on the history of mathematics without including ethnomathematics and Chinese mathematics” (Andersen, 2002). In this sense, the Mexico Congress of 2001 meant at the time the realization of the main motive that inspired the Bucharest Declaration 20 years earlier. It is true that from its beginnings the intention of the signatories received numerous encouragements from influential personalities of the international community of the history of science. One of them deserves special mention, René Taton, director of the Alexandre Koyré Center in Paris and award medalist Kenneth O. May. In his opening speech at the 1981 Cali seminar referred to above, Saldaña remembered an anecdote that Taton used to spread in different circles: “that when he was preparing around the 1960s the four volumes of his outstanding Histoire Générale des Sciences (Taton, 1957–1964), he was unable to find someone to write the chapter on Science in Latin America with an overall vision, since the Latin American historians and scientists he consulted were ignorant of the regional scientific process” (Saldaña, 1986). The testimony of Taton and other personalities in the same sense became a challenge, and the new leadership of the SLHCT undertook two major undertakings: to integrate regional activities and to publish collective works on the history of science in our countries. Dirk J. Struik is another awarded medalist Kenneth O. May whom, per force, we must remember when talking about the visibilization of Latin American History of Science, as an honorary member of the SLHCT and member of the Board of Directors of the journal Quipu since its foundation and in connection with other initiatives led by Ubi. For historians of my generation, Struik’s works were the focal point of an approach to the social history of mathematics in which mathematical ideas are intimately related to their respective sociocultural contexts of production. I refer in particular to Concise History of Mathematics (Struik, 1948b) and in general to Yankee Science in the Making (Struik, 1948a). Over the years we would come to understand (mostly indirectly through his personal relationships with Paulus Gerdes and Ubi) that Struik’s social history approach had a profound relationship to

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an anti-colonialist program of science such as that which Ubi and the SLHCT leadership team were promoting in the region. Struik was part of an increasingly representative group of historians from the northern hemisphere who sought to explain scientific activities in countries like ours, in their own cultural dynamics and not in a restricted way as a simple product of Eurocentrism and dominant exogenous cultures. This direction of ideas was already evident in the article that Struik published in the first issue of Quipu in 1984 on science at the beginning of the colonial period in North America and Mexico (Struik, 1984a, b). Because of its content and the notoriety of its author, this work undoubtedly contributed to the good reception of the newly created journal in the international media. Years later, in 1988, following an invitation from Ubi to give lectures at UNICAMP and USP, Struik became so interested in the scientific development of Brazil during the period of the Dutch occupation of the Northeast (1624–1664) that he wrote a paper on this subject for the Revista da Sociedade Brasileira de História da Ciência (D’Ambrosio, 2004).

4 The Latin American Management of Institutional Spaces in History of Mathematics and Science Struik’s case illustrates Ubi’s important role in engaging academic personalities in the history of Latin American science. In fact, one of the highlights of his work in the history of mathematics, mathematics education, and other fields was his commitment to promote the approach of eminent scientists and educators from the northern hemisphere to studies on science and education in the southern hemisphere. Special mention should be made of Ubi’s promotion of regional activities in history and mathematics education during the period 1984–1988 when he served as president, together with Christian Houzel, of the HPM-International Study Group on the Relations between History and Mathematics Pedagogy. This area of studies and the HPM group itself became better known among historians and mathematicians in Latin America as a consequence of Ubi’s efforts to organize discussion spaces on these relations in several events. It is worth mentioning the creation in the Brazilian Society for the History of Mathematics (1983) of a permanent group on the use of history in the teaching of mathematics (an initiative that was quickly welcomed in several countries), the organization of a special section on this topic at the 17th Congress in Berkeley, as well as the introduction of this area of work at the 2nd Latin American Congress on the History of Science and Technology, chaired by him in São Paulo in 1988, and the holding of the HPM Conference in Campinas in 1990 (Magalhães Gomes, 2006). One aspect to highlight in his lectures of those years is his defense of the need for a new methodology in the analysis of the relationship between history and pedagogy of mathematics situated in diverse sociocultural contexts. By this he meant a new critical approach to the generation, transmission, and dissemination of knowledge among well-identified social groups, that is, groups of people who share in their knowledge practices common and

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specific civilizational characteristics, as in the case of the groups studied by ethnomathematics. The Kenneth O. May medal awarded to Ubi and other distinctions for his scientific work in the history of mathematics and science, including his nomination as a member of the International Academy of the History of Science in Paris, should also be considered as recognition of his efforts to give international visibility to the history of science in our region. The list of conferences and congresses in which Ubi participated is impressive, as well as his stays as a guest professor at numerous universities and research centers. But this was not limited to a particular academic purpose, since the same activity of producing original results in the expansion of the conceptual field of history of mathematics and science at the same time allowed him to contribute to form schools of thought and consolidate institutions in this field. There is no other reasonable way to understand his frenetic presence in so many international events. Finally, I cannot fail to refer to the exceptional merit that the academics who interacted with him recognized above all: his pedagogical disposition, his people skills, and his usual willingness to listen and to relate to the most diverse temperaments and personalities. All this helped him to direct his abilities to the creation and direction of different institutions in the history of science in Brazil and other Latin American countries and at the international level. Among them, it is worth mentioning the presidencies of the Latin American Society for the History of Science and Technology (1988–1992), the Brazilian Society for the History of Science (1991–1993), and the Brazilian Society for the History of Mathematics (1999–2007). Equally noteworthy in this area is his participation as a member of the Executive Committees of the International Commission for the History of Mathematics (1989–1997) and of the International Commission for the History of Science (1993–2009) and of the Advisory Board of the Southern Cone Association of Philosophy and History of Science (2000–2004) of which Ubi was a founding member. When mentioning the Brazilian Society for the History of Mathematics, I am obliged to recall the Festschrift Ubiratan D’Ambrosio organized in commemoration of the 75th anniversary of Ubi (Nobre, 2007). This special issue of the society’s journal was attended by numerous members of the global community of historians of mathematics. I have chosen the epigraph of my contribution to the Festschrift as the closing of this tribute (Arboleda, 2007): Para Ubi quien me convenció firmemente, desde los primeros encuentros de una ya larga amistad de la que me enorgullezco, que la “História e a filosofía da matemática não se separam e somos assim levados a refletir sobre a naturaleza do conhecimento matemático.”

References Andersen, K. (2002). The awarding of the Kenneth O. May prize for the fourth time. Retrieved June 16, 2008 from http://www.unizar.es/ichm/may4.html

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Arboleda, L.  C. (ed.) (1986). Seminario latinoamericano sobre alternativas para la enseñanza de la historia de las ciencias y la tecnología. Cali, 4-10 de noviembre de 1984. Instituto Colombiano para el Fomento de la Educación Superior-ICFES, Universidad del Valle. Arboleda, L.  C. (2007). Modalidades constructivas y objetivación del cuerpo de los reales. En (Nobre, 2007), pp. 215–230. Blanco, H. (2006). La Etnomatemática en Colombia. Un programa en construcción. Revista Bolema- Boletim de Educação Matemática, 19(26), 49–75. D’Ambrosio, U. (1985). Ethnomathematics and its place in history and pedagogy of mathematics. In For the learning of mathematics (Vol. 5). FLM Publishing Association. D’Ambrosio, U. (1986). Etnociencia: Alternativa para la historia y la enseñanza de las ciencias. En Arboleda (1986). D’Ambrosio, U. (1988). Socio-cultural influences in the transmission of scientific knowledge and alternative methodologies. En (Saldaña, 1988). D’Ambrosio, U. (1990). The role of mathematics education in building a democratic and just society. For the Learning of Mathematics, 10(3), 20–23. D’Ambrosio, U. (1992a). For a new historiographical approach of the so-called “traditional knowledge”. In P. Petitjean, C. Jami, & A. M. Moulin (Eds.), Science and empires (Boston studies in the philosophy of science) (Vol. 136, pp. 15–17). Springer. D’Ambrosio, U. (1992b). Ethnomathematics: A research program on the history and philosophy of mathematics with pedagogical implications. Notices of the American Mathematical Society, 39, 1183–1185. D’Ambrosio, U. (1993). Historia social de las ciencias: Aspectos metodológicos. En (Quevedo, 1993, vol. 1, pp. 179–183). D’Ambrosio, U. (1997). Ethnomathematics. Challenging eurocentrism, in mathematics education. In A. B. Powell & M. Frankenstein (Eds.), (pp. 13–24). State University of New York Press. D’Ambrosio, U. (2000). Historiographical proposal for non-western mathematics. In H.  Selin (Ed.), Mathematics across cultures. The history of non-western mathematics (pp.  79–92). Kluwer Academic Publishers. D’Ambrosio, U. (2001). A matemática na época das grandes navegações e início da colonização. Revista Brasileira de História da Matemática, 1, 3–20. D’Ambrosio, U. (2002). Etnomatematica. Pitagora Editrice. D’Ambrosio, U. (2004). A Interface entre História e Matemática: uma visao histórico-pedagógica. Site oficial de Ubiratan D’Ambrosio. http://vello.sites.uol.com.br/ubi.htm (Actualización 2004). D’Ambrosio, U. (2014). Um sentido mais amplo da matemática para a justiça social. Cuadernos de investigación y formación en educación matemática, 12, 35–54. Declaración de Bucarest. (1981). Documento preparatorio de la fundación de la Sociedad Latinoamericana de Historia de la Ciencia y la Tecnología. Adoptada en Bucarest, el 1° de septiembre de 1981, durante la celebración del XVI Congreso Internacional de Historia de la Ciencia. https://issuu.com/cihcytal/docs/declaracion_de_bucarestb, http://www.revistaccuba. cu/index.php/revacc/article/view/219 ICHM-International Commission on the History of Mathematics. (2001). The awarding of the Kenneth O. May prize for the fourth time. ICHM web site: https://www.mathunion.org/ichm/ awarding-­kenneth-­ o-­may-­prize-­fourth-­time Magalhães Gomes, M. L. (2006). Interview: Ubiratan D’Ambrosio-Historian and Pedagogue of Mathematics, Former Chair of the HPM Group (1984–1988). HPM Newsletter, n° 61, March 2006, 1–5. Nobre, S. (ed) (2007). Festschrift Ubiratan D’Ambrosio em Comemoração ao 75° Aniversário. Revista Brasileira de História da Matemática. Especial, n° 1, dezemro 2007. Petitjean, P., Jami, C., & Moulin, A. M. (Eds.). (1992). Science and empires (Boston studies in the philosophy of science) (Vol. 136). Springer. Quevedo, E. (ed.) (1993). Historia Social de la Ciencia en Colombia. 10 volúmenes. Proyecto Colciencias OEA, 1983–1986. Tercer Mundo Editores-Colciencias.

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Quipu. (2011). Revista Latinoamericana de Historia de las Ciencias y la Tecnología y la Tecnología. http://revistaquipu.com/index.html (Sitio actualizado el 30 mayo 2011). Saldaña, J.  J. (1986). Balance general de la historia de las ciencias en América Latina. En Arboleda (1986). Saldaña, J.  J. (ed.) (1988). Cross cultural diffusion of science: Latin America. Vol. 5: Acts of the 17th International Congress of History of Science, Berkeley, California, 31 July-8 August 1985. Cuadernos de Quipu, n° 2. Struik, J. D. (1948a). Yankee science in the making. Little, Brown. Struik, D. J. (1948b). A concise history of mathematics. Dover. Struik, J. D. (1984a). Early colonial science in North America and Mexico. Quipu, 1, 25–54. Struik, J. D. (1984b). Early colonial science in North America and Mexico. Quipu, 2, 323–325. Taton, R. (ed.) (1957–1964). Histoire générale des sciences. 3 tomos en 4 volúmenes. Presses Universitaires de France. Réediton (1966–1983).

The APUA – Ubiratan D’Ambrosio Personal Archive and the Research on the Production of New Knowledge: History of Mathematics, Ethnomathematics and Mathematics Education Wagner Rodrigues Valente and Luciane de Fatima Bertini Abstract  The chapter aims to present the Ubiratan D’Ambrosio Personal Archive through a collective research project. The question guiding the project involves analyzing knowledge production in some of the different fields in which Ubiratan D’Ambrosio was a reference. Through APUA, by following D’Ambrosio’s professional trajectory, we consider that we can enable an entryway for studies on the production of new knowledge, its dynamics, and processes given by the constitution of collectives of researchers in the History of Mathematics, Mathematics Education, and Ethnomathematics. How was reference knowledge created for the constitution of the Sociedade Brasileira de História da Matemática, the Sociedade Brasileira de Educação Matemática, and the groups mobilized around ethnomathematics? Documents from personal collections allow us to get closer to the practices involved in the processes and dynamics of knowledge systematization. Work drafts, intermediate papers that enable organizing the knowledge, letters with contacts and exchanges of ideas about projects, selected studies from international and national congresses, and a whole range of materials that allow the construction of new knowledge are present at APUA.  This production is dispersed, not systematized, and gathered in the hundreds of boxes donated to the Documentation Center by Ubiratan D’Ambrosio’s widow, waiting for the constitution of the next phases of the APUA. Keywords  Personal archive · History · Mathematics · Mathematics education · Ethnomathematics

W. R. Valente (*) · L. de Fatima Bertini Universidade Federal de São Paulo, São Paulo, Brazil e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_7

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1 Initial Considerations The way knowledge is produced has been gaining attention from researchers in recent decades. We try to enter laboratories, offices, spaces prepared for researchers’ conferences, and any other place where there may be traces of scientific activities so that we can highlight processes and dynamics involving knowledge production. Some of the many works that have been dedicated to the subject, starting with the titles, are enlightening: “A vida de laboratório – a produção de fatos científicos” [Laboratory life – the social construction of scientific facts] by Latour and Woolgar (1979), “Ciencia: abriendo la caja negra” [The Very Idea] by Woolgar (1988), “Ciência em Ação – como seguir cientistas e engenheiros sociedade afora” [Science in Action  – how to follow scientists and engineers through society] by Latour (1997), “Ciência tal qual se faz” [Science as it is done] by Gil (1999), “A dimensão material do saber, séculos XVI- XXI” [L’Ordre matériel du savoir. Comment les savants travaillent XVIe – XXIe siècles]; [The material dimension of knowledge, sixteenth to twenty-first centuries] by Waquet (2015), and “Nos bastidores da ciência – técnicos, ajudantes e outros trabalhadores invisíveis” [Dans les coulisses de la science. Techniciens, petites mains et autres travailleurs invisibles]; [Behind the scenes of science – technicians, helpers, and other invisible workers] by Waquet (2022). Latour and Woolgar (1979) explain that their studies on the production of knowledge, on the way they are elaborated, based on research in laboratories, give continuity to what became known as the “strong program” formulated by David Bloor (1976): “Bloor’s original idea was to encourage historians and sociologists who were still hesitant to move from a history and sociology of scientists to a history and sociology of the sciences” (p. 22). Woolgar (1988) continues the discussions by emphasizing “an approach designed to oppose the erroneous and idealized portrayals of science and the scientific method, through the revelation of the ‘most delicate’ of science: science as practiced in the laboratory” (p. 128). With the black box metaphor, Latour (1997) uses controversies to analyze how it was possible to establish specific scientific facts steadily for some time. The author points out the lack of interest in many studies on how knowledge is elaborated: “(...) unfortunately, almost no one is interested in the process of building science. They shy away from the chaotic mix revealed by science in action and prefer the organized contours of scientific method and rationality” (1997, p. 34). In the words of Fernando Gil, regarding the theme that involves scientific production, in the presentation of the book under his organization, “(...) the intention here is to present the doing of science, not facts and theories. If it is not always possible to make known the what, nothing prevents us from understanding the how of scientific knowledge” (1999, p. 9). Be that as it may, the analysis of the processes and dynamics that involve the production of new knowledge having as a strategy ethnographic studies in scientific laboratories – analyses of controversies that precede the dissemination of scientific

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production, among other expedients  – imply historical-sociological works that attempt to understand how it was possible to systematize a given knowledge in a given time, even if the time is the present time. Based on consolidated knowledge, we ask how this elaboration process occurred. How could it pass from a set of dispersed information to the systematization of knowledge? This is a fundamental issue mentioned by Peter Burke (2015) in his work “What is the history of knowledge?” Historical research on processes and dynamics involved in producing new knowledge has shown different dimensions that have been part of this production in more recent works. In this sense, the title of the work of the researcher Françoise Waquet, quoted above, is emblematic: “L’ordre matériel du savoir: comment les savants travaillent, XVIe-XXIe siècles.” Her book includes material forms considered important in elaborating new knowledge. Who could, until recently, include the role of binders, notebooks, and ways of organizing ideas for writing a text as determining elements in the systematization of new knowledge (Waquet, 2015)? In her last work, the author analyzes the backstage of scientific production, highlighting the important role played by characters hitherto considered invisible. She means technicians and helpers (Waquet, 2022). Modes of knowledge production, changes in those modes, and the history of knowledge refer to the interest of this text. How, at a given time, does the passage of dispersed information to a systematized knowledge occur? This question guides the specific analysis of documentation contained in personal collections. Such archives are considered privileged places of study on processes and dynamics of knowledge production. In particular, we are interested in discussing possibilities for conducting research on the production of new knowledge, taking into account the documents that make up the APUA – Ubiratan D’Ambrosio Personal Archive.

2 About Personal Archives The place of analysis of processes and dynamics that involve the production of new knowledge, in this present study, involves research in personal archives. Such archives will be considered as laboratories of scientific production that keep dispersed documentation accumulated during the life of a person who was present in different crucial moments of knowledge systematization. How can a set of dispersed information turn into a systematized organization of new knowledge? For this study, we will seek the answer to this question in the APUA personal archives. From the outset, it is worth mentioning that this text adopts Heloísa Belotto’s (2004, p. 266) writings as a reference for the typification of a personal archive: […] a set of papers and audiovisual or iconographic material resulting from the life and work/activities of statesmen, politicians, administrators, leaders of professional categories, scientists, writers, artists etc. Anyway, people whose way of acting, thinking, and living may be of interest for research in the areas where they developed their activities; or even

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In Brazil, during the great political repression that marked the 1970s, there was a leap in the appreciation of personal archives as research sources, especially in historians’ work. In the same period, “several documentation centers were created to house private archives, as well as the increase in the presence of these documentary sets in institutions such as museums and libraries” (Departamento, 2015, p. 9). Thus, this study is aligned with the movement of investigating the production of new knowledge to understand how a given knowledge could be consolidated, supported by the systematization of the results of investigations carried out in recent times, and, in this case, considering the personal archives as a research laboratory. All sorts of papers, audiovisual material, photographs, letters, congresses, classes imparted, photocopies of texts, records, and course notebooks, among so many other elements, are selected as research sources to answer the question: how did dispersed information turn into consolidated knowledge? This general, broad question gains specificity in dealing with the APUA, turning attention to the study of processes and dynamics that were present in the configuration of knowledge that enabled the establishment of scientific communities such as the Sociedade Brasileira de História da Matemática [Brazilian Society of the History of Mathematics], Sociedade Brasileira de Educação Matemática [Brazilian Society of Mathematics Education], and the groups mobilized around ethnomathematics. How was the initial knowledge of reference of those communities created? This question can be answered through studies that consider the massive amount of documents that make up the APUA.

3 Non-fiction Biography: Another Writing of Ubiratan D’Ambrosio’s Trajectory Taking the personal documents of a particular character to study usually leads us to biographical studies. After all, the researcher has before him/her a whole range of records of the trajectory followed by a given author, scientist, politician, administrator, and professor, among so many possibilities of professional locations of the character. First, however, we must problematize biography writing before discussing the research possibilities that the APUA may make viable. In historical studies, biography writings suffered a severe blow with Pierre Bourdieu’s criticism in the 1980s. In a reference article, written in 1986, entitled “L’illusion biographique,” [The Biographical Illusion] Bourdieu pondered that: Life history is one of those common-sense notions smuggled into the universe of scientific knowledge; initially tiptoeing into ethnological studies, later, and more recently, parading into sociology. To speak of life history is to presuppose at least – and that means nothing – that life is a history. (1986, p. 69)

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Bourdieu warned of the consequences that writing a life story or biography could bring to history writing itself. To think of a character’s life as a story, similar to writing novels, is to treat a character’s trajectory as guided by a previous project, which follows a chronological order, an origin, a logic. In this sense, the biography character and the biographer have “the same interest in accepting the postulate of the meaning of the told existence (and, implicitly, the meaning of all existence)” (1986, p. 69). So, for Bourdieu: to treat a character’s life through his/her biography as a story, i.e., as a coherent narrative of an event-driven sequence of meanings, is to sacrifice writing to a rhetorical illusion, to an ordinary representation of existence, which all a literary tradition has not ceased to reinforce. (1986, p. 70)

Bourdieu’s critique of biographical writing  – comparing it to fictional writing  – shocked the genre, which had been struggling to constitute itself as historical writing. Thus, historians gave up writing biographies. On the other hand, sometime later, biographical writing resurged, seeking to overcome the weaknesses and anachronisms of the previous way of writing that had built the biographies (Offenstadt, 2004). More recent studies in areas such as the history of mathematics, mathematics education, and the history of mathematics education, such as Toillier’s (2019), show us a growing interest in biography writings. This author inventoried works presented at scientific events held from 2010 onward, bringing together about 100 studies that dealt with biographies in one way or another. Toillier’s (2019) conclusions point to a still empirical, non-historical way of writing biographies in those fields. According to the author, the analyzed studies show a perspective that one learns to write biographies by writing them: Thus, sometimes, one may face the risk that some writing about someone’s life seems to be a homage to praise the character’s remarkable achievements. However, we understand that talking about someone’s life is to go beyond that. It is a search for the meanings of experience, for understanding the spaces and times lived by the subject. It is to understand that our character is not alone but immersed in a society that shapes them and that they help to shape and that these are the ideas that should be part of biographical writing. (p. 11)

Thus, it seems that the areas Toillier (2019) mentioned present biographies that have not yet been dealt with historically; they show us chronologies and, with them, the characters’ outstanding achievements. The historical treatment given to a given character, far from seeking to edify them, from considering their trajectory as a coherent sequence of events they lived, leading to an existential, logical, and coherent objective, is something different. Elaborating a biography from the resumption of biographies by historians takes the biographed as a point of reference, an entry into the research for understanding phenomena that occurred throughout the character’s life, whose importance goes beyond the unique aspects that involved the character’s personal trajectory. Thus, far from the fictional writing of a biography – like so many already existing, edifying texts that pay homage to professor D’Ambrosio – in the research that will mobilize the APUA archives, we are guided by the goal of understanding the

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processes and dynamics that were present in the elaboration of reference knowledge for the establishment of new research communities in Brazil. Instead of a fictional work about D’Ambrosio, we want to write a biography based on the character, through his personal archive, to enter behind the scenes of the production of specific knowledge.

4 Knowledge, Scientific Disciplines, Disciplinary Fields... The question that drives the writing of this text – let us mention once again – involves analyzing knowledge production in some of the different fields in which Ubiratan D’Ambrosio was a reference. Through APUA, by following D’Ambrosio’s professional trajectory, we consider that we can enable an entryway for studies on the production of new knowledge, its dynamics, and processes given by the constitution of collectives of researchers in the history of mathematics, mathematics education, and ethnomathematics. How was reference knowledge created for the constitution of the Sociedade Brasileira de História da Matemática, the Sociedade Brasileira de Educação Matemática, and the groups mobilized around ethnomathematics? This issue makes it imperative to discuss conceptual tools such as knowledge, scientific disciplines, and disciplinary fields, among others. In an article published in 1975, Bourdieu dealt with the “La spécificité du champ scientifique et les conditions sociales du progrès de la raison” [The specificity of the scientific field and the social conditions of the progress of reason], and, still, on the same topic, in 1976, he published the article entitled “Le Champ Scientifique” [The Scientific Field]. Later on, in the College de France, reproducing the course he taught at the institution in the academic year 2000–2001, Bourdieu published the book “Science de la Science et Réflexivité” [Science of Science and Reflexivity]. In the book, the author returns to the category of field, according to himself, incorporating theoretical advances and obtaining new implications that this concept could allow. Following Bourdieu’s presentation, the first element to mention indicates that using the notion of field makes it possible to break with assumptions tacitly accepted by most of those interested in science. Thus, a first rupture implies disregarding the idea of the existence of “pure” science, perfectly autonomous and developing according to its internal logic, and also within a “scientific community” (Bourdieu, 2001). For the author, “talking about the field means breaking with the idea that scientists form a unified, homogeneous group” (2001, p. 91). Still, the idea of the field subverts the thinking that the scientific world is a place of generous exchanges in which all researchers collaborate for the same purpose. The notion of field implies taking into account the relative autonomy of the scientific groups considered in relation to the broader social universe. This means, more precisely, that:

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[...] the system of forces that are constitutive of the structure of a field (tension) is relatively independent of the forces exerted on the field (pressure). In any case, it has the “freedom” necessary to develop its own needs, its own logic, its own nomos. (Bourdieu, 2001, p. 95)

In this way, the use of terms such as “science,” “scientists,” “mathematicians,” “mathematics,” and “mathematical educator,” among others, becomes less fertile. Therefore, we are interested, instead, in considering “the mathematical disciplinary field,” “the field of mathematics education,” “the sciences of education,” “the professional field of teaching,” and the relationships that the individuals in these fields share – tensions, as Bourdieu would say, those that the members of a given field maintain with that same field. Furthermore, it is important to study, over time, the relationships established between different fields. The intellectual, scientific knowledge is shown to be systematized within the scope of the scientific disciplines, housed in disciplinary fields. Historian José D’Assunção Barros characterizes scientific disciplines considering that: (...) each discipline is unique, understood here as the set of its defining parameters or as what makes it truly unique, specific, and which justifies its existence – in short: that which defines the discipline in question by opposition or contrast with other disciplinary fields. (2010, p. 207)

On the other hand, in opposition to the singularity that distinguishes a given scientific discipline: (...) it will be necessary to understand the opposite phenomenon: although each field of knowledge certainly presents a singularity that makes it unique and gives it identity, in fact, there is not a single disciplinary field that is not constructed and constantly reconstructed by interdisciplinary dialogues (and oppositions). (2010, p. 207)

Barros (2010) also analyzes the emergence of a new disciplinary field and its dynamics: (...) the process of emergence of a new disciplinary field sometimes seems much more to be a real fight inside the scientific arena than a childbirth. And this struggle, as well as the bonds of solidarity that are also established between the new and old fields of knowledge, all take place within intense and necessary interdisciplinarity, in the face of which what is new must appear before an already established knowledge and sometimes institutionally already consolidated. (2010, p. 207)

With this theoretical-methodological tool, it is possible to work with the APUA to analyze new knowledge to be systematized from all the material collected by D’Ambrosio in his professional trajectory. Certainly, being historical, such knowledge is transformed. However, our interest is to know which references built the research practices within the history of mathematics, mathematics education, and ethnomathematics areas that guided the construction of repertoires for the establishment of researchers’ collectives within those fields. The constitution of references, of repertoires to be used by researchers, is a determining trait in the consolidation of new scientific disciplines. Barros (2010) clarifies that:

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W. R. Valente and L. de Fatima Bertini Of course, a disciplinary field does not develop in the sense of having only a single theoretical or methodological orientation, but to present a specific theoretical-methodological repertoire that must be considered, and that becomes known by its practitioners, generating adhesions and several criticisms. In the same way, the development of a disciplinary field generates a common language through which its exponents, theorists, practitioners, and readers can communicate. There are even disciplinary fields that eventually generate some repertoire of jargon that can be easily recognized, even externally. Anyhow, any disciplinary field, as it is constituted, is also inscribed in a specific discourse modality, sometimes with internal dialects.

Thus defined the reference knowledge, how it is produced constitutes a challenge to the investigators and researchers of the history of knowledge. As mentioned above, we seek to understand specifically the processes and dynamics of the elaboration of reference knowledge in the constitution of researchers’ collectives within the SBMat, SBEM, and ethnomathematics groups. Therefore, it is the analysis of the constitution of a vocabulary that beginners and research practitioners in different groups must have. The analysis of the elaboration of this vocabulary and its references and how they were systematized derives from the historical research that involves the APUA documentation.

5 The GHEMAT Documentation Center: Collections and Examples of the History of the Production of New Knowledge In 2000, encouraged by D’Ambrosio himself, a documentation center was created to gather personal documents from former mathematics teachers. Thus, modestly, in two small adjoining rooms at the Pontifical Catholic University of São Paulo (PUC/ SP), personal documents from professors began to be gathered, such as those of Euclides Roxo (1890–1950), leading to the creation of APER – Arquivo Personal Euclides Roxo [Euclides Roxo’s Personal Archive]. APER was created from the donation of Stélio Roxo, Professor Roxo’s son, in 2000. The documents cover from 1909 to 1955. It comprises a total of 624 documents.1 The production of APER documents has been analyzed considering the international and national contexts of mathematics teaching. In this sense, Roxo’s documents have constituted a gateway to the analysis of knowledge in mathematics teaching. Regarding the international context, in 1908, in Rome, mathematicians showed interest in discussing issues related to teaching at an international mathematics congress. To this end, an international commission was created to study mathematics teaching. Once the commission was constituted, a central committee led by the mathematician Félix Klein was elected (CIEM, 1908, p. 446).   The APER summary handle/123456789/173456 1

inventory

can

be

consulted

at:

https://repositorio.ufsc.br/

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A relatively long time passed between international discussions on mathematics teaching and curriculum changes in Brazil, probably due to the lack of places of representation made up of educators, mathematicians’ associations, or bodies with specificity to deal with educational issues at the national level. From the end of the 1920s onward, debates between different positions on how to treat mathematics teaching intensified. There were tensions between mathematics as a disciplinary field and mathematics teaching. In fact, at that time, it would be an anachronism to call the place occupied by teachers who taught mathematics courses in Brazilian secondary education a “mathematical disciplinary field,” as most were engineers. Due to the creation of philosophy colleges in Brazil, it was only from the late 1930s onward that the country began having teachers graduate in mathematics 2020. The many debates related to mathematics teaching led us nationally to Euclides Roxo, himself an engineer. Roxo held the position of director of Colégio Pedro II in Rio de Janeiro, a model institution for secondary education in Brazil, all aligned with French high schools. His post as the director gave him the status of minister of education when no such ministry existed. Roxo promoted changes in secondary school mathematics at the end of the so-called Old Brazilian Republic, the period between the Proclamation of the Republic (1889) and what became known as the 1930 Revolution. The interlocutor of international proposals, Roxo was also a member of the ABE  – Associação Brasileira de Educação [Brazilian Education Association] (founded in 1924), having maintained close dialogue with exponents and leaders of primary education and teacher education. Roxo also taught classes at the Instituto de Educação do Rio de Janeiro, qualifying primary school teachers. Due to these characteristics, Roxo moved away from the typical high school mathematics teacher, close to the mathematical disciplinary field, who, in general, had little or no didactic-pedagogical training. Due to his professional background, Roxo fought a long battle to include new knowledge in teacher education beyond strictly mathematical knowledge (Valente, 2004). Newspaper excerpts, drafts of proposals for curricular organization, letters to authorities and teachers, and a whole set of documents present in APER attest to the movement of mathematics modification that should be present in secondary education. Euclides Roxo engaged in transforming the curriculum for mathematics teaching, seeking to assert what seems to have been the most important point of the movement Félix Klein led: bringing differential and integral calculus to elementary school through initiation to the study of functions. As an integrating concept of arithmetic, geometry, and algebra, Roxo sought in US textbooks a new organization for teaching and writing works that would integrate those mathematical branches until then taught separately. Through Roxo’s actions, the school subject Mathematics was institutionalized in Brazil, merging the different branches. All the innovations we analyzed from Roxo’s documentation placed him at the center of debates and tensions in which, on the one hand, there were the few secondary teachers with insertion in the field of educational sciences and, on the other hand, engineers with no affinity with the educational discussions, but clinging to mathematics. This triggered, from the first decades of the twentieth century, a public

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debate on mathematics and the teaching of mathematics. This debate was more focused on the field of mathematics teaching, on the professional field. In the spotlight was the mathematics that should be taught. The disputes focused on teaching programs, guidelines for pedagogical work, mathematics textbooks, etc. Who would have the authority to give references for what should be taught in mathematics and how it should be taught? The Brazil of the 1920s – agrarian, with a mostly illiterate population and with few isolated colleges – was beginning industrialization, taking advantage of the opportunities that emerged in the First World War, and aiming to be modernized. Roxo freely used the authority argument by evoking teaching in more advanced countries. He took Félix Klein as a major reference and put together what he considered proposals for US pedagogical practices to make it possible to merge arithmetic with geometry and algebra in the construction of mathematics as a school subject. All those transformations related to the knowledge developed in the 1930s, the knowledge that a mathematics teacher should have to exercise his/her profession, contrast with those disciplinary knowledge that made up the model 3+1, precisely at the time of creation of the mathematics course, where the 3 years brought together the subject matters of the mathematical field (among them the analytical and projective geometry; mathematical analysis; and vector calculus). If, on the one hand, mathematical research spreads through its different branches, the mathematics teacher’s professional knowledge was a fusion of them, in an elaboration of the professional field of teaching in dialogue with international teaching trends. The analysis of the APER documents included letters between Euclides Roxo and political leaders; Roxo’s letters and books exchanged with other professors; documents proving his actions in the elaboration of the first national curriculum for mathematics teaching, made official by the Francisco Campos Reform; newspaper excerpts reporting debates about the new knowledge that should be part of the teaching work, to merge the old mathematical subject matters into a single rubric entitled Mathematics; and many other documents allow us to understand processes and dynamics of production of new knowledge, especially the knowledge of the Brazilian secondary school mathematics teachers from 1930 to 1950. The mobilization of APER documents allowed the elaboration of several studies under the common aegis of analysis of how new knowledge was elaborated from dispersed information contained in Roxo’s documents. Examples of these studies are the works of Duarte (2002) and Braga (2003). Another essential collection we must mention, given the writing of non-fictional biographies, i.e., studies that take a character as a gateway to understanding the systematization of new knowledge, is Osvaldo Sangiorgi’s documentation. Professor Osvaldo Sangiorgi’s daughters donated his personal collection, comprising 1600 files. The documents include photos, books, letters, and intellectual production, among many other papers.2 Similarly to what was done with Euclides

  The APOS summary handle/123456789/173403 2

inventory

can

be

consulted

at:

https://repositorio.ufsc.br/

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Roxo’s documentation, when analyzed, all of Sangiorgi’s documentation needs to be entangled in the national and international context in which Sangiorgi was the icon of mathematics teaching. Then, redemocratization came. On the international scene, the end of the Second World War left a legacy of victory over fascist-oriented totalitarian regimes. Domestically, there was an agglutination of opposing forces to overthrow Getúlio Vargas’s regime in 1945. For the new times, the construction project of Brasília became the icon of political-economic modernity: Brasília, the project, and São Paulo – the city and the state – reality. São Paulo had around 239,820 inhabitants in 1900; half a century later, it had 2,662,786. São Paulo became the largest Brazilian metropolis and, at the same time, the largest industrial center in Latin America, generating more than 50% of all industrial production in the country (Sevcenko, 2000, p. 104). On the other hand, the growth level in São Paulo in the 1950s can be evaluated from the average income per inhabitant. During this period, it was twice the national average. From the 1940s to the 1950s, the state changed its socioeconomic profile, which could be realized in the change in employment from primary to secondary and tertiary activities. The result was massive urbanization, with an increase of around 160% in the population that moved to urban areas (São Paulo, 1962, pp. 17–19). In the educational field, especially in secondary education, the number of enrollments in schools virtually doubled in a decade, reaching a total of 360 thousand students in 1960 (São Paulo, 1962, p. 36). In the state of São Paulo, until the 1940s, the network of state middle schools (ginásio, final years of elementary school) consisted of 37 establishments in the countryside and 3 in the capital; in 1950, there were 143 of them in the countryside and 12 in the capital; and in 1958, the numbers reached 294 schools in the countryside and 65 in the capital (Sposito apud Bontempi Jr., 2006, p. 140). Following this trajectory of enormous growth in the number of middle schools and the school population of this level of education, a growing number of textbooks were produced. Companhia Editora Nacional, founded in the 1920s by Monteiro Lobato, was at the forefront of this escalation, bringing together a group of authors who practically dominated the production of mathematics textbooks. Among the great authors were Jacomo Stavale, Ary Quintella, and Osvaldo Sangiorgi. In São Paulo, in the 1950s, coffee farming began to be replaced by industrialization. It constituted the foundation of São Paulo’s prosperity, with its headquarters in the capital, already in an advanced integration process with neighboring municipalities, in a complex multidirectional expansive process that gave rise to the so-called Greater São Paulo. In parallel, a new emerging social stratum began to compose the local elite, which was basically formed by industrial entrepreneurs linked to families of more or less recent immigration (Sevcenko, 2000, p. 104). These elite’s children were privileged by being tutored by the best teachers their economic conditions could afford. Private classes and preparatory courses became an important source of income for those professionals. Osvaldo Sangiorgi was an example of those excellent teachers disputed by the wealthy families of São Paulo to give private lessons to their children. At the time,

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good teachers, recognized and advertised for their students’ achievements, held the social status of liberal professionals. Teachers from traditional schools interspersed their day-to-day regular classes with private lessons. Publishers, it seems, followed this movement and invited them to write textbooks. This is what seems to have happened with Sangiorgi in the 1940s and 1950s. Osvaldo Sangiorgi was born on May 9, 1921. His training included a degree in mathematical sciences in 1941, as stated in his diploma, awarded by the Philosophy, Sciences, and Letters College, Education Section, of the University of São Paulo. Osvaldo Sangiorgi began his professional life teaching at the Instituto Feminino de Educação Padre Anchieta – called Normal School, a high school level institute dedicated to qualifying teachers for primary school – in the Brás neighborhood, in São Paulo. Sangiorgi organized his mathematics course through Ary Quintella’s books. His memories led him to state that he started writing textbooks because he was welcomed by the Cia. Editora Nacional. Sangiorgi also reiterated that during the 1940s to 1950s, this publishing house “kept an eye” on good teachers, proposing that they write textbooks (Sangiorgi, 2004). Based on his didactic-pedagogical experience with the mathematical training of the Normal School students, Sangiorgi certainly felt motivated to prepare one of his first publications by Cia. Editora Nacional: the book Matemática e Estatística [Mathematics and Statistics], destined to the institutes of education and Normal Schools. In the first edition of April 1955, the book had 10,030 copies, according to the publisher’s Mapa das Edições [Maps of Editions], which, today, belongs to the institution’s historical collection (Valente, 2008). This first publishing success was followed by a collection of works for the middle school: Matemática  – curso ginasial [Mathematics  – middle school]. In the 3 years following the release of the volume for the first grade of the middle school, Sangiorgi’s collection was very well accepted. The circulation did not stop rising, reaching, in 1957, for the first volume, the landmark of 100,000 copies. From then on, it remained with this yearly circulation until 1963, when, according to the files of the Cia. Editora Nacional, the 134th edition of the book was published (Valente, 2008). Considering that the school population of the entire secondary education in the state of São Paulo, from the 1950s to the 1960s, as seen above, doubled, reaching 360,000 students, we can see how expressive the numbers reached by the collection Matemática – curso ginasial by Osvaldo Sangiorgi (Valente, 2008) were. The unprecedented growth in the number of middle schools in the state and capital of São Paulo, combined with a greater degree of flexibility for the states of the federation to organize their own secondary education and the accelerated development of the São Paulo publishing park in the production of textbooks, will characterize the educational environment in the late 1950s. At this point, Osvaldo Sangiorgi was already recognized as the most important reference for mathematics teaching. A great author of textbooks, Sangiorgi carried with him the mathematical, didactic authority, and experience of a great articulator of joint actions between the Cia. Editora Nacional and the Education Department

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for the promotion of meetings and courses for teachers. In those courses, the reference was his didactic works. Between June and August of 1960, Sangiorgi left for the United States to participate in an improvement course, with a scholarship from the Pan American Union and National Science Foundation, in an internship at the University of Kansas. Back in Brazil, Sangiorgi soon promoted articulations between teachers, the media, and the São Paulo State Education Department, aiming to modify the mathematics programs, similar to what he saw in the United States. The newspaper Folha de São Paulo, on October 11, 1960, reported: “São Paulo teachers aim to reform the programs and methods of teaching mathematics.” In the text, they informed that the Education Department, in its general restructuring plan, created a working group to study mathematics teaching, coordinated by Professor Osvaldo Sangiorgi (Nakashima, 2007). After this report, a true bombardment of news about the changes that mathematics would undergo became a topic in the print media. Above all, the newspapers of São Paulo followed each of Osvaldo Sangiorgi’s steps and initiatives around the changes in mathematics teaching toward the so-called modern mathematics. They reported on courses for teachers, with time-off, by the Education Department; the arrival of foreign researchers for lectures; the creation of GEEM – Mathematics Teaching Study Group, coordinated by Osvaldo Sangiorgi; congresses on mathematics teaching and modern mathematics; and Sangiorgi’s interviews and testimonies, among other news about the modern teaching of mathematics (Nakashima, 2007). The entire scenario built for the entry of modern mathematics into Brazilian education culminated in the launching of a collection of textbooks. It was in mid-1963, in middle schools, to be used in the 1964 school year. That year, the Cia. Editora Nacional launched more than 240,000 copies of Volume 1, of the work Matemática – curso moderno by Osvaldo Sangiorgi (Valente, 2008). At a time of enormous growth in the school population along with the publication of textbooks, these works were a fundamental reference for the professional knowledge of mathematics teachers and their practice. The mathematics textbooks guided the teachers’ work in the school routine. In the second half of the twentieth century, Osvaldo Sangiorgi, moving through the most different fields – education departments, publishing houses, print and even television media (TV Cultura), and GEEM, among many others – promoted and disseminated his didactic works, making them true best sellers (Valente, 2008). Initially, such productions were the result of the systematization of their teaching experiences in the use of other didactic works of mathematics. Later, when modern mathematics was in force, they added new knowledge for teaching, taking into account books and courses from the United States. The APOS documents allow the analysis of changes in the professional knowledge of mathematics teachers and their transformations from the 1950s to the 1980s. When elaborating new mathematics teaching programs during the MMM, Sangiorgi’s work left behind what would later become official curricular documentation for mathematics teaching. The professional field of mathematics teaching in

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middle schools allowed this author to systematize new knowledge that made him an icon of mathematics teaching in Brazil due to the success of his textbooks. Results of studies that mobilized APOS can be read in the studies of Nakashima (2007) and Oliveira Filho (2009).

6 The APUA – Ubiratan D’Ambrosio Personal Archive Since the beginning of the constitution of the Center, Ubiratan D’Ambrosio began to donate part of his documents, hundreds of books, and thousands of texts and materials related to his professional and research trajectory, housed in two apartments of his. With D’Ambrosio’s live donation, APUA – phase I and phase II – was created.3 These two phases of inventory of the donated documentation initially demonstrate that the APUA is composed of a mass of documents with a great diversity of topics, including medicine, arts, education, technology, history, and mathematics, and mail sent and received by Ubiratan D’Ambrosio from the 1950s to the present day. The material is cataloged and gathered in around 500 folders that include numerous documents from his participation in conferences, colloquia, symposia, and scientific congresses and articles written by him and by Brazilian and foreign mathematicians and mathematics educators and professionals from other areas. The collection also includes drafts of published books; various projects and teaching programs, theses, and dissertations; overhead projector transparencies of courses that D’Ambrosio imparted in Brazil and abroad and handwritten or typed speeches by himself and others; newspapers and magazines containing articles by him and other authors; photographs and photo negatives of various events with personalities with whom Professor D’Ambrosio came into contact at the congresses; and opinions referring to articles that had been sent to journals, on various topics and by various authors, among others. A tiny part of this entire mass of documents corresponding to phases I and II was taken for research for the production of the work “Ubiratan D’Ambrosio” (Valente, 2007). This work aimed to disclose the chronological biography of D’Ambrosio and his professional relationships with former advisees. The book includes chapters that directly mobilized APUA documents, such as the text by Maria Cristina Araújo de Oliveira entitled “A formação matemática de um matemático e educador matemático” [The mathematical training of a mathematician and mathematics educator]. Oliveira analyzed the files D’Ambrosio prepared as a student in the mathematics course at the University of São Paulo in the first half of the 1950s. From those class sheets, the author could build “a brief genealogy of the mathematics course attended by Ubiratan D’Ambrosio” (2007, p. 72).  The separation phase is linked to the different donation flows of D’Ambrosio’s materials. The APUA inventory – phase I and phase II – can be consulted at https://www.ghemat.com.br/centrode-documentacao. A PDF of the inventory of all documentation can be obtained at https://repositorio.ufsc.br/handle/123456789/173452 3

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Over time, the Documentation Center came under the exclusive custody of GHEMAT – Research Group on the History of Mathematics Education (www.ghemat.com.br). Moved from PUC/SP in 2008, it gained a new space in a more suitable environment, provided by a private school in the west of São Paulo. The increase in the area for storing the collections made it possible for D’Ambrosio to continue donating materials, documents, and books, transforming the APUA into the largest collection in the Documentation Center. A new stage of cataloging the APUA was inaugurated: phase III. After Ubiratan D’Ambrosio died in 2021, his wife, Dona Maria José, contacted the Center and made new and huge donations of D’Ambrosio’s documentation, now tripling the existing material in volume, announcing a new and extensive phase of cleaning, cataloguing, and inventorying thousands of documents (phase IV). In 2022, the space previously given to the Documentation Center was requested by the private school where the collections were kept. GHEMAT then looked for a new and not temporary space and acquired a large commercial room in Santos, a coastal city in the state of São Paulo. The new Documentation Center, with official opening scheduled for May 2023, is in the process of being reorganized, financially supported by institutions such as FAPESP (Fundação de Amparo à Pesquisa no Estado de São Paulo) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico). Such support is fundamental for purchasing special papers that help preserve documents, proper boxes for storing materials, and other essential elements for cleaning and keeping the collections. Technical assistance grants awarded by CNPq made possible improvements in the computerizing and digitization of the collection. All these ongoing activities accredit the Documentation Center beyond its use by projects directly linked to GHEMAT. As a result, the Center is increasingly being recognized as a space open to researchers and those interested in topics linked to mathematics, mathematics teaching, and ethnomathematics, among other areas.

7 Final Considerations The construction of the history of knowledge involves the challenge of answering the question: How has dispersed information become consolidated knowledge throughout history (Burke, 2015)? The work being developed through the GHEMAT Documentation Center privileges documentation contained in personal collections. The analysis of this varied documentation will allow, in a given historical time, to analyze, from the APUA, how the knowledge systematization linked to scientific communities such as SBHMat, SBEM, and ethnomathematics groups, among other collectives, occurred. Documents from personal collections allow us to get closer to the practices involved in the processes and dynamics of knowledge systematization. Work drafts, intermediate papers that enable organizing the knowledge, letters with contacts and exchanges of ideas about projects, selected studies from international and national

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congresses, and a whole range of materials that allow the construction of new knowledge are present at APUA. This production is dispersed, not systematized, and gathered in the hundreds of boxes donated to the Documentation Center by Ubiratan D’Ambrosio’s widow, waiting for the constitution of the next phases of the APUA. Acknowledgments  The authors would like to acknowledge support from CNPq and FAPESP.

References Barros, J. A. (2010). Contribuição para o estudo dos “campos disciplinares”. Revista ALPHA Patos de Minas: UNIPAM, 11, 205–216. Bellotto, H. L. (2004). Arquivos permanentes: tratamento documental (2nd ed. rev. ampl). Ed. FGV. Bloor, D. (1976/2009). Conhecimento e imaginário social. Edunesp. Bontempi, B., Jr. (2006). Em Defesa de ‘Legítimos Interesses’  – o ensino secundário no discurso educacional de O Estado de São Paulo (1946-1957). Revista Brasileira de História da Educação, Campinas, 12, 122–159. Bourdieu, P. (1986). L’illusion biographique. In Actes de la recherche en sciences sociales (Vol. 62–63, pp. 69–72). https://doi.org/10.3406/arss.1986.2317 Bourdieu, P. (2001). Science de la Science et Réflexivité (p. 2001). Éditions Raisons d’Agir. Braga, C. (2003). O processo inicial de disciplinarização de função na Matemática do ensino secundário. Dissertação (Mestrado em Educação Matemática). Pontifícia Universidade Católica de São Paulo. Burke, P. (2015). What is the history of knowledge? Polity Press. CIEM. (1908). Rapport Préliminaire sur l’organisation de la Commission et le Plan General de Ses Travaux. L’Enseignement Mathématique, 10. Departamento de Arquivo e Documentação. (2015). Casa de Oswaldo Cruz. Fundação Oswaldo Cruz- Manual de organização de arquivos pessoais. Fiocruz/COC. Duarte, A. R. S. (2002). Henri Poincaré e Euclides Roxo: subsídios para a história das relações entre Filosofia da Matemática e Educação Matemática. Dissertação (Mestrado em Educação Matemática). Pontifícia Universidade Católica de São Paulo Gil, F. (Coord.). (1999). A ciência tal qual se faz. Edições João Sá da Costa. Latour, B. (1997/2000). Ciência em Ação – como seguir cientistas e engenheiros sociedade afora. Editora da UNESP. Latour, B., & Woolgar, S. (1979/1997). A vida de laboratório – a produção dos fatos científicos. Relume Dumará. Nakashima, M. (2007). O Papel da Imprensa no Movimento da Matemática Moderna. Dissertação (Mestrado em Educação Matemática)  – Programa de Estudos Pós-Graduados em Educação Matemática, Pontifícia Universidade Católica de São Paulo. Offenstadt, N. (2004). Les mots de l’historien. Presses Universitaires du Mirail. Oliveira Filho, F. (2009). O SMSG e o Movimento da Matemática Moderna no Brasil. Dissertação (Mestrado em Educação Matemática) – Universidade Bandeirante de São Paulo. Sangiorgi, O. (2004). Entrevista concedida aos professores Célia Maria Carolino Pires e Wagner Rodrigues Valente no dia 25 de março de 2004. São Paulo (Estado). (1962). II Plano de Ação do Governo, 1963–1966. Imprensa Oficial do Estado. Sevcenko, N. (2000). Pindorama Revisitada – cultura e sociedade em tempos de virada. Peirópolis. Toillier, J.  S. (2019). Como as biografias são tratadas nos eventos brasileiros de Educação Matemática, História da Educação Matemática e História da Matemática. Anais do XV

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EPREM, Londrina, 10 a 12 de outubro de 2019. Disponível em: http://www.sbemparana.com. br/eventos/index.php/EPREM/XV_EPREM/paper/viewFile/1280/867. Acesso: 13 jul. 2022. Valente, W. R. (Org.) (2004). Euclides Roxo e a Modernização do Ensino de Matemática no Brasil. Editora da Universidade de Brasília. Valente, W.  R. (2007). Ubiratan D’Ambrosio  – conversas, memórias, vida acadêmica, etnomatemática, história da matemática, inventário sumário do arquivo pessoal. Annablume/CNPq. Valente, W. R. (Org.) (2008). Osvaldo Sangiorgi – um professor moderno. L F Editorial. Waquet, F. (2015). L’ordre matériel du savoir. Comment les savants travaillent XVIe – XXIe siècles. CNRS Éditions. Waquet, F. (2022). Dans les coulisses de la science. Techniciens, petites mains et autres travailleurs invisibles. CNRS Éditions. Woolgar, S. (1988/1991). Ciencia: abriendo la caja negra. Editorial Anthropos.

Ubiratan D’Ambrosio and His Contribution to the History of Science and Mathematics Sergio Nobre

Abstract  This text is the translation into Portuguese of part of the text published in the Revista Brasileira de História da Matemática in memory of Professor Ubiratan D’Ambrosio shortly after his death. In it we evidence his trajectory in the area of the History of Mathematics, as a pioneer in approaches for its diffusion in countries considered peripheral. Ubiratan’s contributions to the institutionalization of the History of Mathematics area, internationally and mainly in Brazil and Latin America, are also presented in this text. The personal relationship with my professor, advisor, and intellectual mentor, and especially my great friend, is also part of this text. Keywords  Ubi D’Ambrosio · History of Mathematics in Brazil It was during his undergraduate course in mathematics that Ubiratan became interested in the History of Mathematics. His doctoral dissertation contains, in addition to specific mathematics content, a historical review of the Calculus of Variations. During his time at Brown University, he attended seminars and courses at the Department of History of Mathematics (founded by Otto Neugebauer). Ubiratan was a founding member of the HPM/International Study Group on the Relations Between History and Pedagogy of Mathematics, during the ICME-3/Third International Congress on Mathematics Education, in Karlsruhe, Germany (1976), and Chair of the HPM between 1984 and 1988. Ubiratan D’Ambrosio participated intensely in the History of Mathematics sessions at the AMS/American Mathematical Society and the MAA/Mathematical Translated from the text in Portuguese published in Revista Brasileira de História da Matemática, vol. 21, number 41, as Editorial, 2021. S. Nobre (*) São Paulo State University – UNESP Rio Claro, São Paulo, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_8

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Association of America and in numerous events in several countries, especially the ICHS/International Congress of History of Science, ICM/International Congresses of Mathematicians, and ICME/International Congresses on Mathematics Education, held in the United States, in Europe, and in several other countries around the world. His outstanding participation in these international events allowed him to enter the international scene of the History of Science and Mathematics. Over the years, Ubiratan held positions in important academic institutions linked to the area: he was President of the Latin American Society for the History of Science and Technology, SLHCT, 1988–1992; Member of the Executive Committee of the International Commission on History of Mathematics, ICHM, 1989–1997; Founding Member and President of the Brazilian Society for the History of Science, SBHC, 1991–1993; Member of the Advisory Board of the Association of Philosophy and History of Science of the Southern Cone, AFHIC, 2000–2004; and Founding Member and President of the Brazilian Society for the History of Mathematics, SBHMat, 1999–2007. Among his participations in important events in History of Mathematics, we mention the plenary lecture given at the Workshop History of Mathematics in the seventeenth and eighteenth centuries, at the Mathematisches Forschungsinstitut Oberwolfach, in Germany (1979), “Latin American Mathematics in the Conquest and Early Colonization.” His ideas, presented there, certainly showed those in the audience that a new way of writing scientific history, especially the History of Mathematics, was necessary. With this lecture, Ubiratan called for the need for the emergence of communities of historians of science that would work on subjects considered peripheral. In Brazil this gained strength mainly from the 1990s onward. The role of Ubiratan D’Ambrosio as the articulator of the movement that was then beginning was decisive. On his initiative, different people who were starting to do research in the History of Mathematics in Brazil got to know each other. Although distant, because in addition to Brazilian territorial distance, some were in other countries, Brazilian researchers who carried out their research in the History of Mathematics began to establish contacts, and the first scientific meetings of a national dimension, whose focus was the History of Mathematics, were carried out from 1993 onward, the first ones being carried out at the Federal University of Paraná, in Curitiba. But Ubiratan D’Ambrosio had anticipated this process when, a few years earlier, he had established contacts with the community of historians of mathematics in Portugal. Also through him, Brazilians and Portuguese researchers met.1 In 1987, at the event in memory of the 200th anniversary of the death of the Portuguese mathematician José Anastácio da Cunha (1744–1787), Ubiratan D’Ambrosio was one of the invited speakers, and, from then on, he began to strengthen relations between Portuguese and Brazilian Historians of Mathematics, participating in specific events and indicating Brazilians who worked in the area of the History of Mathematics to  I thank my great friend Luis Saraiva, professor at the University of Lisbon, for the detailed information about the presence of Ubiratan D’Ambrosio in the community of Historians of Mathematics in Portugal. 1

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participate in events promoted by the community of historians in Portugal. It was in this way that I, Sergio Nobre, started my contact with the Community of Historians of Mathematics in Portugal, especially as Prof. Luís Saraiva, when, on the recommendation of Prof. D’Ambrosio, I participated in the Summer School on the History of Mathematics, an important international event, held in 1990 in the city of Évora. In 1993, with the holding of the 1st Luso-Brazilian Meeting on the History of Mathematics, D’Ambrosio’s desire to establish lasting links between the Portuguese and Brazilian communities of historians of mathematics was fulfilled. This meeting, held at the University of Coimbra, was strategically scheduled for the week following the International Congress on the History of Science, which took place in the city of Zaragoza, Spain, with the aim of making it easier for Brazilians present at the International Congress to extend their stays in Iberian Peninsula and be able to go to Coimbra and take part in the Luso-Brazilian Meeting. The 1st Luso-Brazilian Meeting on the History of Mathematics showed to the participants in that event how much research in the area still needed to be done and how essential joint work was necessary. This sealed the possibility of holding new meetings in both countries. So far, eight Luso-Brazilian Meetings have been held, alternatively in Portugal and in Brazil. In Brazil, as mentioned above, from the first meetings in Curitiba, the idea of organizing a National Meeting emerged, so that we could analyze the possibility of holding such meetings in the country. The history of the National Seminars on the History of Mathematics, told below, is proof that Ubiratan D′Ambrosio’s dreams could come true and that the emergence of a specific scientific community of researchers in the History of Mathematics in Brazil would become a reality. The first National Seminar on the History of Mathematics took place in 1995 in the city of Recife, State of Pernambuco, and, at that meeting, the scientific community that attended the meeting became aware that the History of Mathematics, as a scientific area, should be developed. In view of this, it was decided that other national events would be held, in order to strengthen the different groups that were already developing work in the area. Every 2  years the National Seminar on the History of Mathematics is staged in a different place in Brazil. The last one, the 14th, took place in 2021. In the third edition of the National Seminar on the History of Mathematics, in 1999, in the city of Vitória, State of Espírito Santo, the Brazilian Society for the History of Mathematics was founded, with Ubiratan D’Ambrosio as its first president (Fig. 1). Among the numerous activities to strengthen the scientific movement of the History of Mathematics in Brazil, Ubiratan left us an important legacy: his vision, as a mathematician and historian, on Mathematics in Brazil. And this is presented in his book A Concise History of Mathematics in Brazil. D’Ambrosio offers the reader a stimulating invitation to delve deeper into historical research on different topics related to the History of Mathematics in Brazil. In addition to presenting a peculiar and very original view of scientific development in Brazil, the author addresses topics that are practically open, which lack the proper investigative depth. The author highlights in the explanatory note of the book that its objective was “to give a

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Fig. 1  Opening of the First National Seminar on the History of Mathematics, UFRPE, 1995. From left to right: Ubiratan D’Ambrosio, Fernando Raul Assis Neto, Sergio Nobre, and Seiji Hariki

panoramic and critical view of the receptivity of a thought conceived and produced in Europe, brought by the conqueror and the colonizer.” In this sense, this panoramic flight allows the reader to identify specific subjects that appear indicated in the text, in order to expand knowledge about details that have not yet been investigated.

1 Our Academic and Personal Relationship Like a father, he took my hands and taught me the paths to follow. Ubiratan D’Ambrosio was present throughout my academic career, and to him I am eternally grateful. The first contact was marked by what was his way of being and treating people: a great supporter of young people! I was in the first year of my undergraduate course at Unicamp, in 1978, and the V Inter-American Conference on Mathematics Education (V CIAEM) would take place at the beginning of the following year, with Ubiratan being the coordinator/organizer. I told the professor of the Mathematics course that I would like to participate in the event, but I didn’t have the money to pay the registration fee. This teacher asked me to fill in the Registration Form and told me to accompany him. He entered the room of Prof. Ubiratan, then director of the Institute, and said “this undergraduate student wants to participate in the V CIAEM, but has no way of paying the registration fee.” Ubiratan took the form that was in my hands and, without questions, wrote on it: EXEMPT. Participation

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in the V CIAEM was decisive for the professional career that I would follow. Ubiratan was still my teacher in some undergraduate subjects, and he was my master’s supervisor. Following his teachings on Ethnomathematics, and the social focus of Mathematics, I developed some works on Mathematics in people’s daily lives. My master’s dissertation was on the Mathematics of the “Cruzado Plan,” an economic plan launched by the government in early 1986. Another work that even gained international repercussion was on the “Mathematics of the Animal Game,” a popular game of chance in Brazil. When Ubiratan saw that I liked History of Mathematics, and knowing my political position, he referred me to do a doctorate with one of the main historians of Mathematics in the world, Hans Wussing, at the Karl-Marx-Universität, today Universität Leipzig, at then East Germany. With my entry into the community of Historians of Mathematics, I was present at several international events, in different countries, alongside Ubiratan D’Ambrosio. By his appointment, I attended the Meeting of the Pugwash Conferences on Science and World Affairs, held in Stockholm, Sweden, in 1992. There were moments of great learning, of meeting new colleagues who were introduced by him. But the learning was not restricted to subjects about mathematics and its history; it was a learning of the world. With Ubiratan I visited several museums and Churches, and he always had the patience to introduce me to old religious buildings and great artists and their works. Leisure moments in restaurants, bars, taverns, etc. were very pleasant. With him I learned to taste regional cuisines and to taste its spices, not to mention the countless appetizers that were presented to me. Whenever we were in Portugal, after dinner, Ubiratan would choose an appropriate place to enjoy a “bagaceira” as a digestive (Fig. 2).

Fig. 2  Ubiratan D’Ambrosio, Sergio Nobre, Hans Wussing, Oberwolfach 1998

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Upon returning to Brazil, after completing my doctorate in 1994, we began the path toward the institutionalization of the scientific movement in the History of Mathematics. Together with colleagues who had also completed a doctorate on historical subjects, we inaugurated the series of National Seminars on the History of Mathematics that culminated in the foundation of the Brazilian Society for the History of Mathematics – SBHMat. Ubiratan was the first president of SBHMat, and I was the Secretary General during the two terms in which he was president (1999–2007). We worked together, naturally counting on the support of the other members of the board, to formalize the newly created partnership. In Rio Claro, in the Department of Mathematics of the Institute of Geosciences and Exact Sciences at Unesp, the Research Group in History of Mathematics  – GPHM – was created in 1995. The students, supervised by Ubiratan in the Graduate Program in Mathematics Education, who developed studies on the History of Mathematics, were members of the Group. There were many activities organized by the GPHM that had the presence of Ubiratan D’Ambrosio. I highlight the events entitled “Jornadas Unespianas de História da Matemática” held at the Department of Mathematics, in Rio Claro, and which took place between 1998 and 2004. In addition to the institutionalization of the research area in the History of Mathematics, which took place with the foundation of SBHMat, Ubiratan saw the importance of having a scientific medium for the dissemination of research results, and his thought was that we should have a scientific journal, but which was not to be restricted to Brazilian researchers. In this sense, we founded the Revista Brasileira de História da Matemática – an international journal on the History of Mathematics. Ubiratan contributed significantly with excellent scientific articles published in our journal. To conclude, I reproduce below the text published in the presentation of the book, which was a Special Edition of the Revista Brasileira de História da Matemática, published in December 2007, in honor of Ubiratan: Festschrift is a German word which does not have an adequate translation into Portuguese. Its meaning can be translated as an academic tribute book. Outstanding academics in the Western world receive this honor normally when they leave their academic activities to enter retirement. The history of this Festschrift to Prof. Ubiratan D’Ambrosio started in 1992 when I participated in the colloquium in honor of Prof. Hans Wussing, my doctoral advisor. Hans Wussing was 65 years old and his friends gave him a magnificent Festschrift. At that moment, I thought that one day we could also organize a book like this for Prof. D’Ambrosio. When Ubiratan turned 65 years old, I tried to organize such a work, but it was not possible, mainly due to the fact that Brazilian publishers are not used to this type of publication. In celebration of its 70th anniversary, I again tried to organize this work, this time with the intention of not depending on publishers, but the greatest difficulty was deciding on the invitations that would be made, because, as many know, Prof. D’Ambrosio works in different areas and it would be difficult to make a selection of those who would be invited to compose the body of authors of the book. One more opportunity passed, but I continued with the fixed idea that we would have to organize a Festschrift for him. With the foundation of the Brazilian Society for the History of Mathematics, and the creation of the “Revista Brasileira de História da Matemática”, the idea arose of organizing a special issue of that journal as the Festschrift. In this case, we would not need to look for a publisher that would be responsible for the graphic part of the book, since the Journal already had a life of its own and a special issue would only be one more to be published, and, as it is a publication

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in the History of Mathematics, we would restrict the group of authors only to those who develop their scientific investigations in this area. I made the invitation to people from different parts of the world and got a favorable response from everyone. Later, some apologized for not being able to meet the manuscript submission deadline, but others sent their contributions so that we could compose this work. Renowned researchers in the History of Mathematics in Brazil and around the world make up the list of authors of this work, which will certainly serve as a source of reference for many who develop their research in the area. To all the authors, I express my thanks. On my behalf and on behalf of the authors, I dedicate this work to Prof. Ubiratan D’Ambrosio. Sergio Nobre (organizer)

Cover of Festschrift as a special issue of the Brazilian Journal of History of Mathematics, published in 2007

Remembering Ubiratan D’Ambrosio (1932–2021) Luís Saraiva

Abstract In this paper, we will remember some key moments in Ubiratan D’Ambrosio activities in the international community in the fields of History of Mathematics and of the History of Mathematical Instruction. We will talk about two particularly important distinctions: the Kenneth O’May medal, which the International Commission for the History of Mathematics awarded jointly to him and to the Singaporean historian of Mathematics Lam Ley Yong in 2001, and the Felix Klein medal, which he was awarded by the International Commission on Mathematical Instruction in 2005. We will show his decisive role in the genesis of the collaboration between Brazilian and Portuguese historians of mathematics. These two communities have been organizing regularly Luso Brazilian meetings on the history of mathematics since 1993. We will write in more detail about his participation in events in the Luso-Brazilian community as well as in events in Portugal. We will end on a personal note, giving some reflections on a friendship of more than 30 years. Keywords  History of mathematics · History of mathematical instruction · Luso-Brazilian community

With a few necessary adaptations, this is the English translation of a paper that was published in Boletim da Sociedade Portuguesa de Matemática, volume 79, pp. 1–10, 2021, and is included here by kind authorization of the Portuguese Society of Mathematics. L. Saraiva (*) Centro Interuniversitário de História das Ciências e da Tecnologia, Departamento de Matemática da FCUL, University of Lisbon, Lisboa, Portugal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_9

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Professor D’Ambrosio with Luís Saraiva, on his right, and Sergio Nobre (Professor at UNESP Rio Claro, Brazil, founding member of the Brazilian Society for the History of Mathematics and its president between 2007 and 2015), on his left, during the 6th Luso-­Brazilian Meeting on History of Mathematics, in S. João del Rei, Brazil, 2011

Professor Ubiratan D’Ambrosio, a great friend of Portugal and of the Portuguese historians of Mathematics and Mathematics Education, died on 12 May, in S.  Paulo, Brazil. He established himself in the international community of historians of mathematics and mathematics education through his works and talks, opening new horizons and ways of problematizing historical and educational situations through Ethnomathematics, an area of research that implies a historical, socio-cultural, and anthropological perspective on mathematics and mathematics education. In Portugal, he not only actively participated in events on the history of mathematics and mathematics education but also played an important role in the establishing of working relations between the Portuguese and the Brazilian communities of historians of mathematics.

1 Action in the International Community The work of Ubiratan D’Ambrosio has been recognized internationally many times, so I will not detail here his immense work,1 published not only in Brazil but also in other countries. I will only highlight two deeply significant moments, two honors of

1  On the work of Ubiratan D’Ambrosio, see the website http://ubiratan.mat.br. An In Memoriam about this researcher was published in the International Archive of the History of Science, vol. 71, 187, pp. 178–200, 2021, written by Sergio Nobre and Luís Saraiva, which contains detailed biographical and bibliographic information about Professor D’Ambrosio.

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great value – the Kenneth O May Award in 2001 and the Felix Klein Medal in 2005. I will transcribe extracts from the presentation of these two awards, which, in my opinion, summarize the essential of the great value of the work (actions and writings) of this citizen of the world. Kenneth O’May (1915–1977) was an American mathematician and historian of mathematics who in the later years of his life orientated his work toward building a unified community of historians of mathematics. He was one of the influential personalities in the creation of the International Commission for the History of Mathematics (ICHM) and in the emergence of the international research journal Historia Mathematica, still today the most prestigious journal in the field of the history of mathematics. In recognition of the value of this mathematician and to perpetuate his memory, ICHM created the Kenneth O’May Award, designed to acknowledge and reward outstanding works in the field of the history of mathematics. The first prize was awarded in 1989, during the 18th International Congress of the History of Science (ICHS) in Hamburg and Munich, jointly to Dirk Struik (1894–2000) and to Adolph P. Youschkevitch (1906–1993), both with connections to Portugal – Struik because his book A Concise History of Mathematics was one of the first books on the history of mathematics to have been translated in our country2 and Yuschkevitch for having been instrumental in the international dissemination of the works of the Portuguese mathematician José Anastácio da Cunha (1744–1787). Since then, the prize has been awarded every 4  years, during the International Congress on the History of Science, which, from 2001 onward, also became a Congress on the history of technology. In 2001, during the 21st ICHST, held in Mexico City, the prize was jointly awarded to Ubiratan D’Ambrosio and to the Singaporean historian of mathematics Lam Lay Yong (born in 1936). In the presentation text of these awards, the reason for the choice was stated: The two scholars were awarded the May Prize because they have contributed significantly to enlarge history of mathematics by opening new research fields which actually also soon found their ways into textbooks. Thus today no serious historian of mathematics would write a general book on the history of mathematics without including ethnomathematics and Chinese mathematics.

At the request of ICHM, D’Ambrosio indicated five works of his that he considered significant in his work on Ethnomathematics: –– Ethnomathematics and its place in History and Pedagogy of Mathematics, in For the Learning of Mathematics, vol. 5, 1985, LJM Publishing Association, Canada –– Ethnomathematics. Challenging Eurocentrism, in Mathematics Education, eds. Arthur Powell and Marilyn Frankenstein, State University of New York Press, Albany, 1997, pp. 13–24 –– Historiographical Proposal for Non-Western Mathematics, in Mathematics Across Cultures. The History of Non-Western Mathematics, ed. Helaine Selin, Kluwer Academic Publishers, Dordrecht, 2000, pp. 79–92  Translated by João Cosme Santos Guerreiro (1923–1987), a professor of the Faculty of Sciences of the University of Lisbon, and published by Gradiva. 2

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–– A Matemática na Época das Grandes Navegações e Início da Colonização, Revista Brasileira de História da Matemática, vol. 1 (1), 2001, pp. 3–20 –– Etnomatematica, Pitagora Editrice, Bologna, 2002 The International Commission on Mathematical Instruction (ICMI) is a commission of the International Mathematical Union (IMU) created at the International Congress of Mathematicians in Rome in 1908. Initially it had the objective of analyzing and comparing the teaching of mathematics in different countries. Over the years, ICMI has expanded its objectives, and today it encompasses all levels of education, from primary to higher education, constituting an important forum for debate and exchange of ideas and experiences on teaching and learning mathematics. ICMI created the Felix Klein Medal to reward scholars who produce outstanding work in mathematics education. The first Felix Klein Medal was awarded in 2003 to the French academic and researcher Guy Brousseau (born in 1933) and has since been awarded every 2 years. Ubiratan D’Ambrosio was the second recipient, in 2005. It is worth including a significant part of the speech attributing this medal, to have a better understanding of the scope of the work developed by Ubiratan D’Ambrosio: The second Felix Klein Medal of the International Commission on Mathematical Instruction (ICMI) is awarded to Professor Ubiratan D’Ambrosio, Brazil. This distinction acknowledges the role Ubiratan D’Ambrosio has played in the development of mathematics education as a field of research and development throughout the world, above all in Latin America. It also recognises Ubiratan D’Ambrosio’s pioneering role in the development of research perspectives which are sensitive to the characteristics of social, cultural, and historical contexts in which the teaching and learning of mathematics take place, as well as his insistence on providing quality mathematics education to all, not just to a privileged segment of society. His role in promoting mathematics education research and development in Latin America, both as regards priorities and content and as regards institutional and organisational frameworks, can hardly be over-estimated. His focus on providing graduate and post graduate programmes for young researchers exemplifies his contribution. […] Ubiratan D’Ambrosio belongs to a generation that helped to found the field of mathematics education. His contribution to research is essentially as a philosopher  – in the classical broad sense of that word – of mathematics education reflecting on its role in a complex world characterised by unrest and by an uneven distribution of goods and privileges across regions, countries, and societies. By focusing his attention on developing cultures, Ubiratan D’Ambrosio broadened our conception of mathematics education. More than that, he has helped to open the eyes of the mathematics education community to an understanding of how mathematical ideas are generated and how they evolved through the history of mankind. This work made a significant contribution to our appreciation of the field of scientific invention and its relation to ad hoc practices that occur in different cultures and subcultures. His contribution has played a key role in legitimating alternative forms of mathematical activity and in elaborating the now-familiar idea that the quasi-mathematical knowledge of the learner can be built upon rather than rejected.

2 Action in the Luso-Brazilian Community 1987 was a pivotal year for the history of mathematics in Portugal. During the commemorations of the bicentennial of the death of the mathematician José Anastácio da Cunha (1744–1787), the highlight was an international colloquium held in

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Lisbon, in which prestigious historians of mathematics such as Ivor Grattan-­ Guinness3 (1941–2014), Enrico Giusti (born in 1940), Jean Mawhin (born in 1942), Jésus Hernández (born in 1944), and Ubiratan D’Ambrosio participated as guests of the organization. This first direct contact between Portuguese academics interested in the history of mathematics and renowned researchers in this area made them more deeply aware of the void that existed in Portugal in research in the history of mathematics, which, since the late 1940s, had only been carried out occasionally by some mathematicians4 who worked in isolation, although sometimes the result obtained was of great value. This international colloquium served to establish the first international contacts of this group of scholars interested in the history of mathematics. At the end of the colloquium, there was a working meeting with Professor Ivor Grattan-­ Guinness, in which the main seeds of what would be the future National Seminar on the History of Mathematics (SNHM), created shortly after, in January 1988, were established. The basic principles formulated then continue to be followed today, including the essential need to maintain regular international contacts. Correspondence with Professor Ubiratan D’Ambrosio continued, and he was the first international guest of our National Seminar on the History of Mathematics, whose first meeting took place at the University of Minho, in Braga, in April 1988. The seminar progressed, and knowing that international contact was a crucial condition for its growth and for the development of its members, it promoted and organized a Summer School in Évora in 1990. It had as lecturers Jean Dhombres (born in  1942), Enrico Giusti, Giorgio Israel (1945–2015), and Ahmed Djebbar (born in 1941), who brought some of their Ph.D. students. Professor D’Ambrosio, being a researcher who kept an eye on the current events in the history of mathematics, became aware of this event and advised one of his students to participate in this Summer School, knowing that it should prove very useful for his training, due to the quality of the historians who were involved in its structure. The student in question was Sergio Nobre, who was preparing his Ph.D. in the history of mathematics in Leipzig, then East Germany, under the supervision of Professor Hans Wussing (1927–2011), a German historian and future Kenneth O’May medal in 1993. The arrival of Sergio Nobre to Évora had the effect of strengthening the links between the community of historians of Portuguese mathematics and the corresponding Brazilian community. This was so especially since Sergio Nobre became an essential figure in the revitalization and the progression of this community, not only as an organizer of events but also as an essential contributor to the foundation of the Brazilian Society for the History of Mathematics (Sociedade Brasileira de História da Matemática – SBHMat) in 1999, of which Ubiratan D’Ambrosio was the first president, and to the creation of the Brazilian Journal for the History of Mathematics (Revista Brasileira de Historia da Matemática – RBHM) in 2001, thus  Kenneth O’May Medal in 2009.  Among them we have José Vicente Gonçalves (1896–1985), Luis de Albuquerque (1917–1992), José Joaquim Dionísio (1924–1999), José Tiago de Oliveira (1928–1992), and Fernando Roldão Dias Agudo (1925–2019). 3 4

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having a journal for the publication of papers in Portuguese by historians of mathematics. Later, in 2014, SBHMat created the Journal of the History of Mathematics for Teachers (Revista da História da Matemática para Professores – RHMP) and, in the following year, the Journal of the History of Mathematics Education (Revista da História da Educação Matemática – HISTEMAT). The contacts established also led to the creation of the Portuguese-Brazilian Meetings on the History of Mathematics, the first of which took place at the University of Coimbra in 1993. Since then, these meetings have been held alternately in Brazil and Portugal, always with many participants. The ninth meeting took place in Setúbal, Portugal, in October 2022. With the exception of the third meeting, held in Coimbra in 2000, all these meetings have their proceedings published, the Setúbal Proceedings should be available by early 2024. We can therefore say that the action of Ubiratan D’Ambrosio was decisive for the establishment of ties between the Portuguese and Brazilian communities of historians of mathematics, corresponding, in fact, to an old aspiration of his. But Ubiratan was not just influential at an organizational or institutional level. His great experience of life, the simple way in which he communicated his fundamental ideas, and his calm and friendly speech were always motivating elements for Portuguese researchers, who always found in him, in addition to an experienced and knowledgeable researcher in his area, a friend, someone always ready to collaborate or inform, and a person always available to share useful contacts with those who requested them. We can say that Ubiratan was always a source of inspiration and motivation for all the researchers who contacted him. Ubi, as he was affectionately known, continued to come regularly to Portugal to give lectures or participate in colloquiums, with talks on the history of mathematics or on the history of mathematics education or on ethnomathematics. He participated in four SNHM meetings: in addition to the aforementioned 1st meeting, in 1988, he also presented talks at the 8th meeting, at the University of Porto, in 1996; at the 9th meeting, at the University of Coimbra, in 1997; and at the 19th meeting, at the University of Aveiro, in 2006. In this last meeting, he also participated in a Round Table with the theme “Teaching the History of Mathematics at the University: the challenge of the Bologna Process.”

3 Other Participations in Portugal In 1994, he was present at ProfMat94 – National Meeting of Mathematics Teachers – promoted by the Association of Mathematics Teachers (APM), which took place in Leiria. At this meeting, he delivered the closing plenary lecture The Research in Mathematics Education: from Theory to Practice or from Practice to Theory? and another lecture Assessment: to eliminate or to maintain? Or to reconceptualize? He also was part of the panel The Reforms in Mathematics Teaching and the Role of Teachers and Researchers (developments in several countries). It is worth mentioning that Beatriz D’Ambrosio, daughter of Ubiratan, gave a talk in another of these

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meetings, ProfMat96, held in Almada in 1996, with the conference The changes in the role of the mathematics teacher in the face of the education reforms. In 1996, he attended a satellite meeting of the International Congress of Mathematics Education (ICME) in Braga, which took place jointly with the 2nd European Summer University of History and Epistemology of Mathematics Education.5 In 2000, he was invited by the University of Aveiro to give the inaugural conference of the International Year of Mathematics, entitled Ethnomathematics: a proposal in History, Epistemology and Pedagogy of Mathematics.6 Ubiratan also participated in two meetings in honor of Paulo Abrantes (1953–2003), an important researcher in Mathematics Education. In the first, Mathematics Education: paths and crossroads, in honor of Paulo Abrantes, held in Lisbon in 2005, he had the closing conference of the meeting – Paulo Abrantes: in memory – ; in the second, held in 2013 also in Lisbon, Remembering Paulo Abrantes 10  years after his death, he took part in the symposium Mathematics in the Curriculum with the talk Mathematics and other subjects. In 2006, in Viana do Castelo, he gave the talk Ethnomathematics and Education at the Second International Meeting of Elementary Mathematics Education.7 And in 2012 he was a guest speaker at the Second Meeting on the History of Mathematics and Science, held at the University of the Azores, Ponta Delgada, where he presented the talk Mathematical relations between Portugal and Brazil, especially after World War II. With regard to publications, in addition to the texts of his talks published in the Proceedings of the respective meetings, Ubiratan D’Ambrosio published (in Portuguese) in the Educação e Matemática journal the articles Homage to Paulo Freire, EeM n.° 43, 1997, and Mathematics Education for Citizenship and Creativity, EeM no. 125, 2013.8 He participated in the first six Luso-Brazilian Meetings on the History of Mathematics, the last of which took place in 2011 in São João del Rei, Brazil, and presented papers in all of them, as well as having participated in some Round Tables. In 2013, medical problems prevented him from traveling by plane, making it impossible for him to attend meetings away from S. Paulo. However, the organizers of the next Luso-Brazilian Meeting, in Óbidos, in 2014, asked him to record his presentation on video, as this would be a way of continuing to participate in an event where his contribution had been essential for its existence. So it was, and the lecture he

 I thank Professor João Caramalho Domingues for this information.  I thank Professor Helmuth Malonek for this information. 7  I thank Professors Helmuth Malonek and Mária Almeida for this information. 8  I would like to thank Professor Henrique Guimarães for the information on the participation of Ubiratan D’Ambrosio and his daughter Beatriz in events that took place in Portugal and on their publications included in the paragraphs relating to 1994. I also thank him for the information on the tributes to Paulo Abrantes and on the publications in the Educação e Matemática journal. 5 6

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recorded, about his reminiscences about José Sebastião e Silva9 (1914–1972), was presented at that meeting. Ubiratan D’Ambrosio had an influence that was reflected in many other countries. I only mention as an example a Portuguese-speaking country, Mozambique, where Paulus Gerdes (1952–2014), a professor at the Eduardo Mondlane University until the end of the 1980s, who later taught at the Pedagogical University, developed influential and important work, applying the ideas and concepts of Ubiratan D’Ambrosio’s Ethnomathematics. Using them as a starting point, he then developed them according to his own perspective. Gerdes founded in Maputo, in 1989, the Research Center in Ethnomathematics – Culture, Mathematics and Education.10 Personal Notes I met Professor Ubiratan D’Ambrosio for the first time in 1987, at the International Meeting held in Lisbon on José Anastácio da Cunha. Since then, we have met many times, either in Portugal, or in Brazil, or in other countries, where we were both going to present papers at conferences. The common interest in the history of mathematics, and in particular the history of Portuguese-Brazilian mathematics, brought us together, and over the years we forged a solid friendship. It was an immense pleasure for me to meet him, often accompanied by his wife, Maria José, his great support since 1958. In particular, I keep as very special memories the visits to his house in S. Paulo and the moments we spent together talking about a wide range of topics. In this way, little by little, Maria José also became part of what I usually call my family in a broad sense. Ubiratan D’Ambrosio was part of the group of researchers for whom culture is not just a word; his knowledge and wisdom went far beyond his field of research and allowed him to have a global vision and a critical stand in relation to problems as diverse as they could be, from the arts and literature to the media and politics. But this knowledge was accompanied by a great humanity in his speech, putting any well-meaning interlocutor at ease in his dialogue, because for him everyone deserved respect and attention; the mistakes or failures that they could have were inherent to their human condition. Being cultured also means knowing to listen, and Ubiratan had this great quality: he listened and answered, calmly and slowly, taking into account what he had been told. Added to this characteristic was the ability to contextualize situations in broader frameworks, where they had their true weight and meaning. Sometimes what looked very serious seemed to evaporate once it was integrated in a wider picture. Thanks to his deep knowledge of Brazilian culture, I owe him some non-­academic discoveries that would have passed me by if he hadn’t called my attention to them.

 2014 was the centenary year of the birth of Sebastião e Silva, an event celebrated in Portugal with several meetings, and the Luso-Brazilian Meeting had a symposium dedicated to this essential figure in Portuguese mathematics. 10  I thank Professor Henrique Guimarães for the data on Mozambique, as well as for his collaboration in the final revision of this text. 9

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One of the many he gave to me was that of singer-songwriter Dolores Duran (1930–1959), a tragically short life for such an immense talent. Sometimes I managed to get him things that only existed in Portugal, and that showed his interest in Portuguese culture and history. I remember, among others, offering him the DVD of the Leitão de Barros’ film Camões (1946). Ubi, I will miss you. But there’s always something about our friends that remains with us when they leave. Dear Ubi, great Palmeirense, warm hugs to you now and forever!

Professor Ubiratan with his wife, Maria José, in S. Miguel, Azores Islands, 2012

Part II

Ethnomathematics in Action

“Ethnomathematics Has Worked, and VEm Brasil Is Proof of That” Olenêva Sanches Sousa

Abstract  This essay addresses Ethnomathematics with Ubiratan D’Ambrosio. It covers the idealization and holding of the Virtual Etnomatemática Brasil (VEm Brasil) meeting, which, from April 2020 to July 2021, brought together ethnomathematicians, sympathizers, and curious people from all over the world. The hyperdocument aims to disseminate the bases of the Program Ethnomathematics and D’Ambrosian thinking and behavior and guide and encourage similar or innovative actions. Choosing a time frame, Ubiratan’s last month of life, an illustrative scheme of the VEm Brasil, with email messages and oral history, is used to bring out and express feelings and memories of this event that occurred in collaboration with its leading theorist. Freely, the text brings enchantments, tensions, and expectations that permeated the trajectory before VEm Brasil and a perspective of D’Ambrosian ethnomathematics manifestation. It is based on D’Ambrosio’s Program Ethnomathematics, supported by conceptions and strategies for the dissemination and popularization of this general epistemology, according to the author’s doctorate, supervised by Ubiratan, and on the e-Almanac EtnoMatemaTicas Brasis, a product of VEm Brasil, of which he was an editorial consultant, author, and collaborator. This work hopes to signal subjective, political, and socioculturally strengthening characteristics of the Program Ethnomathematics, besides inspiring the implementation of events and movements referenced therein. Keywords  Program Ethnomathematics · Ubiratan D’Ambrosio · VEm Brasil · Epistemology · Dissemination and popularization of science

O. S. Sousa (*) EtnoMatemaTicas Brasis, Bahia, Brazil Red Internacional de Etnomatemática, coordenação Brasil (RedINET-Brasil), São Paulo, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_10

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1 Ubiratan D’Ambrosio and VEm Brasil: Scenes, Scenarios, and Backstage In the opening lecture, D’Ambrosio announced: “Ethnomathematics has worked, it is being used and worked on all over the world, and VEm Brasil is proof of that.”1 Ubiratan D’Ambrosio was a proponent of Virtual Ethnomathematics Brazil (VEm Brasil), as a consultant, collaborator, presenter, and spectator. Aware that national and international actors and actions would be gathered there, the inspiring theorist of the VEm Brasil envisioned it as an “example of how promising it is that we are virtually reunited, all of humanity and all cultures.” This chapter presents an overview of the VEm Brasil 2020 scenario, conceived collaboratively with D’Ambrosio and theoretically guided by the Program Ethnomathematics. It also has an introductory character of an essay on specific aspects of the event. It will be complemented with some backstage experiences and some scenes, which will be the focus of the following two sections, which will precede the final considerations, Inspiration for the future. Coinciding with the beginning of the suspension of face-to-face activities due to the COVID-19 pandemic, VEm Brasil 2020 pioneered an academic form of communication and gave great visibility to ethnomathematics. And even for Ubiratan, who had already a vast curriculum of recordings and live in-person and distance appearances and a special interest in the relationship between epistemologies, cultures, and technologies, the proposal was new. It was worth experiencing it! Enchantments, tensions, and expectations have permeated the entire trajectory for VEm Brasil. Those situations were revealed in comments and questions. Applicants and viewers who signed up made contacts filled with “whys” that they usually could not answer – but that they soon would. I liked to tell Ubiratan about some of those whys, those stories, expressing my fears, my anxieties, and my dreams of a successful future. He was always ready to support me. He had the modesty of a person who was there to stir up a sense of enchantment, overcome tensions, and work to make expectations happen. Therefore, the next subsection is entitled: Enchantments, Tensions, and Expectations on the Path to VEm Brasil. Its objective is to bring out this set of feelings, focused mainly on Ubiratan D’Ambrosio’s, in the process that began in a telephone conversation about the possibilities for a virtual movement with reference to the Program Ethnomathematics and extended to the real experience of the Virtual Ethnomathematics proposed and coordinated in Brazil. In this essay, this process of emergence of feelings is developed from a process of immersion in the memory of some facts that reveal the creative action of VEm Brasil with the collaboration of its main theorist of reference. To this end, elements of communication and dissemination and some products available on the Internet are considered.  “Ubiratan D’Ambrosio, VEm Brasil on the VEm Brasil- EtnoMatemaTicas Brasis YouTube channel, available at: https://youtu.be/we1zbAh-fxg 1

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As a proponent and coordinator of VEm Brasil, I believe that the importance of having had D’Ambrosio – theorist, advisor, and friend – in the construction of the conception of the event is undeniable. Thus, I start to register episodes of our oral history, still clear in memory, mixed with messages exchanged by email, during the trajectory that preceded VEm Brasil 2020. That is why, throughout the text, I often felt obliged to mix the person of the verb between the I and the we. I understand that, in some way, these writings can strengthen some subjective, political, and sociocultural characteristics of the Program Ethnomathematics. There are sensitive points of academic research and production that are limited to the space dedicated to motivational aspects. Regarding mathematics, conceptions of rigor and neutrality of science, research, and the researcher seem to extend to education, the school, and the teacher. Despite the studies of ethnomathematics and the efforts of mathematics education, we still see much more emphasis on the application of school mathematical contents aiming at the learning of procedures already consolidated in the ethno2-academic field of mathematics than in the critical and creative development of the conceptual and attitudinal possibilities of mathematical knowledge in the different ethnos. For me, one of Ubiratan’s most commendable efforts in the process of intellectual and social organization of the Program Ethnomathematics, was his determination to highlight  – and exercise  – the human impulses of transcendence in the generation and sharing of knowledge and establishing relationships with major issues and causes that contribute to the humanization of the human being, even if this seems paradoxical. He always drew attention to the distinction between the noun “being” and the action verb “to be.” He was involved in actions in favor of peace, sustainability, and social justice, and, in living with others, he was flexible, intimate, considerate, affectionate, and loving. Both “being” a scientist and “being” a philosopher, he organized the Program Ethnomathematics considering holism, transculturalism, transdisciplinarity, and other factors that, due to their subjectivity, cannot be prescribed, quantified. Those factors still find itself in arid soil where to sprout in pedagogical practice. They do not find great opportunities to be externalized in the scientific literature either. But to D’Ambrosio (2005, p. 105), it is worth the “effort to contextualize our actions, as individuals and as a society, in the realization of the ideals of peace and a happy humanity. […] How to be an educator without a utopia?” After all, “reality is in permanent transformation through our creative action. Our fundamental action is to try to bring the current reality –which is presented to us with a fact– closer to a reality that is part of our utopia” (D’Ambrosio, 2009, p. 118). So far, exhaustively, we brought as priority those issues that make us human, that give meaning to education, that seek to eliminate hierarchies of knowledge, and that seek the ethics of diversity, respect, solidarity, cooperation, awareness of incompleteness, peace, and social justice that underlie the Program Ethnomathematics and  From the perspective of the Program Ethnomathematics, ethno: one of the terms of the conceptual word Ethnomathematics; forms a natural and sociocultural reality or context. Mathematics as science is an ethnomathematics in the ethno academic sense. 2

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D’Ambrosian thought and behavior. Certainly, this has a very significant weight in the legacy left by Ubiratan to science and education that think about sustainability in the future. Next, in the title VEm Brasil as a D’Ambrosio’s ethnomathematical manifestation, I would like to summarize the VEm Brasil project and briefly contextualize the Program Ethnomathematics as a research program and general epistemology that underlies it as a manifestation of D’Ambrosio’s thought. Other email messages will be included to illustrate this process. We expect this essay will provoke reflections about possible rigidities and flexibilities of academic models that, even disagreeing with or contradicting what most of them defend, end up contributing to the maintenance of hierarchies and valorization of vanities, by making it difficult for each individual to achieve creative potential and promoting citizenship harmed by an education that omits or denies its political force – or omits or refuses to see it or to fight for it – to form subjects who have wills, feelings, and intelligences. As a consequence, schooling is established – I allow myself not to call it education – simply for the subservience of the people to the prevailing power. Furthermore, I hope that this brief history of the first edition of VEm Brasil inspires the construction of other virtual scenarios for ethnomathematics. This would represent  – using the verbal tense with more emphasis on cultured norms than on my expectation and my feeling that it will represent – Ubiratan’s legacy on a trajectory of transcendence to a possible near future. He was the first reader of the first version of the project and reflected on options for names and acronyms. He soon realized that the VEm would be invariable in “Virtual Etnomatemática,” “Virtual Ethnomathematics,” and “Virtuales Etnomatemáticas” and that diversity would be in the proponent or the performer. Then, other “VEm” may come, in Brazil and around the world.

2 Enchantments, Tensions, and Expectations on the Path to VEm Brasil At 8:00 am on Saturday, April 25, 2020, I presented the VEm Brasil proposal to its audience. My main concern was to highlight the program well to the participants. At the time, being completely virtual was a novelty. It was also necessary to detail the dynamics of the event that would run uninterrupted until midnight Sunday, 26. There was also the specificity of scientific communications and experience reports occurring precisely and sequentially. All this required the coordination, lecturers, and audience to behave very differently than they were used to in face-to-face academic events. From the beginning, Ubiratan looked forward to seeing “concretely” what the event would be like and participating in it. He had the whole schedule given to him and explained in detail, and he had already watched my introduction, recorded a few

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days before. This anxiety was completely acceptable because, at the time, despite the YouTubers and what we call in Brazil “lives,” we were unaware of academic events of that nature. We never imagined that we would so soon be imbued entirely with this cyberculture in academia and education. There would be live and recorded presentations, and the format would depend on the proposal initially made by the lecturer, based on the technological resources they had, they knew, or simply because they preferred to risk new experiences. Although most opted for video screenings as scheduled premieres on YouTube, there were live lectures on the channel, on Facebook and Instagram. In both cases, recorded or live, the lecturer should interact with the audience, either by answering live questions directed to them or ensuring, during the premiere of the video, an interaction via chat, answering the questions pertinent to the presentation. Therefore, both the lecturers and their viewers were qualified as interactive. From its sketch, or initial outline, the project envisaged 40 h of activities over a weekend. Except for the topic, ethnomathematics, and the base time of the talks, 15 min minimum or a multiple of up to 90 min, everything else would be free. The initial idea included the development of a product that also expressed this freedom to communicate using science, art, culture, education, philosophy, and anything else that the Program Ethnomathematics could involve. In September 2019, Ubiratan received a phone call delivering him a “package” of ideas not yet linked together. As he was a great listener, it was easy to develop an interaction filled with parallel topics, something like the hypertextuality that should be manifested in VEm Brasil. We liked the hypertexts! D’Ambrosio saw the obvious advantage of accessing the various subjects covered in a text in the text itself, and I had already, consciously, started trying to produce academic ethnomathematics hyperdocuments since 2012. In those submissions, some were accepted with a mandatory review for withdrawal of hyperlinks, and some were rejected because part of the reasoning was not explicit or just because hyperlinks were not allowed. In 2020, however, hypermedia was already part of the academic culture, and VEm Brasil used them countless times. It is worth mentioning that in July and August of the same year, we gathered at Ubiratan D’Ambrosio’s home in São Paulo. Amid conversations about various subjects in the company of his wife, Maria José, we told D’Ambrosio of our interest in contributing to the dissemination – and popularization – of actual ethnomathematical concepts and principles. This was and is, in fact, an interest of mine directly aligned with the object of my doctoral research on the Program Ethnomathematics, advised by him and defended in 2016. Obviously, it was also his wish to see and enable all this in practice. In this sense, a virtual event for ethnomathematics would be a good experience to live. Ubiratan has always feared that some powerful interest could make disciplining and institutionalization might hamper or even worse, thwart, “ethnomathematical freedom.” Thus, I understand that the practices assumed as ethnomathematics, like VEm Brasil, should contribute to understanding the research program and the general epistemology intellectually organized by D’Ambrosio. At the same time, they should be both sensitizing and experiential, revealing ethnomathematical concepts

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using a theoretical set that he defended and disseminated – or even revealing others based on the same principles. I learned from Ubiratan the importance of recognizing and disseminating ethnomathematics as a research program. Particularly, I think his having built this philosophical conception for ethnomathematics based on Lakatos was inspirational. In 2016, I considered that this recognition “implies admitting theoretical contestations and a dynamic that mobilizes his research, (re)constructing concepts and perspectives continuously, without losing sight of what strengthens it  - the knowledge cycle - and gives it the title of ‘general theory of knowledge’” (Sousa, 2016, p. 38). VEm Brasil seeks to embrace the different conceptions of ethnomathematics and promote the integration and interaction of those involved, people from theory and practice, sympathizers, and inquisitive people.3 A sense of hypertextuaness characterizes the organization of VEm Brasil, as it was and is reflected in its memories, openness, and products. The program of the event was interrelated in various media contexts. Therefore, attendees and the general public could access whatever they wanted to watch from various social media platforms, both during and after the event. The event site had a specific tab that allowed organizers to display posters, titles, abstracts, authors, and presentation schedules.4 The cloud was the complementary resource for disseminating a program5 on the site. There were still two scheduled albums with posters of the lectures and descriptions, one in the EtnoMatemaTicas Brasis a Facebook community,6 promoter of the event, and another on Google Photos.7 And on the chat, authors and viewers dialogued during the talks. But, like Ubiratan, some liked to visually follow the dialogues between the participants and subjects. Ubiratan was ecstatic with the project! This enchantment was very peculiar to him! This feature was posthumously highlighted in the Mathematical Cruise 2022,8 when he was honored. Alexandre, his son, who was on the crew, commented that most people lose over the years what his father never seemed to have lost: the ability to be dazzled by things. “He was passionate about what he did, and until the last days of his life, he continued to show enthusiasm and be delighted with new ideas and discoveries.”9

 Objective on VEm Brasil site: https://doity.com.br/vem-brasil-virtual-etnomatematica-brasil   Program on the site of the event: doity.com.br/vem-brasil-virtual-etnomatematicabrasil#schedule> 5   Program in the cloud: drive.google.com/file/d/1ESSqjKY6WGfDbNPzHr8rR7oxbT2qgo_R/ view> 6  Program on Facebook: facebook.com/media/set/?set=a.1012666882452384&type=3 7  Program on Google Photos: photos.app.goo.gl/ZnBJbbTjxBsSZYVc8 8  A virtual, international, live event, promoted by Red Internacional de Círculos y Festivales Matemáticos [International Network of Mathematical Circles and Festivals] (CYFEMAT) that involved all of Latin America, Spain, and the United States. 9  “Alexandre D’Ambrosio habla sobre su padre Ubiratan D’Ambrosio (en portugués)” [Alexandre D’Ambrosio speaks about his father Ubiratan D’Ambrosio (in Portuguese)], on the CYFEMAT YouTube channel, available at youtu.be/ThdkfFWeWos 3 4

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Ubiratan’s enchantment was contagious; it was all-encompassing and inclusive. He could see excellence in projects – and in people – and he multiplied his vision for others, whenever possible, embracing them. The memories of D’Ambrosio’s presence in face-to-face events showed this magical potential of his to bring people and ideas together, always attentive to anyone who approached him. Thus, somehow, the Vem Brasil fascinated him. It put a spell on him, and the project turned into the magic that became our first VEm Brasil. Strong qualitative and subjective aspects characterize our trajectory toward VEm Brasil. For me, a “D’Ambrosian trajectory” is the one that seeks to bring into harmony what, due to several factors that we will not deal with here, is separated, in opposition, or loaded with hostilities, exclusions, etc. Tensions are inevitable! D’Ambrosio’s trajectory was fundamental to combining rigor with delicacy and experience with initiation, among other disparities that often seem incoherent to many of us. Thus, at the very beginning of the measures regarding public health, in the middle of the COVID-19 pandemic, we felt confident in coordinating social distancing and the hashtag #stayhome with the slogan “All together at VEm Brasil.”10 Anyone in the academic environment with Ubiratan knows how much he valued his speech. This theoretical-practical coherence was one of its strong characteristics. One could see in him the courage to contest conceptions and academic paradigms; and ethnomathematics is an example of this. On the other hand, without drastically violating norms and ingrained values, he had enough creativity, balance, and wisdom to face and foster whatever came his way. In this case, I think a good example is the intellectual organization of the Program Ethnomathematics as a research program and a general epistemology. I believe the book “A Era da Consciência” [The Era of Consciousness], a transcription of the master class of the first graduate course in science and human values in Brazil, shows well this being (verb) human who came out of epistemological cages to see the mathematical sciences with other eyes, to see greater and better the mathematical knowledge spread across all contexts and actions. Thus, Ubiratan eventually sewed hope, support a sense of struggle, educational practice, and pedagogical and research projects all over the world. This book was introduced to me by Ubiratan in 2010. Something in the speech transcribed impacted me a lot and still does: “Young people no longer want to be deceived by a school, an obsolete institution, by teachers who no longer know how to repeat the old. They want to find people who, together with them, look for the new” (D’Ambrosio, 1997, p. 11). I got the “tip.” I started to consider it in projects and activities for my students in the public school system, I mentioned it in my thesis, and I quoted it in some of my lectures. And I could not fail to bring it when it comes to our trajectory for VEm Brasil.

 “Todos Juntos no VEm Brasil” [All together at VEm Brasil] is an audiovisual production with collages of excerpts from short videos recorded by interactive proponents of VEm Brasil. It was used as an invitation to the event, published on YouTube on April 20, available at https://youtu.be/ ynglrVETHp4 10

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Fig. 1  Ubiratan was ready to learn anything new including VEm Brasil

After reading the project and the draft letter of invitation sent to ethnomathematicians, Ubiratan sent me an email message, one of many that complemented our telephone conversations. I bring this one specifically (Fig. 1) due to his declaration that he was ready to learn anything that crossed his way until VEm Brasil. Written in Portuguese, the message11 shows how intimate Ubiratan was with VEm, because, as we mentioned before, he envisioned that other countries could promote the event under the same acronym. In addition, it shows how much he believes that “ad hoc practices to deal with problematic situations arising from reality are the result of the action of knowing. That is, knowledge is triggered from reality. Knowledge is knowing and doing” (D’Ambrosio, 2005, p. 101). Aware of the immeasurable risk zone, we were curious to know how the event would be received by its lecturers and viewers. From my side, several questions arose on the path to VEm Brasil. How could such a novel event contribute to the consolidation of the Program Ethnomathematics? Would it be possible to identify and recognize conceptions of ethnomathematics that are developed in education and research? But before registration opened, the project already had many connections. Representing the virtual community EtnoMatemaTicas Brasis and the coordination of the International Network of Ethnomathematics in Brazil (RedINET-Brasil), I sent the invitations for participation in the VEm Brasil to ethnomathematicians, extended to their guests. And in the same month, in October 2019, as shown in Fig. 2, I already thanked D’Ambrosio, as I called Ubiratan personally and warmly: According to Fig. 2,12 we exchanged matters that ranged from academic to personal and family affairs. I think there were intimacy and equality between Ubiratan and his interlocutors, especially with the advisees. And as can be seen in Fig.  3, many messages were very informal.

 The VEm is moving forward. My opinion is that the people involved should talk. I don’t know how it will work. I’m “old,” virtual meetings are a mystery to me. But I will learn. 12  My dear D’Am. How are you? I’ve called twice and haven’t found you. But I will call again. I can only thank you for being the usual stimulator of the madness and inventions of your advisees, in a more appropriate language: Creative Insubordination. VEm Brasil only receives praise, for the proposed form and mainly for the innovation. Today, we already have acceptances from 16 states and 6 countries. We are posting on Facebook/Instagram daily, updating with new partners joining. 11

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Fig. 2  Thanks to Ubiratan for VEm Brasil, in October 2019

Fig. 3  Ubiratan’s informality

In this message,13 besides using a totally informal expression, Ubiratan also played with the verbs to go and come. Obviously, we had no idea that, soon after, the reality of VEm Brasil would be a routine in the lives of educators and researchers, a turning point in education and research in school and academic contexts. I had some teaching experience in distance education (DE) and knew its pedagogical possibilities. Volunteering in the struggles for the demarcation of indigenous land, he “introduced” me to Paulo Freire and the DE via the regular post office. Twenty years later, the experience was expanded in specialization courses and, especially, in a degree in a mathematics course via streaming that allowed me to produce content on ethnomathematics or, based on it, and develop pedagogical practices based on D’Ambrosio’s ethnomathematics thinking. The idea of holding an entirely virtual event for ethnomathematics seemed strange or absurd to some of the closest researchers. The biggest arguments were the “coldness” of the environment, the inability to deal with technological resources, and the belief that there would be no participation due to the Sixth Brazilian Congress of Ethnomathematics (CBEm6) that would take place in May 2020. In the beginning, the positive reception of my son, my experience in distance education, and the desire to concretize conclusions from the results of my theoretical-­ philosophical thesis on the Program Ethnomathematics weighed heavily. But Ubiratan’s encouragement and support from the so-called package of unconnected ideas were crucial to the continuity of the project. Furthermore, I was sure that Ubiratan saw warmth and vast possibilities in using technological resources to promote an event without any planning for face-to-face moments.

 Hi, Olenêva, ok, Let’s go! And VEmBrasil is also going to (translation that does not illustrate the informality of the original Portuguese text so well). 13

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I understand that an experience in the DE degree in 2006 made Ubiratan and I establish our mutual intimacy and trust. I invited him to a video streaming interview, and he accepted. I asked him whether he had any resistance to DE. He answered immediately and assuredly that, for him, the medium did not matter much if it had to do with education. The interview had a much larger audience than expected, as students from all degrees at the institution also attended. The institution for reasons particular to it did not allow us to share the talk Ubiratan gave but kept the talk on a CD and only came to use it in a tribute to him, with his consent, and the part we managed to recover was only shown after his death.14 What fascinated Ubiratan so much about the VEm Brasil proposal? What made him believe when he, in advanced age, recorded his speech stating that the event was proof that ethnomathematics had worked? His text for the RedINET-Brasil Bulletin,15 year 4, number 18, was sent to me by email. There were ten lines for VEm Brasil lecturers who wanted to answer a specific question. Figure  4 brings Ubiratan’s question and answer that came to my email. I believe this message16 in Fig.  4 presents his vision of the VEm Brasil very briefly as an opportunity for the ethnomathematics community and illustrates the

Fig. 4  Ubiratan’s view of VEm Brasil

 Excerpt from the interview retrieved for the tribute: https://youtu.be/zu3S5d1ZmLk?t=895  RedINET-Brasil Bulletin is a bimonthly publication of the RedINET coordination in Brazil. All numbers are available at https://docs.google.com/document/d/1Zj4GkrmlqCPkcb6FXaEkq8LCJt 44z5GDf45Pr0iasnw/edit?usp=sharing 16  What can the VEm Brasil represent as an opportunity for researchers and Ethnomathematics actors to meet? Ethnomathematics, although its main objective is to understand and explain traditions, which originate in different ethnos, therefore a LOCAL approach to knowledge, has as its quintessence the recognition of the species Homo sapiens sapiens as a single species, inhabiting and evolving throughout the planet. This gives ethnomathematics a GLOBAL character. This local+global duality, which some call glocalization, is one of the main characteristics of the 14

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Ubiratan that asked for – and accepted – opinions about his productions and who, despite not praising norms, knew very well how to summarize and comply the requests. His class assessments were reports with a limited number of lines, and on compliance with rules, he once told me, “know how to please the inquisitors, Galileo!” After this the outline of VEm Brasil was consolidated. I will now try to point out some enchantments. However, as we will see below, all the facts clearly denounce the ethnomathematical thinking – attentive and sensitive – of D’Ambrosio, at the age of 87. A big advantage was that everything was recorded after the event. He wanted to watch a little bit of everything and review some lectures. Ubiratan had the gift of oratory, he could talk for hours, but it seems that nothing made him tenser at VEm Brasil than recording 15 min of speech. However, he surprised me, as Fig. 5, with message17 subject “test”: It worked! Together, the very young Maria Eugenia and her grandfather succeeded. His channel has only three subscribers, but it served him to overcome the tension of recording independently and, of course, not to be ashamed of the event that hid the “mystery” of total virtuality. The news of the achievement was shared with Carlos Mathias, a mutual friend who was also at VEm Brasil and who has a YouTube channel, Matemática Humanista [Humanist Mathematics]. Carlos offered to record the speech remotely. We both felt that Ubiratan was relieved: “Just enter the link and talk?”. That was it! And less than a week before the event, Ubiratan recorded his lecture in 15 min, without the need for editing, the same lecture after which we named this article. Happy but concerned, he sent an email immediately after recording, as we can see in Fig. 6.

Fig. 5  Ubiratan’s YouTube channel

Program Ethnomathematics. Vem Brasi, making extensive use of new communication and information resources, the Internet and the variety of media available, is the realization of this characteristic. 17  Dear Oleneva, see if you can find me on YouTube under the name udambro. Maria Eugênia, 10 years old, taught me. If it worked, I’ll record the speech like this for VEm Brasil.

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Fig. 6  Ubiratan recorded his lecture for VEm Brasil

The importance of the family appears very clearly in his message, but, very like him, he never imposed his opinion. And his friend Carlos agreed with him.18 After watching the recorded presentation, Ubiratan still modestly asks if there is anything more to be done, as in Fig. 7, which also includes my late reply approving Maria José’s presence.19 In this way, the tension that preceded the recording was overcome, and Ubiratan could dedicate part of his time to understanding the details of the final organization and his participation during the exhibition of his lecture and throughout the event. Another thing that enchanted Ubiratan was the program. The content had already led him to conclude that ethnomathematics worked, but the format was something he observed, as seen in Fig. 8, after trying out its functionality on the morning of the first day of the event. Although he did not use Facebook or Google Photos, he recognized that it would make it easier for many to have the program in several media. However, he preferred the program table in the cloud, where he could clearly identify the lecturer and the lecture title and simply click on the link to access it at the

 Ubiratan: I managed to record the 15 min for VemBrasil. Maria José was listening to me standing by my side, and she decided to sit down to rest… and she was recorded, too. Let’s see if we should remove her at the end. I don’t think you need it, because it gives a family touch. Let’s see what Carlos and you think. Thank you very much, Carlos. Hugs, Ubiratan. Carlos: Dear Professor Ubiratan and Oleneva, I thought it was perfect… really. Maria José’s timing was great; it was really beautiful! If you want to see the final video, please download it from: [Note. The link has been removed because it is in a private dropbox]. 19  Ubiratan -In fact, it came out good. Thank you very much. Something else I should do? Hugs, Ubiratan Olenêva -Hey! Why cut Maria Jose?! Poor her! That’s so bad! She is a supporting player in everything: co-producer, co-advisor etc.. 18

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Fig. 7  Ubiratan asks if there is anything more to be done

Fig. 8  The program is easy “even for techno-dummies” like Ubiratan

opening or later. He told me that he shared the program table with many people. Besides Ubiratan’s emphatic and encouraging answer,20 Fig.  8 also displays the attachment of the message I sent him, the PDF image of the program that was made publicly available on Google Drive. The program that “works well” and is easy “even for techno-dummies” was created from a timetable distributed in intervals of 15 min for each lecture, which was edited by all their proponents. During the reception of the proposals, applicants would send an abstract and fill in the table with their names and time of preference, respecting the times already filled in by other applicants. Although he did not fill out this editable table, Ubiratan was one of the few who did not doubt that ethics would prevail in that collective construction of the program. There was no code of ethics; nobody even thought about it! D’Ambrosio observed the sequenced lectures, as  What a beauty! How well it works. It’s easy, even for little techno-dummies like me. Congratulations, Olenêva. You made your dream come true in the most brilliant way possible. Hugs, Ubiratan. 20

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everyone would have their audience moments. Renowned researchers, PhDs, and post-doctors would not be privileged to the detriment of beginning and young researchers. There would be no parallel presentations, which usually makes it so difficult to choose which one you should watch. One of the most delicate points of VEm Brasil was the acceptance of all works, including the applicants’ right to present their works during the event and to publish them in its proceedings and the final work foreseen in the planning. Of course, all abstracts were read, and all applicants received feedback, being kind and flexible and open to adjustments. Ubiratan really liked the risk of not having a scientific committee with the power to disapprove of works, and we were both very happy with the result. It effectively supported and contributed to the main product of the event, a virtual almanac, the e-Almanaque EtnoMatemaTicas Brasis (Sousa, 2020), which had enchanted him for years, caught his interest. The e-almanac had already been rejected for funding but saw its opportunity at VEm Brasil. Very optimistic about the event, he thought there would be many people involved, among applicants and viewers. He invested R$ 5.00 for the use of a paid service for the site and enjoyed that besides the low price, everything would be open on YouTube to all interested. He could not understand well how the premiere of a presentation on YouTube with the lecturer’s participation in interactions via chat would happen and preferred to wait to understand it by experiencing it during my talk, just before his. Behind the scenes, this trajectory for VEm Brasil was for Ubiratan  – and for me – a mixture of many enchantments, some tensions, and great expectations. His willingness to learn and to get involved with “the new” proved to be very important at VEm Brasil. The public had the opportunity to virtually “be” with him during the event in the YouTube environment. Expectations outweighed the tensions we faced. Bringing to memory one of his funniest statements about this overcoming always makes me laugh: “I will not be ashamed, nor will I let you down!” I never imagined this, and, in fact, he was brilliant, as we plunged together into VEm Brasil. Before the event, Ubiratan D’Ambrosio praised the 40-h burst, said he believed I would handle it well, but apologized for not being able to fully follow it during its occurrence. He promised to watch it later, and we could exchange ideas about some aspects that caught his attention. And so, it happened. Those dialogues were very relevant in the actions that emerged or unfolded from VEm Brasil, which we will address in the following section. All those considerations in this title sought to highlight the singular being (verb) human that Ubiratan was and the importance of this being (action) for the success of VEm Brasil. Cited in 100% of the event’s presentations, he is the biggest reference in ethnomathematics in the world, a builder of a wide-ranging research program, and a philosopher that proposed an epistemology that is easy to understand and that is attentive to the individual and the collective, to the real and the imaginary, to the sociocultural and environmental, and to the pedagogical and political. This internationally recognized thinker had the brilliance to reveal his weaknesses, to assume his incompleteness, to be someone who allowed himself to be met by young people, sure that “they want to find people who, together with them, seek the new.”

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3 The VEm Brasil as a D’Ambrosio’s Ethnomathematics Manifestation April 12, 2021, this date is a milestone to VEm Brasil 2020. We celebrated the anniversary of our actions. The event unfolded intensely until June of the following year. There was a proposal for an upcoming event, which has not yet taken place, but it was necessary to fully assess the first edition. I drew up a scheme that illustrated the entire process, abusing the colors and their tones, and sent it to Ubiratan (Figs. 9 and 10). The schema should be published in PDF because it was a hyperdocument with free access to the history of the event’s actions. It covered from the opening of the YouTube channel to the planned VEm Brasil 2020 Special Edition on Journal of Mathematics and Culture (JMC). It was used to produce a commemorative video of the event’s 1-year anniversary,21 but the attempt to publish the hyperdocument was frustrated, given that the small article was published in International Study Group on Ethnomathematics22 (ISGEm), in the ISGEm Newsletter,23 without the hyperlinks. VEm Brasil is a D’Ambrosian ethnomathematics manifestation, starting from the April 12 schedule, when we had so many expectations for a second edition soon. However, I resume the growing chronology of the process. Brief considerations about the Program Ethnomathematics, activity schedules per period, and email messages exchanged between Ubiratan and me constitute this section. I also resume the hypertextual feature of VEm Brasil, which Ubiratan liked so much because of the ease of immediately satisfying his curiosity as he read.

Fig. 9  VEm Brasil: I drew up a scheme that illustrated the entire process. (Dear D’Am, with the rains, without bathing in the sea for three days, the distraction was to make a scheme of the VEm Brasil 2020 process, which should serve to think about the 2022 event. I made the same scheme in two backgrounds, a black and a gray one, which are in the two attached images.)

 “VEm Brasil 2020: vídeo-celebração de 1 ano” [1-year anniversary], available at https://youtu. be/LpL0STPzKMI 22  Grupo Internacional de Estudos em Etnomatemática (Tradução). 23  “VEm Brasil 2020: o processo”, ISGEM Newsletter, v. 19, n. 1, mai. 2021, p. 28–29. Available at https://www.etnomatematica.org/home/wp-content/uploads/2021/05/ISGEm_Newsletter_ 19_1.pdf 21

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Fig. 10  VEm Brasil: first scheme that illustrated the process. (Source: Author’s elaboration)

VEm Brasil was theoretically guided by the Program Ethnomathematics, the intellectual legacy of Ubiratan D’Ambrosio. In the previous sections, I tried to prioritize aspects of D’Ambrosian feeling and behavior that were essential to the project design. I wanted to expose the backstage, its trajectory of enchantments, tensions, and expectations experienced until April 25, when the curtains opened to the VEm Brasil 2020 appearance. Also in 2019, the site doity.com.br/vem-­brasil-­virtual-­etnomatematica-­brasil> was created. Almost 150 ethnomathematicians made proposals (individual and coauthors) for communications and knew that there would be space for interaction between them and the viewers. As a Lakatosian research program, the diversity of conceptions, contexts, methodologies, practices, etc. was very welcome. VEm Brasil showed to be promising for the exchange and dissemination of research and experiences in ethnomathematics and for the consolidation of this program. Actions leading up to April 25–26 (Fig. 11) signal this potential.

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Fig. 11  VEm Brasil: actions leading up to April 25–26. (Source: Author’s elaboration)

The channel VEm Brasil -­EtnoMatemaTicas Brasis was launched. In the video description “VEm Brasil: bem-­vindo ao canal!” [VEm Brasil: welcome to the channel!] and on the website, the project was justified by the need to allocate a free virtual moment to ethnomathematics that would constitute a free virtual collection. Thus, the VEm Brasil community should deal with differences with “respect, solidarity, and cooperation,” inside an “ethics of diversity.” This spirit was signaled in the Special Edition of the RedINET-­Brasil Bulletin, year 4, number 18, when presenters reflected on what the event could represent as an opportunity for ethnomathematicians to meet, and Ubiratan wrote the text shown in Fig. 4. The dynamics of the cultural encounter is an important D’Ambrosian key concept. With applicants from 22 Brazilian states, the Federal District and the three Americas, Africa, Asia, and Europe, cultural differences and transculturality were evident in this meeting, with a dynamic favored by technologies.24 During the month of the event, I accepted two invitations to promote interest in the object of VEm Brasil: the interview with RedINET with its founding director, Hilbert Blanco-Álvarez, and the chat Vem Brasil in Humanistic Mathematics with Carlos Mathias, on the program “Conheça o VEm Brasil! O primeiro evento virtual aberto sobre Etnomatemática” [Come to VEm Brasil! The first open virtual event on ethnomathematics]. Both were valuable to the success of the event, nationally and internationally. And as the invitations were extensive, we celebrated when professors Daniel Orey and Milton Rosa brought three researchers from Nepal, who experienced and participated in much of the process.

 I suggest reading my article “EtnoMatemaTicas de Etnomatemáticos: Dinâmica do Encontro (Ciber)cultural” [EthnoMathematics of ethnomathematicians: dynamics of the (cyber) cultural encounter], 2021, available at https://journalofmathematicsandculture.files.wordpress. com/2021/05/article_1.pdf 24

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We had Ubiratan’s credibility, but the amplitude of the VEm Brasil can be seen as proof that ethnomathematics has worked and is being consolidating as a research program. Furthermore, the technologies enabled a multimedia profile, as Fig. 12, and the audience could search @etnomatematicas.brasis and access information from various environments. Actions followed after the base event, as shown in Fig. 13. In May 2020, the proceedings of VEm Brasil were published with the abstracts and links of the videos of the lectures and the viewers’ mini-accounts. After that, the RedINET-­­ Brasil Bulletin, year 4, number 20, brought articles signed by the RedINET-Brasil coordination and by two collaborators. All expressed their views on different aspects of the VEm Brasil. This issue had an attachment with the transcript of Ubiratan’s lecture, the link to his video subtitled in Portuguese and another in his honor, showing readings of excerpts from his works by ethnomathematicians

Fig. 12  VEm Brasil: a multimedia profile. (Source: Author’s elaboration)

Fig. 13  VEm Brasil: actions followed after the base event. (Source: Author’s elaboration)

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of VEm Brasil 2020, “Ubiratan D’Ambrosio: saber-­fazer e transcender” [Ubiratan D’Ambrosio: know-­how-­to-­do and transcend]. The Virtual EtnoMatemaTicas Humanistas (VEm Humanistas) [Virtual Humanistic EthnoMathematics (VEm Humanists)] was not foreseen. With similar purposes, the communities EtnoMatemaTicas Brasis and Matemática Humanista established a partnership to involve researchers of VEm Brasil in presentations and debates on ethnomathematics. Ubiratan opened the debate from his lecture text based on the text “Visão historiográfica da Etnomatemática como empreendimento humanista” [Historiographic view of ethnomathematics as a humanist enterprise], written for this purpose. The event highlighted theoretical, philosophical, educational, political, and investigative aspects of the Program Ethnomathematics. Experienced and leading researchers in the area such as Ubiratan and Gelsa Knijnik shared the space with leaders of research groups and educators, mixing different concepts, contexts, and interests. Streamed live on both channels, memory is in the playlist VEm Humanistas. At the invitation of professors Milton Rosa and Daniel Orey, we worked on the VEm Brasil 2020 Special Edition at JMC. The publication of volume 15, number 1 and number 2, was posthumously dedicated to Ubiratan. I intentionally left the e-Almanaque EtnoMatemaTicas Brasis for last, DOI https://doi.org/10.51361/9786586592139, final product of VEm Brasil. Published in December 2020, it had Ubiratan as an editorial consultant, author, and effective collaborator with stories, presentations, work drafts, and curiosities. But, as mentioned above, the idea of an e-almanac based on the Program Ethnomathematics marveled Ubiratan long before VEm Brasil. As he declared at the Authors’ Meeting, he was surprised by the number of pages and attributed the work to “ambitious and crazy ideas that come from my head,” making those present laugh. Ubiratan contributed a lot to the e-book and even nicknamed it, as shown in Fig. 14. With 456 pages, I contributed to the organization and editing of the “e-big almanac” and the Federal Institute of Piauí to the coediting and editing. The trilingual work of 100 authors had releases that lasted until April 22, 2021, justifying the creation of the playlist e-­Almanaque EtnoMatemaTicas Brasis. With the theme ethnomathematics and without norms, there were typically academic articles, many

Fig. 14  Ubiratan contributed a lot to the e-almanac.

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from mathematics education, but other literary and artistic expressions and various focuses, signaling “creative potentials,”, “transdisciplinarity,” and escapes from “epistemological cages.” Several actions to date emerge from the “e-big almanac,” such as the course “Introduction to ethnomathematics” from the Eskada platform of open courses at the Universidade Federal de Maranhão. Figure 15 presents an access code to the e-Almanaque and a view of resulting actions. After the publication, we met virtually at the Authors’ Meeting. Ubiratan was with Maria José, and both were surprised by their son’s presence as one of the special guests. It was a festive, emotional event that led me to believe that an assumption of ethnomathematics implies recognizing the importance of sociocultural aspects in the “Knowledge Cycle.” Immediately after the meeting, I received a message, Fig. 16, that made me very happy. Imbued with the happiness provided by the Authors’ Meeting and Ubiratan’s sweet words, I evaluated the recording of the event for the production of the

Fig. 15  e-Almanaque EtnoMatemaTicas Brasis: a view of resulting actions. (Source: Author’s elaboration)

Fig. 16  Ubiratan loved the Authors’ Meeting! (How beautiful. Thanks, too bad I don’t have my glass [here] to toast. You really make a difference. You can fulfill our hope. You are a winner. Thank you. Hugs, Ubiratan)

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premiere video of the General Launch of the e-­Almanaque, to take place the following week, when authors and audience would meet at the chat. At this point, Ubiratan and I had an argument. He had said a very shy hello in the opening chat of VEm Brasil and expressed his joy at seeing so many responses with affection but explained that he could not keep up with the speed of typing. Upon understanding that the launch would be in the chat, he sent the message from Fig. 17, to which I answered immediately. But after thinking about it, 3 min before the event, he revealed that he did not know what he should do and confirmed his presence. This conversation made us laugh a lot afterward. As soon as the video started, I called to see if he was at the launch. He was watching and really enjoying it. About the chat, Ubiratan was delighted with the speed, the people he knew, the messages he received when they knew he was there, the interactions, etc. Returning to Fig. 15, there are two sections of the e-almanac that were proposed to have their own “life” and continuity, fed by ethnomathematicians. There were special releases: the Calêndricas EtnoMatemaTicas, a differentiated system of calendaring events in the world of ethnomathematics, and the EthnoMatemaTicas Digital Library (BDEm), which gathers and receives publications on ethnomathematics, providing access links for download or purchase. e-almanac was designed to be a totally free publication, both in terms of the types of production and its access and insertion in libraries, repositories, etc. With so many authors, we encouraged them to promote their own launches, the Lançamentos Itinerantes [Itinerant Launches]. In all, there were ten launches inside

Fig. 17  Ubiratan did not know what he should do but confirmed his presence (Chatting in chat means what? Just watching? Yes, Ubiratan, in part. Chat is a synchronous online textual interaction, there where you see everyone writing and that you have also written in some events. As soon as YouTube starts showing the video, scheduled for 10:00 am, the chat is activated, until the end of the video. At 10, I enter. I don’t understand what you mean by a synchronous online textual interaction. What a complicated phrase. But let’s go)

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and outside Brazil, which lasted until April 2021. They were autonomous actions, and their recordings are in the playlist of the e- Almanaque of the VEm Brasil-­ EtnoMatemaTicas Brasis channel. Finally, bringing together the sub-schemas presented here, Fig. 18 displays an overview of VEm Brasil 2020, highlighting sub-events, actions, activities, proponents, and partners. I go back, then, to April 12, 2021. On this date, I received the last email from Ubiratan. As seen at the beginning of this topic, I started the message talking about the panoramic scheme of VEm Brasil 2020. This motivated us to spontaneously consider the subject on the agenda of our conversations from then on, which intensified and occurred almost on a daily basis. When I sent him an email to update him on a topic or event that might interest him, he already knew he did not need to answer; we could talk over the phone. During this time, we had a lot to talk about, as we were organizing the event “Ubiratan D’Ambrosio: pessoa e contribuições” [Ubiratan D’Ambrosio: the person and contributions], an international tribute in which he would participate. It would be another partnership between EtnoMatemaTicas Brasis and Matemática Humanista. I enjoyed telling him details about the event and seeing how, instead of being flattered, he was amazed and grateful. Ubiratan knew me well and knew that everything I said would not correspond to the whole plan for the event, so he could fully enjoy the surprises. The tribute should represent yet another expression of affection and recognition of its theoretical-philosophical and human potential and of its contributions to science, to education, and to society. Ubiratan passed away in the middle of the trajectory. Aware of the importance of his legacy for ideals of a better future, the tribute took place as planned, with a doubled workload, and its title was changed to

Fig. 18  VEm Brasil: another scheme that illustrated the process. (Source: Author’s elaboration)

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Fig. 19  Ubiratan: the last email. (The angel who watches over me does everything to ease the pain. I feel so lucky. There is always something good to do. I will prepare the next presentation. Hugs and go back to the beach soon. Ubiratan)

“Ubiratan D’Ambrosio: pessoa, contribuições e memórias” [Ubiratan D’Ambrosio: the person, contributions, and memories], the same name as in the playlist. In April 2021, Ubiratan explained that he had already said everything he wanted to say. I felt responsible for spreading the Program Ethnomathematics, as he organized it, as a research program and a general epistemology. It was not a doctoral commitment on an uncaged conceptual set capable of establishing dialogues and promoting conceptual interfaces with the most diverse areas of knowledge, in different contexts. It was a commitment to the trust he placed in me in these actions to spread his ideas. He sometimes expressed his satisfaction that there were many of us who did indeed understand the Program Ethnomathematics. Ubiratan was already tired, and the physical pains were getting worse. He did not lose the ability to be marveled by the good and the beautiful, which were in his eyes. He was always grateful and attentive to others, respectful of individualities. On April 12, exactly 1 month before his death, in response to the VEm Brasil scheme that I used here as a key element of reflections and memories of our trajectory and experience in the project, he writes me the message shown in Fig. 19, which highlights this incredible human being that life allowed me to follow his path, to get involved in his ideas and who makes me responsible for taking his great legacy to the future.

4 Inspiration for the Future The theme VEm Brasil has already been discussed much, but it is not exhaustive, as the diversity of actors and contexts that an ethnomathematically grounded project can contemplate is enormous. Its contents are available in the proceedings, in the e-Almanaque EtnoMatemaTicas Brasis and in the videos of the playlists VEm Brasil, VEm Humanistas, and e-Almanaque EtnoMatemaTicas Brasis, and some more in-depth reflections can be found in the articles of the Special Edition VEm Brasil 2020 of the JMC.

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Some concepts and principles of the Program Ethnomathematics are essential to a project inspired by it, guided by it. Therefore, I have highlighted in bold some of them throughout the text. In addition, VEm Brazil must be developed considering the dynamics of reality. By conception, it must be virtual and have ethnomathematics as its main motivational element and reference. It must be open to the new and seek to embrace the various conceptions presented to it by those who call themselves ethnomathematicians. Ubiratan was very happy when he realized the complexity of the set of concepts that he was tying and firming to each other and qualified it as a Lakatosian research program. As such, he could pass through several epistemological cages without being trapped in any of them, as he realized that transdisciplinarity and transculturality are essential conditions for those who seek to direct their research and pedagogical practices in trajectories of greater questions and objectives, such as social justice, peace, ethics, and sustainability. Of course, any event I organized would not have been so special. Supposedly, it would not have attracted so many experienced researchers in the area, or it would not have multiplied into so many actions and sub-events, or it would not have unfolded for so long, or, or… However, VEm Brasil 2020 had the seasoning of the D’Ambrosian ethnomathematical thinking. The rain continued for more than a month until a rainbow appeared, bringing the sun, the sea bathing, and the awareness that D’Ambrosian ethnomathematical thinking must be kept alive, as it represents Ubiratan’s very transcendence. For the opportunity to relive these moments with Ubiratan at VEm Brasil, my sincere thanks to Springer and its collaborators, friendships built through ethnomathematics and Ubiratan. I also acknowledge the trajectories of hopes for a better future that can come from this meeting of intellectual production guided by the legacy of Ubiratan D’Ambrosio. They are worth all our efforts! “Ethnomathematics came to stay,” as Ubiratan observed in his lecture on April 25, 2020, and “to evolve and to expand all its reach to shape a happy future.”

References D’Ambrosio, U. (1997). A era da consciência: aula inaugural do primeiro curso de pós- graduação em ciências e valores humanos no Brasil. Fundação Peirópolis. D’Ambrosio, U. (2005). Sociedade, cultura, matemática e seu ensino. Educação e Pesquisa, 31(1), 99–120. D’Ambrosio, U. (2009). Transdisciplinaridade (2nd ed.). Palas Athena. Sousa, O. S. (2016). Programa etnomatemática: interfaces e concepções e estratégias de difusão e popularização de uma teoria geral do conhecimento. Tese (Doutorado em Educação Matemática). Programa de Pós-Graduação em Educação Matemática, Universidade Anhanguera de São Paulo, São Paulo. Sousa, O.  S. (Org.). (2020). e-Almanaque EtnoMatemaTicas Brasis. https://doi. org/10.51361/9786586592139.

Influences and Contributions of Ubiratan D’Ambrosio in the Development of Ethnomodelling as a Research Concept Related to Ethnomathematics and Modelling Milton Rosa Abstract  This chapter aims to show the role of Ubiratan D’Ambrosio and his contributions to the development of ethnomodelling. His work promoted the advancement of interactions between members of distinct cultures by valuing and respecting the diversity of mathematical knowledge that is developed locally in different contexts. He both sought and encouraged new directions of investigation that allows us to provide innovative and relevant approaches in mathematics education. It is particularly interesting to further the exploration of his reflections about globalization, localization, and glocalization in regard to ongoing investigations related to the development of ethnomodelling that values and respects different forms of tics of diverse mathema rooted in different ethnos. One of the main contributions of D’Ambrosio to the development of ethnomodelling is related to the comprehension of unique connections between modelling, ethnomathematics, and cultural anthropology, which is related to the cultural dynamics of members of distinct communities by valuing and respecting diverse ways of mathematizing procedures and practices found across cultures. This approach enables the study of ethnomodelling in order to show that members of distinct cultures play an important role in the evolution of humanity, which leads them toward the development of a D’Ambrosian approach of building a planetary civilization that rejects the inequality, arrogance, and the prejudices that violate these dimensions of peace. Keywords  Ethnomathematics · Ethnomodelling · Modelling · Program · Ubiratan D’Ambrosio

M. Rosa (*) Universidade Federal de Ouro Preto, Ouro Preto, Minas Gerais, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_11

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1 Initial Considerations The life story of Ubiratan D’Ambrosio represents many facets, and each one contains stories to be revealed, gradually, through the eyes of historians, educators, researchers, and his former colleagues, friends, and students. His story also reveals the way he displayed his brilliant life lessons. In my meetings with him at national and international congresses and events related to mathematics education, as well as in my valuable conversations during meals with him and his wife Maria José, in their apartment in São Paulo, the richness of his life story was revealed. Sharing D’Ambrosio’s thoughts and memories enabled me to follow his personal, professional, and academic trajectory, as well as his contributions to the social, cultural, and political fields of education and mathematics education, which enabled Professor Daniel Orey and I to develop the theoretical foundation of ethnomodelling. For example, D’Ambrosio (2017) stated that: (…) Rosa and Orey (2010) are the pioneers in introducing the concept of ethnomodelling (...) and since we met, about twenty years ago, both have enthusiastically embraced the idea of ethnomathematics and of mathematical modelling and contributed significantly to the development of both areas of research. In this way, they synthesize that ethnomodelling is the intersection between cultural anthropology, ethnomathematics and mathematical modelling. (p. 13)

In general, D’Ambrosio’s national and international contributions to the evolution of education and mathematics education are mainly related to the development of the ethnomathematics program and the valorization of mathematical knowledge directed toward total peace, which have as their objective the search for social justice. As parallel contributions, I highlight his contributions to the development of the connection between ethnomathematics and modelling through ethnomodelling. In this way, the fact of being close to one of the most important and influential mathematics educators of the twentieth and twenty-first centuries was a privilege for me, mainly in relation to his guidance, support, and encouragement for educators and researchers who, through their investigations directed to social, political, economic, and environmental issues, became aware of the importance of valuing and respecting the sociocultural characteristics of mathematics. Thus, D’Ambrosian productions promoted interactions between different social classes by valuing and respecting the diversity of mathematical knowledge developed locally and dialogically in different contexts, such as school/academic environments. From this perspective, D’Ambrosio’s concern was primarily with the well-being of members of distinct cultural groups and also with the preservation of natural and cultural resources. This approach can be summarized as Peace in its various dimensions, such as inner peace, social peace, environmental peace, political peace, and military peace, which are directed toward the promotion of total peace. According to D’Ambrosio (1998), this approach is essential for the development of a civilization that rejects inequality, arrogance, and prejudice, which are considered as violations of these dimensions of total peace, and, as a consequence, promotes the impediment of the

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development of social justice. What is unique is that he developed this concept from the perspective of a mathematics educator, something truly unique and different.

2 My Personal Relations with Ubiratan D’Ambrosio: A Lasting Friendship My personal relations with D’Ambrosio, over the last 25 years, have helped me to understand his role as a person, a professional, an academic researcher, and an educator in the fields of Education and Mathematics Education and his role in the development of ethnomathematics as program and ethnomodelling as a pedagogical action and to understand the philosophy committed to a more just social order with dignity and a quality of life for humanity. Thus, the events that I share in this chapter refer to our time together, during these years, which only emphasize my increased admiration for D’Ambrosio, who made it possible for me to participate in his personal, professional, and academic life, which we shared in mentoring, in cooperative projects, and in many national and international events. For example, in 1996, when I was teaching mathematics at a public school in Amparo, in the interior of the state of São Paulo, Brazil, our school library received new books from the state government and made some of the old volumes available for free to the teachers. These books were placed on a table in the teachers’ lounge, and one of these volumes was entitled Ethnomathematics: Art or Technique of Explaining and Knowing, written by Ubiratan D’Ambrosio, in 1990. So, I took the book. I became really interested in this book and I read it in less than 2 h, and, later in the school year, as a mathematics teacher, I tried to apply some of the ideas in it about ethnomathematics and its pedagogical action in the classroom by developing mathematical curricular activities contextualized in the daily life of my students. In 1998, I joined a Specialization Course in Mathematics Education at the Pontifícia Universidade Católica de Campinas (PUCCAMP), in Ethnomathematics and Mathematical Modelling, promoted by Ubiratan D’Ambrosio, Geraldo Pompeu Júnior, and Rodney Bassanezi, among others. In this course, D’Ambrosio was a professor in the History of Mathematics course. At that time, we spoke about his book and also about Ethnomathematics, and then we exchanged some ideas for the development of a pedagogical action that I could use with my students in the classrooms. During this time, D’Ambrosio introduced me to Prof. Daniel Orey, who was serving as a Fulbright Scholar in the second semester of 1998. In 1999, with the help of Prof. Orey, I was invited to participate in an Exchange Program for Teachers of Mathematics, sponsored by the Department of Education of California, from September 1999 to January 2011.

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It is important to highlight that during this period, I worked as a mathematics teacher in a public high school in Sacramento, the capital of California, mainly teaching and working with immigrant students, many of whom were refugees or from families that were politically persecuted, as well as others were there for a diversity of religious and economic reasons. In 2000, I finished my master’s degree in Mathematics Education at California State University, Sacramento. The problem statement of this study was related to the development of a mathematical curriculum based on ethnomathematics and on the sociocultural and sociocritical perspectives of mathematical modelling for immigrant students in California. This thesis was entitled From Reality to Mathematical Modelling: A Proposal for Using Ethnomathematical Knowledge. In many ways this work launched the beginning of our study related to ethnomodelling. During my course work, I exchanged several ideas and emails with D’Ambrosio, and which were related to the development of this mathematical curriculum from an ethnomathematics and modelling perspectives, and he was the external reader of this thesis. In 2002, while working and studying in California, I helped to organize an event entitled Supper with Ubi D’Ambrosio, promoted by the University of California, Davis, which happened during D’Ambrosio’s stay at our house in Sacramento for 10 days. During his stay in California, he gave talks at schools and universities, visited the school where I worked and spoke with my immigrant students in different languages; as well, we discussed our ideas related to the connection between ethnomathematics and modelling. During the time I worked and lived in California, I met Ubi at various congresses and conferences in the United States, as well as in Brazil when visiting my family on vacation. We then collaborated on writing articles, interviews, and book chapters; as well, I conducted various interviews about diverse issues related to ethnomathematics, modelling, and the connections between the two knowledge areas. For example, when on vacation in Brazil, we would meet in São Paulo, in his apartment, with his wife Maria José, for lunch, coffee, and/or dinner, in which we would talk about life, mathematics education, our families, our friends, and, of course, ethnomathematics and other issues related to the sociocultural perspectives of this program and its connection to mathematical modelling. In 2010, I completed my doctorate in Education and Educational Leadership at California State University, Sacramento, which allowed me to study the influence of language and culture on the development of mathematical knowledge of immigrant students as English Language Learners. This dissertation was related to the connections between ethnomathematics and modelling, according to the perceptions of the leaders (principals and vice principals) of high schools in a suburban school district of Sacramento. As usual D’Ambrosio helped us to develop and understand the concepts of ethnomodelling and its cultural dynamism as well as its relation to dialogical approaches of his program and pedagogical action. During my stay in the United States for 11 years and, after my return to Brazil, in February 2011, to start my career and research work at the Universidade Federal

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Fig. 1  D’Ambrosio and I and D’Ambrosio and his wife in their apartment in São Paulo in November 2019 before the pandemic. (Source: Rosa’s personal file)

de Ouro Preto (UFOP), I continued to exchange ideas and emails with D’Ambrosio about ethnomathematics as program and its influence on the teaching and learning process in mathematics of students from minority cultural groups, as well as in relation to the development of Ethnomodelling. Thus, the visits, coffees, lunches, and dinners at D’Ambrosio’s apartment, in São Paulo, with his wife Maria José, were more frequent. Figure  1 shows D’Ambrosio and I and D’Ambrosio and his wife Maria José in their apartment in São Paulo, in November 2019, in our last face-to-­ face visit before the COVID-19 pandemic. Until his passing on May 12th, 2021, D’Ambrosio continued to offer honest, healthy, and relevant mentoring and guidance to my preparation as a professor, educator, and researcher.

3 Personal, Professional, and Academic Life of Ubiratan D’Ambrosio The life story of Ubiratan D’Ambrosio is a reflection on education, mathematics education, mainly ethnomathematics, which was developed in Brazil and internationally, and he “has been instrumental in helping to make sure that the socio-­ cultural context of mathematics and its teaching and learning are considered in conferences, publications and the day-to-day work of thousands of mathematics education around the world” (Scott, 2011, p. 1). D’Ambrosio was born on December 8, 1932, in the district of Belenzinho, in the city of São Paulo, in the state of São Paulo, Brazil. In 1943, after completing 4 years of elementary school at the Liceu Coração de Jesus, he was approved in the entrance exam to enter the secondary school (currently middle school). In 1946, he was enrolled in the scientific course (currently high school), at Colégio Visconde de Porto Seguro, in the city of São Paulo. In 1950, he signed up for the entrance exam for the licentiate and bachelor’s degree in mathematics (Borges et al., 2014). In 1951, D’Ambrosio entered the Faculty of Philosophy, Sciences and Letters of the University of São Paulo (USP), where he studied for a bachelor’s degree in

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mathematics. In the 3rd year of his studies, he began teaching in elementary and high school in both classical and scientific courses. After his graduation, in 1954, D’Ambrosio also taught at the Pontificia Universidade Católica de Campinas (PUCAMP), and in 1955 he completed the Licentiate Course in Mathematics at the same Faculty of USP. In 1958, D’Ambrosio married Maria José and became a full-time instructor at USP’s Escola de Engenharia de São Carlos (EESC), and in January 1960, they had their daughter, Beatriz. In 1961 he was transferred from EESC to the Faculty of Philosophy, Sciences and Letters, in Rio Claro, in the state of São Paulo, from the Department of Education, where he taught the subject of Algebra and Higher Analysis in the Mathematics Teacher Education Course. From 1960 to 1961, he was awarded with an Italian Government Scholarship, from the Instituto Mathematica dell’Università di Genova, in Italy. In January 1962, D’Ambrosio and Maria José had their second child, Alexandre. In 1963, D’Ambrosio completed his doctorate in Pure Mathematics by defending the thesis entitled Generalized Surfaces and Finite Perimeter Sets, from EESC, at USP.  In January 1964, he was invited to develop his postdoctoral study, from 1964 to 1965, as Research Associate, in the Department of Mathematics, at Brown University, in Providence, Rhode Island, in the United States. In 1964, D’Ambrosio moved to the United States accompanied by his wife Maria José and their two children, Alexandre and Beatriz. Although his intention was to move away from Rio Claro, São Paulo, for just 1 year, the Brazilian military coup encouraged him to remain in the United States, where he obtained the position of full professor at the State University of New York, in Buffalo, where he worked as a professor in undergraduate and graduate courses in mathematics and also as a researcher and an advisor. During his stay in the United States, D’Ambrosio dedicated himself to the study and research in pure mathematics, and in 1970, he was responsible for the Mathematical Analysis Sector, of a project proposed by the United Nations Educational Organization, Science and Culture (UNESCO) that was implemented in Bamako, capital of the Republic of Mali, Africa. This project was developed and implemented in 1971 at the Center Pédagogique Supérieur (CPS Project) as well as it was designed to prepare and instruct PhD students in Mathematics. This project is exemplified as a successful approach to higher education in a developing country by considering that this Center was created to respond to the needs for the training of personnel with higher education in the academic and professional areas, as it aimed to provide an intensive training program at the postgraduate level, which largely depended on the collaboration of visiting professors. By 1977, this program had trained 20 docteurs de spécialité with a level comparable to that of the French third cycle, but with its own characteristics through an absolute identification with the educational problems of Mali, as well as regarding to its role in the developmental process of this country (D’Ambrosio, 1977). In this context, D’Ambrosio traveled to Africa every 3 months and resided in Bamako. During this period, he was also a Consultant and Visiting Professor at the Graduate Program at the Center Pédagogique Supérieur, in Bamako, from 1970 to 1980. It is important to

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note that D’Ambrosio’s first doctoral student to obtain the title of Doctor in this program was Bakary Traoré, in 1973, with the dissertation entitled Théorie du controle optimal et courbes generalisées de Young. In 1972, D’Ambrosio returned to Brazil to teach at the Universidade Estadual de Campinas (Unicamp), where he served as Director of the Institute of Mathematics, Statistics and Scientific Computing (IMECC), from 1972 to 1980. His work at this university enabled him to develop research projects in pure mathematics, yet he also continued his participation in the doctoral projects he developed in Africa. Consequently, he supervised several doctoral and masters students at UNICAMP. Upon returning to Brazil, D’Ambrosio began his interest in conducting research and developing human resources and training courses for the teaching of science and mathematics. In 1974, he developed a project equivalent to which he had participated in Africa, at the master’s level, for all Brazilian states and countries in Latin America and the Caribbean, with ample funding from the Ministry of Education of Brazil and the Organization of American States (OAS). It is important to highlight that, from 1975 to 1980, D’Ambrosio was the director of this Master’s Program in Science and Mathematics Teaching. According to Chassot and Knijnik (1997), from 1978 onward, D’Ambrosio began to participate in the annual meetings of the Pugwash Conferences on World Science and Business, through which, in general, members of this organization discussed topics related to nuclear issues and peace. It is noteworthy that its members are elected by the active participants of that organization, and D’Ambrosio was elected a member of the committee of these conferences in 1987 and re-elected in 1992. In 1979, D’Ambrosio was elected President of the Inter-American Committee on Mathematics Education (IACEM) from 1979 to 1987, which allowed him to begin his investigations in history, sociology, and education, mainly in science and mathematics by enabling him to conduct investigations related to the connections between mathematics, society, and culture, as well as to establish the initial concepts of ethnomathematics as a program. It is important to emphasize that D’Ambrosio was a signatory of important documents in the world of science, such as the Venice Declaration of 1986 that was developed at the symposium entitled Science and the Frontiers of Knowledge – The Prologue to Our Past Cultural, which was organized by UNESCO and encouraged a greater reflection in a spirit of transdisciplinary and universality. In this symposium, the relevant theoretical and methodological procedures related to transdisciplinary, and universality, were presented, which enabled the understanding of the context in which they are developed, as well as helped to verify the conditions in which these ideas can be replicated and/or adapted in other sociocultural contexts. In 1995, the Nobel Peace Foundation, in recognition of services rendered in the name of world peace and social justice, awarded half of the Nobel Peace Prize to the Pugwash Conferences, the other half was given to Professor Joseph Rotblat, who was the president of this organization. Consequently, this double award honored all Pugwash members, including D’Ambrosio (Chassot & Knijnik, 1997).

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It is necessary to state here that, 1 year earlier, in 1994, D’Ambrosio retired from UNICAMP and was awarded with the honorary title of its Emeritus Professor. However, despite of his retirement, he continued to pursue his professional and academic career by developing his research agenda and advising master’s degree and doctorate students, in diverse investigative areas, in other universities, yet with more intensity at PUC-SP, in the Postgraduate Programs in Mathematics Education and History of Science. In this regard, D’Ambrosio also advised students in the Faculty of Education, at USP, and at the Institute of Exact and Geological Sciences, at Universidade Estadual Paulista (UNESP), in Rio Claro, São Paulo. It is important to state here that he was also the President of the International Study Group on Ethnomathematics (ISGEm) from July 1996 to June 2000, which is a research group he helped to found in 1985.

4 D’Ambrosio’s Role for the Development of Ethnomathematics D’Ambrosio’s personal, professional, and academic journey was remarkable and permeated by a diversity of experiences that led him to the creation of the movement related to the Ethnomathematics Program in the mid-1970s. Thus, in 1976, he organized and chaired the section entitled Why teach mathematics? of the Thematic Group: Objectives and Goals of Mathematics Education, during the Third International Congress on Mathematics Education (ICME-3), in Karlsruhe, Germany. In this section, he discussed about the cultural roots of mathematics in the context of mathematics Education (Rosa & Orey, 2014). In 1977, the term ethnomathematics was used for the first time in a lecture given by D’Ambrosio at the Annual Meeting of the American Association for the Advancement of Science, in Denver, Colorado, in the United States. In 1984, the term ethnomathematics was consolidated in the opening lecture entitled Sociocultural Basis of Mathematics Education, given by him at the Fifth International Congress on Mathematics Education (ICME-5), in Adelaide, Australia. This event was an important element for the development of mathematics education, as it was when he officially instituted the Ethnomathematics Program as a Lakatosian research field (Rosa & Orey, 2014). In 1985, D’Ambrosio published an article entitled Ethnomathematics and its Place in the History and Pedagogy of Mathematics, for the journal: For the Learning of Mathematics. According to Powell and Frankenstein (1997), this article represents the first comprehensive and theoretical treatise, in English, on Ethnomathematics as a Program. Therefore, these initial concepts of the Ethnomathematics Program stimulated the development of this research field internationally. In the first decade of the twenty-first century, Carpenter et al. (2004) selected this article to compose the book of the National Council of Teachers of Mathematics (NCTM), entitled Classics in Mathematics Education Research, because of its

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positive influence on international investigations in mathematics education, mainly related to the connection between the cultural aspects of mathematics and cultural anthropology through ethnomathematics. The publication of this book was a response to a request from the Educational Materials Committee (EMC) to develop a compilation of articles that would reflect the history of educational investigations, as well as to discuss about the international pedagogical and methodological influences that directly impacted the evolution of mathematics education internationally. On the other hand, the results of the study conducted by Shirley (2000) showed that D’Ambrosio was elected as one of the most important mathematicians of the twentieth century, mainly, in relation to social, political, cultural, historical, and anthropological issues in local and global communities through the development of the Program Ethnomathematics. For Rosa and Orey (2014), with the international expansion of the Ethnomathematics movement, in 1985, the International Study Group on Ethnomathematics (ISGEm) was created officially launching the Ethnomathematics Program internationally. In this regard, since its inception, ISGEm has promoted the appreciation and respect for the cultural diversity of ideas, procedures, techniques, strategies, and mathematical practices developed locally by seeking to apply this knowledge in education and mathematics education, as well as in sustainable development and social justice, in search of total peace. The ethnomathematics program values and respects the mathematical knowledge (knowing/doing) of members of peripheral cultures, as it seeks to understand the cycle of generation, diffusion, intellectual, and social organization, as well as the dissemination of this knowledge and practices in diverse contexts (D’Ambrosio, 1985). It is necessary to emphasize that, in the encounter of distinct cultures, there is a dynamic of adaptation and reformulation that accompanies the development of this cycle, including the cultural dynamism between distinct cultures and mathematical knowledge through dialogue that is triggered with symmetry and alterity (Rosa & Orey, 2007). Consequently, D’Ambrosio’s contributions in social, cultural, political, economic, and environmental areas established a deep relationship between mathematics, culture anthropology, and society. In this direction, he offered guidance, encouragement, leadership, and dissemination of ideas, concepts, and innovative perspectives involving Ethnomathematics as a program internationally, as well as their applications in education, mathematics education, and in other research and knowledge fields. Hence, D’Ambrosio is still the main leader in the dissemination of this research field, as his broad and holistic view of ethnomathematics sought to explain the dialogical transformation of mathematical knowledge developed by members of distinct cultural groups in their communities and societies. In this perspective, the epistemology of the ethnomathematics program is consistent with Freirean approaches that seek an educational process aimed at autonomy and freedom, as it is aligned with the current survival needs of members of different

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cultural groups and with the development of critical consciousness through reflections that aim at reaching its transcendence. In this approach, the development of mathematical knowledge is cooperative, collaborative, dynamic, and interactive because mathematics is the result of human activity, and, in this case, this study field is not static, linear, ordered, and/or universal, as it is humanist. In 1983, D’Ambrosio was honored with the title of Fellow of the American Association for the Advancement of Science (AAAS) for his imaginative and effective leadership in the evolution of mathematics education in Latin America and for his efforts toward the development of the international cooperation. In a previously established agreement, Gerdes (1997) and Powell and Frankenstein (1997) stated that D’Ambrosio is regarded as the intellectual Father of the Ethnomathematics Program. In 2005, D’Ambrosio was honored by the International Committee of Mathematics Instruction (ICMI) with the second Felix Klein Medal in recognition of his contributions to the development of Mathematics Education. In 2016, he was awarded with the title of Emeritus Member of the Brazilian Society of Mathematics Education (SBEM) for his contributions to the development of ethnomathematics and mathematics education in Brazil. In this context, D’Ambrosio contributed to the development of investigations in ethnomathematics by guiding researchers, educators, teachers, and students from several national and international universities to the study of the cultural aspects of mathematics and its use in diverse educational contexts. Thus, his personal, professional, and academic life was exemplary for encouraging thousands of scholars around the world in the search for peace and social justice through the appreciation and respect for mathematical knowledge and practices developed locally by members of distinct cultures. According to this perspective, D’Ambrosio paved the way for the development of investigations that are sensitive to the social, cultural, and historical characteristics of students and of internal and external school environments in which the teaching and learning process in mathematics is triggered. He also discussed the need for the entire population to have access to a quality educational process, which cannot be only directed to privileged segments of society. In this regard, D’Ambrosio has always defended the principles, ethics, and values that seek to promote quality education that provokes the development of critical, reflective, and active citizens in society who also seek the transformation of society.

5 D’Ambrosio’s Influences and Contributions to the Development of Ethnomodelling One of D’Ambrosio’s main contributions to ethnomodelling was to enable the development and connections of mathematical knowledge with cultural dynamics by valuing and respecting the different forms of mathematical thinking that are

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found in diverse cultural contexts. Consequently, he showed that mathematics is a broad and holistic field of knowledge, as well as a humanistic endeavor. In this context, according to D’Ambrosian contributions to its development, ethnomodelling enabled the study of ideas, procedures, and mathematical practices found in distinct contexts, as well as raised awareness that members of distinct cultures play an important role in the evolution of humanity. This unique context provided the connections between various mathematical knowledge, educational systems, and local and school/academic communities (global). In 2010, along with Professor Daniel Orey, we developed the concept of ethnomodelling to establish mathematical modelling as a relevant pedagogical action for the development of an ethnomathematics program, which helps members of distinct cultures to understand how cultural origins and linguistic, social values, morals, and lifestyles influence the evolution of mathematical knowledge developed in diverse context. In this context, it is necessary to outline the D’Ambrosian influences in the development of the concepts and the theoretical basis of ethnomodelling. For example, in his masterpiece entitled Ethnomathematics and its Place in the History and Pedagogy of Mathematics, published by the journal For the Learning of Mathematics, D’Ambrosio (1985) established his initial thoughts about the connection between ethnomathematics and modelling, which concept presents a broader interpretation of the nature mathematics and its association with culture. In this regard, it is important to state that: (…) culture manifests itself through jargons, codes, myths, symbols, utopias, and ways of reasoning and inferring. Associated with these we have practices such as ciphering and counting, measuring, classifying, ordering, inferring, modelling, (…), which constitute ethnomathematics. (p. 46)

In this regard, D’Ambrosio (1985) also stated that: This is a very broad range of human activities which, throughout history, have been expropriated by the scholarly establishment, formalized, and codified and incorporated into what we call academic mathematics. But which remain alive in culturally identified groups and constitute routines in their practices. (p. 45)

In accordance with this assertion, D’Ambrosio (1990) comments that everyday problems are solved through the use of locally developed mathematical information that requires the management of strategies and techniques through the elaboration of models. For example, he argues that ethnomathematics is characterized as a way of understanding mathematical thinking of the members of distinct cultural groups, while mathematical modelling functions as a tool that becomes important for them to act and interact in the world. This approach considers the environment of members of different cultural groups who collect local information regarding the resolution of a given problem or phenomenon and then develop specific procedures to model them through mathematization to transform the results into actions that can benefit the community. Therefore, modelling real situations is the most appropriate method for working with

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diversities because it provides the development of pedagogical actions in the classrooms by raising awareness about the cultural aspects of mathematics (D’Ambrosio, 1990). In his lecture entitled Ethnomathematics and Modelling, at the First Brazilian Congress on Ethnomathematics (ICEm-1), held at the Faculty of Education, USP, from November 1 to 4, 2000, D’Ambrosio (2000) stated that it is important to examine the cycle of knowledge, mainly, how its generation and organization occurs. Knowledge results from information received from reality through the senses, memory, and genetic codes. Information is processed in order to generate knowledge. In this regard, D’Ambrosio (1993) affirmed that the information is captured thanks to the intellectual and material instruments available in the sociocultural environment, which is organized as representations of reality through the elaboration of models. This process is named modelling, whose development is linked to the use of ethnomathematical instruments available in this context. From this perspective, D’Ambrosio (2000) states that everyone is developing modelling, for example, applied mathematicians are using the mathematical knowledge of academic circles (global), while members of distinct cultural groups are using mathematical practices developed in their own communities (local). Thus, the members of each culture use their own intellectual and material resources, that is, their own ethnomathematics. It is important to emphasize here that D’Ambrosio’s readings on the connections between ethnomathematics and modelling influenced our investigations in these fields of study in a vigorous and profound way, as we sought to understand them epistemologically, socially, and culturally, so that I could have a holistic view of reality and enhance this understanding toward the development of ethnomodelling. This context enabled me to conduct, in 2000, my master thesis in which I proposed the development of a mathematical curriculum based on ethnomathematics and mathematical modelling for immigrant students in California in order to achieve their educational objectives in this pedagogical action. The results of this study showed that it is necessary to develop modelling processes by using mathematizations developed by the students in their cultural contexts in order to respect and value their own traditions and cultures. Previously, Scandiuzzi (2002) reported in the article entitled Água e óleo: modelagem e etnomatemática? (Water and oil: modelling and ethnomathematics?) that there are philosophical and epistemological aspects that would make impossible the development of the relation between ethnomathematics and mathematical modelling by pointing out that these trends apply distinct research methods. In this same year, Bassanezi (2002) addressed the relation between ethnomathematics and mathematical modelling in the book entitled Ensino-aprendizagem com modelagem matemática (Teaching-learning with mathematical modelling) by coining the term ethno/modelling in which its meaning is related to the position of assuming mathematics as a knowledge field present in people’s daily lives that enables them to consider it as strategies and techniques for action and interpretation of their own realities.

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Conversely, based on the readings and investigations conducted by D’Ambrosio during his professional and academic life, we (Rosa & Orey, 2003) published the article entitled Vinho e queijo: Etnomatemática e Modelagem Matemática! (Wine and cheese: ethnomathematics and mathematical modelling!) in which we discussed the relation between these two trends in mathematics education by affirming that this combination has the potential to be developed in the pedagogical action in the classrooms. In search of continued understanding and deepening the conceptualization of the mutual interaction between ethnomathematics and modelling, as well as to comprehend the conceptualization of ethno/modelling, we, Rosa and Orey (2010), published an article entitled Ethnomodelling: An Ethnomathematical Holistic Tool, in which we developed the theoretical and methodological basis of ethnomodelling by publishing several articles and book chapters in English related to the conduction of investigations in this area. Two years later, we, Rosa and Orey (2013), wrote our first article in Portuguese on ethnomodelling by defining it as a pedagogical/methodological approach that considers it as a practical application of ethnomathematics, which adds cultural perspectives to the mathematical modelling processes. It is necessary to highlight here that D’Ambrosio’s advices and comments helped us to further develop the concept of ethnomodelling. In 2017, along with Professor Daniel Orey, we wrote our first book entitled Ethnomodelling: The Art of Translating Local Mathematical Practices, in which we discussed the interpretation of mathematical knowledge developed by members of culturally distinct groups by explaining the advancement of the theoretical foundations and reflections on these mathematical procedures and practices. In this context, D’Ambrosio (2017) stated that Rosa and Orey (2017) selected examples by making a theoretically accessible and academically rigorous approach to local and global mathematical knowledge by guiding researchers who develop investigations in other cultures in which they find fewer familiar aspects of reality. In this context, members of every culture share myths, a common language, a set of knowledge, acceptable behaviors, and actions that are, in general, subordinated to parameters called values. For D’Ambrosio (2017), this book was also an important contribution to the research methodology, as we, Rosa and Orey (2017), were able to show that investigations in ethnomodelling are related to the acquisition of emic knowledge, which is essential for intuitive and empathic understanding of the mathematical ideas, procedures, and practices, and etic, which is essential for conducting cross-cultural comparisons through comparative categories. Thus, he showed that there can be a complete understanding and a broad understanding of the scientific and mathematical knowledge developed by members of distinct cultural groups. In this regard, D’Ambrosio (2017) stated that emic and etic knowledge have a long history. For example, since the mid-nineteenth century, when theoretical reflections on linguistics, anthropology, and the social sciences began, the emic (local) approach arises from internal concerns with the psychology of popular beliefs and the concern of cultural anthropologists to understand the culture of natives. The etic

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(global) approach attempts to understand external influences and relationships, such as economic, commercial, political, environmental, and/or ecological conditions, on global practices, often not perceived as relevant to members of distinct cultures. The emic approach recognizes experiences and subjective mathematical knowledge from an internal point of view, while the etic approach seeks to explain cultural phenomena from external points of view by using categories that only have meaning for researchers. In this context, the relation between the emic and etic approaches is complementary in a dialogic manner (glocal), which favors the development of cultural dynamism. However, it is important to emphasize that investigators are not able to develop purely emic (internal) or purely etic (external) conclusions when conceptualizing culture and its influences on cognition (D’Ambrosio, 2017). This means that the theoretical approach of ethnomodelling develops the concepts of emic and etic, introduced by the American linguist/theologian Kenneth Pike (1954), as the essence of the concepts of phonemics and phonetics, which in addition to the linguistic phenomenon, enable investigators to analyze sociocultural phenomena, as emic studies the subjective experience acquired internally by members of a particular cultural group, while etic deals with the objective explanation of a sociocultural phenomenon from the point of view external to the culture. Consequently, it is important to use both emic and etic approaches to deepen the understanding of important issues related to cultural psychology, education, and the power relations existing between the members of distinct cultural groups by developing a dialogic approach in which mutual understanding is developed with respect to the differences. For example, D’Ambrosio contributed to the development of ethnomodelling by showing that members of distinct cultural groups react in different ways to the events that occur in their daily lives, as well as to the sociocultural conditions in which they survive and to the natural and cultural phenomena, as well as to the international health, environmental, political, religious, and war crises that afflict humanity. In this direction, D’Ambrosio (1995) states that the evolution of mathematical knowledge is influenced by the diversity of cultures and their components, such as language, religion, customs, economy, political, and social activities, which they influence and stimulate the development of creativity in the process of solving local and global problems and situations. These creative processes enable members of distinct cultural groups to develop their own way of mathematizing phenomena that occur in their realities. Another influence of D’Ambrosio to the development of ethnomodelling as a research field was to show that modelling offers the use of cognitive strategies that can assist members of distinct cultural groups to interact with situations and problems present in their own ethnos, which also deals with their every day and imaginary lives. Thus, dealing with situations and problems is a response to daily survival, which is closely linked to understanding and explaining larger existential and cosmological issues, which is a response to the transcendence of this knowledge. It is important to point out that D’Ambrosio’s invaluable contributions to the conceptualization of ethnomodelling enabled us to understand that the insertion of

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researchers in the sociocultural context of a given community has unpredictable effects that can affect the texture of society. In this regard, the generic, abstract, and symbolic concepts, which are fundamental to elaborating and expressing thoughts in the investigated communities, are often not recognized by the investigators, who are concerned with finding, in these communities, the specific concepts of their culture and for their training and tradition. Finally, ethnomodelling values and respects the knowledge and practices developed locally by members of distinct cultural groups, which were acquired in their daily experiences, behavioral concepts, and coexistence relations. Therefore, it is necessary to value this kind of (mathematical) knowledge in order to relate it to diverse knowledge systems, such as school/ academic systems, so that they can contribute to the formation of critical and reflective citizens so that they can actively act in society in search of total peace and social justice.

6 Final Considerations Currently, in the second decade of the twenty-first century, there is a growing cultural sensitivity toward understanding and comprehension of ideas, procedures, and mathematical practices developed by members of distinct cultural groups. Due to the conduction of ethnomodelling studies, investigations in this research field show the possibility of internationalization of mathematical practices in distinct cultural contexts through dialogues. From D’Ambrosio’s contributions to the development of ethnomodelling, it can be concluded that members of distinct cultural groups build and apply their own mathematical knowledge that was developed through different cognitive processes, which enabled the development of mathematical competences that include the actions of counting, locating, measuring, drawing, representing, playing, understanding, comprehending, explaining, mathematizing, inferring, and modelling. This context enabled D’Ambrosio (2020) to expand the national and international discussion on cultural plurality, as well as regarding the nature of mathematical knowledge that is linked to the social, cultural, economic, political, and environmental aspects of distinct cultures. However, the great challenge for humanity is understanding how to deal with the conflicts and confrontations that are imminent in social relationships, policies, and governments, which aim to solve everyday problems that are intrinsic to life itself. Hence, there is a need for humanity to become aware of the priority for the search for a civilization with dignity, in which the effects of iniquity, arrogance, and fanaticism are eradicated so that members of distinct cultural groups can achieve a peaceful world with social justice. In this regard, it is important to encourage new directions for the development of a planetary peace through the deepening of critical reflections on the processes of localization and globalization of mathematical practices and their role in the development of society (D’Ambrosio, 2007).

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Throughout his history, D’Ambrosio developed and encouraged the use of a critical and reflective sense that valued and respected the different ways of mathematizing reality by seeking to raise the self-esteem of members of distinct cultural groups by promoting creativity and the dignity of their cultural identity. Thus, he showed that humanity’s survival depends on its relations with nature and the environment, which can be regulated by ecological principles. According to his personal, professional, and academic life trajectory, D’Ambrosio sought total peace and social justice by proposing the development of an awareness process about the relationship between mathematics, culture, and anthropology, which aims to develop a fairer society through the development of ethnomodelling. In this regard, he sought harmony between peoples, so that together the members of these cultures can transcend to a planetary civilization with peace and dignity for humanity. Consequently, D’Ambrosio’s influences and contributions to ethnomodelling validate, respect, and legitimize the experiences of members of distinct cultural groups for their active participation in society as critical and reflective citizens, who seek to understand colonialism, power relations, and oppression, in a critical way when considering the effect of culture and language on the development of mathematical knowledge. These influences and contributions form the basis for the development of ethnomodelling, which helps us to examine and reconceptualize pedagogical actions related to the contextualization of mathematical practices that emerge in diverse cultural contexts. Thus, in the context of the development of national and international dialogues, D’Ambrosio expanded the discussion on the possibilities of including diverse and innovative perspectives and contexts on ethnomodelling in relation to the sociocultural diversity of members of distinct cultural groups in their quest for survival, transcendence, social justice, and total peace. This approach is related to the development of understanding, sharing, solidarity, respect, and appreciation of differences through the realization of other dialogues with the main objective of minimizing/reducing and/or avoiding domination and oppression. In conclusion, very important influence and contribution of D’Ambrosio to the development of ethnomodelling are related to the valuable message of respect and solidarity of our many diverse sociocultural differences, because in today’s world, while migration and immigration grow in intensity, the encounters between members of distinct cultures are frequent. Thus, mutual respect is the answer to prevent these differences from being resolved through confrontation and violence. Therefore, it is important to thank Ubiratan D’Ambrosio for having offered us this message of peace during his personal, professional, and academic life.

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References Bassanezi, R. C. (2002). Ensino–aprendizagem com modelagem matemática: uma nova estratégia [Teaching-learning with mathematical modelling: A new strategy]. Editora Contexto. Borges, R.  A. S., Duarte, A.  R. S., & Campos, T.  M. M. (2014). A formação do educador matemático Ubiratan D’Ambrosio: trajetória e memória [The formation of the mathematician educator Ubiratan D’Ambrosio: Trajectory and memory]. Bolema, 28(50), 1056–1076. Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (2004). Classics in mathematics education research. National Council of Teachers of Mathematics (NCTM). Chassot, A., & Knijnik, G. (1997). Conversando com Ubiratan D’Ambrosio [Talking with Ubiratan D’Ambrosio]. Episteme, 2(4), 9–25. D’Ambrosio, U. (1977). The project ‘CPS-Bamako’: An option in post-graduate training for developing countries. Educafrica: Bulletin of the UNESCO Regional Office for Education in Africa, 1(2), 79–83. D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44–48. D’Ambrosio, U. (1990). Etnomtemática: arte ou técnica de explicar e conhecer [Ethnomathematics: Art or technique of explaining and knowing]. Editora Ática. D’Ambrosio, U. (1993). Etnomatemática: um programa [Ethnomathematics: A program]. A Educação Matemática em Revista, 1(1), 5–11. D’Ambrosio, U. (1995). Multiculturalism and mathematics education. International Journal on Mathematics, Science, and Technology Education, 26(3), 337–346. D’Ambrosio, U. (1998). Mathematics and peace: Our responsibilities. ZDM, 98(3), 67–73. D’Ambrosio, U. (2000). Etnomatemática e modelagem [Ethnomathematics and modelling]. In Proceedings of the first Brazilian Congress on ethnomathematics – ICEm1 (p. 142). FE/USP. D’Ambrosio, U. (2007). Peace, social justice and ethnomathematics. Monograph 1. The Montana Mathematics Enthusiast, 25–34. D’Ambrosio, U. (2017). Prefácio [Preface]. In M. Rosa & D. C. Orey (Eds.), Etnomodelagem: a arte de traduzir práticas matemáticas locais [Ethnomodelling: The art of translating local mathematical practices] (pp. 13–16). Editora Livraria da Física. D’Ambrosio, U. (2020). Etnomatemática e matemática humanista: uma conversa com Ubiratan D’Ambrosio [Humanistic mathematics: a conversation with Ubiratan D’Ambrosio]. Série Debates sobre Matemática., Cultura e Escola. Programa Matemática Humanista ao vivo com Carlos Mathias. Program aired on April 9th, 2020. Universidade Federal Fluminense. Gerdes, P. (1997). On culture, geometrical thinking and mathematics education. In A. B. Powell & M. Frankenstein (Eds.), Ethnomathematics: Challenging Eurocentrism in mathematics education (pp. 223–247). SUNY. Powell, A.  B., & Frankenstein, M. (1997). Ethnomathematical knowledge. In A.  B. Powell & M. Frankenstein (Eds.), Ethnomathematics: Challenging eurocentrism in mathematics education (pp. 5–13). SUNY Press. Rosa, M., & Orey, D. C. (2003). Vinho e queijo: etnomatemática e modelagem! (Wine and cheese: ethnomathematics and modelling!). Bolema, 16(20), 1–16. Rosa, M., & Orey, D. C. (2007). Cultural assertions and challenges towards pedagogical action of an ethnomathematics program. For the Learning of Mathematics, 27(1), 10–16. Rosa, M., & Orey, D. C. (2010). Ethnomodelling: An ethnomathematical holistic tool. Academic Exchange Quarterly, 14(3), 191–195. Rosa, M., & Orey, D.  C. (2013). Ethnomodelling as a methodology for ethnomathematics. In G. A. Stillman & J. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 77–88). Springer. Rosa, M., & Orey, D. C. (2014). Fragmentos históricos do programa etnomatemática [Historical fragments of program ethnomathematics]. In S. Nobre, F. Bertato, & L. Saravia (Eds.), Anais/ Actas do 6° Encontro Luso-Brasileiro de História da Matemática [Proceedings of the 6th Luso-­ Brazilian meeting on the history of mathematics] (pp. 535–558). SBHMat.

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Rosa, M., & Orey, D. C. (2017). Etnomodelagem: a arte de traduzir práticas matemáticas locais [Ethnomodelling: the art of translating local mathematical practices]. Editora Livraria da Física. Scandiuzzi, P. P. (2002). Água e óleo: modelagem e etnomatemática? (Water and oil: modelling and ethnomathematics?). Bolema, 15(17), 52–58. Scott, P. (2011). The intellectual contributions of Ubiratan D’Ambrosio to the ethnomathematics. In Proceedings of the 13 Interamerican conference on mathematics education (pp. 1–6). Universidade Federal de Pernambuco Shirley, L. (2000). Twentieth century mathematics: A brief review of the century. Teaching Mathematics in the Middle School, 5(5), 278–285.

The Importance of Ubiratan D’Ambrosio in Latin America Armando Aroca

and Maria Cecilia Fantinato

Abstract  This chapter aims to analyze the importance of Ubiratan D’Ambrosio for the development of the ethnomathematics program in Latin America. The text begins with a brief historical and geopolitical characterization of Latin America. Then, it presents the biography of the mathematical educator Ubiratan D’Ambosio, creator and promoter of the ethnomathematics program. From a survey in the Proceedings of the International Meetings on Ethnomathematics (ICEm), the growth in the number of authors and the diversity of Latin American countries in recent years is evident. To understand this development, it was decided to conduct interviews with representatives from different Latin American countries. The results point to the importance of D’Ambrosio in this process, for being the thesis supervisor of Latin American researchers, for being an inevitable reference in any research on ethnomathematics, and for the strength of his ideas of the ethnomathematics program, which are in favor of a mathematical education that respects the cultural diversity of the countries. Keywords  Ubiratan D’Ambrosio · Life history · Ethnomathematics program · Latin America

A. Aroca (*) Universidad del Atlántico, Barranquilla, Colombia e-mail: [email protected] M. C. Fantinato Fluminense Federal University, Niterói, RJ, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_12

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1 Presentation In this chapter, some topics will be presented that will allow us to understand the importance of Ubiratan D’Ambrosio for Latin America. It begins with the question, what is Latin America? In sequence a second question is addressed: who was Ubiratan D’Ambrosio? We go through the development of ethnomathematics in Latin America, and then we addressed the following question: what did Ubiratan D’Ambrosio mean for Latin America? Subsequently, the responses of ten ethnomathematical researchers from Latin America are presented. In the final topic, conceptual elements about Ubiratan D’Ambrosio and its importance in Latin America are discussed.

2 What Is Latin America? Latin America, according to the literal connotation of the term, are territories where Romance languages are spoken; this also includes the provinces of Quebec and New Brunswick in Canada; the US states of Louisiana, Florida, California, Texas, Arizona, Nevada, Colorado, and New Mexico plus Puerto Rico, one of the US dependencies; and the French territories of French Guiana, Clipperton, Guadeloupe, Martinique, Saint Barthélemy, Saint Martin, and Saint Pierre and Miquelon. For a better understanding of this territory, see Bohoslavsky (2009). So Latin America goes far beyond what we understand by the classic region of Central America, South America, and the Antilles. We saw the need to make this geographical description, because often titles about the impact of people in territories are used, without really knowing what the territory is. Having now an approximation to what Latin America is, a deeper investigation would tell us about the importance of Ubiratan D’Ambrosio in each of these territories and countries. In this text, we intend to present some elements about the importance of this math educator in ten countries of this region called Latin America. These countries are Peru, Chile, Panama, Mexico, Ecuador, Costa Rica, Venezuela, Guatemala, Brazil, and Colombia.

3 Who Was Ubiratan D’Ambrosio? Ubiratan D’Ambrosio was a noble person and a great researcher, who contributed to the history of mathematics and science in general. He was a tireless fighter for equality and peace based on the acceptance of other people’s ways of thinking, which can be inferred from his main contribution to the field of mathematics education, such as the ethnomathematics program. The ethnomathematics program helps us understand the nature of mathematics in and among the diverse sociocultural contexts of the world. Perhaps one of the mathematical historians who has best

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described Ubiratan D’Ambrosio as a person and as a researcher has been his friend Professor Luis Carlos Arboleda Aparicio, in Arboleda (2012). In Arboleda (2012) three descriptions of Professor D’Ambrosio are made, namely: • In the first place, what is most obvious to the international community: his contribution to the renewal and expansion of the field of history of mathematics by introducing the epistemological perspective of ethnomathematics (D’Ambrosio, 2002). • Second, the program of its creation and in which he has been working for more than 25 years, on the pedagogical appropriation of the history of mathematics based on the ethnomathematics approach (D’Ambrosio, 1985, 1992). • Finally, his research on the history of non-Western scientific cultures and, in particular, on the history of Latin American science, based on his radical criticism of Eurocentrism (D’Ambrosio, 2000, 2001) (p. 236). We also find a great review of the teacher D’Ambrosio by another of his great friends, Professor Milton Rosa, who declares the following in D’Ambrosio and Rosa (2008, p. 92): The Role of Ubiratan D’Ambrosio in the Development of the ethnomathematics. It Is important to highlight the importance of the Brazilian mathematician and philosopher Ubiratan D’Ambrosio, in relation to the development and evolution of the field of ethnomathematics. D’Ambrosio is also one of the most important theoreticians in this field. By offering encouragement, leadership, and dissemination of new ideas, concepts, and perspectives involved in ethnomathematics around the world and its applications in mathematics education he is without a doubt the primary leader. Powell and Frankenstein (1997a) stated: D’Ambrosio’s broader view of ethnomathematics accounts for the dialectical transformation of knowledge within and among societies. Moreover, his epistemology is consistent with Freire (1970, 1973) in that D’Ambrosio views mathematical knowledge as dynamic and the result of human activity, not static and ordained (p.  8). D’Ambrosio’s studies in the area of socio-political issues established a strong relationship between mathematics, anthropology, culture, and society. In 1983, D’Ambrosio was honored with the title of Fellow of the American Association for the Advancement of Science (AAAS) for his imaginative and effective leadership in Latin American Mathematics Education and his efforts towards international cooperation. Gerdes (1997) and Powel & Frankenstein (1997b) have considered D’Ambrosio “the intellectual father of the ethnomathematics program” (p. 13). D’Ambrosio was also selected as one of the most important mathematicians of the twentieth century in the area of sociopolitical issues and ethnomathematics (Shirley, 2000). In 2001, D’Ambrosio was the recipient of the Kenneth O.  May Medal of History of Mathematics granted by the International Commission of History of Mathematics (ICHM). Andersen (2002) stated that “The ICHM has awarded the May Medal to D’Ambrosio for his never ending efforts through writing and lectures to promote Ethnomathematics and thereby contributing intensely to make the field established” (p. 1). In 2005, D’Ambrosio was awarded with the second Felix Medal of the International Commission on Mathematical Instruction (ICMI) that acknowledges his role in the development of mathematics education as a field of research.

The profile of the master Ubiratan D’Ambrosio exceeds any description that can be written about him; we only dare to say that his contributions to mathematics education have been exceptional to understand the ways of doing, thinking, and

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communicating mathematics of other cultures, other peoples, and other communities and that this is one of the paths to peace.

4 The Development of Ethnomathematics in Latin America According to most ethnomathematics studies, the pioneering work in the area was carried out in the second half of the twentieth century, as ethnomathematics became a genuine disciplinary field in the early 1980s. By briefly pointing out the historical contributions of other researchers from the first half of the twentieth century to the emergence of the term, Paulus Gerdes (1994) highlights D’Ambrosio and his first definition of ethnomathematics: “[…] we will call ethnomathematics the mathematics which is practiced among identifiable cultural groups, such as nationaltribal societies, labor groups, children of a certain age bracket, professional classes, and so on” (D’Ambrosio, 1985, p. 45). Powell and Frankenstein (1997) consider the Brazilian mathematics educator as the “intellectual father of the Ethnomathematics Program.” On the other hand, Gelsa Knijnik (2004) acknowledges that: “Ethnomathematics owes the beginning of its development as an area of mathematics education to Ubiratan D’Ambrosio” (p. 20). Ethnomathematics has grown significantly throughout the world since its inception. In 1985, a group of mathematics educators, including the Brazilian Ubiratan D’Ambrosio, founded the International Study Group on Ethnomathematics (ISGEm). This association was articulated by geographic regions, such as the North American Ethnomathematics Study Group (NASGEm), responsible for the first academic journal in this field, the Journal of Mathematics and Culture. In Brazil, ethnomathematics is a particularly strong and established field, first, because the main theoretician of ethnomathematics is a Brazilian, Ubiratan D’Ambrosio (Powell & Frankenstein, 1997). Although he was not the first to use this expression, D’Ambrosio is internationally recognized for having founded the Ethnomathematics Program at the Fifth International Congress on Mathematics Education (ICME-5) in Adelaide, Australia, in 1984. For the most part, studies in ethnomathematics around the world recognize the ideas of D’Ambrosio and his influence on the production of the field (Aroca-Araujo, 2016). D’Ambrosio also contributed to Brazilian educational policies in the 1990s: he actively participated in the formation of the Brazilian Society for Mathematics Education (SBEM), introducing ethnomathematics as a line of research in mathematics teaching. The first issue of the SBEM magazine, Educação Matemática em Revista, launched in 1990 and republished in 2002, was entirely dedicated to ethnomathematics. Research and educational practices from an ethnomathematical perspective have been growing in this country over the years (Fantinato, 2013). In 2018 alone, three qualified journals, respectively, Educação Matemática em Revista, Ensino em Revista, and Educação Matemática em Foco, published a thematic issue entirely dedicated to the topic. In addition, since 2000, specific ethnomathematics events have been organized in

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Brazil, such as the Brazilian Congress of Ethnomathematics (CBEm), which is held every 4 years and which had its sixth meeting in 2022. More recently, Latin America has been standing out in this area. Initially created in 2003 in Colombia, the Latin American Network of Ethnomathematics (RELAET) has expanded as a space for exchange among Ethnomathematics researchers and has launched its own journal, Revista Latinoamericana de Ethnomathematics. Currently, RELAET has more than 780 members from America, Europe, Africa, and Oceania. Since 2019, during the Segundo Encuentro Latinoamericano de Etnomatemática (ELEm2), this association officially became the International Network of Ehnomathematics (RedINET), with coordinators from different continents of the world.1 The development of ethnomathematics in Latin America can be evaluated by the participation of researchers from this continent in the International Conferences on Ethnomathematics, the ICEms. The partial results of an ongoing investigation2 (Fantinato & Alves, 2021) indicate the gradual and expressive growth of Latin American representation in these events. In a first stage of the research, we consulted and shared information on the latest International Conferences on Ethnomathematics (ICEms): ICEm3 (New Zealand, 2006), ICEm4 (United States of America, 2010), ICEm5 (Mozambique, 2014), and ICEm6 (Colombia, 2018). These events were chosen because they are the most important in the area and because they bring together the international community of researchers in ethnomathematics. We then conducted a survey of authors who had published in the ICEms´ Proceedings. These were then classified by country of origin and continent. In the investigation in question, it was decided to differentiate the Americas, opting for the cultural-historical roots of this classification, and not only the geographical ones.3 Table 1 indicates the number of authors found per congress, per continent, and points to the gradual growth of Latin American participation. At ICEm3, the seven representatives from Latin America represented 22% of the total authors of the event, but they were exclusively Brazilian. At the 2010 event (ICEm4), the significant expansion of ethnomathematics research on this continent already begins to appear, possibly as a consequence of the expansion of the Latin American Network of Ethnomathematics. Both ICEm4 and ICEm5 had the participation of 27% of Latin American authors, and the diversity of countries grew, joining the list of countries: Argentina, Brazil, Chile, Colombia, Costa Rica, Mexico, and Venezuela. The last international congress analyzed was the ICEm6, which had a high representation from Latin America (165 authors, representing 81% of the total). This  https://www.etnomatematica.org/home/?page_id=4307 Access 11.08.20.  Research project entitled “Marks of Brazilian academic production in European ethnomathematics research,” supported by a Scientific Initiation Scholarship from the National Council for Scientific and Technological Development (CNPq), from 2020 to 2023. Scholarship holder: Carolina Luiz Alves. 3  We adopted the Anglo-Saxon America and Latin America categories, instead of North America, Central America, and South America. 1 2

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Table 1  Total ICEms authors by continent ICEm3

ICEm4

ICEm5

ICEm6

Africa

Conferences

3

1

6

0

Anglo-Saxon America

4

20

7

10

Latin America

7

12

13

165

Asia

2

3

1

1

Europe

7

8

14

20

Oceania

9

1

8

7

32

45

49

203

Total

Source: Authors’ elaboration

significant growth in Latin American participation refers to the location of the event, held in Medellín, Colombia, but it also seems to be the result of the great recent development of ethnomathematical research in countries of this continent. Brazil, which was already internationally recognized in the area, began to share space with countries such as Argentina, Bolivia, Chile, Colombia, Costa Rica, Ecuador, Guatemala, Mexico, Panama, Peru, and Venezuela. Many factors can explain this significant recent advance in ethnomathematics research in Latin America. Among them, we can highlight the role of Ubiratan D’Ambrosio in this process and the influence of his work on the academic production of the area. According to Fantinato et al. (2018), D’Ambrosio was a reference for 87% of the works published in a 2014 event held in Rio de Janeiro.4 In the following topic, we analyze the mark and the importance of D’Ambrosio for Latin American countries, based on the testimonies of some researchers from this continent.

5 What Did Ubiratan D’Ambrosio Mean for Latin America? As already mentioned, we consulted ten Ethnomathematics researchers from ten countries. These researchers were chosen based on some criteria: for being coordinators in the Latin American Network of Ethnomathematics and/or for being recognized researchers in the field of Ethnomathematics in their respective country. We know that we could have consulted more people, but we trust on the description of these researchers about the importance of the Professor Ubiratan D’Ambrosio in their respective Latin American country. Thus, the ten researchers were asked the following question: What was the role and importance of Ubiratan D’Ambrosio in

 Encontro de Etnomatemática do Rio de Janeiro (ETNOMAT-RJ).

4

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your country? The main ideas were taken from the text sent by each researcher that we present below. María Del Carmen Bonilla – Pontificia Universidad Católica del Perú, Perú

Research began in the 1970s, with the study and revaluation of the mathematics of the native peoples. In the twenty-first century, the “Intercultural Mathematics” project advances in the implementation of interculturality and ethnomathematics new research approaches aligned with community expectations. Ethnomathematics has given scientific support to the various studies related to the ancestral mathematical knowledge of the native Peruvian peoples, contributing to recognize and revalue their cultures and raise their self-esteem. Anahí Huencho – Universidad Católica de Temuco, Chile

Professor Ubiratan played a fundamental role, first of all liberating and secondly hopeful for many indigenous people, teachers, academics, and/or researchers who, in Chile, are looking for a way to recognize the mathematics present in traditional practices, the understanding of the cosmos, etc. and bring them to a status of present (visible) and flexible “knowledge” for today’s demands, in terms of territorial and, above all, educational needs. These investigations have currently had an impact on School Classroom and on Initial and Continuous Teacher Education.

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The Ethnomathematics Program in Chile has been established as a consequence of the search for equity, justice, and peace of historically dominated groups and that has begun with Peasants, Mapuche, and Aymara indigenous people and which we hope to extend, nurture, and strengthen from their virtues. Violorio Ayarza – Universidad Especialidad de las Américas, Panamá

We face a century of teaching from a foreign culture that practically decimated mathematical knowledge and acculturated it. In the last two decades, through the EBI.guna project, we have been recovering and systematizing to revitalize mathematics, aiming at teaching in an intercultural context in the classroom. In the last decades, the study of Ethnomathematics and other cultural contexts gained strength. Ethnomathematics, just as it was conceived, with a broad conceptualization as defined by Master Ubiratan D’Ambrosio. Miriam Micalco Méndez – Universidad Autónoma de San Luis Potosí, México

Research on teacher training, on intercultural and bilingual education, on pedagogical intervention in the classroom and in the community, with ethnomathematics as its center. Spaces are opened in favor of Ethnomathematics in some public universities and governmental educational instances, such as the Ministry of Education (SEP).

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Professor D’Ambrosio’s approach inspired the construction of spaces that go beyond the mathematical fact and include the diverse knowledge that is built between quantities and activities typical of cultural groups. Therefore, a critical, sharp, and inclusive look is built. Professor Ubiratán D’Ambrosio contributed with his vision of mathematics, to think and rethink the ways in which we seek to understand its teaching and learning. Another fundamental aspect that he planted in our hearts was to awaken the ability to think and rethink ethnomathematics and to be an active part in the construction of the broad project. Roxana Auccahuallpa – Universidad Nacional de Educación en el Ecuador, Ecuador

The work carried out by Ubiratan from Ethnomathematics becomes a fundamental part of research in mathematics education. Ethnomathematics is worked from the basic education curriculum for the 14 nationalities, as an essential part of attending to the mathematics of the peoples. The integration of cultural practices typical of a multicultural and intercultural country, that seeks the quality of Bilingual Intercultural Education-EIB, is considered. Based on the ideas presented by Ubiratan, the aim is to revalue the knowledge and cultural knowledge of these Kichwa, Shuar, and Achuar peoples, among others. Educators, teachers, and researchers from universities, the Ministry of Education of Ecuador and the Bilingual Intercultural Education System, seek in Ethnomathematics the methodological alternative for teaching and learning mathematics in a contextualized way, discovering, reviving, and incorporating traditions and learnings developed within our own culture.

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Ana Patricia Vásquez – Universidad Nacional de Costa Rica, Costa Rica

Among D′Ambrosio’s contributions to Costa Rica is the sharing of the foundations for the recognition of multiple social contexts, their practices, and their associated knowledge, betting on social inclusión processes from the Educational Dimension of the country, changing the idea that it was only about indigenous knowledge or their vestiges. Based on research by Vásquez and Gavarrette (2005), the Ethnomathematics Program began to spread across the country. Before 2005 there was a strong tendency to consider ethnomathematics as a line of research associated only with indigenous populations and their vestiges, evidenced in published works, mainly by anthropologists-archaeologists and scarcely by works with the participation of mathematicians (Vasquez & Trigueros, 2015). The edges of research in Ethnomathematics are broadened, by exploring other social contexts, their practices, and their knowledge. Ubiratán provides a conceptual basis for the Educational Dimension of Ethnomathematics. Costa Rica has recent evidence of the application of these foundations in the incorporation of new values and knowledge, based on an ethic of respect, for strengthening the cultural roots of Costa Rican societies. Examples of this are the works of Vásquez et  al. (2020), and Gavarrette et al. (2020). Oswaldo Martínez – Universidad Pedagógica Experimental Libertador, Venezuela

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The role that Ubiratan D’Ambrosio has played in the development of Ethnomathematics in Venezuela has always been and will be preponderant due to the very solid foundations that he laid with respect to this program. Some ideas were accepted at the El Mácaro- Venezuela Rural Pedagogical Institute. Several undergraduate theses have been programmed and developed, highlighting the inclusion of a compulsory subject of ethnomathematics that is part of the curriculum of the Bilingual Intercultural Education Program (PEIB), at the Universidade Pedagógica Experimental Libertador. This course is developed with an intercultural approach, paying attention to cultural diversity and identity processes that are built to the sound of the culture that is created, to the implicit request of ways of life and as required by the environment, to the knowledge and knowledge underlying the daily practices of sociocultural groups served by the program. Andrea Morales – Centro Universitario de El Progreso de la USAC, Guatemala

Dr. Leonel Morales, a distinguished Guatemalan researcher, had the experience of having D’Ambrosio as advisor for his doctoral thesis. During this process, Dr. Morales reports that one of his advisor’s advice was that, upon finishing his doctorate, he should return to his country because his country needed him. He did so, he returned to Guatemala as a researcher, writer, speaker, and project advisor as an independent consultant, vital activities for the development of education in Guatemala, in which the influence of his advisor was present. Ubiratan D’Ambrosio is that professional who inspired some Guatemalans through his writings, through his presence and his wise words, aimed at asserting “cultural relevance, not as a discourse but as an action of dialogic thought” (Yojcom, 2018).

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Olenêva Sanches Sousa – Instituto Federal de Educação, Ciência e Tecnologia do Piauí, Brasil

Ubiratan D’Ambrosio conceived the “Ethnomathematics Program” as a “general theory of knowledge,” transdisciplinary, and transcultural. He always maintained political struggles for human rights and social justice, such as decoloniality. He reflected on the means that favor and inhibit the conception/reference of ethnomathematics in different areas: he defended the “ethics of diversity” and the goals of peace; he pointed to “epistemological cages” and power interests. Let us consider that ethnomathematics has its roots in Brazil. But his journey described the path of ethnomathematics around the world. D’Ambrosio contributed to opening the doors of the cages and the eyes of science and education, so that the knowledge of diversity is revealed and adds identity values to the country. Not only in Brazil, the importance and role that Ubiratan D’Ambrosio has played for the development of respect for Ethnomathematics and for the concomitant development of ethnomathematics in research and education is evident. Hilbert Blanco Álvarez – Universidad de Nariño, Colombia

Ubiratan was a promoter of ethnomathematics studies in the country. Starting in 1984, Ubiratan D’Ambrosio was invited to several events or training processes in

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Colombia: invited by the Colombian Academy of Physical, Exact and Natural Sciences, opening conference “Mathematics and Citizenship” of the First Conference on Mathematics Education, organized by the Mathematics Education Group, Universidad del Valle. Today there are several researchers, research groups, and universities where theoretical and methodological processes are developed within the Ethnomathematics Program. It was Ubiratan D’Ambrosio who sowed this interest in the country’s academics at the time. It should also be recognized that he has always followed research projects in the country, for example, being a master’s thesis evaluator and other activities that he developed with universities.

6 D’Ambrosio and Its Importance in Latin America Ubiratan D’Ambrosio deeply marked educational and research policies in Latin America. The diffusion of the Ethnomathematics Program proposal has been gradually taking place in the last decades. The repercussion of his ideas occurred directly or indirectly. In some cases, D’Ambrosio acted as director of doctoral thesis, and his influence was direct. In this way, recent doctors returned to their countries of origin and became trainers of other ethnomathematicians, as well as implementers of educational policies aligned with the ethnomathematics proposal. On other occasions, Ubiratan D’Ambrosio participated in boards of master’s or doctoral works or, very often, was an invited speaker at events held in Latin America. The impact of his ideas was so strong that it had repercussions among professors and researchers from different countries, sometimes even without his physical presence. The same question asked to ten Latin American researchers, about the importance of Ubiratan D’Ambrosio in their country, resulted in a diversity of answers, depending on the historical specificities of each place, which converged to some common categories, listed in Fig. 1. One of the most recurrent aspects in the interviewees´ narratives was the role played by D′Ambrosio’s Ethnomathematics – as a scientific support – in valuing the ancestral knowledge of the peoples, pointing out ways to integrate their own cultural practices in school curricula schoolchildren. This aspect is associated with the ethics of diversity, so important in multicultural, multilingual countries with a large indigenous population, such as Latin American countries. In these countries with a history of European colonization, indigenous school education has sought to incorporate “among its principles the overcoming of paradigms of modern European thought, advocating the recognition, appreciation and promotion of the knowledge of each people, their languages, identities, histories and cultures” (Leite & Camargos, 2021, p. 4). In this sense, ethnomathematics, in its political dimension, is part of this decolonization project (D’Ambrosio, 2001) of the Latin American peoples. Another aspect addressed by the interviewees refers to ethnomathematics in its educational dimension (D’Ambrosio, 2001). This theoretical approach, brought by D’Ambrosio and his followers, promoted a new educational approach, a rethinking of ways of teaching and learning mathematics, generating an impact on both

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Fig. 1  Influence of Ubiratan D’Ambrosio in Latin America. (Source: Authors’elaboration)

curricular proposals and teacher education in Latin American countries. We agree with the Colombians Blanco-Alvaréz and Molano-Franco (2021, p. 3) when they affirm that the teacher training proposals inspired by ethnomathematics “support the processes of struggle and demand for mathematical knowledge of the peoples of the Americas, the important curricula repercussions that may have on undergraduate mathematics programs.” In some Latin American countries, such as Venezuela, Ethnomathematics has been gradually integrated into the curriculum for initial teacher education. Likewise, the introduction of ethnomathematics in the initial training of mathematics teachers “can represent a position of resistance, aiming to raise innovative strategies and attitudes towards Eurocentric and exclusive mathematics that still predominates in some teacher training courses” (Soares & Fantinato, 2021, p. 21). Finally, some interviewed researchers also mentioned D′Ambrosio’s Ethnomathematics Program and its broad perspective, as a general theory of knowledge, as a research program in search of peace and social justice. In D’Ambrosio (2009, p. 19) the math educator pointed out the risk of epistemological caging, even among ethnomathematicians, that “seek to explain and understand the knowledge and actions of other cultures according to categories specific to academic mathematics.” Latin American ethnomathematics research, which has developed so much in recent years, needs to pay attention to these lessons from its main inspirer.

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Ethnomathematics and Complexity: A Study of the Process of Elaboration of a Peruvian Andean Textile María del Carmen Bonilla-Tumialán

«There is an extremely expressive and very common term in Quechua chanka; when an individual wants to express that in spite of everything he still is, that he still exists, he says: Kachkaniraqmi!» I am still! ¡Sigo siendo! José María Arguedas (1911–1969) Peruvian writer, poet, translator, professor, anthropologist and ethnologist

Abstract  The objective of the study is to unveil the mathematical knowledge that underlies the process of weaving on a four-stake loom used in the region of Puno, Peru. From the theory of complexity, the research has addressed various dimensions, political, anthropological, historical, epistemological, mathematical, and educational, articulated by ethnomathematics, to solve the problem posed. The methodology used is qualitative, applying first the ethnographic method, field work in Puno, participant observation, and semi-structured interviews with key informants. Subsequently, the data recorded by audiovisual means are analyzed. First, the phases of the Andean weaving process are identified. In the phase in which the loom base is built, based on an anthropological vision, gestures made by the weavers are related to some mathematical notions and properties used in the process. The research shows that the Quechua culture possesses mathematical knowledge, transmitted from generation to generation, which is used by the inhabitants in the process of loom weaving, knowledge that could be used in teaching and learning processes of mathematics. Keywords  Ancestral mathematical knowledge · Quechua · Loom weaving · Epistemologies of the South

M. d. C. Bonilla-Tumialán (*) Comunidad de Educación Matemática de América del Sur, Lima, Peru © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. C. Borba, D. C. Orey (eds.), Ubiratan D’Ambrosio and Mathematics Education, Advances in Mathematics Education, https://doi.org/10.1007/978-3-031-31293-9_13

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1 Introduction The present research work is related to the study, recognition, and revaluation of the ancestral mathematical knowledge developed by the Quechua-Collao culture, specifically those that underlie the process of textile elaboration in four-stake loom, knowledge that has been systematically invisibilized by official culture since the Spanish invasion. From the theory of complexity, in order to find a solution to the problem posed, several dimensions have been approached, such as political, anthropological, historical, epistemological, mathematical, and educational, articulated by ethnomathematics. The objective is to unveil some mathematical notions and properties that emerge from the process of making textiles, with the future objective of giving didactic orientations and incorporating them into the design of learning and teaching processes of mathematics for students of basic education in the Puno region. The methodology used is qualitative since the ethnographic method is applied first, field work in Puno, participant observation, and semi-structured interviews with key informants. Subsequently, the data recorded by audiovisual means are analyzed. The analysis of the first phases of the textile elaboration process is carried out, and some notions and mathematical properties used by the weaver in the process are identified, as well as mathematical notions that can be visualized in the weaving process. The research shows that the Quechua culture possesses mathematical knowledge, transmitted from generation to generation, and that it is used by the inhabitants in the process of loom weaving (Bonilla, 2019).

2 Diversity and Interculturality in Peru The Peruvian population is changing, the 2017 National Censuses have revealed that 25% of the Peruvian population self-identifies as Indigenous, Quechua, Aymara, or Amazonian, despite the fact that only 16% have a native language as their mother tongue (National Institute of Statistics and Informatics, 2018). Since the beginning of the century, there has been a strong and growing trend toward self-identification as a native people. Interculturalism and bilingualism, as state policies, are present in various sectors, such as health, the justice sector, commerce, communications, and education. At the university level and in scientific institutions, research is increasingly being promoted on the ancestral knowledge used by native peoples to solve the problems of their environment, in order to put them into effect, because they are effective, economical, and in harmony with nature. The idea is to try to articulate the solutions found with official science and technology. In Peru since the 1970s, laws and sectoral policies aimed at the development of Interculturality and intercultural bilingual education (IBE) have been progressively being developed (Ministry of Education, 2018). Currently, the Peruvian State aims to provide a relevant and pertinent educational service, which guarantees a better learning of the school population of native peoples, implementing IBE in all stages,

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forms, and modalities of Peruvian education, as well as Intercultural Education throughout the educational system, including in all urban schools so that schoolchildren learn about diversity, value its richness, respect it, and thus be able to build a healthy citizenship, without discrimination.

3 A Multidisciplinary Research from Complexity Within the framework of diversity, in an initial teacher training program for Quechua, Shipibo, and Aymara students of the intercultural bilingual education program of the Faculty of Education of the Cayetano Heredia University, a research project was developed and obtained funding by winning the 2015 Interuniversity and Multidisciplinary Research Award promoted by the Consortium of Peruvian Universities. The project involved educators, anthropologists, and mathematicians from the Cayetano Heredia University and the Pontifical Catholic University of Peru. They joined forces to find solutions to the mathematics learning problems of indigenous students. Traditionally, there is a gap between the mathematics learning achievements of students in urban areas and the achievements of students in rural areas, who have the lowest performance (Ministry of Education, 2017). In order to try to find a solution to the stated problem, it is necessary to consider the sociocultural differences that exist between students in urban and rural areas, an aspect that is considered by ethnomathematics. Ethnomathematics is a research program in the Lakatosian sense (Lakatos, 1983) of the generation, organization, institutionalization, and dissemination of knowledge that drives a pedagogical action (D’Ambrosio, 1993). It proposes an epistemological approach that starts from reality and has a historical, cultural, social, political, cognitive, and pedagogical character. It seeks the consolidation of the relativistic paradigm in mathematics (RPM), based on a cultural epistemology that explains mathematical knowledge taking into account the contextualization of the sociocultural group of the producing subjects (Oliveras, 2006, 2015). The RPM is a conception opposed to positivism, the dominant paradigm in Western culture, where principles are absolute, based on universal criteria based on reason. The new epistemologies suggest that it is more significant to accept the existence of diversity, at all levels, even in science, as there is scientific knowledge, professional knowledge, and everyday knowledge, which have different validation criteria. The ancestral knowledge of the native peoples is part of everyday knowledge, which corresponds to different areas of knowledge and often does not correspond to any specific discipline. D’Ambrosio (1993) also highlights the importance of the sociocultural and political dimensions of mathematics education, which question Eurocentric scientific priorities and compare the process of developing scientific knowledge in peripheral countries that have undergone colonialism and neocolonialism, with the scientific and technological development of industrialized countries. In both processes, the mathematics of ethnic minorities belonged to an educational universe different from

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that of Western mathematics, which led him to propose ethnomathematics as an epistemological alternative more adequate to the diverse sociocultural realities. Mention has been made of the historic, cultural, social, political, cognitive, and pedagogical dimensions of ethnomathematics, which give it a holistic character. The need to resort to multidisciplinarity to solve the problem posed brings ethnomathematics closer to the theory of complexity developed by Edgar Morin (1999).

4 A Rapprochement Between Ethnomathematics and Complexity Theory In universities, from a classical perspective of science, the segmentation into watertight disciplinary areas obeys a dogma that underlies classical scientific knowledge, the desire to explain what is visible and complex through the invisible and simple (Jean Perrin, quoted by Morin, 1999). In the midst of a reality where agitation, dispersion, and diversity reign, there is a desire to legislate and to discover laws, principles of classical science, laws that govern the fundamental elements of matter, of life, of the social, and to achieve this, one must disunite, isolate, and reduce the objects subject to the laws. At present, scientific work based on these principles is insufficient. It is necessary to introduce complex thinking in scientific work, to make visible the historical character of knowledge, and to argue that what is important is not only to identify the elements of a science, but to study how they interact. It is necessary to identify the principles of order, but also those of disorder and organization, and other principles as well. From the vision of complexity, in the reality inherent to intercultural bilingual education, diverse phenomena converge, educational, linguistic, political, ethnic, historical, cultural, and economic, which make it more complex to find the solution to the mathematics learning problems of the students of the native peoples, who live mainly in rural areas. As can be seen in Fig. 1, the complex reality is conditioned by the colonial character of the Peruvian political system, in which, despite having achieved an apparent political "independence" in the nineteenth century, there still persist political and economic practices of subordination to the command of foreign centers of power, mainly Western (Mariátegui, 2007; Quijano, 2000).

4.1 Colonialism and Epistemicide Since the Spanish invasion, there has been an epistemicide, a systematic destruction of the knowledge of the indigenous American peoples (Grosfoguel, 2022), of their culture. A single example reveals the truth that is not disclosed in Peruvian history books. The destruction of the quipus was ordered by the Council of Lima in 1583 (Vargas Ugarte, cited by Ascher & Ascher, 1997). The destruction of culture was

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Fig. 1  Complexity present in intercultural bilingual education

more violent after the defeat of the Túpac Amaru Revolution in 1781 (Valcárcel, 1945). The Spaniards regretted having educated the Inca nobility and took measures against Andean education and culture, thinking that this would prevent future independence. Genocide, exclusion, inequity, and the domination of the Spanish over the Andean and Peruvian were characteristics that persisted in the Republic. Not only were cultural practices made invisible, but also the way of learning of the native Peruvian peoples, their epistemology, was also ignored and underestimated, giving it a character of backwardness and inferiority. The American peoples also developed an indigenous science; they also investigated to find the solution to their problems, but the process of knowledge production is different from the Western one. Knowledge was not recorded by means of linguistic signs; in the case of the Inca and pre-Inca culture, the quipu was used as a means of writing.

4.2 The Quipu: Inca Writing System Quipu is a Quechua word meaning knot (Bonilla, 2016); it was an Inca writing system containing statistical information, demographic data, and also a record of information of abstract topics, such as political systems, myths, etc. (Zapata, 2010). The Incas were not an exclusively oral culture. They knew how to preserve historical memory through an annotation system with knots. The quipu is not a phonetic writing system. It is a writing system using visual or tactile signals containing meaning. It is a system of three-dimensional annotations that makes sense through the

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combination of knots, colors, shapes, and rope twisting, made with camelid wool or cotton fibers or a combination of various fibers. The societies that spoke languages with the same root, as root of Indo-European languages, developed a common path toward phonetic writing, despite being mutually unintelligible. In ancient Peru, those nearby territories, where languages from various roots interacted, developed ideographic writing, because the signs did not represent sounds, but concepts  – such as the knots in the case of quipu  – and it allowed people from different languages to read without translation. Hence, the information recorded in the quipu could be interpreted by different quipucamayocs (the one who can with the quipu), despite the fact that they spoke different languages. The quipus were objects created by cultures prior to the Incas, but in the government of the Incas, they became part of the art of governing (Ascher & Ascher, 1997), along with the Qhapaq Ñan or Great Inca Road (Ministry of Culture, n.d.-a; Zapata, 2010). The quipus were a system for recording complex information, were the language of the Inca empire, and were associated with the expansion and strengthening of the Tahuantinsuyo. The Incas ruled by means of two instruments: a network of roads and the quipus system. There were mathematical quipus and non-mathematical quipus elaborated and interpreted by the quipucamayoc, the one who can with the quipus. Mathematical quipus functioned on the basis of a decimal and positional numbering system, just like Western mathematics (Ascher & Ascher, 1997). Having a basic positional system simplifies arithmetic, since there is a group of symbols and rules that convert them into other symbols, elements on which the key principles and concepts of arithmetic, which were known by the Incas, are developed. The quipu is formed by a thicker main string, from which hang several thinner strings, each one representing a number. At the base of the rope, there is a knot that represents the units (100); at the next higher level of the rope another knot represents the tens (101) and so on; the knots go up; each knot in a higher position or level represents a higher power of 10 (102, 103, …). The absence of a knot represents zero. Long knots are used in the units position, and single knots can have from one to nine loops, and in each case, a digit from 1 to 9 is represented. The numbers of the upper strings in a quipu are usually the sum of the numbers of the hanging strings with which they are associated. A quipu held by a quipucamayoc can be seen in Fig. 2.

4.3 An Inca Mathematical Tool: The Yupana Yupana was used as a tool for calculating mathematical operations; it comes from the Quechua word that signifies to count (Bonilla, 2016). The yupana is a type of abacus for performing arithmetic operations. They were made of clay, stone, or wood. The numbers were represented with corn kernels, seeds, or pebbles. A yupana can be seen in the lower left corner of a drawing of “Primer Nueva Corónica y Buen Gobierno,” a book written by the indigenous chronicler Felipe Guaman Poma de

Ethnomathematics and Complexity: A Study of the Process of Elaboration of a Peruvian… 185 Fig. 2  Quipu and yupana. (Source: Guamán Poma de Ayala, 1615)

Ayala (1615) (Fig. 2). The illustration suggests that the quipucamayoc calculated with the yupana and then recorded the data on the quipu. The yupana was known and appreciated by Spanish colonial administrators. The Spanish priest José de Acosta wrote about this artifact in his book Historia Natural y Moral de las Indias and considered that the Incas were superior to them in calculating with the yupana. Ascher and Ascher (1997) point out that the way in which the Incas conceived the concepts of number, geometric configuration, and logic, as well as the quipu, had no parallel with other cultures; the Incas did not go in the same direction as other cultures; it is not possible to know where they would have taken their ideas. In addition, they had a different way of recording their knowledge, not with linguistic signs but with strings, in a three-dimensional and tactile way. Therefore, it can be affirmed that the way of producing their knowledge, their epistemology was different from other cultures, also from the Western one. So, what arguments support the imposition of Western culture and sciences on Peruvians in general and on native peoples in particular? No scientific basis, only the reason of the weapons that was shielded in an apparent superiority and modernity, but that basically only had the pretext to appropriate their natural resources to develop capitalism and modernity in

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Europe. This raises an ethical and moral problem that invades the field of law, of human rights.

4.4 Epistemologies of the South and Epistemic Rights In 2009, the United Nations Organization (UNO, 2019) agreed on a mandate on cultural rights, which propose the protection of human creativity; the freedom to choose, express, and develop an identity, to belong to a specific group; the rights of individuals and groups to participate, or not, in the cultural life of their choice; the right to interact and exchange opinions with others; the right to enjoy and access the arts and knowledge; and the right to participate in the interpretation, elaboration, and development of cultural heritage. But the proposed cultural rights do not contemplate the right of peoples to have an original way of producing and valuing their knowledge, of researching, of developing an epistemology different from the hegemonic one, specifically, as Boaventura de Souza Santos (2010, 2014) puts it, the right of peoples to have the Epistemologies of the South recognized. And if they have an epistemology different from the Western one, the processes of teaching mathematics and student learning are also original, different from the Eurocentric perspective. In this order of ideas, it is necessary to fight for epistemic justice, which means recognizing within cultural rights, the epistemic right of peoples, defined as the right to exercise processes of construction and validation of knowledge emanating from the cultural practices of each community. That is to say, the right of each native people to develop a distinct epistemology that corresponds to their historical evolution; relate to nature according to their worldview; face and solve problems that arise in community practice taking into account their norms and principles; research and build bodies of knowledge or theories about the various areas of human activity that they put into practice, according to their validation criteria; record autonomously, not necessarily through linguistic signs, the information that is collected from social practices; and analyze the information, through relevant and proper cognitive processes, with the purpose of producing knowledge.

4.5 The Science of Weaving in Peru In the case of the Quechua-Collao people, achieving epistemic justice implies recognizing, vindicating, and revaluing their ancestral knowledge. Peruvian weaving is an ancestral knowledge that has been studied since the last century by social scientists, historians, archaeologists, anthropologists, foreigners, and nationals. The sophisticated technique used in its manufacture, the exquisiteness of its finish, and the beauty of its designs are characteristics of Peruvian weaving. The most ancient

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antecedent of Peruvian weaving is the fragments of fabric made with vegetable fibers found in the Guitarrero Cave (Callejón de Huaylas, Ancash), which date back to 5780  BC, proving that the textile art of pre-Hispanic Peru predates ceramics (Solanilla Demestre, 1999). Based on the foundations of ethnomathematics, supported by the relativistic paradigm of mathematics, we start from the premise that the original Quechua-­ Collao peoples that live in the Puna or Jalca, Peruvian Andean region located between 4000 and 4800  meters above sea level, developed their mathematical knowledge in a particular, natural and own way, in order to be able to face and solve the problems that arose in their constant adaptation to nature, to satisfy their basic needs for food, housing, and clothing, using the resources of their environment, the Andean Puna or Jalca. It is in this process of occupation and human adaptation to the ecosystem of the Andean Puna or Jalca that the settlers relate with South American camelids (llama, alpaca, vicuña, guanaco), animals that were domesticated between 4000 and 3500 BC (Wheeler, 1988), and which are covered with precious wool, a material that served as raw material for the weaving of their garments. Due to inclement weather, with an average annual temperature ranging between 7 °C and 0 °C, and a minimum temperature varying between −9  °C and −25  °C, it was necessary to cover themselves with thick, dense clothing that allowed them to conserve the warmth of the human body. At least since the Early Horizon (900–200 BC), a stage in the history of ancient Peru, the inhabitants of the high Andean zone progressively developed loom weaving to satisfy basic clothing needs and also as an artistic expression (Desrosiers, 2013), knowledge that later reached the coast and probably the jungle, using cotton as raw material. Textile activity, developed by the pre-Inca and Inca cultures, occupied an important place in society, a situation that was recorded by various mestizo and European chroniclers and scholars such as Guamán Poma de Ayala, Ludovico Bertonio, and Diego González de Holguín, among others (Desrosiers, 1986, 2013; Arnold & Espejo, 2013). Because of the characteristics mentioned above, Peruvian weaving is an ancestral knowledge that was used in various areas of knowledge. In Peru, bridges were woven (Ministry of Culture, n.d.-b) and the roofs of houses, and writing “was woven” through quipus and tokapus. However, it is an activity that has not been valued or studied in basic education or in higher education, and even less and less Peruvians practice it, despite the fact that it is still in force in its three regions, coast, highlands, and jungle, from the north to the south. That it has survived in spite of the indifference turns the Peruvian weaving into the symbol of cultural resistance; the weavings are the books that the colony could not burn; in the weaving underlies deep knowledge of the Andean, Amazonian, and coastal communities. Thanks to the climate of the Peruvian coast, it has been able to survive in the archaeological excavations, in spite of the passage of the centuries, to give faith of the high technology used by Peruvian culture.

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4.6 Ethnography and Participant Observation in the Province of Melgar in Puno In order to identify the mathematical knowledge of the Quechua-Collao people in loom weaving, an exploratory study was carried out using the ethnographic method. The fieldwork techniques used were participant observation, non-participant observation, and interviews with key informants in a bilingual context. The methodology is participatory as meetings were held with community groups. Ethnography is an inductive process of theoretical construction, with a non-­ ethnocentric perspective, which seeks to understand the other, within a process of involvement in which it is necessary to participate, not to observe from afar. In ethnography, the object of study is the “other” (Guber, 2001). In the process, an epistemological tension is established between objectivity, subjectivity, and neutrality. Regarding the distinction between the subject that investigates and the object that is studied, De Souza (2010) proposes a criticism that comes from quantum mechanics, from Werner Heisenberg, who points out that when we observe phenomena, we change them. Therefore, from this perspective, there cannot be an absolute distinction between subject and object. From the field of action of anthropology, what is desired is to observe the process of elaboration of Andean weavings and at the same time participate in the work learning to weave, with the intention of identifying elements of the mathematical bodies of knowledge of the Quechua people that possibly function through an internal rationality different from Western mathematics and that can be visualized in the cultural practices carried out by the members of the communities. By participating in the elaboration of the four-stake loom weaving, it has been possible to observe the activity from the inside and understand the process more easily. The recording of the information was audiovisual, not written. The weaving is produced by the interweaving of the warps and weft. Figure 3 shows the elements that make up a loom deployed vertically: the sticks or awas, the warps, mini or weft, the illawa. During the field work, four weavers were observed while they were weaving different types of fabrics. In the process of weaving on the loom, it was possible to distinguish the following phases: (1) preparation of the wool for weaving, (2) construction of the base of the loom structure, (3) the allwido or warping, (4) preparation of the loom for weaving, and (5) the actual weaving (Fig. 4). Once the wool is prepared, the base of the rectangular loom is fixed to the ground with four stakes. A stick or awa is tied to two stakes that delimit the width of the rectangle, and at the other end another stick is tied to the two front stakes. Using the two sticks, the warps are stretched, intertwining them and forming a figure eight. This is how the warp or allwido is formed. Subsequently, the warp is crossed with the weft. The weaving technique is warp-faced. What is seen is the warp and the weft is not seen. According to anthropological studies, it is one of the most difficult techniques in the world (Arnold & Espejo, 2013; Desrosiers, 1997). Figure 4 shows

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Awa or sck Alwido or warps

sck that separates

Awa or sck Mini or we Fig. 3  Common Andean loom with an illawa. (Source: d’Harcourt, 1934; quoted by Desrosiers, 2010, p. 266)

Fig. 4  Unqhuña and phases of the weaving process

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the image of an unqhuña. The unqhuña is a medium square cloth used to transport coca leaves for chacchado (chewed). Of the five phases of the weaving process observed, the second phase is analyzed below, when the weavers drive the four stakes into the ground to build the base of the loom structure. In the second phase, mathematical notions and techniques were identified.

5 Construction of the Base of the Loom Structure The process of constructing the loom base was visualized in the work done by a weaver from Puno. The process observed is similar to the work done by three weavers from Melgar, province of Puno, although differences can be found among them. The idea is to observe the weaving process with the purpose of analyzing it in order to identify notions and mathematical techniques used by the weavers in the practice of weaving. Next, the steps used by the four weavers will be explained and then analyzed from a mathematical perspective. The idea is to start from an anthropological point of view, describing the gestures made by the weavers when they assemble the loom base; then to point out the definitions and properties related to the rectangle; and, finally, to relate the gestures with the definitions and properties, representing them symbolically.

5.1 Construction of Rectangles When Making the Loom Base In order to carry out the analysis, four different processes carried out by the weavers are described. In two of them, the first and fourth cases, the rectangle is constructed from the width of the loom; in the second case, it is constructed from the length of the loom; and, finally, the third case is constructed from the diagonals. In the four videos that record the process of building the loom base, it can be seen that the weavers intuitively use parallelism and perpendicularity between sticks and ropes, but no instruments are used to verify the accuracy of the constructions.

5.2 Construction of the Rectangle from the Width The construction process can be seen in the following video. Mrs. Isidora places the two stakes A and B in the ground with a distance of two and a half hand spans between them (Fig. 5), thus fixing the side ΑΒ. Next, she transfers the measure of two and a half hand spans to the rope, doubling it, since she is going to weave two unqhuñas. She makes a loop at each end of the rope. She uses two sticks or awas P1 and P2 that have the same dimensions.

Ethnomathematics and Complexity: A Study of the Process of Elaboration of a Peruvian… 191 Fig. 5 Width ΑΒ of the rectangle

A

B

5 hand spans measurement of P1 = measurement of P2

|| P1 || P2

= 5 hand spans < ABC = 90° = two hand spans and a half = five hand spans

Fig. 6  Points A, B y C of the parallelogram

Place the ends of the stick P1 next to stakes A and B. At one end of P1, next to B, insert the loop of the five-hand span rope. In the other loop of the rope, insert one end of P2 which is in a position parallel to P1. With the help of P2, extend the rope, intuitively making sure that the rope is perpendicular to P1, moving it from right to left or vice versa until perpendicularity is found. With P2, tighten the rope and drive the third stake C at the intersection of the rope and P2 (Fig. 6). Through P1 and P2, she moves the ends of the five-hand spans length rope, in one end from B to A and the other end from C to a point where it is intuitively appreciated that the rope preserves perpendicularity with respect to P1. Thus, she determines the position of stake D at the intersection of P2 and the rope and nails stake (Fig. 7).

5.3 Construction of the Base of the Loom from the Measurement of the Length The process of building the loom base can be seen in this video. Mrs. Isabel begins to build the parallelogram by first determining the measure L of length in a rope. She plants the stake A. The rope L has loops at both ends, and through each loop she inserts the sticks P1 and P2 that have the same measure P. Intuitively she places P1 in a position parallel to P2, taking care that both are perpendicular to the rope.

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