Oral History and Mathematics Education [1st ed. 2019] 978-3-030-16310-5, 978-3-030-16311-2

Table of contents :
Front Matter ....Pages i-xx
Oral History in Mathematics Education: An Overview (Antonio Vicente Marafioti Garnica, Carlos Roberto Vianna)....Pages 1-19
Oral History and Mathematics Education: Historiographical Research (Maria Laura Magalhães Gomes)....Pages 21-34
Oral History and Mathematics Education: The Dialogue with/among Different Theoretical and Philosophical Perspectives (Marcelo Bezerra de Morais, Filipe Santos Fernandes)....Pages 35-45
Oral History in Mathematics Education: Possibilities for Effective Intervention in Teaching (Heloisa da Silva)....Pages 47-66
History of Mathematics Education and Oral History: Possibilities for the Classroom (Vinícius Sanches Tizzo, Antonio Vicente Marafioti Garnica)....Pages 67-84
Oral History in Mathematics Education: On Non-historiographical Research (Maria Ednéia Martins Salandim)....Pages 85-99
Narratives and Their Power Against Silencing in and by Scientific Research (Luzia Aparecida de Souza)....Pages 101-113
Back Matter ....Pages 115-128

Citation preview

History of Mathematics Education

Antonio Vicente Marafioti Garnica Editor

Oral History and Mathematics Education

History of Mathematics Education

Series Editors Nerida F. Ellerton Illinois State University Normal, IL, USA M.A. (Ken) Clements Illinois State University Normal, IL, USA

More information about this series at http://www.springer.com/series/13545

Antonio Vicente Marafioti Garnica Editor

Oral History and Mathematics Education

Editor Antonio Vicente Marafioti Garnica Mathematics Department Sao Paulo State University Bauru, São Paulo, Brazil

ISSN 2509-9736     ISSN 2509-9744 (electronic) History of Mathematics Education ISBN 978-3-030-16310-5    ISBN 978-3-030-16311-2 (eBook) https://doi.org/10.1007/978-3-030-16311-2 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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, express 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 to the Series Books in Springer’s series on the history of mathematics education comprise scholarly works on a wide variety of themes, prepared by authors from around the world. We expect that authors contributing to the series will go beyond top-down approaches to history, so that emphasis will be placed on the learning, teaching, assessment and wider cultural and societal issues associated with schools (at all levels), with adults and, more generally, with the roles of mathematics within various societies. In addition to generating texts on the history of mathematics education written by authors in various nations, an important aim of the series will be to develop and report syntheses of historical research which has already been carried out in different parts of the world with respect to important themes in mathematics education—like, for example, “Historical Perspectives on how Language Factors Influence Mathematics Teaching and Learning,” and “Historically Important Theories Which Have Influenced the Learning and Teaching of Mathematics.” The mission for the series can be summarized as: • To make available to scholars and interested persons around the world the fruits of outstanding research into the history of mathematics education; • To provide historical syntheses of comparative research on important themes in mathematics education; and • To establish greater interest in the history of mathematics education. The present book is an important addition to the series. There are seven chapters emanating from members of the “Oral History and Mathematics Education Research Group” (GHOEM)—a Brazilian group of researchers whose work is based on contemporary philosophies, theories, and methods of oral history. A unique feature of the book is that the authors see themselves as part of a developing oral history tradition which embraces researchers working in a wide-range of disciplines. We are confident that this book offers readers access to a rapidly developing and increasingly important approach to researching, interpreting, and documenting the history of mathematics education. We hope that the series will continue to provide a multi-layered canvas portraying rich details of mathematics education from the past, while at the same time presenting historical insights which can support the future. This is a canvas which can never be complete, for today’s mathematics education becomes history for tomorrow. A single snapshot of mathematics education today is, by contrast with this canvas, flat and unidimensional—a mere pixel in a detailed image. We encourage readers both to explore and to contribute to the detailed image which is beginning to take shape on the canvas for this series. Any scholar contemplating the preparation of a book for the series is invited to contact Nerida Ellerton ([email protected]), in the Department of Mathematics at Illinois State  James, at the Springer New York office. University or Melissa Normal, IL, USA

Nerida F. Ellerton M.A. (Ken) Clements

v

Preface to the Book The “Oral History and Mathematics Education Research Group” (GHOEM) is a Brazilian collective of researchers interested in the possibility of using oral history as a methodological resource in mathematics education. GHOEM was created in 2002, and nowadays, more than 15 years after its creation, all researchers agree that the main interest of the group is the study of school culture and the role of mathematics in this culture. This book is, from a certain perspective, the history of this Group and the history of the research which its members have conducted. It is not unexpected that there are some misunderstandings about what oral history is. Some researchers in mathematics education understand oral history as naturally related to historiographical research, and this is one point, among other points, that this book must discuss. Using oral history implies the use of narratives, but narratives are not necessarily used in order to develop historiographical research—although narratives are, always, historic in the sense they are created in a specific time, in a specific place, and in specific conditions, and they are the register of the memories of those that tell us their stories and, doing that, create themselves at the same time they create their narratives about themselves. Narratives, therefore, are not only a story or tale (narratives have not, so to speak, a status only in literature and a place only among literary discussions): narratives are a way to create things (narratives have, so to speak, an ontological status) and allow us to know ourselves and all those people we are related with, those people who share their memories with us (narratives have, in that sense, an epistemological status). The use of oral narratives in the field of mathematics education started more emphatically in the early 2000s. Since then, interest in this approach has spread, and one might say that nowadays the use of oral narratives in mathematics education research has become much more accepted in this and in other areas of knowledge. It was in Brazil that these studies linking narratives, oral history, and mathematics education, began to be used powerfully. Mathematical educators dialogued with anthropologists, sociologists, historians, psychologists, artists, and philosophers and appropriated oral history developed in those areas to study aspects related to the teaching and learning of mathematics. I am not talking about the use of interviews, in general, and other oral information in research, but about a specific kind of interviews and oral information, developed under the parameters of oral history taken as a methodological resource, according to some specific philosophical perspectives and following specific procedures. The creation of GHOEM, in 2002, was a landmark for the development of this methodology. By using oral history, the researchers in the group have been able to conduct investigations on the history of mathematics education, as well as non-historiographical research, such as that whose objective is to implement alternative models for mathematics teaching and teacher formation. Research in mathematics education, either theoretical or the initiatives with more pragmatic objectives—focusing on specific interventions in spaces where mathematics is learnt and taught—is supported by several grounding approaches, usually rooted in philosophy. This characteristic is also present in investigations in mathematics education developed with the aid of oral history resources. Thus, a panoramic view of the research in oral vii

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Preface to the Book

history in mathematics education requires the examination of works based on Foucault, Nietzsche, Deleuze and Guattari (the so-called philosophy of difference, in general), Ludwig Wittgenstein, and Pierre Bourdieu. When research is viewed as an interpretative plot, its hermeneutic foundation is usually given by Paul Ricoeur and John B. Thompson, either derived from Ricoeur’s hermeneutical phenomenology or from hermeneutics of depth (attributed to John Thompson and clearly linked to sociology and the philosophy of Gadamer and Ricoeur). However, in the field of history, there are works that dialogue with studies by Hartog, Le Goff, Certeau, and Chartier, for example, as well as several other scholars who support concepts of spatiality and historicity, such as Jean  Schimitt  and Doreen Massey. This arsenal of authors and works is joined by those who have investigated biographical and autobiographical narratives, such as Bruner and Lejeune. The focus on the concept of “narrative”—a fundamental ingredient in the comprehension of effectivity and the potential of oral history in mathematics education—has fostered recent research initiatives with (and about) narratives, which have been increasingly explored in the humanities, in general, and mathematics education in particular. We believe that all the aspects mentioned above are contemplated in the book we are presenting here. The aim is to describe the foundations of both concepts and procedures related to oral history, as well as the way it has been applied in the field of mathematics education, always giving examples of research already conducted and, whenever possible, suggesting possible research exercises. Linking theoretical discussion and real examples of studies and themes not only helps the reader but also argues in favor of the legitimacy and the potentialities of the methodology which this book intends to explore. Our proposal was to write a seven-chapter book in which each of the themes briefly discussed above is addressed. The chapters were elaborated by researchers from the Oral History and Mathematics Education Research Group, all of whom have significant experience regarding the use of oral history in the field of mathematics education. The first chapter, Oral History and Mathematics Education: An Introduction, will provide an overview of the role of oral history in mathematical education and how it has been procedurally mobilized. Through a brief history, the chapter will describe the status of oral history in mathematics education in the broad oral history scenario. The purpose of the chapter is also to discuss historiographic implications (i.e., the link between oral history and history) and make some considerations about the rationale which supports the methodology, including an overall discussion about narratives, truth, method, historiographic sources, conceptions of history. Finally, the chapter discusses not only how oral narratives are produced but also, in a general way, how such oral narratives have been analyzed in several works. In short, the aim of this first chapter is to present an overview which provides an introduction to the topics to be explored in further detail throughout the book. Oral History and Mathematics Education: Historiographical Research, the second chapter, explores the use of oral history in mathematics education research which has a historiographical nature (more specifically, in works about the history of mathematical education and teacher formation). The exercises presented derive from the Project “Mapeamento da Formação e atuação de professores que ensinam/ensinaram Matemática no Brasil” (Mapping the Formation and Practices of Mathematics Teachers in Brazil), a broad-spectrum research project whose aim is to understand how mathematics teachers are/were trained in different historical periods and how they have acted in the most diverse levels of education. Ultimately, the objective of this project is to create, and study, historical sources which will enable intervention in public policies related to teacher training. Examples of research work which mobilizes oral history in historiographic studies not linked to the “Mapping” Project will also be examined in this chapter, including biographies and autobiographies.

Preface to the Book

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The aim of the third chapter—Oral History and Mathematics Education: Dialoguing with Different Philosophical and Theoretical Perspectives—is to bring about a comprehensive discussion of some of the different theoretical-philosophical perspectives mobilized in mathematics education research, which employs oral history. These theoretical approaches have been used both to support the concept of method and history/historiography, as well as to support the analyses of the narratives produced using oral history. The fourth chapter—Oral History and Mathematics Education: Effective School Intervention—deals with the possibilities and potential of oral history as a formative strategy in mathematics education, specifically in teacher formation courses or in places and situations where mathematics is taught and learned, in general. In the case of teacher formation, we will discuss the creation of the course “History of Mathematics Education” in higher education courses specifically for mathematics teachers, which will, in turn, foster implementation, independently of any specific discipline, of exercises in oral history in such courses. This chapter is closely related to the next chapter—History of Mathematics Education and Oral History in Mathematics Classrooms: Examples. Initiatives for using the history of mathematics education in teaching are still not very widespread. This is due mainly to the fact that even though his line of research in mathematics education has grown vertiginously in the last decade, it is still new compared to the other research trends. Moreover, the work conducted often has a more theoretical approach, focused in the study of historical themes, without much concern for the mobilization of such studies in school practices. However, in Brazil, some researchers have developed and proposed historiographical exercises in the classroom—whether using oral history or not—to discuss aspects of teaching and learning mathematics and to discuss mathematical content, though with less emphasis. Examples of such exercises are (a) conducting interviews with former teachers and school administrators, about school, educational legislation, and mathematics education; (b) the development of questions to guide interviews with students about their relationship with mathematics; (c) conducting studies about specific moments in mathematics teaching, such as the new mathematics movement and how such mathematics were distinctively appropriated by different institutions at several educational levels; (d) conducting interviews with textbook authors; and (e) the study of old mathematics textbooks, interviewing (or not) their authors. Such exercises are conducted mainly in teacher formation courses (particularly in mathematics teacher-formation courses). The objective of this chapter is to present and discuss such experiences. The sixth chapter—Oral History and Mathematics Education: On Non-historiographical Research—explores the possibilities of oral history in mathematics education research, which, while producing historical sources, does not aim to develop (or mobilize the resulting sources) for studies of a strictly historiographical nature. Oral History and Mathematics Education: Narratives, the final chapter, deals with some theoretical aspects of narratives and how they have been viewed and mobilized in research in mathematics education which draws on oral history. It is an exploratory study of how narratives have generally been used in research in mathematics education and how oral history contributes to this new international research trend. Bauru, Brazil  Antonio Vicente Marafioti Garnica Summer of 2018 

Contents Preface to the Series���������������������������������������������������������������������������������������������������������������������   v Preface to the Book������������������������������������������������������������������������������������������������������������������������  vii Contents�����������������������������������������������������������������������������������������������������������������������������������������   xi Overall Abstract ���������������������������������������������������������������������������������������������������������������������������  xiii Individual Chapter Abstract�������������������������������������������������������������������������������������������������������  xv About the Authors�������������������������������������������������������������������������������������������������������������������������  xix 1 Oral History in Mathematics Education: An Overview �����������������������������������������������������   1 Antonio Vicente Marafioti Garnica and Carlos Roberto Vianna Some Procedures and Justifications �����������������������������������������������������������������������������������������   5 Choosing the Interviewees and the Moment of the Interview �������������������������������������������������   6 Transcriptions and Textualizations�������������������������������������������������������������������������������������������   7 About the Analysis of the Sources �������������������������������������������������������������������������������������������  11 Oral History and Mathematics Education: Motivations and Examples �����������������������������������  12 Teacher Education Programs ���������������������������������������������������������������������������������������������������  13 From the Center to the Periphery���������������������������������������������������������������������������������������������  13 About the References���������������������������������������������������������������������������������������������������������������  18 References���������������������������������������������������������������������������������������������������������������������������������  19 2 Oral History and Mathematics Education: Historiographical Research �������������������������  21 Maria Laura Magalhães Gomes Historical Panorama of the Formation of (Mathematics) Teachers in Brazil���������������������������  23 Oral History, History, and Historiography of Mathematics Education�������������������������������������  24 Mapping the Training and Practice of Mathematics Teachers in Brazil: What Research has Shown�������������������������������������������������������������������������������������������������������  28 References���������������������������������������������������������������������������������������������������������������������������������  32 3 Oral History and Mathematics Education: The Dialogue with/among Different Theoretical and Philosophical Perspectives���������������������������������������������������������  35 Marcelo Bezerra de Morais and Filipe Santos Fernandes Introduction�������������������������������������������������������������������������������������������������������������������������������  35 Theorizations in Oral History and Mathematics Education�����������������������������������������������������  37 First Scenario: The Object/Process Under Investigation Guides the Choice of Theoretical/Philosophical Perspectives �����������������������������������������������������������   37 The Second Scenario: Different Theoretical and Philosophical References are Mobilized in Order to Understand the Object/Process Under Investigation in a Multiple Sense�����������������������������������������������������������������������������������������������������������������   40 Final Considerations�����������������������������������������������������������������������������������������������������������������  43 References���������������������������������������������������������������������������������������������������������������������������������  44 4 Oral History in Mathematics Education: Possibilities for Effective Intervention in Teaching���������������������������������������������������������������������������������������������������������  47 Heloisa da Silva Introduction�������������������������������������������������������������������������������������������������������������������������������  47 History of Mathematics Education: Place, Affiliation, Identity, and Present Time �����������������  49 xi

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(Oral) History as an Approach to Teaching and Training of Mathematics Teachers ���������������  51 Contributions of Oral History to the Training of Mathematics Teachers���������������������������������  59 References���������������������������������������������������������������������������������������������������������������������������������  64 5 History of Mathematics Education and Oral History: Possibilities for the Classroom �������������������������������������������������������������������������������������������������������������������  67 Vinícius Sanches Tizzo and Antonio Vicente Marafioti Garnica Introduction�������������������������������������������������������������������������������������������������������������������������������  67 History in Mathematics Education and Initial Training: Possibilities for Integration �������������  69 History of Mathematics Teaching: An Introduction�����������������������������������������������������������������  71 Articulating Oral History and Elementary Education: Possibilities for Work with Children�������������������������������������������������������������������������������������������������������������  73 Oral History in the Initial Training of Mathematics Teachers �������������������������������������������������  75 Oral History as Support for Problematizing Public Policy in Education and Mathematics Education�����������������������������������������������������������������������������������������������������  76 Oral History Approach in the Discipline “Content, Methodology, and Teaching Practice of Mathematics” in a Pedagogy Course�����������������������������������������������  77 History in Mathematics Education: A Discipline and a Pedagogical-Methodological Contribution for Teaching and Research�����������������������������������������������������������������������������������  79 Oral History: Other Exercises Conducted in Mathematics Teaching Courses�������������������������  81 Some Considerations ���������������������������������������������������������������������������������������������������������������  83 References���������������������������������������������������������������������������������������������������������������������������������  83 6 Oral History in Mathematics Education: On Non-historiographical Research���������������  85 Maria Ednéia Martins Salandim Introduction�������������������������������������������������������������������������������������������������������������������������������  85 About Oral History in Mathematics Education �����������������������������������������������������������������������  87 About Themes and Research Intentions�����������������������������������������������������������������������������������  90 On Oral History in flux in Mathematics Education: Some Examples���������������������������������������  94 Final Considerations�����������������������������������������������������������������������������������������������������������������  96 References���������������������������������������������������������������������������������������������������������������������������������  97 7 Narratives and Their Power Against Silencing in and by Scientific Research����������������� 101 Luzia Aparecida de Souza On “Inappropriate” Questions for the Narrative����������������������������������������������������������������������� 105 Sources, Formation, and Narrative Aesthetics as Possibilities of Ethical Commitment in Academic Work����������������������������������������������������������������������������������������������� 107 Narratives as Sources ��������������������������������������������������������������������������������������������������������������� 107 Narratives as Possibility for Analysis��������������������������������������������������������������������������������������� 110 Form-Content��������������������������������������������������������������������������������������������������������������������������� 111 References��������������������������������������������������������������������������������������������������������������������������������� 112 Combined Reference List������������������������������������������������������������������������������������������������������������� 115 Author Index��������������������������������������������������������������������������������������������������������������������������������� 125 Subject Index��������������������������������������������������������������������������������������������������������������������������������� 127

Overall Abstract This book tells how a group of Brazilian educators have responded to enormous challenges in mathematics education. In particular, the authors draw attention to the creation, and work of those involved in the Oral History and Mathematics Education Research Group (denoted by GHOEM, which was created from the Portuguese-language version of the Group’s name—Grupo de Pesquisa História Oral e Educação Matemática). As its name suggests, GHOEM, has gathered large amounts of data, in the form of narratives, from a wide range of stakeholders who have learned, or have taught, or are learning, or are teaching, mathematics. Brazil is the world’s fifth-largest country by area and, with over 200 million people, is the fifth most populous. Although long inhabited by indigenous peoples, it was reached by the Portuguese in 1500 CE, and today its national language is Portuguese. The nation has many remote areas, and many different groups of peoples. It also has large cities such as São Paulo and Rio de Janeiro (with populations about 12 million and 6 million, respectively). The challenges of creating and maintaining high-standard mathematics education programs, suitable for all learners, has been, and continues to be, immense. This book, Oral History and Mathematics Education, edited by Antonio Vicente Marafioti Garnica, documents an approach to mathematics education in Brazil, which, over recent years, has carved its own unique pathway through an often-challenging terrain. It is timely that this pathway be made known to the rest of the world. In each of the seven chapters of Oral History and Mathematics Education the deeply philosophical, ethnomathematical, and historiographical bases of the work of GHOEM and its members are on display. It is this backdrop that adds special significance to the book. Data for the research described have been gathered from both “bottom-up” and “top-down” vantage points, and the chapters display much thought-provoking analyses of what forms of “narratives,” and how narratives can offer unique perspectives on issues under consideration. The authors provide access to details related to an important development in education research, which, until now, has been largely hidden from the rest of the world—possibly as a result of language and geographical factors. Every chapter has something of significance to say to educators, mathematicians, and historians—no matter where they are, and no matter the particular contexts in which they find themselves. This book offers readers a chance to become acquainted with groundbreaking Brazilian efforts to develop a grassroots, and fascinating, research methodology, which has much to offer to a wide range of researchers. In particular, it invites mathematics educators, worldwide, to move beyond the textbook-oriented, top-down, pure-mathematical and, one might say, colonialist, versions of the history of, and current practices in, mathematics education. Nine authors have contributed to the seven chapters. Each chapter presents an emerging story, and this book has the potential to change not only the way mathematics education research is conducted across the world, but also how the teaching and learning of mathematics can be better linked to the needs of individuals, communities, and nations.

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Individual Chapter Abstracts Chapter 1: Oral History in Mathematics Education: An Overview

Abstract: This chapter aims to provide a panorama of what we understand as oral history

and how it can be used for research in mathematics education. To this end, we advocate oral history as qualitative methodology designed to create narratives that may assist the development of historiographic projects; however, every narrative created is a historiographic source. The most commonly used procedures used in oral history are interviews, transcriptions, and textualizations. In this chapter, special attention is given to the possibilities of analyzing narratives generated through oral history. As a whole, this chapter is meant to serve as an introduction, an overview, to the other chapters in this book, which seek to problematize and exemplify further what is discussed here.

Chapter 2: Oral History and Mathematics Education: Historiographical Research

Abstract: This chapter focuses on historiographic research in mathematics education carried out with the use of the oral history methodology. To this end, a panorama of the investigations of the Oral History and Mathematics Education Group is presented. The group works in a project intended to clarify how the training and activities of mathematics teachers have been developed in Brazil within different institutions, various school levels, and completely different times and environments. A general outline of the history of education and of teacher formation in Brazil is also presented. Key theoretical and methodological issues are discussed as to the participation of oral history in history research, and particularly in the history of mathematics education. Finally, this chapter presents a summary of what the research about mathematics teachers’ training and activities in Brazil have been showing through time. At the same time that the oral history methodology highlights and values peculiarities and individualities, this methodology has enabled the perception of characteristics, which are common to many scenarios such as the lack, the urgency, and the discontinuity of training activities of mathematics teachers in Brazil.

Chapter 3: Oral History and Mathematics Education: The Dialogue with/among Different Theoretical and Philosophical Perspectives

Abstract: As a result of the effort to produce and articulate several narratives, the research in mathematics education that mobilizes oral history has encouraged dialogue with and among several theoretical and philosophical perspectives and developed investigations characterized by conceptual and procedural plurality. In this chapter, starting from examples of research in

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­ athematics education, we attempt to debate two scenarios that demonstrate different ways of m fostering interdisciplinary dialogue in the investigation process: In the first scenario, the object/ process under investigation guides the choice of the theoretical/philosophical perspectives mobilized by the researcher. The conceptual and procedural constructions are a consequence of the issues that arise from the narratives created/gathered for the research. In the second scenario, the mobilization of different theoretical/philosophical perspectives enables multiple glances at the object/process under investigation and develops understandings that lead to a multiplicity of aspects, which characterize the object/process in its complexity. These scenarios show how oral history methodology in mathematics education research, when open to several theoretical/philosophical perspectives, enables links with different forms of research, thus emphasizing a practice in which dynamism and heterogeneity become the core of the investigation.

Chapter 4: Oral History in Mathematics Education: Possibilities for Effective Intervention in Teaching

Abstract: By presenting and advocating the potential of oral history as a pedagogical strategy in mathematics education and in teacher formation, this chapter describes the aspects and possibilities of this approach in the treatment and problematization of issues relating to educational cultures and mathematics education. This chapter also presents a panorama of history of mathematics education in Brazil as a line of research and its contributions for training mathematics teachers. Finally, there is a debate of a theoretical and methodological nature about the key issues and strategies to be considered in interventions that use oral history in teaching; some examples are given as reference.

Chapter 5: History of Mathematics Education and Oral History: Possibilities for the Classroom

Abstract: Besides presenting some experiences conducted with oral history in mathematics teaching (focusing on teacher formation courses), this chapter also proposes practical possibilities for classroom practice. Examples of several natures are shown, particularly relating to topics associated with the history of mathematics education. We believe that history, seen as a potent aid in the building of knowledge and because of the richness and diversity of the elements involved in a historiographic operation, enables problematization of some fundamental circumstances that lead us, both teachers and students, to become what we are. Therefore, these historiographic practices, together with oral history, are essential aids in teaching and learning processes (in general and, particularly, in mathematics) when transposed to the educational ­scenario in a promising and instigating manner. The examples shown must be understood as possibilities for action that trigger perspectives so that each reader, with their own interests, conceptions, and curricular demands, may adapt, transform, and transcend these suggestions, thus creating their own sequences and didactic scripts so as to include history, oral history, and history of mathematics education in teaching and learning processes.

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Chapter 6: Oral History in Mathematics Education: On Non-historiographical Research

Abstract: The purpose of this chapter is to demonstrate how oral history has been mobilized as a research methodology in the area of mathematics education in Brazil, more specifically in research that has no historiographical approach. We describe how oral history became incorporated into the area with the emergence of a research group with a specific interest in problematizing the potential of this methodology for mathematics education. We also describe protocols, procedures, and topics that have been explored in the research. Moreover, we list the contribution of this methodology for the understanding of topics, practices, and scenarios of Brazilian mathematics education through accounts of experiences of people who are or have been involved in it.

Chapter 7: Narratives and Their Power Against Silencing in and by Scientific Research

Abstract: The key role of oral history is to identify narratives that may be used in several ways. This chapter deals with the experience of the Oral History and Mathematics Education Research Group (GHOEM), which works with oral history and, as such, uses narratives as the core element of its investigation practices. The objective of this chapter is to debate the concept of “narratives,” as well as the ways they can be created/built and how they can be analyzed. Other topics are also approached such as sources, attentive listening, multiplicity of truths, form, and content (form-as-content), and the need to motivate/promote different sensitivities that are not common within academia when scholars search for new ways of thinking and building knowledge.

About the Authors Filipe  Santos  Fernandes  is a professor in the School of Education of the Federal University of Minas Gerais (UFMG). His doctoral degree was obtained in the Graduate Program in Mathematics Education of the State University of São Paulo (UNESP), Rio Claro campus. He is a member of the Oral History and Mathematics Education Research Group (GHOEM) and of the Research Group on Education in/for Rural Areas (NEPCampo). His main concerns in mathematics education research are how individuals became historical beings, mathematics education as a social practice, and the educational processes in Brazilian rural areas, especially in relation to peasant education. Antonio  Vicente  Marafioti  Garnica  is a professor in the Mathematics Department, School of Sciences, State University of São Paulo (UNESP), Bauru campus, and of the Graduate Program in Mathematics Education (UNESP-Rio Claro). He is editor of BOLEMA (Mathematics Education Bulletin, created in 1985—the oldest journal in this field in Brazil) and leader of the Oral History and Mathematics Education Research Group (GHOEM). His master’s dissertation focuses on the possibilities of a hermeneutical interpretation of mathematical texts, and his doctoral thesis discusses the role of formal proofs in undergraduate courses for mathematics teachers’ formation. Both of these studies were carried out in the Graduate Program in Mathematics Education of the State University of São Paulo (UNESP), Rio Claro campus. His main research themes are the history of mathematics education in Brazil, oral history in mathematics education, and the development of mathematics teachers. Maria Laura Magalhães Gomes  is a professor in the Mathematics Department of the Federal University of Minas Gerais (UFMG) where she was, in 1977, student of the undergraduate program. Her doctoral thesis was developed in the State University of Campinas (UNICAMP), and her large experience in mathematics education includes teaching mathematics in undergraduate courses and having, as the main issues of her research agenda, the history of mathematics education, history of mathematics, and analysis of mathematics textbooks. Marcelo Bezerra de Morais  is a professor in the Department of Education and also of the Graduate Program in Teaching (POSENSINO), both branches of the State University of Rio Grande do Norte (UERN), where, in 2010, he finished his undergraduate formation. Both his master’s and doctoral theses were developed in the Graduate Program in Mathematics Education of UNESP, Rio Claro campus, having experienced a period in the University of Lisbon during his doctoral studies. He is a member of the Oral History and Mathematics Education Research Group (GHOEM), and his main research themes are the history of mathematics education, mathematics teaching, narratives, and the constitution of human beings according to a philosophical point of view.

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Maria Ednéia Martins Salandim  gained her PhD degree in Mathematics Education at the Graduate Program in Mathematics Education of the State University of São Paulo (UNESP), Rio Claro campus. She now works in the Bauru campus of the same university, as professor of the Department of Mathematics and as researcher, advising students, at the Graduate Program in Science Education. She is a member of the Oral History and Mathematics Education Research Group (GHOEM) and has experience in education, with emphasis in mathematics education, focusing, in her research practices, on the formation of mathematics teachers and the analysis of mathematical textbooks. Heloisa da Silva  is a professor in the Department of Mathematics Education of the same university where she developed her master and doctoral thesis—the State University of São Paulo (UNESP), Rio Claro campus. She teaches mathematics for undergraduate student and advises students of the Graduate Program in Mathematics Education. Her experience in mathematics education includes the study of themes such as history of mathematics education in Brazil, oral history, and practices and policies related to mathematics teacher education. Luzia Aparecida de Souza  ained her undergraduate degree at the State University of São Paulo (UNESP), in 2003, and presented her master’s dissertation (in 2006) and her PhD thesis (in 2011), both, in the Graduate Program in Mathematics Education of UNESP, Rio Claro campus. Since 2009, she is the Mathematics Institute in the Federal University of Mato Grosso do Sul (UFMS, Campo Grande campus) and coordinator of the Graduate Program in Mathematics Education of the same UFMS. Her experiences in mathematics education research focus on teacher education, history of mathematics education in Brazil, oral history, and narratives. Vinícius  Sanches  Tizzo  developed his doctoral thesis in the Graduate Program in Mathematics Education of the State University of São Paulo (UNESP), Rio Claro campus. His undergraduate studies were completed at the Federal University of Mato Grosso do Sul (UFMS). He is a member of the Oral History and Mathematics Education Research Group (GHOEM), working mainly in the following subjects: mathematics education, teacher education oral history, narratives, and educational policies. Carlos Roberto Vianna  gained his undergraduate degree at the Federal University of Paraná (UFPR) and prepared his doctoral thesis at the University of São Paulo (USP). He is a professor in the Department of Mathematics, a branch of the Federal University of Paraná (UFPR), and his research emphasizes the philosophy of mathematics education, school inclusion, teacher education and the history of mathematics education.

Chapter 1 Oral History in Mathematics Education: An Overview Antonio Vicente Marafioti Garnica and Carlos Roberto Vianna

Abstract:  This chapter aims to provide a panorama of what we understand as oral history and how it can be used for research in mathematics education. To this end, we advocate oral history as qualitative methodology designed to create narratives that may assist the development of historiographic projects; however, every narrative created is a historiographic source. The most commonly used procedures used in oral history are interviews, transcriptions, and textualizations. In this text, special attention is given to the possibilities of analyzing narratives generated through oral history. As a whole, this chapter is meant to serve as an introduction, an overview, to the other chapters in this book, which seek to problematize and exemplify further what is discussed here.

Keywords:  Oral history and mathematics education · Procedures · Concepts · Panorama There is no voice which should not be heard, a story which could not be told—this is the motto of this chapter whose intention is to present a panoramic view of oral history and to make some remarks on the ways it can be used in mathematics education, focusing on what we consider fundamental concepts and issues. The history of oral history can be told in many ways, some leading back to Herodotus, others stemming from the invention of portable voice recorders. The breadth of this practice can be proven using Internet search engines which compile results in areas as diverse as art, medicine, and anthropology. The use of interviews in oral history, for either academic or nonacademic purposes, alludes to authors as diverse as Svetlana Alexievich (Nobel Prize in Literature in 2015), Alessandro Portelli, and Studs Terkel. The 51st Annual Meeting of the Oral History Association, held in Minnesota in October 2017, drew attention to the prominence of the oral history movement. The theme of the

A. V. M. Garnica (*) University of São Paulo State, UNESP, Bauru/Rio Claro, Brazil e-mail: [email protected] C. R. Vianna Federal University of Paraná, UFPR, Curitiba, Brazil © Springer Nature Switzerland AG 2019 A. V. M. Garnica (ed.), Oral History and Mathematics Education, History of Mathematics Education, https://doi.org/10.1007/978-3-030-16311-2_1

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event was “Engaging Audiences: Oral History and the Public,” which highlights two important aspects of the movement—first, its institutional longevity and second, the growing public impact of the collective production of researchers who have embraced different approaches to oral history. The folder of the event showed the diversity of areas of knowledge represented: “Program participants throughout the conference will draw upon their own work with oral history in the arenas of education, theater, art, radio and podcasting, film, museums, archives, and other community spaces, and instigate creative thought about how oral history’s real and perceived audiences inform our work during and after the interview itself.” Oral history is seen to produce sources, record memories, and document experiences. It is not only mobilized in the academic field, but in a myriad of human activities, giving this approach both a democratic and ecumenical character, as each of the distinct groups who use it contributes in their own way, according to its particular mindset. According to Philippe Joutard (1983, 1999), the original inspiration for oral history consists was the need to heed the voices of those excluded, showing realities that writing cannot transmit, and witnessing situations of extreme abandonment. In mathematics education certainly many are excluded, there are countless situations which could be called indescribable, in addition to an almost inexhaustible variety of fields unchartered by researchers in education. It is worth giving some examples: although we are proficient in researching classroom practices, with often meticulous observations, it is somewhat difficult to find records of teachers’ statements about how they plan their classes, how they evaluate their classroom performance, how they prepare assessments, etc. One of the reasons for the scarcity of this type of register derives from one of the oldest and most persistent theoretical discussions in oral history: What if the interviewee lies? Another reason for the lack of records of this nature may be the neglect that both researchers and teachers have shown in relation to these everyday, routine, usual, ordinary practices which have only very recently been considered as objects of legitimate research, whose study is as important as it is necessary. Oral history provides ways of investigating what is said, what is left unsaid and, sometimes, even tapping into the unspeakable. It is therefore a matter of investigating and recording each version presented by interviewees, constructing sources which are taken as “enunciations in perspective,” which preserve voices which are often alternative and dissonant to what is conventionally labeled as “historical fact.” Under this perspective, we deny “historical fact” and work with “versions of history.” This enables us to navigate a scenario in which multiple rationalities intersect. Oral history, as a qualitative research methodology, though very close to the intentions of written history, is not necessarily used to develop research whose themes are historiographical. This is one of the characteristics which distinguish oral history from other qualitative approaches used in mathematics education: when employing oral history, the researcher intentionally creates historical sources and labels them as historical sources, although not necessarily using such historical sources in historiographical investigations. It is true that any work made public becomes a potential historical source. However, the difference we want to stress is that those who draw upon oral history are “source creators” who know that they are creating and therefore are prepared to recognize, from the beginning, there is no single and definitive truth. They know that the clash between different points of view begets several possible explanations for each event and that they have a responsibility to admit that all sources cannot encompass all possible viewpoints and will always give a partial notion of the events, situations, or realities they wish to explain or understand. Researchers using oral history know—and must be prepared to defend this option—that the sources they construct are lacunar and partial. This implies that, whether or not they are willing to conduct research of a historiographical nature, researchers who use oral history must be aware of the conception of history that it entails. This requires, for

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instance, previous studies on the history of historiography, on the elaboration of historical studies (the writing of history), and on the themes and approaches practiced in history. The oral history of which we speak is an academic practice, which almost always results from (and in) academic research projects. However, there is a myriad of other oral history practices, often not validated by academia. Examples are the efforts of groups of people, in specific communities, which record interviews with former residents to keep records of the history of neighborhoods and cities, as well as interviews with artisans about crafts such as artistic masonry, some forms of embroidery, basket weaving, and ceramics, which are almost extinct today. In the realm of the school environment, the most common targets are teacher practices, deriving from interviews with parents or other education professionals. Although such practices seldom undergo a ritual of academic validation, they should not be overlooked, nor should it be assumed that the results do not generate appropriate sources for future reference. However, the practices described in this chapter have always resulted from projects developed and institutionally monitored by universities. Thus, oral history will be defined as a methodology, a set of well-founded procedures adopted by researchers. An effective method gives clues to understanding situations, solving problems or directing responses to some challenges. But the effectiveness of the method is always measured against theoretical assumptions and experiences of the researcher, which implies the impossibility of separating the use of a methodology from a worldview and the theoretical principles adopted by the researcher. Thus, the methodology should not be viewed as a fixed set of procedures. It forces researchers to question, confront, and evaluate their assumptions continuously. The methodological question is: one must search for a method which is effective and appropriate to research questions, a method whose presuppositions can be— and truly are—tested continuously in the public sphere, shedding light on achievements and embarrassments. Among the assumptions adopted, those we believe characterize qualitative research are as follows: (a) the results are transient; (b) there is not an a priori hypothesis to be proven or refuted; (c) the non-neutrality of the researcher is shown by the research narrative itself; (d) understanding occurs in the multiple pathways which can be continuously reconfigured; and (e) it is impossible—or rather inadvisable—to establish prior systematic procedures and generalizations. To take such assumptions as a starting point implies acknowledging that, ultimately, the same assumptions can be modified considering the development of the research, mainly because the context and the object of research require specific ways of applying certain strategies. Thus, it is also important for each work to describe the ways in which the method has been used, constructing what could be called “methodology in action,” which shows obstacles and successes so that other researchers can be aware of such cases and create strategies to minimize obstacles and maximize successes in their own course of action. They in turn will also be evaluated and carefully described in the future, thus producing a cycle of methodological reflection which results in more stable strategies, which, as previously stated, cannot be seen as exclusive or final. In view of these basic principles, the Research Group on Oral History and Mathematics Education (Grupo de Pesquisa “História Oral e Educação Matemática”—GHOEM), a group of Brazilian researchers who develop the potential of oral history as a theoretical-­ methodological resource for research in mathematics education, has been using and examining oral history as a qualitative research methodology in mathematics education. GHOEM has developed studies in mathematics education and history of mathematics education. On the one hand, it seeks to understand the fundaments of oral history and, on the other, to contribute to an understanding of the relationships between mathematics, teaching, and learning. It is important to contextualize the projects about the history of mathematics education which map the training of mathematics teachers in Brazil, highlighting

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the production of groups—institutionalized or not—in mathematics education. Examples worth mentioning are the Núcleo de Estudo e Difusão do Ensino de Matemática (NEDEM) (Center for Study and Dissemination of Mathematics Teaching), linked to the process of implementation of new mathematics in the state of Paraná, Centro de Educação Matemática (CEM) (Mathematics Education Center) in São Paulo, and the research group História Oral e Educação Matemática (GHOEM) (Oral History and Mathematics Education). Other examples include projects which analyze institutions and agencies responsible for curricular guidelines to national education, such as Coordenadoria de Estudos e Normas Pedagógicas da Secretaria da Educação do Estado de São Paulo (CENP) (Coordination of Studies and Pedagogical Norms of the Education Department of the State of São Paulo). In the general field of mathematics education (investigations which use oral history but do not necessarily have the intention to produce historiographical studies) include projects which study the school-family-mathematics dynamic, professionalization, resistance, utopias, as well as prejudice related to mathematics teachers’ practices and profession. Specific chapters in this book will discuss such approaches, themes, and research in more detail. In summary, it is important to reiterate that the oral history methodology is treated here not as a series of regular and definitive procedures, but as a set of well-founded, publicly divulged, and continually evaluated procedures which give us an overview on how to use the methodology, constituting some stability with respect to its mobilization. This methodology is described in the elaboration and configuration of research results which also show the researchers’ perspectives. This requires abandoning the assumption that the greater the distance a researcher assumes while conducting the investigation, the more rigorous it will be. Thus, using oral history implies reintroducing the subjectivity of interviewees and researchers into academic domains, denying the positivist assumptions which insist on infiltrating our environments, even when discourse to the contrary is openly propagated. Oral history, seen as a research methodology, requires us to consider those who implement it and the scenario in which it is implemented. Furthermore, under these assumptions, oral history is a research methodology of a qualitative nature inscribed in the field of humanities, whose breeding ground is based on descriptions. Humanities can only exist if we consider how individuals or groups represent something to themselves; how they compose their speech, revealing or concealing what they think or mean. The concepts on which humanities are based, in terms of qualitative research, are produced by descriptions. Descriptions and subjective narratives cannot be labeled as right or wrong or true or false. Descriptions imply at least one speaker and one listener. The description always comes from someone who talks about something which is not fully known to the interlocutor. However familiar the situation described may be to the listener, the living experience of the situation cannot be transferred to another person. Thus, even if two or more interviewees describe the same situation, the description will always be different, always new to those who listen carefully. In this respect, the vantage point of the person who describes it is “better” than that of the listener. The breadth, precision, and balance of a description, however, can only be evaluated by the researcher and their group, in view of the sociocultural-political and theoretical context in which the research is being conducted. Therefore, to accept descriptions as essential to knowledge, as primordial and fundamental vectors in the research conducted according to oral history parameters, and as residues of enunciation from which the researcher intentionally creates historical sources implies accepting a conception of historiography according to which all history is a version. One does not seek historical truth, but the history of truths. Biographical and narrative research, particularly in education, is based on the “hermeneutic turn” of social sciences. This is a movement through which social phenomena are seen as text, whose value and meaning are given primarily by how subjects, through first-person

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narratives, interpret themselves and their circumstances. The Spanish researchers Bolivar, Domingos and Fernandes (2001) claim that in sociology a reflexive orientation is adopted, focusing particularly on biographical orientation; in anthropology-ethnography, more decisively than in other social sciences, the narrative approach (culture as text) is adopted in the same way as the metaphor “life as narrative” emerges in psychology. An exception to this is the Chicago School, well-grounded, for example, in the work of Paul Thompson (1988) about oral history, whose exercises with biography date back to the 1920s. With this hermeneutic turn in the social sciences, through which human experiences and the world itself, in its various aspects, are viewed as text, and therefore open to interpretation, we feel the need for different methodological tools and strategies to account for these readings of the world. Understanding the world, interpreting it as text, implies recognizing the collection and analysis of biographical and narrative data, particularly texts, enunciated by specific individuals, as a legitimate form of intervention, from which an entire investigation arises. This implies that we must consider narratives as gateways to the narrators’ experiences, and ultimately recognize the narratives of others as a means of constituting other individuals and as a possibility for understanding and constituting ourselves from experiences which are not always rationally established. The “narrative” is understood as experience structured as a story, as a “tale.” As a focus of research, narratives are a path for constructing meaning: meaning for the individual, as narrators describe themselves while narrating something, and meaning of what is being narrated, as communication is a human endeavor, even though it is always somewhat frustrated in its objective to communicate actions limited by time as fully as possible, using descriptions of things, individuals, and ourselves (biography). The argumentative plot, temporal sequence, characters, and situation, all constitute a narrative. The narration of life through self-reporting is a means of inventing one’s self, of creating an identity (a narrative identity). Jerome Bruner argued that the object of the narrative is the peculiarities of human intentions. As a means to knowledge, the account captures the richness of details and meaning in human experience (motivation, desires, feelings, or purposes) which cannot be expressed in factual statements or abstract propositions as logical-­formal reasoning does. Narrative is essential for research because individuality cannot be explained solely by external references. Subjectivity is a necessary condition of social knowledge. Narrative not only expresses important dimensions of experience, but, more radically, it mediates the experiences themselves and shapes the social construction of reality. For Paul Ricoeur, all meaningful action is a text to be interpreted, and all human time is articulated in a narrative manner: narratives of interviewees and narratives of researchers fuse productively to understand social reality. If positivism implanted the correlation between depersonalization and objectivity, the point of narrative investigation is precisely to deny this presupposition: the interviewees speak of themselves, denying the silence of subjectivity, and it is up to the researchers who use oral history, a research approach based on subjective voices, to make the voices of the interviewees echo in spaces where these voices are often inaudible. 

Some Procedures and Justifications

We have already pointed out that methodology is not a mere set of steps, but a series of possible procedures applied in a well-grounded manner. We have already stated that the methodology of oral history that we have practiced in mathematics education research is subject to continuous evaluation, from which a more stable set of procedures arises, which can be applied in the studies, but which are never definitive or crystallized.

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This set includes considerations about (a) the choice of interviewees, the composition of an interview script, and the timing of interviews; (b) the transcription of interviews—the transition from the spoken to written record; (c) the textualization of the interviews—an exercise in editing the initially transcribed text whose elaboration occurs in several phases; and (d) the textualization review after which the interviewee signs a letter of assignment of rights so that the final text can be divulged and studied by the researcher. We will discuss each of these items in more detail below. 

Choosing the Interviewees and the Moment of the Interview

We understand that no one sets out to investigate any phenomenon if they are not aware of their intentions regarding the research they wish to undertake. Although it is known that with the development of the work, the initial intentions can be reformulated, an initial configuration is necessary for the research project. From these initial queries, a range of possibilities for interviews can be drawn. As Paul Thompson suggested, it is the multiplicity of points of view gathered from the interviews which will enrich the plot of the narrative which can be constructed. The search for people who agree to give interviews is often a process of forming a network; as the community becomes interested on the topic, it is common for a respondent to suggest names of others who can talk about the same subject. The investigator records these suggestions and makes the necessary contacts, explaining to each respondent the nature of the research and the procedures to be implemented in the investigation. The interviews are scheduled in advance and the nature of the research and procedures must be clarified, and the interviewee may even be given a script of the conversation in advance. Some may say that anticipating the nature of their intentions impairs the analysis of narratives, but we believe that in addition to being an ethical issue, the clarification of the intentions of the researcher and the prior knowledge of the questions by the respondents will later enable the analysis of “absences” or “silences”: knowing the path that the researcher intends to take, why doesn’t the interviewee mention the subject? Why do they approach the theme in this manner and not another? In the cases described, the interviews are dialogues about the object of research and the greater the atmosphere of confidence and trust between interviewer and interviewee, the richer they will be. For this bond to form, the researcher must be aware of several aspects of what the interviewee recounts. The interview is a time when attempts to overcome the incommunicability of experiences as much as possible are made. This is built on a dialogue which accentuates details and perceptions, reconstituting and sharing the views about the subject of investigation. An interview is also a time of conflict: the interviewer may challenge the interviewee, requiring an explanation of their assumptions or positions. In such cases, there is a possibility of interruption of the interaction, and, therefore, the researcher must carefully evaluate the situation before opting for confronting the interviewee. There are several strategies to trigger the narratives that we intend to obtain from interlocutions, among which are photographs; presentation of previously collected accounts, books, and mementos common to the interviewer and interviewee; small handwritten cards with words or expressions to serve as triggers; etc. Retrieving memories or creating an atmosphere of dialogue often requires more effort from the interviewer than the search for information on the topic of conversation with the interviewee. During the interview, depending on the interviewer’s sensitivity and ability, the conversation may take a more informal and uninhibited tone. In such nuances of language may reside one of the researcher’s perspectives of analysis, as the text of the interview, in addition to data, manifests discourse which not only reconstitutes what is being narrated but is in itself an opportunity for constituting situations and individualities. Respondents go beyond telling a story: they narrate them-

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selves and, by doing that, express their subjectivity. This is precisely why the question of anonymity rarely arises, in the research conducted by GHOEM according to the parameters of oral history, except for very few exceptions. We will revisit this point later on, when dealing with the ethical issues involved in oral history research. Interviews may be focused on a specific topic or may be more general in character. The first case consists of a thematic project in which the researcher’s interest is more specific and focused on a moment or situation which is familiar to the interviewee in some way. The interview is organized around this theme, and all the questions in the script will be linked to or revolve around it. The more the interviewer is accepted as the interlocutor, and their legitimacy is recognized, the richer the narrative will be. The second case is a Life History Project, in which the respondent is asked to make an account without resorting to a specific focus or, if such focus exists, it is approached amid memories of their life as a whole. Thus, the narrator talks about childhood, adolescence, profession, or parents, and this makes it possible to outline the circumstances, mishaps, or contributions associated with an individual or group of people. The script may also lead to memories and suggest starting points which may evoke narratives. Some researchers form support groups for the interview, specifying the role of each participant: one is responsible for the questions, the other for circumstantial notes (elaboration of a field diary, as in anthropology, for example), and another for controlling the recorder and, if applicable, camcorder, as interviews can be audio-recorded or videoed. There are both positive and negative implications to having a team of interviewers: several people may render the interview environment more confusing and disrupt the formation of closer bonds between the interlocutors; however, it can prevent very common technical problems in interviews, such as abrupt interruptions of people alien to the situation, problems with recording quality, etc. The researcher must decide whether to form such teams. In any case, the conditions in which the interview will be conducted must be discussed with the interviewees, who will choose dates and places, and may impose conditions for the encounter. It is common for interviews to be conducted with more than one respondent at the time. Not infrequently, the interviewee invites or suggests that another interlocutor with whom they live be invited, so that they are together at the time of the interview. There are cases in which the interviewer himself chooses to hold collective interviews, or even individual interviews which are later repeated with a group of interviewees. Such decisions are a prerogative of the researchers, but in agreement with the interviewees, and are linked to the way interlocutors relate to each other the purpose and theme of each research. 

Transcriptions and Textualizations

For some researchers who use oral history as a methodological resource, the quintessential source is the recorded tape. For others, it is the written text with the recording of the moment of the interview. Attempting to decide which is the most representative “reality” of the moment of the interview renders this discussion innocuous, as any support whatsoever is a record of the moment and, like any record, a filter. A moment experienced cannot be fully grasped by any instrument, by any kind of record. Even the language through which narrative manifests itself has been filtered by historical time, by sociocultural conditions, and even by the natural limitations of communication. There is no definitive record of facts since there are no facts; there is always a perception, a communication of intentions, “something” which is shown in perspective. It is this perspective (the speaker’s) which is most often understood from another’s perspective (the listener’s). Indeed all perspectives, whatever they may be, elude attempts of apprehension by the mesh of either image, sound, or writing.

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Although this is a fluid medium in which listener and speaker, significantly, exchange positions, none of them with the prerogative of dominating the speech of the other, and although none of them can record or control the ways in which meanings are attributed to what has been said or heard, something remains: records remain; residues remain. It will be up to researchers to choose those records which they deem most useful, given the limitations imposed for the dissemination and conservation of the various possible records. Therefore, the choice between recorded speech and written text is technical, not philosophical. All current electronic resources have made it simpler and cheaper to store oral records and images through different media, but the preservation of such media requires more care, for example, than that required for the preservation of written text (historically, our institutions have prepared themselves for receiving, storing, and delivering written records, while there is still a rather slow movement toward the organization and treatment of other media). More important than deciding the appropriate media to disseminate the sources that research has gathered is to ensure that such sources be made public in some way. Researchers play an important role in allowing the dissemination of sources created from their interviews. Once the question of the prevalence of one medium over another has been solved, how should one proceed after the interviews? The next step is the transcription of recorded speech. Decoding (or transcription) is the transference of the interview medium (either magnetic or digital) to paper. It is the moment in which the dialogue is fixed by means of graphic characters. Generally, transcription is a time-consuming and meticulous process, through which the researcher records the moment of the interview, whose record will then undergo other treatments. Post-transcription treatments are known as textualizations (even if, of course, the actual decoding process is already textualized—the latent meaning of which being “transform into (written) text”). The distinction in nomenclature is an unnecessary but common preciosity among those who use oral history as a resource. So, it seems appropriate to configure the moments of textualization. Textualization is a process through which the researcher dives into the text of the interview (in hardcopy form, which greatly facilitates the work), no longer as technically as it did during the transcription phase (there are devices and applications specifically designed for transcriptions). There are no devices or applications for textualization, as importing meaning is essentially a human action. There are levels of textualization: the researcher can choose to exclude from the transcription only some traces of orality (usually called “expletives,” “mannerisms,” or “language degradation”) and fill in a few gaps which will make reading more fluent. Respondents themselves often demand this “cleaning,” since they cannot recognize themselves in the transcription and since spoken and written language are very different modes of expression. After this initial textualization, the researcher can choose to reorder the discursive flow of the interviewee, and the new order can be thematical or chronological. Some researchers choose not to change the order, but insert subheadings, highlighting the subtopics in the order in which they appear. There are researchers who maintain the questions and answers, while others include the questions in the answers and constitute a single text, without a marked intervention of the interviewer. There are innumerable ways of achieving textualizations, and certainly the boldest of such forms is transcreation, which consists in creating a situation from the information available in the transcription (which may be totally fictitious regarding its characters and time). Transcreation consists in the elaboration of a text in which the author assumes the speech of the interviewee, working according to their interests. It is the construction of the narrative by an author, that is, the researcher, who shapes the text toward the goals

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of their work. This intervention occurs with the intention of perfecting the text and it should be clear to the reader. So far, the transcreation shows elements which are at the heart of the process of textualizing. However, there are differences: transcreation is established as a process in which the author assumes the voice of the interviewees and creates their speech from what has been said, making cuts in turns, changing the sequence of subjects, to render the text more cohesive, according to the researcher’s point of view. However, this process requires sophisticated techniques and a familiarity with literary creation which, if mastered, adds value the narrative and enables the reader to dialogue with the text, extracting their own analyses and conclusions from it and giving meaning to phrases, words, and expressions only insinuated in a gesture, or a look, captured during the interview. Fiction, in this case, does not have the connotation of “fabrication,” but of “creation,” of incorporation of a character to mediate the testimonies obtained. Transcreation has seldom been used in oral history research in mathematics education, as it is a bold and difficult method to implement. It is more common in oral history exercises in other areas, such as literature and the arts, in general. Transcription, textualization, and transcreation are procedures. However, it would be foolish to consider such work with the text as a mere technique, concerned essentially with style. In addition to its technical content, transcription, textualization, and transcreation are methodological strategies, instances of familiarization in relation to the narrative. Much of what happens during the interview may go unnoticed by the participants. Therefore, further work with the oral source, transformed into a written source, thus, more easily manipulated (from a technical point of view, but also from the ethical point of view, as the written text is more easily available for revisions and verification), is an essential exercise for attributing meaning to what was said/heard. As a final remark regarding the elaboration of textualization, it should be noted that the researcher, although free to edit the text, should maintain what is called the “vital tone” of the interviewee as much as possible. This means preserving the manner of speaking and characteristic expressions, so that both interviewer and interviewee recognize themselves in the textualization. There is always the possibility that the researcher may remove too much of what he considers to be excesses, and/or fill in the text with information that he deems necessary so that it is better understood by the reader, and/ or still maintain vices of language that the interviewees themselves reject. Adjustments regarding possible misunderstandings will be made when the interviewer revises the text. The resulting text is the version which will be ultimately kept and/or used as the source for research. The interviewee has full right over their memories and, therefore, they take the lead in this verification process, although the interviewer can and should interfere when deemed necessary. For the verification phase (or phases), it is important that the researcher provides the interviewee with not only the textualization, but all the sources prior to textualization (recording and transcription) obtained in process. However, our experience shows that usually, for lack of interest, or time, or a number of other reasons, the interviewee chooses to review only the textualization. Whatever the case, it is the interviewee who will define which of the source(s) may be made public. In our research, we make available in the corpus of the work the integral version of the textualizations, taken as sources and made public, only the textualizations, not the recordings and/or transcriptions. It is up to the interviewee and the interviewer to agree upon such issues, recording this decision, clearly, in the letter of assignment of rights, which will be discussed below. The researcher’s interference in the text generated from the recording is often questioned. There are, for example, researchers who take the transcription as a source, choosing not to conduct any textualization. According to our oral history research practices, interviewees tend not to recognize themselves in transcription or audio segments. This is because

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spoken language and its crude transcription are very different from written language, which is seen as the “desirable” communication standard by the speaker, within the educated norm, and is defined by its own grammar and terminology, although minimal. Thus, they usually request that the revision is based solely on the textualization, in which they can fully recognize themselves. We believe that textualization must be understood not as an intervention in a text which belongs to another individual, but as an elaboration, by the researcher, of what has been narrated and, therefore, already infused with interpretations and biases. Although we do not often speak of coauthoring of textualization (the authorship shared between interviewer and interviewee), it so happens that the writing of the researcher already manifests interpretations initiated in the moment the very possibility of the research was proposed. The researcher’s perspectives, in one way or another, are inscribed in the written record of that fleeting moment of face-to-face interaction between interviewer and interviewee. Thus, it is not a question of the interviewee recognizing that the textualization is “exactly what was said” or the “exact expression of what occurred at the time of the interview”: however, it is vital that the interviewees know what is being registered and recognize themselves in that register, whether in terms of their expression or in the information given. We must assume that there is no neutrality in research. Even if we preserve and divulge only recorded versions, even if the transcripts suffice in their roughest state, a whole range of possibilities and interpretations have already been insinuated. As a strategy to give the interviewee some degree of control (or knowledge) of what will be recorded, there is a moment reserved for corrections and revision, in the end of which a letter of assignment of rights is signed. This letter contains all the terms according to which the dissemination and use of texts (written and spoken) may occur. It is a measure of both ethical and legal nature. The letter of assignment of rights is an instrument which will be best elaborated according to the decisions and possibilities of those involved. For some, an oral agreement of the interviewee is enough to allow the interview to be used; for others, the document must be elaborated in detail. Some request a more formal wording, with legal jargon; others are satisfied with an objective description of what will be done with the recording of the interview. There are no previously stipulated rules to follow in such cases: in the surveys that we have followed, several situations were observed, which show that there is no way to standardize these moments. There was a case when the simple, correct, and clear wording of the agreement listed in the assignment letter suggested by the researcher was rejected in detriment of an almost illegible and grammatically incorrect piece elaborated by a “family lawyer.” There was a case when the family suggested that “perhaps” the letter of assignment should be paid. One finding which has been constant is that elderly people, especially those who, throughout the research project, find themselves in a center-stage position, narrating their experiences (which, by the way, they initially dismissed as uninteresting, even if they do not always think so), take longer in the reviewing stage and demand countless comings and goings, rework, and corrections. Some researchers argue that this is because the respondents want to stay in this prominent position as long as possible. Finally, it is necessary to consider the need for alternative forms which attest to the assignment of rights of the interviewee to the interviewer. While working with indigenous communities, for example, or with illiterate, deaf, or visually impaired people, such alternative forms of legal assignments, which may be oral, in Braille, videotaped, or in the interviewee’s mother tongue, will be more appropriate. There are still no concrete results—although there have been concerns from researchers in this direction—in the parameterization of the use of oral history with speech-impaired interviewees. The development of lines of research dealing with ethnomathematics and inclusive mathematics education has made it increasingly necessary to adapt oral history methodology.

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About the Analysis of the Sources

There is a significant difference between “practicing oral history” and “using oral history to understand certain topics in specific areas of knowledge.” An oralist (as those who work with oral history are usually called), is, briefly speaking, interested in creating and divulging sources. There are numerous initiatives developed by oralists in specific communities, just as there are specific centers which publicly divulge oral sources generated by such initiatives. In Brazil, for example, there is the Museu da Pessoa (The People’s Museum), a virtual and collaborative project of life stories founded in the city of São Paulo, in 1991. The objective of Museu da Pessoa is to record, preserve, and transform into information life stories of all members of society. Its collection consists of oral records of people who are willing to tell their life experiences which, once recorded, can be accessed by anyone who is interested. It is believed that this initiative reveals, in a personal and human way, an enormous diversity of experiences which can be used in projects related to institutional memories, culture, the development of specific communities and education, thus exposing people to a multitude of themes and situations. The oralist, in this case, carefully creates the necessary conditions for the oral record to occur and promotes broad access to the records. On the other hand, oralists who use oral history as a way of creating sources for specific research, such as oral history in mathematics education, for example, create these sources to understand a specific theme in their field. That is to say, they create sources to ground their research. In this case, it is natural to analyze the sources created to meet research objectives more directly, although creating the sources is always one of the objectives of research which uses oral history. One of the most common misconceptions when dealing with oral history resides exactly in this distinction between “creating/making sources available” and “creating/making sources available for specific research.” Since every oralist intentionally creates sources and the primary objective of this practice is the careful recording of experiences and memories that are reported by specific individuals, every oralist is involved with work of a historiographical nature, as the sources are purposely created historiographically. Since these sources are records of the ways people see themselves and live in society at a given time and space, they are historiographical sources. It so happens that, having created the sources, oralists do not necessarily have to conduct a historiographic operation based on them. That is to say, they do not need to mobilize the source created for research in history. In short, the oralist creates historiographical sources, and that is it. With the sources at hand, oralists can, if they wish, take advantage of them for very different purposes, for example, to create literary text (as Svetlana Alexievich did, on her book about Chernobyl); to study the history of a particular institution; or even to sensitize themselves with stories in order to produce art, generate land occupation projects for communities, to rethink the use of certain architectural spaces, to promote certain didactic-pedagogical strategies, etc. Such purposes may or may not be historiographical (i.e., sources may or may not serve to trigger a historiographical operation) and, as the starting point of such an operation, projects may or may not be included in the field of scientific knowledge, as historiographical operations can also be performed outside academia. In light of these observations, an issue which may arise is whether any manner of creating oral sources can be called oral history. We could say that it is, provided that certain technical and ethical measures are adopted. By technical and ethical measures, we mean the agreements signed by the communities which are responsible for the creation of the sources. The continuous process of negotiation within a specific community is called methodological regulation, as opposed to a methodological proscription which creates rigid forms of action

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and overrides the wishes of those who implement it. In this sense, in our view, it is possible for nonacademic communities to develop oral history projects legitimately. However, when ­participating in a certain academic-scientific field (in our case mathematics education), the mobilization of oral history must follow a certain order of discourse, which is accepted by the academic-scientific community, even if it entails an obligation to establish methodological proscriptions which stiffen, if not hinder, the continuous negotiation of procedures and arguments among those who practice oral history in the scientific domain. Thus, the legitimacy of oral history as a scientific methodology is something which concerns the community who practices it, provided the criteria by which this group proceeds are made public, explicit, and open to criticism, as well as the arguments which support these criteria. This is the option which enables us to assume oral history as a democratic methodology, as it constitutes continuous agreements negotiated within a community, whether they are academic or not. The procedures which legitimize the choice of oral history, as previously stated, are focused on the choice of interviewees; on the interviewing processes, recording, transcribing, and textualizing; on the ongoing reflection on procedures; and on the defense of non-­ neutrality in the creation and study of sources. The research theme will certainly have definitive implications in this process, since only with a defined theme is it possible to elaborate scripts and seek out those who will be interviewed. It is the object of research which will define the nature of the study, whether it will be properly historiographic or not. Other chapters in this book provide more detailed examples of the mobilization of oral history, for example, in historiographical research for training of teachers who teach/taught mathematics in Brazil, and in interventional work in mathematics classrooms. Having argued that the analysis of sources is part of the mobilization of oral history in academic projects, several theoretical resources can support this analytical exercise. This book also contains a specific chapter on foundation theories which have been implemented in our studies, which allow us to state that the oral history that we defend is, from this point of view, ecumenical, as it does not limit the ways in which the researcher attributes meaning to the experiences reported by interviewees. It is worth pointing out that when developing historiographic research in oral history, the sources created from the interviews should not be considered sufficient. Oral sources bring a dimension which is usually absent from sources classically used in historiographical work but are essential both to reintroduce subjectivity into the scientific domain and to examine the ways in which meanings have been attributed to experiences which have been lived or are thought to have been lived, showing that the past holds many other futures beyond what has actually occurred. But oral sources can, and often do, beget support, supplementation, and/or confirmation which can only be given by other available resources, such as the so-called official sources. Interviews with former teachers can tell us how they faced daily classroom challenges, their doubts, their anguish, and their accomplishments, relationships, and failures, school textbooks, student attendance records, school buildings, uniforms, minutes of meetings, as well as any other information about school life. These interviews would also reveal whether such information is in line with the perspectives that manifest in the voice and memory of those education agents. Researchers would then face the task of understanding these different perspectives, creating their own points of view from these multiple data sources. 

Oral History and Mathematics Education: Motivations and Examples

Other chapters in this book will discuss and present themes and research exercises in mathematics education developed according to the oral history approach. In this chapter which was meant as a general introduction to this theme, we present some thoughts which

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may serve as an initial example of how oral history has been used, by studying the formation of mathematics teachers from a historiographical point of view, and the motivations and results of this study. 

Teacher Education Programs

Conceptions and practices are terms which are viscerally interrelated. Conceptions are informed by practices and become explicit and are rearticulated in these practices, thus generating and reinforcing conceptions. The study of the practices and education of teachers would therefore seem to be a fundamental ingredient to be discussed in a course in which the main objective is educating teachers who will explore practices and conceptions—many of which are already established and, therefore, comfortable, demanding revaluation, and destabilization. However, the study of practices will be much more effective if based on concrete subjects who are close at hand, actors in the school process (teachers, students, administrators, fathers, mothers, neighbors, and friends) as well as actors excluded from the school process. Understanding the mechanisms of inclusion and exclusion in the educational system, paying attention to educational moments occurring outside the school, learning about methods of management and evaluation (past and present) of the so-called educational systems, reconstituting didactical approaches to sound out their pedagogical bases, and understanding how strategies for the education of teachers and students developed historically—all of these are important themes in teacher education programs. Oral history can play a fundamental role in such discussions, as it focuses on real subjects close to the students’ contexts (which makes it possible for the future teachers to see that the practices being discussed are not from an abstract realm which is distant and foreign to them). If we conceive of pedagogical theories, didactic methods, and the philosophical, political, and axiological aspects of education as elements of abstraction, the proposed themes— specifically, oral history as an essential resource for the study of these themes—could link the disciplines classically known as “pedagogical disciplines” to concrete situations, seeking articulation between theory and practice. 

From the Center to the Periphery

Just as mathematics history suffers from an already well-analyzed Eurocentrism, the history of mathematics education is characterized by a similar fascination: a fascination with the center. To talk of mathematics teacher education in Brazil has necessarily implied speaking of the creation, in 1934, of the first Brazilian university, the University of São Paulo (USP). To speak of distinguished former mathematics professors almost always implies talking about those who have been portrayed as the great heroes of educational issues: legislators; the erudite of Colégio Pedro II,1 in Rio de Janeiro (the nation’s capital until the mid-twentieth century); the well-known authors of textbooks and teaching manuals, with their innumerable editions; and the great pedagogical movements. Although we acknowledge the importance of all these, we point to the need to reconfigure the history of mathematics education to include, more clearly and properly, the anonymous people who contributed significantly to building this complex system of mathematics education in Brazil.

 Created in 1739 and transformed into the Institute of Secondary Education in 1837, and assuming the role of educating the elite politicians and intellectuals of the country, the Colégio Pedro II became the model for other teaching institutions, which followed its educational guidelines. 1

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In this nation of continental proportions which, “discovered” by the Portuguese at the beginning of the sixteenth century, would only begin to build a national public education system in the second half of the twentieth century, those who study the history of Brazilian mathematics education generally overlook the importance of the teachers who, through a great effort, taught in the more remote, inland regions, the arid heartland, far from the larger cities and better-known institutions. We seldom inquire about the education of these ­teachers who, unable to attend any of the few universities established by the mid-twentieth century, taught mathematics classes2 in secondary schools, which were only effectively created with the amplification of the few existing schools in the 1950s. What are the consequences of this Brazilian model of teacher education? In a nation in which, at the beginning of the twentieth century, only 10% of the population resided in urban areas—a reality which would change abruptly in the 1970s during a period of economic miracle and mass rural exodus—what were the practices in mathematics education, and where were they rooted? We have attempted to fill some of these gaps using the resources of oral history, writing a complementary record of the Brazilian history of mathematics education. One of the projects developed with this intent traces the education and practices of mathematics teachers in upstate São Paulo, in a community whose culture is labeled “backward” or “hick” (in Portuguese, caipira). In Brazilian sociology, the term “rustic” is associated with the universe of traditional cultures of people from the countryside, and is conceived of as approximating the expression “country culture.” In order to designate sociological aspects of people who inhabit the interior of the state of São Paulo, the adjective caipira is used to express a manner of being, a way of life that differs from that of the big cities, without reference to race. Thus, we inquired into the education and activities of caipira teachers at a moment in history in which institutions of higher education were located far from the interior, in the capital and the larger cities. The process of educating caipiras was always seen as an heroic mission; studying was seen as relatively unimportant compared to the project of moral regeneration, since the caipira was conceived of as being contrary to study, accommodated and not very ambitious, as nature provided him/her with the basics for subsistence with little effort. A few studies in the history of education which focus on the period of the First Republic3 point to the perverse manner in which this regeneration—deemed necessary due to the disinterest of the caipira with respect to school—was implemented. A study by Martins (2003)4 examining the education and activities of teachers and students in rural schools in the 1950s and 1960s (decades which preceded the rural exodus which intensified in the 1970s) showed that little has changed. Despite the great interest and attention with which rural communities viewed the possibility of and need for formal education (precisely the opposite of what was stated in the documents of educational policies at the time and in much of Brazilian sociology), rural schools followed the criteria of providing classes only until the third year, impeding rural  University-level teacher education programs in Brazil began to be constituted with the creation of the first university (University of São Paulo, USP, in 1934), although for a long time, they followed not the USP model, but the model of normal schools, which were secondary education institutions established in the nineteenth century to educate elementary school teachers. 3  The period called the “First Republic” in the history of Brazil includes the years from 1889 (the year the imperial regime was changed to a Republican regime) to 1930. 4  Martins’ (2003) study was an Oral History and Mathematics Education Research Group investigation. It analyzed the depositions of teachers, students, and administrators in a geographical region which included 34 municipalities. 2

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children from completing the primary school, composed of four to five years of academic study. Only the School Groups (Grupos Escolares)—institutions created at the end of the nineteenth century, generally in the cities, and extinct in 1975—offered classes through the “fourth year.” Obligated to work to contribute to the family income, children often abandoned their studies after completing the third year of primary school. The discourse of equal opportunity so intensely proclaimed by teachers and republican politicians was not complemented with efforts to guarantee equality of means. Teachers who taught in the rural schools were educated in the normal schools5 where no specific guidelines regarding the habits, experiences, and clients in the rural areas were discussed. Together with this inadequate preparation, the natural difficulties faced by these teachers are to be considered. Accustomed to urban life—even in small cities—they were precipitously taken to live on farms and ranches where, faced with the difficulties of transportation and lack of teaching materials, they often needed to turn to the rural landowners6 for assistance. In addition to these difficulties, teachers had to endure the extreme vigilance of the teaching inspectors (who, restricting themselves to the role of technical control, gave final examinations to the students and made surprise visits to check the class diaries) and the lack of pedagogical support, leading them to seek transfers to the nearest School Group as soon as possible. Painting an idyllic picture, in their depositions, of the docile peasant with healthy habits and an authentic, natural way of being—the “good savage”—rural school teachers, missing life in the city, used the countryside as a “land of passage,” leaving engraved in their memories of this short passage a vivid distinction between the comforts of the city in relation to the difficult life in the country. Their depositions allow us to comprehend which the discourse of modernization, of well-being and access to consumer goods, which they spread so arduously and widely, acted to favor the disconnection of the caipira from his/her roots, being one of the driving elements of the rural exodus. What was taught in the countryside, teachers of that time proudly point out, was the same as what was taught in the city, judged to be fundamental for the entry of country-dwellers into urban life once they had abandoned the country. This abandonment was expected to occur sooner or later given the precarious conditions of rural life—conditions which were always emphasized, directly or indirectly, by the teachers to their students. Just like the official documents of the time, the teachers’ discourses confuse the need to minimize differences in life conditions between rural and urban areas with the need to remove people from the country. Regarding the relation between school, family, and community, Martins (2003) showed that it was common for the community itself to take responsibility for building the schools. Refusing to simply submit to the conditions they were given, they developed strategies to guarantee access to school knowledge, attempting to bypass the precariousness of the system offered by the State. These initiatives, however, had the undesirable side effect of contributing to the government’s exemption from responsibility for the maintenance and development of rural school centers. The study also emphasized the decisive participation of the family in the formal education of their children. In this sense, Martins detected, with respect to the teaching of mathematics, that the initial processes of counting and rudiments of basic operations were often developed in the home. This interest manifested by the rural community leads us to rethink the image of the caipira as being averse to formal education.

 The available literature shows us that these institutions, when they began, were really more assistentialist than pedagogical in character. 6  This disposition of the farmers/ranchers, often characterized as “good will,” masked political interests, such as the need to prepare the farmhands to form groups of voters, and the desire to increase the value of their property. These same practices were also common during the First Republic in Brazil (1889–1930). 5

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The education of the teachers who taught mathematics in the elementary schools appears to have been full of gaps, but the contents to be taught were quite varied, albeit not always covered completely. Few teachers were able to address mathematical themes in any great depth—to teach as far as the study of fractions and percentages, for example. There was much emphasis on the decimal system (restricted almost entirely to counting), the four basic operations, simple problem solving, and memorizing multiplication tables from two to nine. The greatest difficulties with respect to the teaching and learning of the “problems” were not in the operations to be solved, but in the comprehension of their content—by teachers as well as students—and what strategies to use to solve them. Greater difficulties were encountered, however, when dealing with the operation of division (mainly by three algorisms) and the inability to memorize the multiplication tables. Students were expected to “know” (usually meaning “memorize”) the tables by the third year, which would increase their agility in working with the operations and progress with the “problems.” Even in the face of these difficulties and gaps, the rural students who continued their studies evaluated their education, in their depositions, as adequate. But the fact that some overcame, at least in part, the conditions of inequality obscures what, for the majority, was a determining factor in their decision to abandon their studies. In the 1950s, the rural educational system was not yet showing signs of debilitation (rural schools today are rapidly becoming extinct in the interior of the state of São Paulo). At that time, they served as a kind of test for initiation into the profession,7 when the rural exodus and the general phenomenon of urbanization in the state of São Paulo were not yet causing great concern, and when, therefore, agrarian reform was a political discourse or theme of specialized economic and agronomical investigations, an expansion of secondary education into the interior began. This led to a wave of construction of new school buildings. This expansion did not include elementary schools, however, but only secondary schools. How, then, could teachers be provided for these schools built in cities in the interior, far from the capital and other large cities in the state where there were teacher education programs? Out of this urgent need to educate secondary school teachers emerged the Campanha de Aperfeiçoamento e Difusão do Ensino Secundário—CADES (Campaign for the Improvement and Diffusion of Secondary Education), created in 1953. Despite providing emergency treatment for the problem, the lack of teachers to fill the needs of the schools continued to be a pressing matter almost a decade later (Baraldi, 2003). The literature specialized in education and the history of education in Brazil has remained unjustifiably silent with respect to this teacher education campaign. For the interior of the state of São Paulo—and the rest of the country, as well—the CADES played a much more important role in teacher education than the faculties of philosophy. The effective implementation of the CADES courses began in 1956 (Baraldi, 2003).8 These intensive courses, designed to prepare aspirants for the sufficiency examinations. Conferred upon those who passed the title of secondary teacher and the right to teach wherever there were no teachers educated in Faculties of Philosophy. They generally lasted 1 month (January or July, which coincided with school vacation periods in Brazil) and were designed to address the deficiencies of the teachers—who were, until then, laypersons or people with a level of education far below that desired—with respect to pedagogical aspects and specific contents of the disciplines they would be teaching or were perhaps already teaching. The material studied regarding the CADES allows us to note the strong humanistic  It is significant to note that the teachers interviewed for Martins’ (2003) research considered the beginning of their careers to be when they began teaching in urban areas. 8  Baraldi’s study was prepared in 2003 in the Oral History and Mathematics Education Research Group, as a doctoral dissertation. 7

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thrust of the Campaign, not infrequently revealing a radical and explicit intervention of the ideology of the Catholic Church. Soon after having obtained their certificates through the CADES, the mathematics teachers would be confronted with a radical change in the teaching programs which were part of the new mathematics movement. At the end of the 1960s, the first institutions of higher education began to appear in the interior of the state of São Paulo, rendering unnecessary the courses and exams offered by the CADES, given that their function as facilitators of teacher education was insufficient to ­transform them into a vehicle for continuing education—which was not yet being talked about much at the time.9 The CADES examinations lost their validity definitively in 1971. Continuing on the subject of the education of teachers living far from large urban centers, our work in oral history has led us to study the region of Nova Alta Paulista, located in the far western region of the state of São Paulo, the last region to be colonized by the white man. In a study by Galetti (2004), depositions from mathematics teachers who taught in Nova Alta Paulista during the period from 1950 to 1970 were collected and analyzed. They are pioneer teachers, from the “older regions” of the state of São Paulo who, upon assuming their teaching positions in Nova Alta Paulista, became integrated into the expansionist project in effect in the region since the first decades of the twentieth century. They became actors not only in the educational and cultural process but also in the process of colonization of a region distant from the large cities. At the time, transportation and communication systems were precarious in this region—where small- and medium-sized coffee plantations predominated. These employed a workforce composed mainly of Italian and Japanese immigrants who had not yet mastered the Portuguese language. The teachers who came to the region had different levels of education, as can be seen clearly in their depositions. Into a universe dominated by elementary school teachers educated at the normal schools and in the CADES, came teachers educated at universities in the state capital. In these central universities, these teachers had interacted with experienced professors, among them foreigners and renowned researchers in the field of mathematics10 who used a wide variety of references and had access to good libraries. Upon their arrival in Nova Alta Paulista, they encountered a new “passage” which demanded a “transposition” of the culture they dominated. They needed to be creative, as the new situation required practices for teaching mathematics which could not be those used in the “older” regions. The mathematics teaching that developed in the region was thus peculiar, surprisingly similar and different compared to the “older” regions of the state. Although the mathematical contents traditionally taught in the schools were maintained, new practices were created aiming to transpose them for the new space. Colonization implies acculturation: the teachers from the large urban centers elaborated and maintained a “project” for teaching mathematics for the children of the migrants and immigrants. They were teacher-colonizers. The agents of the process of colonization cannot be limited to those who worked and cultivated the land. A large group of workers, including workers in the field of education, is also considered colonizers. The ancient Latin verb colo means taking care of. Mathematics teachers take care of teaching of mathematics: they are colonizers.  Systematic continuing education projects aiming to provide courses to train in-service teachers emerged, with great emphasis, in the late 1970s, oddly at the time of the most significant decrease in teachers’ salaries. The origin of these courses thus appears to have been the need to hold the teachers responsible for the precarious conditions of the educational system which, in the end, justified their low salaries. 10  With the creation of the University of São Paulo, in 1934, many foreign professors traveled to Brazil on missions to work in São Paulo, and later, Rio de Janeiro. Among them were the mathematicians Albanese, Fantappiè, Weil, Dieudonnè, Zariski, Grothendieck, Desartes, as well as many others (Garnica, 2007; Pires, 2006). 9

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The bandeirante11 and the teacher (albeit in a different manner compared to the bandeirantes of São Paulo and the Jesuit priests who engaged in a deadly battle in the eighteenth century) conciliated their practices, each participating in his own way in the project of colonization. While some planted and commercialized, others built roads and railroads, while still others contributed to educational and cultural development projects. These teachers dedicated themselves not only to the teaching of school mathematics, but also to the development of an expansionist project with respect to the teaching of mathematics. They guided the schooling of the colonizers’ children, seeking to provide them with access to traditional secondary schools and universities in the capital. Thus, a cycle of exclusion similar to that mentioned above with respect to the rural schools is repeated: unequal conditions—in this case, the lack of schools and universities in the region—led the children of the colonizers to seek schools in the large urban centers, a distortion which can, to a degree, also be explained by the need to differentiate a given group from the others. To facilitate the participation of students from the rural areas in school activities, the teacher-colonizers of Nova Alta Paulista went so far as to take them into their own homes. They also cared for the complementary education of the other teachers in the region, taking advantage of their personal friendships with nationally renowned mathematicians to organize classes and meetings with them during visits to the region, during which another kind of interaction was established: a sort of dialogue between the academic mathematics known and produced in the large urban centers and that taught in the schools by the teachers of the region. These brief considerations regarding the education of the “caipira teacher,” examined from the perspective of oral history, reveal distinct tones in these investigations of themes which have been privileged by the history of Brazilian education and, more specifically, the history of Brazilian mathematics education. 

About the References

This chapter was written from the experiences of authors who were members of a research group which has used oral history in the field of mathematics education since the beginning of the year 2000. The productions of this group, however, have been almost entirely elaborated and disseminated, to date, in Portuguese, the mother tongue of the researchers.12 There are, however, some reference texts which can be suggested to the reader for a first approximation with oral history practices in other areas of knowledge.13 Such texts have served as initial reference to Brazilian researchers who, little by little, have constituted an oral history for mathematics education. The objective of this oral history—which certainly dialogues with the ways in which oral history has been used in other areas such as anthropology, history, sociology, and social and cultural studies—is to meet the specificities, objects, and research themes of those who study the teaching and learning of mathematics, in its numerous subareas, such as the history of mathematics education, the philosophy of mathematics education, ethnomathematics, problem solving, mathematical modeling, the use of technologies in education, etc.

 Bandeirantes were members of one of the main groups of Brazilian colonizers in the eighteenth century who carried out, in a country with considerable territory as yet unexplored, expeditions inland to appropriate natural riches and to found cities. 12  For those who read Portuguese, we suggest the website www.ghoem.org where all dissertations produced in GHOEM can be found. 13  These texts, even those not mentioned in the body of this chapter, are included in the list of references, presented below. 11

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References Baraldi, I.  M. (2003). Retraços da Educação Matemática na região de Bauru (SP): uma história em construção. Tese de doutorado em Educação Matemática. IGCE, UNESP, Rio Claro, Brazil. Bolívar, A., Domingos, J., & Fernandes, M. (2001). La investigación biográfico-narrativa em Educación: enfoque y metodologia. Madrid, Spain: La Muralla. Cohen, J. J. (2000). A cultura dos monstros: sete teses. In T. T. da Silva (Ed.), Pedagogia dos monstros: os prazeres e os perigos da confusão de fronteiras. Belo Horizonte, Brazil: Autêntica. Dunaway, D. K., & Baum, W. K. (Eds.). (1996). Oral history—An interdisciplinary anthology. New York, NY: Altamira Press. Galetti, I. P. (2004). Educação Matemática e Nova Alta Paulista: orientação para tecer paisagens. Dissertação (Mestrado em Educação Matemática). IGCE-UNESP-Rio Claro. Garnica, A. V. M. (2003). História Oral e Educação Matemática: Do inventário à regulação. Zetetiké, 11(19), 9–55. Campinas: FE/CEMPEM. Garnica, A.  V. M. (2004). História Oral e Educação Matemática. In M. de C.  Borba & J. de L. Araújo (Eds.), Pesquisa Qualitativa em Educação Matemática (pp. 77–98). Belo Horizonte, Brazil: Autêntica. Garnica, A.  V. M. (2005). Um tema, dois ensaios: método, História Oral, concepções, Educação Matemática. Tese (Livre-docência)—Departamento de Matemática—FC, Universidade Estadual de São Paulo, Bauru. Garnica, A.  V. M. (2007). Resgatando Oralidades para a História da Matemática e da Educação Matemática Brasileiras: a Faculdade de Filosofia, Ciências e Letras da Universidade de São Paulo. Revista Brasileira de História da Matemática. Rio Claro: SBHMat, 07, 247–279. Jenkins, K. (2005). A história repensada. São Paulo, Brazil: Contexto. Joutard, P. (1983). Ces voix qui nous viennent du passé. Paris, France: Hachette. Joutard, P. (1999). Esas voces que nos llegan del passado. Buenos Aires, Brazil: Fondo de Cultura Económica. Langemann, E. C. (2000). An elusive science: The troubling history of educational research. Chicago, IL: The University of Chicago Press. Martins, M. E. (2003). Resgate histórico da formação e atuação de professores de escolas rurais da região de Bauru (SP). Relatório de Iniciação Científica. FAPESP/Departamento de Matemática-UNESP-Bauru. Miorim, M. A., & Miguel, A. (2001). A constituição de três campos afins de investigação: história da Matemática, Educação Matemática e História & Educação Matemática. Teoria e Prática da Educação Maringá: UEM, 4(8), 35–62. Pires, R. da C. (2006). A presença de Nicolas Bourbaki na Universidade de São Paulo. Tese. Doutorado em Educação Matemática. Pontifícia Universidade Católica de São Paulo. Portelli, A. (1991). The death of Luigi Trastulli and other stories: Form and meaning in oral history. Albany, NY: State University of New York Press. Portelli, A. (2003). The order has been carried out: History, memory and meaning of a Nazi massacre in Rome. New York, NY: Palgrave. Portelli, A. (2016). História Oral como arte da escuta. São Paulo, Brazil: Letra e Voz. Santamarina, C., & Marinas, J. M. (1994) Historias de vida e historia oral. In J. M. Delgado & J. Gutiérrez (Eds.), Métodos y técnicas cualitativas de investigación en ciencias sociales (pp. 257–285). Madrid, Spain: Editorial Síntesis. Thompson, P. (1988). The voice of the past. New York, NY: Oxford University Press.

Chapter 2 Oral History and Mathematics Education: Historiographical Research Maria Laura Magalhães Gomes

Abstract: This chapter focuses on historiographic research in mathematics education carried out with the use of the oral history methodology. To this end, a panorama of the investigations of the Oral History and Mathematics Education Group is presented. The group works in a project intended to clarify how the training and activities of mathematics teachers have been developed in Brazil within different institutions, various school levels, and completely different times and environments. A general outline of the history of education and of teacher formation in Brazil is also presented. Key theoretical and methodological issues are discussed as to the participation of oral history in history research, and particularly in the history of mathematics education. Finally, this chapter presents a summary of what the research about mathematics teachers’ training and activities in Brazil have been showing through time. At the same time that the oral history methodology highlights and values peculiarities and individualities, this methodology has enabled the perception of characteristics which are common to many scenarios such as the lack, the urgency, and the discontinuity of training activities of mathematics teachers in Brazil.

Keywords: Oral history · History and historiography of mathematics education · History of training and activities of mathematics teachers in Brazil In this chapter, we approach historiographic research in mathematics education using the oral history methodology. We focus on the contributions that this methodology has brought to the field of history of mathematics education in general. In Brazil, research on the history of mathematics education has invested in various fields, which can be summarized in the following main topics: history of the training and practice of mathematics teachers; history of mathematics teaching, at any level; history of teaching artifacts related to mathematics education; history of cultural groups involved in mathematics education; and history of the scientific-academic production in mathematics education.

M. L. M. Gomes (*) Federal University of Minas Gerais, UFMG, Belo Horizonte, Brazil © Springer Nature Switzerland AG 2019 A. V. M. Garnica (ed.), Oral History and Mathematics Education, History of Mathematics Education, https://doi.org/10.1007/978-3-030-16311-2_2

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A lot of research has been developed focusing on the pathways of the training and practice of mathematics teachers in Brazil over time, among which the research which has been conducted for more than 15  years by the research group História Oral e Educação Matemática (Oral History and Mathematics Education). Extremely diversified, the works produced by the group, based on oral history methodology, converge on the task of “elaborating a (historical) map of the training and practice of the mathematics teachers in Brazil” (Garnica, Fernandes, & da Silva, 2011, p. 241). At first glance, from a practical standpoint, the studies developed in this specific project may not be perceived as relevant for teacher training, which is imperative for delivering education to large portions of the population. What would history contribute, anyway? What contributions would it offer to the problem of teacher training, in mathematics, in a country such as Brazil? Indeed, when we reflect on the contributions of history, the consideration of Lopes and Galvão (2005) that history, from a pragmatic perspective, is useless knowledge, which has long renounced the past judgment and the attempt to draw lessons for the present and the future, is highly appealing. However, it has certainly been contributing to a small measure to the understanding of what the present is insistently portraying as a problem. Even today, Brazilians insist on highlighting the problem of training and monitoring the professional development of mathematics teachers, in view of the large school population in an immense and extremely physically, politically, economically, socially, and culturally diverse country. The indelible role of the past in the constitution of the activities of human groups, often emphasized by eminent historians,1 seems crucial to the field of mathematics education, as well as teacher training and practice. What happens today in school is undeniably the result of what happened before, and this seems an unquestionable reason for this “before” to be known and understood. The objective of this chapter is to present a panorama of the historiographical investigations of the Oral History and Mathematics Education Group, whose objective is to investigate the circumstances of the training and practice of mathematics teachers in Brazil, through a large project without a definite termination date. All such studies, developed according to the oral history methodology, focus on different institutions and academic levels, at several different times and locations. In these preliminary considerations, it is important to draw attention to an aspect which may go unnoticed when focusing on the training of mathematics teachers: in general, while addressing this theme, we tend to forget that the initial years of schooling is the time when the fundaments of mathematical knowledge are taught and learnt. Alongside the mother tongue, this knowledge is the main component of the literacy process and is part of the celebrated “reading, writing, and arithmetic” triad. Therefore, the quest for understanding the historical dimensions of teaching mathematics cannot disregard general knowledge about the past of elementary levels of Brazilian education, commonly referred to as elementary education or primary school. Teachers at these early education levels usually do not have specialized academic backgrounds for each of the subjects they teach, and this circumstance makes their training completely diverse from that of teachers at the more advanced levels of education, often referred to as secondary schools. In Brazil, since the 1930s, educational legislation establishes that

 Examples are the Frenchman Jules Michelet (1798–1894), who said that those who want to be circumscribed to the present will not understand the present, and the British scholar, Eric Hobsbawm (1917–2012), who has drawn attention to the past as a permanent dimension of human consciousness, an indispensable ingredient in the composition of institutions, values, and other standards of society (Hobsbawm, 1998). 1

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these teachers must have degrees in area-specific courses. They obtained teaching certificates in their respective areas of interest. For instance, to teach mathematics, teachers have to complete a teaching degree in mathematics. We stress this fundamental difference in teacher training for primary education (currently encompassing the first 5 years of elementary school) and secondary education (last 4 years of elementary school and 3 years of high school) because in the present chapter we will refer to both types of teachers as teachers of mathematics. Thus, although the teachers who teach in the first 5 years of elementary school are not exclusively “mathematics teachers,” as they teach other disciplines at the same time, this is how we will designate them. A general presentation of the Brazilian educational system and its connections to mathematics education over more than 500 years would require more space than is available in this chapter. However, we will dedicate a small portion of the text to familiarize the reader with some aspects we consider important for the understanding of the historiographical research conducted with the oral history methodology.2 Thus, the following is a general picture of the development of education in Brazil and the trajectory of the institutional training of mathematics teachers. 

Historical Panorama of the Formation of (Mathematics) Teachers in Brazil

In the long interval between the time that Brazil was a Portuguese colony (1500–1822) and the present day, there have been many changes in the educational opportunities offered and in the objectives, contents, and the ways mathematics has been taught. In relation to the period when Brazil was a colony of Portugal, in general, it can be said that there were few initiatives for schooling the population. Of particular interest are the actions of religious institutions, especially the schools founded and maintained by the Jesuit priests. After the declaration of independence (1822), the Portuguese prince D. Pedro I became the first emperor of the country. Only five years later, on October 15, 1827, the first specific law regarding national public education was issued. This established elementary schools, called infant schools, in all cities, towns, and populated areas. Mathematics was present in these infant schools, which were meant, after all, to teach “reading, writing, and counting.” At the time, for the first time, universal education of the populace was treated as a social right, and there was an explicit concern with issues regarding teacher training (Saviani, 2009). In 1834, the imperial government transferred responsibility for infant schools to the administrations of the provinces which made up the Brazilian state. This change made it impossible to establish a school system capable of serving the entire population. As the political, social, economic, and cultural conditions of the provinces were widely diverse, they started to pass specific educational legislation and tackle their educational mandate in many different ways. After this new legislation, the imperial provinces began to create the so-called normal schools, which were institutions whose objective was the preparation of teachers to work in infant schools. Even though such institutions had trained very few schoolmasters and operated in an erratic way, they constituted the model of teacher training for primary education in the period between 1827 and 1890. Normal schools contemplated in their curricula the contents to be taught by the future teachers in infant schools, and therefore included mathematical knowledge (Tanuri, 2000).

 There are many general studies about the history of education and the history of mathematics education in Brazil which can be consulted: see, for example, Miorim (1998), Romanelli (2005), Saviani (2007), Valente (1999, 2003), Garnica and Souza (2012), Veiga (2007), and Lopes, Faria Filho, and Veiga (2003). 2

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Institutions specifically geared to preparing teachers for teaching secondary level were only implemented in Brazil in the 1930s.3 Most teachers working in elementary or high school, during the Empire and First Republic (1889–1930), were professionals who had graduated from higher education institutions which existed in the country at the time,4 as well as people who had secondary degrees, such as those from normal schools, and selfeducated individuals, who were authorized to teach through proof of knowledge ascertained through examinations (Vicentini & Lugli, 2009). In secondary schools which were attended only by very small portions of the population, mathematics was mainly taught by engineers and military personnel (Valente, 2008). The first Brazilian university course to prepare teachers for secondary mathematics education was established in 1934, at the University of São Paulo, in the Faculty of Philosophy. Soon after, in 1935, the University of the Federal District was established in the city of Rio de Janeiro, at the time the capital of the country, where a course to train mathematics teachers, to teach in secondary schools, was also created. In the decades which followed, new institutions were gradually created which offered undergraduate courses in mathematics. Slowly but steadily institutions and training courses for primary and secondary teachers of many types were created until the 1960s. After that, the pace steadily increased.5 Such courses trained teachers who taught mathematics in many places across Brazil. The study of the ways in which these teachers have been trained and worked, from the 1930s to the present, is the core of the investigations of the main project of the Oral History and Mathematics Education Group, which will be discussed later. The project is concerned with investigating aspects related to mathematics teachers of different regions, with different sociocultural backgrounds, and teaching levels, at different times, with the use of different types of textual composition in the research reports, the participation of researchers with various levels of education, and the use of various types of documents (Garnica, Fernandes, & da Silva, 2011). The hallmark of this type of research is the participation of oral sources, even though the work includes sources of various natures, such as institutional documents, legislation, photographs, meeting minutes, school publications, and textbooks. 

Oral History, History, and Historiography of Mathematics Education

The oral history methodology can be briefly characterized as the accomplishment of recorded interviews with people who participated in or witnessed past or present events and circumstances. Oral history was the “initial binding agent of a group of researchers interested in understanding the potential of speech and memory—core features of a method which not without debate, is called oral history for mathematics education” (Garnica et al., 2011, p. 231).

 In 1930, an armed movement took over the presidential government in a coup. The new government created a specific ministry to manage Brazilian education. The first minister in office was Francisco Campos. The government organized a national educational system and created a structure for secondary, commercial, and higher education throughout the country. From that moment on, national legislation for teacher training throughout the country was passed (Romanelli, 2005). 4  These were the first university courses created in Brazil, mainly Law, Medicine, and Engineering. 5  In the state of São Paulo alone, from 1930 to 2007, 73 cities created courses. From 2000 to 2007, 92 courses were implemented, with 88 Education Courses (“licensure”) (Martins-Salandim, 2012). 3

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In order to draw a historical map of the training and practice of mathematics teachers in Brazil, the researchers of the Oral History and Mathematics Education Group have resorted to testimonies of students, teachers, employees, authors of textbooks and teaching proposals, and education managers. Below, we briefly present a set of procedures common to the members of this group, which are also discussed in other chapters of this book. The interviews are based on a script of questions or topics, previously prepared by the researchers, which undergo three stages: recording, transcription, and textualization. Transcription converts spoken language, digitally recorded during the interviews, to reproduce it in written form. Textualization is an edition of the transcription, which “attempts to maintain the tone of the respondent and the interactions occurred in the dialogue between respondent and researcher as much as possible” (Cury, 2014, p. 184). There are several ways of textualizing; for example, “there are cases in which the researcher chooses to omit the questions and/or separate the speech by topics” (Cury, 2014, p. 184). It is possible, though less common, to create a literary text based on the researcher’s understanding of the statement, in an exercise called transcreation. The purpose of textualization is not simply to make the narrative artistic or aesthetically pleasing, but to enable the researcher to appropriate what has been said to produce meanings. Textualization supports the analyses of researchers, although it is recognized that the interview itself, the recording, and the transcription have repercussions in the analyses (Garnica et al., 2011). The consideration that there is a possibility of distance or approximation between the texts and what each interviewee acknowledges to have said in the interviews leads to another of the group’s regular activities—the negotiation with participants in relation to the actual text which will be used in the investigation. Thus, interviewees have access to the recording, transcription, and textualization of the interviews and can propose changes, addendums, restrictions, and exclusions of excerpts, so that an agreement can be reached. It is essential to emphasize that the final text resulting from this agreement is the work of at least two authors6—the interviewee and the interviewer. After agreements have been reached, interviewees sign a waiver in which they grant researchers rights over textualizations, and, in most cases, authorize the disclosure of their names. From the theoretical-methodological standpoint, it is fundamental to discuss some important aspects of the participation of oral narratives, later converted into written statements, in historiographic work. The memoirs of the respondents, materialized in the documents resulting from the interviews, along with other sources, are transformed by the actions of the researchers into historical narratives. This set of actions constitutes the historiographical operation (Garnica et al., 2011). Michel de Certeau, in his book The Writing of History, emphasizes that to regard history as an operation is to attempt to understand it as the relation between a place, procedures of analysis, and the construction of a text: a social place, a practice, and writing. To view history as an operation will be to attempt to understand it, in a necessarily limited way, as a relation between a place (recruitment, medium, profession, etc.), analysis procedures (discipline) and the construction of a text (literature). It is to admit that it is part of the “reality” it deals with, and that this reality may be appropriated “as a human activity,” “as a practice.” In this perspective, I would like to show that the historical operation regards the combination of a social place, of scientific practices and writing. (Certeau, 2010, p. 66, italics in the original)  In a broader sense, such a text is the result of many other personal, social, and cultural instances linked both to the respondent and the researcher. In the case of textualization conducted in the research of GHOEM, there is indisputably a significant repercussion of the group and their activities. 6

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Ricoeur (2007) also uses the term “historiographical operation,” but his approach distinguishes three components in the operation which are not chronologically different— archiving, explanation/understanding, and representation7—emphasizing the interpretive character of all three. Simply put, archiving consists in the construction and use of files, of any kind, with traces of the past; explanation/understanding consists in identifying the way what has been documented is linked, answering the question “why?,” without precedence of one mode over another; representation is the activity of showing a part of the past as a text. For Ricoeur, the whole historiographical operation is characterized by interpretation. Hence, the selection and production of documents, the elaboration of comprehension and explanation, and the writing of the text are marked by the individuals involved. Moreover, as a result of the historiographical operation, a hermeneutical cycle is established, since the writing of historians, when made public, can be a part in the archiving and provide material for the explanation/understanding of other historiographic operations. In addition, such texts are always subject to new interpretations and revisions by other historians. The historiographical operations conducted in the scope of the mapping of the training and practice of mathematics teachers in Brazil, undertaken by the Oral History and Mathematics Education Group, are in line with Ricoeur’s historiographical operation perspective, in relation to the recognition of the interpretative mark and the opportunity for new interpretations of the results of the representation phase, which is of the reports of investigations produced in writing. Oral history accounts express the appreciation of subjectivity in modern historiography. Beatriz Sarlo (2007) states that, for some time now, historians and social scientists have turned their attention to the so-called “ordinary” respondents, “due to the realization that they not only follow the social routes established, but also engage in negotiations, transgressions and variants” (Sarlo, 2007, p.  17). Changes in the perspective of history are often accompanied by changes in research sources. This “subjective shift” requires an “ideological and conceptual reordering of past society and its actors, which concentrates on rights and on the truth of subjectivity” (Sarlo, 2007, p. 18), through which autobiographical sources acquire new value, particularly the testimonies used in oral history. In the case of the research project which is the focus of the present chapter, the relevance attributed to the accounts of respondents who are not always considered in the research in history of education is evident. For a long time, such research was almost exclusively based on written documents from the past. Above all, we refer to mathematics teachers, who constitute the majority of those interviewed by the Oral History and Mathematics Education Group. Much has been said about the central role of the subjective dimension in the oral history methodology. Portelli (1996) points to the impossibility of exorcizing subjectivity, stating that we must “identify rules and procedures that enable us to understand and use it to some extent” (pp. 61–62). He argues that if we can do it, subjectivity, rather than interfere, this “will be the greatest treasure, the greatest cognitive contribution derived from memories and oral sources” (p. 62). Portelli (1997) emphasizes that memory is intrinsically fragmented and partial. Thus, “wrong” statements are still psychologically “correct,” and this truth is as important as others considered reliable.  Ricoeur prefers not to designate the third phase or component as writing of history, “but of literary or record phase, when it comes to the mode of expression, representative phase, when it comes to the showing, the exhibition, the revelation of the historian’s intention considered in the unit of its phases, namely, the present representation of absent things from the past.” The author goes on to say: “Writing, in fact, is the threshold of language which historical knowledge has always transposed, distancing itself from memory to live the triple adventure of filing, explanation and representation. History is, from beginning to end, written” (Ricoeur, 2007, p. 148). 7

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Regarding the historiographic practices used in mathematics education, whose objective is to elaborate past constructions of teacher training, pedagogic movements, teaching practices, and materials, the consideration given to the individual brings about significant changes: A movement, a legislation, a specific teaching practice can no longer be seen according to a fixed view, but in their most diverse manifestations, in their appropriations by individuals. It is no longer the past “in itself,” but the past that is invented in the present and the various futures projected into various pasts invented by respondents. (Garnica, 2015, p. 185–186) When studied, narratives show that “very distinct appropriations of what could be conceived as ONE legislation, ONE movement, ONE practice” can coexist at the same time and in the same scenario (Garnica, 2015, p.  186). Thus, research using autobiographical narratives may not provide new information; however it is very important because it reveals peculiarities which cannot be unveiled without the use of such sources. Narratives originating from the use of oral history offer valuable contributions to the historiographical investigation related to our understanding of the values, motives, and feelings of respondents at different moments in the past (even the recent past) regarding mathematics education, especially the training and practice of mathematics teachers. The narratives present a particular view, under certain specific perspectives, of the individuals interviewed, as much as the content of written documentation is particular and specific. However: Due to the subjectivities at play when organizing the ideas of the research, oral sources warrant reflections of a different order, such as the condition of the individuals who share (therefore, construct and reconstruct) their memories in reports that meet the demands of the researcher, and not necessarily the free flow of memory. (Esquisani, 2012, p. 220) This shifts the focus to the all-important issue of the participation of the interviewer and all their support structure in the elaboration of all documents arising from the interview situation. By support, we mean the researcher, research group, the institutions to which they are linked, the studies conducted, and the pragmatic objective of the research, linked to the obtaining of degrees, diplomas, or other academic benefits. Academic research of historiographical nature with the use of oral history must be conducted with the admission of the impossibility of scientific neutrality and the awareness of the fractured nature of the knowledge constructed. In fact, such characteristics are by no means unique to this type of research, although it seems that the use of original oral testimonies as sources for writing history calls for more eloquent explanations. Therefore, it should be considered that the procedures integrating oral history methodology into historiographical investigations have crucial implications for the development of the interviews and elaboration of the resulting texts. Let us keep in mind, on the one hand, that interviewees do not recall or recount spontaneously, but respond, in specific occasions and in specific circumstances, to what is asked of them. Their account is therefore driven by the interview situation and the presence of the interviewer and the technological apparatus such as tape recorders and/or video cameras. At that moment and under those conditions, the narrative they produce comes from the nuances of their memory which emerge and are uttered in the language they deem appropriate to the context. A different time and another interviewer would give rise to other memories, other narratives. On the other hand, let’s think about the interviewers. In the case of the project of mapping the training and practice of mathematics teachers in Brazil, these are researchers, with a project of their own, who have specific knowledge, not necessarily tuned to those of the

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respondent. Janotti (2010), in commenting on the roles of interviewer and interviewee, reminds us of a third party involved, the public: When deciding to reveal their recollections, respondents conceive the content and form of the language, in view of certain hidden or evident purposes, and believe their account to be true, even if they deliberately disguise experiences. The interviewer (social scientist, reporter, lawyer, prosecutor and judge) has outlined a plan that should be met by the testimony obtained. They possess a specific and conjunctural knowledge that, the respondent might not possess. They have mastered techniques and methods, which need to be verified for the acceptance of the audience to which the work is intended. In general, there are not many coincidences between the worldview of the interviewee and the interviewer. (Janotti, 2010, p. 13) The discussion regarding the theoretical-methodological aspects involved in the historiographic works in mathematics education with the leading presence of oral history could be much more encompassing. Nonetheless, we believe that this chapter has covered the most relevant aspects of the understanding of the research project for mapping the training and practice of mathematics teachers in Brazil and of what they have revealed. 

Mapping the Training and Practice of Mathematics Teachers in Brazil: What Research has Shown

The development of institutional actions for training mathematics teachers in Brazil, as previously explained, began on a large scale in the 1930s. The greater need to offer formal education to a growing population and the urbanization of the country has continually led to the creation of diversified courses and programs to prepare teachers in many different places. Brazil is a continental nation with many distinct landscapes (climate, geography, vegetation) and economic, political, social, and cultural conditions. In addition, educational legislation has undergone many changes since 1931, when the decrees of Minister Francisco Campos represented the first effective attempt toward a nationwide organization of education, to this day. National regulations on educational levels, curricula, teacher training, and education management have been modified many times, and there are still different legislations regarding some of these aspects and the career in teaching at municipal and state level. There is great socioeconomic inequality and in human development, not only in the country as a whole but also within the same state or region.8 Inequality is also apparent in the offer of training opportunities for teachers. Thus, while in parts of the Southeast and South, the creation and implementation of courses took place in the 1930s and 1960s, where such courses have long been consolidated, in the Northeast, Midwest, and North, this development is more recent, and, as a result, the demand for academically qualified teachers is higher in some locations to this day. Such characteristics of Brazilian education, although described briefly and incompletely, suffice to indicate an immeasurable spectrum of possibilities for a project of historically investigating the training and practice of mathematics teachers. The word “mapping” was used in referring to the project of the Oral History and Mathematics Education Group, and the  Brazil is divided into 26 states and the Federal District, where Brasília, the capital, is located. The states and the Federal District are grouped into five regions according to physical, economic, political, and cultural similarities (North, Northeast, Midwest, Southeast, and South). 8

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subheading of this chapter “Mapping the Training and Practice of Teachers Who Teach/ Taught Mathematics in Brazil” alludes to the idea of drawing up a map. A map or geographic chart is a graphical and conventional representation of data relating to terrain, a country, and a territory. A map locates, informs, and describes, and, by extension, the word is used to mean list, picture, catalog, and relationship. Research on the training and practice of teachers of mathematics, in various contexts and periods, discussed in this chapter can be seen as studies involving the science, art, and techniques for producing “maps.” This project shares with cartography the importance of the notion of scale. Its aim is not to study the training and practice of mathematics teachers in Brazil as a comprehensive and single theme. The investigations are based on the variation of scales: teacher training and practice are researched in multidimensional scenarios. The notion of scale, a loan made by history from cartography, architecture, and optics (Ricoeur, 2007), supposes the existence of an external referent, the territory which the map represents. Different scales allow for a change in the level of information depending on the level of organization, but there is no opposition among them due to the proportionality involved. If the territory remains the same, moving to a larger scale implies a loss of detail, complexity, and information. As the mapping of the training and practice of mathematics teachers in Brazil has a historical nature, it is relevant to consider the type of modification achieved in the association between scale variation and history. For Ricoeur (2007, p. 222), “we do not see the same things when they are larger or smaller, in large or small characters.” What we see are different things and “not the same threads that are visible when we change scale” (Ricoeur, 2007, p.  221), but connections which may not be revealed on a larger scale. Focusing on very diversified scenarios and several points of view, human, political, economic, sociocultural, and academic, allows us to see connections which are not evident when the training and practice of mathematics teachers in Brazil are approached under a generic and all-encompassing perspective, which displays the situation throughout the country simply as a mirror of the first instituted models of teacher training. The mapping project focuses on the concept of history advocated by Levi (2014): “History is the science of general questions, but local answers.” According to that historian, historiographical work must preserve “the particularities; preserve the place—understood as a specific situation” (p. 1). The project has shown that the general question about the training and practice of mathematics teachers in Brazil has different answers which are significantly conditioned by the location where the research is conducted. However, by highlighting and valuing peculiarities and singularities, oral history methodology has made it possible to detect many differences. The accounts and their interpretation, combined with the analyses of varied written documentation and the comparison of the results of several investigations, also help detect common ground. In fact, there are aspects which stand out in the accounts of respondents in many surveys. In diversified scenarios, three common characteristics are revealed, which we will further comment: scarcity, urgency, and discontinuity of the training efforts of mathematics teachers in Brazil. Within the scope of the mapping project, approximately 50 masters and doctoral theses were completed up to 2018. To a large extent, these investigations focus on specific institutions and programs for initial or in-service training of mathematics teachers, with the aim of contributing to the history of the creation and development of academic institutions and programs. Other works, however, are not aimed at any particular instance of academic training for teachers, but to investigate more broadly, from accounts of teachers of certain locations or regions, how they were qualified to teach. In this case, it is about identifying the opportunities teachers had to prepare themselves to teach mathematics, not necessarily in conventionally

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structured courses, such as the undergraduate degree offered by a college or university. One example is the doctoral thesis of Baraldi (2003), which points to the existence of such a model for teacher training. These were the courses sponsored by Cades-Campanha de Aperfeiçoamento do Ensino Secundário (Campaign for the Improvement of Secondary Education), created by the Brazilian federal government, in 1953, to disseminate and improve higher education. In many states during the 1960s and 1970s, it promoted numerous courses during school holidays whose goals included the preparation of those individuals who were already teaching for a sufficiency test which would enable them to teach in several areas, including mathematics. This was an opportunity for emergency training provided by the ­campaign’s pedagogical missions, which brought teams of teachers and education professionals to remote corners of the country (Fernandes, 2011). The existence of such courses resulted from the scarcity of institutions to meet the demand originated by the expansion of secondary schools occurred in the 1950s. Nowadays, Cades no longer exists, but similar efforts have been implemented to issue certificates enabling a large number of teachers who, without having taken any specific teaching courses in mathematics, have worked in regions far away from large urban centers, in which university education has been consolidated since before the 1970s. Many such actions are undertaken because, in addition to the growth in population and number of schools, over the years, several legislative bills by the federal government have altered the requirements for teaching, which now demand that teachers have teaching certificates before they can work in elementary or high schools. However, research has shown that even in the case of institutions which created courses to train mathematics teachers in a conventional way (Almeida, 2015; Martins-Salandim, 2012; Paiva, 2016), the creation and the subsequent development of the courses aimed at meeting an urgent need, due to the great expansion of schools which took place in Brazil, mainly since the 1960s. Thus, the interviewees’ accounts in the research of the mapping project have eloquently shown the signs of the scarcity and urgent need of training for mathematics teachers. However, especially in the emergency actions similar to Cades, described in works such as Cury (2011), Fernandes (2011), and Gonzales (2017), the third component has not been functional. In general, such actions take place during school holiday periods and are not intended as preparation for examinations for a temporary certification, as was the case of Cades courses. Their purpose is to confer teachers in office with a permanent certification which will legally entitle them to teach. These programs are developed through agreements between municipal or state governmental agencies and universities. Universities or higher education institutions are responsible for the courses and instructors, and municipal or state governments bear the costs of transportation, lodging, and meals of the student teachers, as well as provide the physical facilities necessary for the training activities. However, as Cury (2014) points out in his work in the state of Tocantins, the courses have often been conducted in an improvised manner, with inadequate instructional materials and insufficient physical structure. Such characteristics are also evident in the investigations of Fernandes (2011) in the state of Maranhão and Gonzales (2017) in Mato Grosso do Sul. The scenario is completed by the discontinuity in teacher-training programs, which are interrupted by changes in government, as respondents in Gonzales’ (2017) research declared, “governments change, agreements are cancelled, financial support withdrawn” (p. 446). For a new government, it does not make sense to continue projects which would strengthen the image of the previous government, rather than their own. Besides the two components previously mentioned— scarcity and urgency—discontinuity is another hallmark of teacher-training actions in mathematics in Brazil, as shown by their transient and provisional nature.

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In relation to the simultaneous presence of regular teacher-training courses and remedial programs offered during school vacations, a relevant issue becomes evident; the discourse about the equanimity of conditions for the training of mathematics teachers coexists with the disparity of opportunities for such training. Even though, apparently, there are many similarities, regarding the curricula of such courses (which could lead to the assumption that the training conditions are equal), accounts clearly show an unbalanced consummation of such actions, even within the same Brazilian state. It becomes clear that, as far as teacher training is concerned, the process creates centers and fringes which are continuously and dynamically related. The mapping project also reveals a fundamental contribution of the narratives; shedding light on the way teachers receive and put into practice, over time, the guidelines originating from pedagogical movements or educational reforms. For example, Souza’s (2011) research in a grade school9 in the countryside of the state of São Paulo showed the appropriations made by teachers regarding the pedagogical guidelines of the movement generally known as New School, which had repercussions in Brazil between 1930 and 1970. New School strongly opposed memorization. However, although the practices narrated by the teachers show how their memories registered, at the time of the interview, their adherence to several characteristics of the New School sourcebook, the accounts also show that, for them, the memorization of the multiplication table was nonnegotiable. Thus, while the pedagogical discourse was contrary to rote learning, it continued to occur in primary school classrooms because teachers believed that memorizing the table was indispensable for learning arithmetic. Historiographical research with the oral history methodology has generally emphasized that educational agents “attribute meanings and take ownership of the various influences to which they are exposed, gauging their actions on such attributed meanings” (Souza & Garnica, 2013, p. 371). The attentive examination of mapping research reveals the absence of a unique identity for mathematics teachers in Brazil. In fact, considering all the political, economic, social, cultural, environmental, educational, psychological, and human diversity which emerge from the interviewees’ accounts, we must agree on the impossibility of considering a broad-brush identity of mathematics teachers in Brazil. Teaching mathematics in Brazil is a singular activity, due to the many and diverse levels and types of schools. The investigations of the mapping project comprise elementary, secondary, and higher levels of education but have also focused on different modes such as those offered by vocational schools, rural schools, and schools providing adult education. Research has also shown that the variability also occurs in regard to the academic preparation of teachers, due to the great disparity in the conditions among states and regions. However, it has become clear from the investigations that the training and practice of mathematics teachers needs to be seen as a continuous process which goes beyond the experience in academic institutions and professional training guidelines. There are important aspects of the personal lives and professional paths of teachers which are paramount for understanding their mathematics teaching practice, about which normative documents, school textbooks, and pedagogical guidelines cannot reveal any information.

 Grade schools were schools created in the beginning of the Republic with the consolidation of previously isolated schools. Such institutions emerged in São Paulo in 1893 and gradually spread to other states, especially in urban areas. They were schools responsible for primary education, taught over four years, and offered physical, moral, and intellectual education. The grade schools were legally extinguished in Brazil in 1971 (Souza, 2004). 9

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Accounts resulting from the oral history methodology provide precious data for investigations. Morais and Garnica (2016) point to the role of space in the different regions of Brazil in the training and practice of mathematics teachers. They raise questions related to the possibilities and characteristics of the different life circumstances of teachers: Is training in the North of Brazil the same as training in the Northeast, South, Southeast and Midwest? Is training in one state the same as training in another, if educational indicators and assumptions about education are so diverse? Is the training a teacher receives on the coast the same as in the countryside, the inland? Do different climates, terrain and vegetation matter? Is the training in the suburbs the same as in urban centers? Is the education one receives in a rammedearth construction, without proper school materials the same as that received by those studying in a room with modern and appropriate physical structures and material? (Morais & Garnica, 2016, p. 91) It seems relevant to investigate how spatial experiences impact the training and practice of teachers. In the work of Morais (2017), the accounts of teachers of mathematics, in the state of Rio Grande do Norte, are intensively marked by the countryside and urban areas, and also different cities. Many accounts show that teachers moved from their hometowns, where they spent their childhood and adolescence, to other places so that they could continue their studies. It is also apparent that some respondents have a very strong bond with some locations, revealing a sense of belonging which may reverberate in their performance. The light shed by these questions on the impacts of spatial experiences contributes to the perception of the complexity of the training and practice of mathematics teachers. They clearly show that this process is not limited to academic institutions or courses and is carried out “in everyday life, on the street, in the cafeteria; at home, with family, with friends; in games, in the park, trips and tours” (Morais & Garnica, 2016, p. 93). In short, it is a process characterized by incompleteness. The possibility of understanding how personal and professional meanings are interwoven in the research on the training and practice of mathematics teachers in Brazil, conducted through historiographical investigations with oral history, brings new challenges for the mapping project.

References Almeida, S. P. N. de C. (2015). Um lugar: muitas histórias: o processo de formação de professores de Matemática na primeira instituição de ensino superior da região de Montes Claros/norte de Minas Gerais (1960–1990). Tese de Doutorado em Educação. Belo Horizonte, Minas Gerais: Universidade Federal de Minas Gerais. Baraldi, I.  M. (2003). Retraços da educação matemática na região de Bauru (SP): uma história em construção. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Acesso em 7 de junho de 2018. Disponível em: http://www2.fc.unesp.br/ghoem/index.php?pagina=trabalhos.php Certeau, M. D. (2010). A escrita da história. Rio de Janeiro, Brazil: Forense Universitária. Cury, F. G. (2014). Histórias da formação de professores, em Goiás e no Tocantins, em duas pesquisas empreendidas pelo Ghoem. In A. V. M. Garnica (Ed.), Cartografias contemporâneas: mapeando a formação de professores de Matemática no Brasil (pp. 179–193). Curitiba, Paraná: Appris. Cury, F. G. (2011). Uma história da formação de professores de Matemática e das instituições formadoras do estado do Tocantins. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Acesso em 7 de junho de 2018. Disponível em: http://www2.fc.unesp.br/ghoem/index.php?pagina=trabalhos.php

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Esquisani, R. S. S. (2012). Entre percursos, fontes e sujeitos: pesquisa em educação e uso da história oral. Educação e Pesquisa, 38(1), 217–228. Fernandes, D. N. (2011). Sobre a formação do professor de Matemática no Maranhão: cartas para uma cartografia possível. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Acesso em 7 de junho de 2018. Disponível em: http://www2.fc.unesp.br/ghoem/index.php?pagina=trabalhos.php Garnica, A. V. M., & Souza, L. d. (2012). Elementos de história da educação matemática. São Paulo, Brazil: Editora UNESP. Garnica, A. V. M., Fernandes, D. M., & da Silva, H. (2011). Entre a amnésia e a vontade de nada esquecer: notas sobre Regimes de Historicidade e História Oral. Bolema, Rio Claro, 25(41), 213–250. Garnica, A. V. M. (2015). O pulo do sapo: narrativas, história oral, insubordinação e educação matemática. In B. S. D’Ambrosio & C. E. Lopes (Eds.), Vertentes da subversão na produção científica em Educação Matemática (pp. 181–206). Campinas, São Paulo: Mercado de Letras. Gonzales, K.  G. (2017). Formar Professores que ensinam Matemática: uma história das Licenciaturas Parceladas no Mato Grosso do Sul. Tese de Doutorado em Educação Para a Ciência. Bauru, São Paulo: Universidade Estadual Paulista Júlio de Mesquita Filho. Acesso em 7 de junho de 2018. Disponível em: http://www2.fc.unesp.br/ghoem/ index.php?pagina=trabalhos.php Hobsbawm, E. (1998). Sobre história. São Paulo, Brazil: Companhia das Letras. Janotti, M.  L. M. (2010). A incorporação do testemunho oral na escrita historiográfica: empecilhos e debates. História Oral, 13, 9–22. Levi, G. (2014). O trabalho do historiador: pesquisar, resumir, comunicar. Tempo, 20, 1–20. Disponível em http://www.scielo.br/pdf/tem/v20/pt_1413-7704-tem-20-20143606.pdf Lopes, E. M. S. T., & Galvão, A. M. d. O. (2005). História da Educação. Rio de Janeiro, Brazil: DP & A. Lopes, E. M., Faria Filho, L. M. & Veiga, C. G. (Orgs.). (2003). 500 anos de educação no Brasil. Belo Horizonte, Minas Gerais: Autêntica. Martins-Salandim, M. E. (2012). A interiorização dos cursos de Matemática no estado de São Paulo: um exame da década de 1960. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Acesso em 7 de junho de 2018. Disponível em: http://www2.fc.unesp.br/ghoem/index.php?pagina=trabalhos.php. Miorim, M.  A. (1998). Introdução à história da educação matemática. São Paulo, Brazil: Atual. Morais, M. B. (2017). Se um viajante… Percursos e histórias sobre a formação de professores de matemática no Rio Grande do Norte. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Acesso em 12 de junho de 2018. Disponível em: http://www2.fc.unesp.br/ghoem/index.php?pagina=trabalhos.php. Morais, M. B. d., & Garnica, A. V. M. (2016). Da duração situada: um estudo sobre historiografia, espaço e Educação Matemática. Revemat: Revista Eletrônica de Educação Matemática, 11, 77–95. Paiva, P. H. A. de. (2016). Entre as memórias do Campo das Vertentes: uma história da formação de professores de Matemática da Fundação de Ensino Superior de São João del-­Rei (FUNREI) no período de 1987 a 2001. Dissertação de Mestrado em Educação. Belo Horizonte, Minas Gerais: Universidade Federal de Minas Gerais. Portelli, A. (1996). A filosofia e os fatos: narração, interpretação e significado nas memórias e nas fontes orais. Tempo, 1(2), 59–72. Portelli, A. (1997). O que faz a história oral diferente. Projeto História, 14, 25–40. Ricoeur, P. (2007). A memória, a história, o esquecimento. Campinas, São Paulo, Brazil: Editora da UNICAMP.

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Romanelli, O. (2005). História da educação no Brasil. Petrópolis, Rio de Janeiro, Brazil: Vozes. Saviani, D. (2007). História das idéias pedagógicas no Brasil. Campinas, São Paulo: Autores Associados. Saviani, D. (2009). Formação de professores: aspectos históricos e teóricos do problema no contexto brasileiro. Revista Brasileira de Educação, 14(40), 143–155. Sarlo, B. (2007). Tempo passado, cultura da memória e guinada subjetiva. São Paulo: Companhia das Letras/Belo Horizonte: Editora UFMG. Souza, L. A. d., & Garnica, A. V. M. (2013). As matemáticas modernas: Um ensaio sobre os modos de produção. Relime, 16(3), 369–393. Souza, R. F. de. (2004). Lições da escola primária. In D. Saviani et al. (Eds.), O legado educacional do século XX (pp. 109–162). Campinas, São Paulo: Autores Associados. Souza, L.  A. (2011). Trilhas na construção de versões históricas sobre o Grupo Escolar Eliazar Braga. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Acesso em 8 de junho de 2018. Disponível em: http:// www2.fc.unesp.br/ghoem/index.php?pagina=trabalhos.php Tanuri, L. M. (2000). História da formação de professores. Revista Brasileira de Educação, 14, 61–88. Valente, W.  R. (1999). Uma história da matemática escolar no Brasil (1730–1930). São Paulo, Brazil: Annablume/FAPESP. Valente, W. R. (2003). A disciplina Matemática: etapas históricas de um saber escolar no Brasil. In M. A. T. Oliveira & S. M. Ranzi (Orgs.), História das disciplinas escolares no Brasil: contribuições para o debate (pp.  217–254). Bragança Paulista, São Paulo: EDUSF. Valente, W.  R. (2008). Quem somos nós, professores de Matemática? Cadernos CEDES, 28(74), 11–23. Veiga, C. G. (2007). História da educação. São Paulo, Brazil: Ática. Vicentini, P. P., & Lugli, R. G. (2009). História da formação docente no Brasil: Representações em disputa. São Paulo, Brazil: Cortez.

Chapter 3 Oral History and Mathematics Education: The Dialogue with/ among Different Theoretical and Philosophical Perspectives Marcelo Bezerra de Morais and Filipe Santos Fernandes

Abstract: As a result of the effort to produce and articulate several narratives, the research in mathematics education which mobilizes oral history has encouraged dialogue with and among several theoretical and philosophical perspectives and developed investigations characterized by conceptual and procedural plurality. In this text, starting from examples of research in mathematics education, we attempt to debate two scenarios which demonstrate different ways of fostering interdisciplinary dialogue in the investigation process: in the first scenario, the object/ process under investigation guides the choice of the theoretical/philosophical perspectives mobilized by the researcher. The conceptual and procedural constructions are a consequence of the issues which arise from the narratives created/gathered for the research. In the second scenario, the mobilization of different theoretical/philosophical perspectives enables multiple glances at the object/process under investigation and develops understandings which lead to a multiplicity of aspects which characterize the object/process in its complexity. These scenarios show how oral history methodology in mathematics education research, when open to several theoretical/philosophical perspectives, enables links with different forms of research, thus emphasizing a practice in which dynamism and heterogeneity become the core of the investigation.

Keywords: Theorization · Interdisciplinarity · Philosophical foundation, Research methodology Introduction For more than 15 years, the Research Group of Oral History and Mathematics Education (GHOEM, Grupo de Pesquisa História Oral e Educação Matemática) has relied on the diversity of theoretical, philosophical, and methodological references to produce its research. Although oral history has preserved some conceptual and procedural unity and

M. B. de Morais (*) State University of Rio Grande do Norte, UERN, Mossoró, Brazil F. S. Fernandes Federal University of Minas Gerais, UFMG, Belo Horizonte, Brazil © Springer Nature Switzerland AG 2019 A. V. M. Garnica (ed.), Oral History and Mathematics Education, History of Mathematics Education, https://doi.org/10.1007/978-3-030-16311-2_3

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stability within the Group, particularly in the scope of historiographical research, the Group seeks to examine how the issues raised and the objects/processes investigated make it possible, with every new project, to outline conceptual foundations, methods, and criteria for analysis, revealing a practice in which dynamism and heterogeneity are central for research. However, this intention requires breaking up with perspectives of production of knowledge which advocate the applicability of a certain theory or use a set of systematically and previously fixed regulations and procedures which determine the paths to be followed by the researcher. Instead, the Group advocates the elucidation of philosophical and theoretical alliances, assessment of the limitations of theories and methods employed, and continuous evaluation of the results obtained to serve as guidelines for the Group’s production. Therefore, the idea is to embrace an attitude of continuous review of the procedures adopted, thus allowing mobilization of conceptual references which are not defined beforehand, but rather arise from the conjunction of a set of narratives—philosophical, theoretical, memorialistic, historic, and literary, among others—designed to produce other narratives in a creative process in which the understanding and discourse about a certain topic are a guide, rather than a starting point. In this sense, oral history enables us to build scenarios which clarify this intention even further. When we use and articulate different narratives, either orally produced particularly through interviews or gathered via other methodological procedures, such as investigations involving archives, books, autobiographies, memorials, and diaries, among others, we create forms of investigation which are ethically committed to dynamism and heterogeneity, positively affirming their possibilities and boundaries. We are concerned with the knowledge produced and the subject matter and form of expression used to divulge them; the different paths that we followed and those that we abandoned; authorization and facilitation so that other researchers can build their considerations (even if they disagree with what we have produced and published); conceptual adjustments and clarification of obstacles and the effort to overcome them; participation of “sociological, anthropological, cultural, literary and pictorial aspects; … monsters, myths, ailments, famous and unknown authors, crime and punishment, metaphors, anxieties, anguish, sacred and profane references…” (Garnica, 2008, p. 112). This movement which we call theorization “is characterized as the intention to produce knowledge through a systematized account of experiences. When we theorize, we tell a story, we produce meaning and create objects in one direction” (Silva & Viola dos Santos, 2012, p. 126). Theorization is the process through which we nourish ourselves with ideas, procedures, and approaches which help us make sense of the questions we face. When theorization is understood in this manner, it is the result of reading, experiences, visualizations, points of view, quotations, impressions, discussions, and dialogue. It is the process through which we present the meanings and the objects that we have built, and that have shaped the foundation of such construction, while providing paths for other readers, equipped with their perspectives and filters, to produce their own meanings and make a critical evaluation of the objects that we have created. Surely, theorization is both the result of an “argumentative logical and formal composition (if… then, if… then) and an overwhelming set of chaotic points of view” (Garnica, 2015, p. 12). It is the result of individual rationalization, as well as of collective formation and negotiation. Therefore, we try to work toward theorization: always open to elements which help us develop and speak about what we intend to achieve and how, using the subject matter and our own peculiar forms of expression. This text is an attempt to demonstrate how the production and articulation of different narratives, made possible via oral history, enable the configuration of a variety of exercises in theorization, intentionally fostering the dialogue with/among different theoretical and philosophical perspectives. In order to do that, we will discuss the movements of theorization of four works in mathematics education which adopt oral history as their methodology, divided in two scenarios as described below.

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Theorizations in Oral History and Mathematics Education

Starting from the conceptions briefly outlined above, we selected four works in mathematics education developed using the framework of oral history as investigation methodology. These provide good examples of how oral history enables the development of research of a broader nature and the inclusion of multiple theoretical and philosophical perspectives. In their movements of theorization, these investigations mobilize different theoretical and philosophical references and promote interdisciplinary dialogue in order to understand the problems posed by the objects/processes under study. Thus, they mobilize authors and perspectives strongly connected to fields such as sociology, anthropology, history, philosophy, geography, linguistics, and mathematics education. These are the themes of the investigations discussed hereafter: (a) how mathematics teachers become the teacher that they are; (b) how the training and practice of a mathematics teacher took place in a specific type of school in Brazil, specifically technical schools geared to the rural and agricultural area; (c) the identity of a group of mathematics teachers who train other teachers; and (d) the qualification process of mathematics teachers in a specific Brazilian state. As a consequence of these theorizations, we introduce understandings which are sometimes new or different from those which are generally accepted in our field of knowledge and lead to other understandings and perceptions. These four research papers are presented herein in two scenarios which, in practical terms, show how the diversity of theoretical and philosophical perspectives has permeated the work with oral history. In the first scenario, the conjunction of investigation and different narratives determines the choice of theoretical and philosophical perspectives to explore concepts which are not generally approached by historiography, thus enabling approximation of research with fields such as sociology, anthropology, history, philosophy, geography, or linguistics. In the second scenario, the mobilization of different theoretical and philosophical perspectives allows us to examine the same object/process under different views, developing understandings which grant us access to different aspects which constitute the object/process. First Scenario: The Object/Process Under Investigation Guides the Choice of Theoretical/Philosophical Perspectives As we have pointed out, working with oral history enables the joining of different narratives either produced via interviews or gathered from archives, books, autobiographies, personal journals, and diaries, among other sources. This diversity often exposes a variety of intentions and ways to produce and record the past. This requires the researcher to dialogue with theoretical-­philosophical perspectives which account for this variety without neglecting the conceptual assumptions which subsidize the researcher’s historiographical activities. Therefore, the dialogue with different references arises from an open movement of understanding such narratives, thus allowing concepts to arise which seem significant when read together. This intentional position of the researcher shows concern with a historiographical exercise that does not fixate on previous, static conceptual filters but rather allows us to resort to other fields of knowledge which help the researcher develop understandings about the object/process under study. To demonstrate this theorization method, we will examine the investigations of Martins-­Salandim (2007) and Morais (2017). In both works, which used oral history as methodology, we notice the approximation of theoretical/philosophical perspectives to deal with concepts arising from narratives produced and/or gathered by the works. This creates the need to develop understandings when such narratives are analyzed not through their peculiarities, but rather taken as a whole, creating what we could call an archive of narratives which nourish research.

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The first study, by Martins-Salandim (2007), produced seven narratives of teachers who worked in an agricultural school in the countryside of the state of São Paulo (Brazil). The objective was to investigate the training and practice of mathematics teachers in the agricultural vocational school.1 Agricultural teaching is integrated into the Brazilian education system with the purpose of providing technical qualification in agriculture. Although it is a regular type of education,2 Martins-Salandim sensed a certain disrepute of this type of teaching/institution in the narratives collected for research; such debasement varies depending on economic, urban, industrial issues and even on changes relating to agricultural practices, influenced by different contexts. Therefore, the author points to marginalization as a key element to understand the processes the respondents underwent while working in these schools. In order to study these indications, Martins-Salandim reviewed the concept of marginalization developed by historian Jean-Claude Schmitt (2006), who interprets this process— and as a consequence the labeling of individuals as marginal—as a series of dynamic events which change according to time and context. Martins-Salandim also resorted to Bertaux (1979) to endorse the perspective that social identity is also appraised according to a person’s profession. This led to the realization that rural/agricultural professions enjoyed marginal standing in comparison to urban activities. One of the key understandings pointed out by Schmitt (2006) on marginalization is that it is a dynamic notion which changes depending on the scale of observation and context. For this reason, when mobilizing this notion, Martins-Salandim (2007) demonstrated that evidence of marginality was perceptible in different ways in agricultural schools, in the 1970s. On the one hand, this marginalization is minimally detected, if at all, in agricultural institutions when compared to urban schools during a period in which activities relating to agriculture are also valued as part of the industrialization process. On the other hand, marginality is noticeably perceived in professional relationships inside these agricultural vocational institutions, in which those who did not work specifically with technical disciplines, as was the case of mathematics teachers, were considered less important. Based on these theoretical/philosophical perspectives, both in the oral interviews and studies on the topic, Martins-Salandim (2007) claims that: 1. The marginal position experienced in agricultural institutions was not accepted passively, as there are records of countless strategies developed inside the schools aiming at challenging this condition of neglect toward vocational agriculture courses. There are also records of efforts to expand the activity of professionals and institutions, thus making them more visible. 2. It is possible to notice the flexibility of the marginal position of agricultural schools in two different ways: when the institutions and wider contexts are examined and when the internal experiences within institutions are analyzed —this is what MartinsSalandim call external and internal marginalities, respectively. 3. Vocational schools and, more specifically, vocational schools geared to agriculture are also marginal topics in Brazilian research. Oral history is a methodology which has proven significant to promote this topic and many others (whether marginal or not) as it fosters contact with social actors who have experienced processes seldom examined and, as a consequence, hardly understood.  Vocational schools are very traditional institutions in Brazil. Their purpose is professional and technical training. There are special technical schools for several fields (i.e., agriculture, technology, industry, pharmaceutics, economics, navigation, etc.). Agricultural vocational schools rank among the oldest types. They were established in Brazil at the end of the nineteenth century. 2  In other words, vocational schools are geared to specific disciplines, but they are part of the regular school system. Students may choose to follow either a regular secondary course (similar to American high schools) or the vocational secondary course. 1

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The second research work described herein as an example of this scenario is that of Morais (2017). Morais attempted to constitute historical versions on the training of mathematics teachers in the state of Rio Grande do Norte, in the northeastern region of Brazil. To do so, Morais constituted and analyzed 12 narratives. In the second analysis, he mobilized 8 other memory narratives from former research. Therefore, Morais used 20 documents originated from interviews. All such sources were reviewed in an investigation based on oral history principles, along with other sources from written and pictorial files mobilized by the researcher. Morais (2017) initially tried to understand how migration movements made by several teachers while they studied and worked might influence these processes. With this in mind, Morais resorted to Massey’s (2008) studies. Massey understands space as the result of interrelations with/among everything which occupies and forms space, thus enabling the existence of multiplicities, and existing due to the continuous formation of multiplicities. Under this perspective, space is not fixed. It is rather an ongoing, constantly flowing production process. Morais (2017) realized that migration could be strongly linked to the teachers’ training processes, and tried to broaden his understanding of the relationship of individual/construction of subjectivity, dialoguing with Morin (1996, 2014), Deleuze and Guattari (1995), Guattari and Rolnik (2005), and Halbwachs (2006). Morin (1996, 2014) does not view the concept of “individual” as easy to understand, but rather as something which exists depending on the context, in which one must seek “interreactions,” the reciprocity among the whole and the parts. Morin questions how local changes affect the whole and how changes to the whole affect the parts. In turn, Halbwachs (2006) advocates the existence of a “collective memory,” considering that individuals are a set of collective events, intentions, and inspirations. We are always inserted in a context, a society, and a group, and that makes us remember certain things, know other things, and live in a certain way. That is to say, Halbwachs understands the individual as continuously shaped by the environment. Halbwachs’s understanding not only corroborates Morin’s ideas but also the ideas of Deleuze and Guattari (1995) and Guattari and Rolnik (2005). For these authors, subjectivity is not something which can be centralized or totaled on an individual person; it does not result in grouped, classified mass production. These authors advocate the strength of the conjunction “and,” that is, they advocate that the formation of individuals and of subjectivities is always flowing among many things, and it makes no sense to look for beginnings or ends. With a focus which lies particularly on the formation of mathematics teachers and defending the premise that we are multiple beings in different relationships with each other, with the environment, and with space/time, Morais (2017) started to question the crystallized notion of the concept of “mathematics-teacher formation,” generally treated as a process carried out intentionally and solely in schools. Even when approaching other aspects, research on teacher formation—at least the way we see it in Brazil—tends to focus on formal educational spaces, both for training and practice. The author found support on Foucault’s understanding of language (1999a, 1999b). For Foucault, it is natural to think that words are directly related to things: according to this naturalized method, words, or their combinations, always say the same things to everyone. However, he advocates that words or their combinations do not always say the same thing to everyone; they do not speak of the same objects, since each individual understands and attributes meanings based on their background, which is always subjective and unique. Therefore, different experiences cannot be unified in a homogeneous linguistic community. In this sense, the author helps us understand that there are power relations in communication processes, in which there will always be dominant discourses, permeated by forces which, with their intentions, can determine what is or is not plausible to approach when

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dealing with certain topics, and make it impossible to perceive certain nuances of these themes. This way, when there is a movement of stabilization of language, of crystallization of concepts, there is a trend to fix perspectives and understandings, making them limited and limiting. By understanding that words and their combinations speak about things or actions in the way they are mobilized and not in a previously determined manner, Morais (2017) claims that the expression “training of mathematics teachers” refers to subjective truths and truths negotiated by a collectivity. This term is related to streams which lead individuals to become teachers or not, and, if they do, these streams determine the possibilities and impossibilities to continue working as teachers; it speaks of truths which define individuals as the teachers they are or believe they are. The term speaks of ways of acting and communicating and about what teachers should and can communicate. Supported by oral and written documents, and working with the oral history methodology, the researcher claims that the process of training mathematics teachers is not restricted to formal institutions or only intentional training events. For the researcher, the formation of mathematics teachers takes place together with the process of individual growth, the construction of subjectivity. Moreover, this process is surrounded by conceptions about “formation of mathematics teachers” which change according to subjective and collective discourses and negotiations, and, when those change, teachers develop within other conceptions and, therefore, in different manners. Morais (2017) points out that it is only possible to understand and advocate this concept of formation if we consider subjective narratives. Taking these subjective narratives as a starting point, the author conceived the teacher-training process as a historic event: the countless movements of training experienced by individuals along their entire lives in different environments bred the teachers that each of them had become. As we can conclude from the considerations above, oral history plays a relevant role in research as it provides a variety of intentions and ways of producing and recording the sources to supply researchers with direction to choose the theoretical/philosophical perspectives which dialogue with the problems which arise from the research. In the works of Martins-­Salandim (2007) and Morais (2017), the movement of theorization suggests the need to treat concepts such as marginality, space, subjectivity, and language, and researchers are invited to look for references which help them express these dimensions of the objects/ processes investigated. The topics of these works are not frequently investigated by research in history of mathematics education in Brazil; thus oral history presents itself as a significant methodology since it enables investigation of historic objects/processes which are not part of a considerable set of research work which give precedence to urban centers. Oral history also brings a new meaning to the movements of training and practice of mathematics teachers in educational, social and cultural, political, economic, and spatial contexts in a scenario as diverse as Brazil. In these works, historiography is conducted differently from the historiography based on the “official history,” in a positivistic sense, a linear and teleological manner. It involves research practices which, mediated by the possibilities of reading the narratives that are created for each project, enable the problematization of great universal topics which have permeated historiography, such as truth, method, and causality.  The Second Scenario: Different Theoretical and Philosophical References are Mobilized in Order to Understand the Object/Process Under Investigation in a Multiple Sense In the previous scenario, we discussed how the oral history methodology makes it possible for narratives about the objects/processes under investigation to influence the choice of theoretical/philosophical perspectives. This choice, guided by the issues which arise from reading the set of narratives created for each research project, leads to mobilization of a new

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set of references, every time a new reading is made, both by the researcher and by the readers. This dialogue with different references can arise when different individuals have contact with the research—the researcher, a first reader, a second reader, and so on—but also when the same individual, usually the researcher, intends to demonstrate, by means of different theoretical/philosophical perspectives, the complexity of the object/process under investigation. This is the perspective of the second scenario. In the second scenario, the dialogue with/among different theoretical/philosophical perspectives is a key tool used by researchers to express the difficulty to approach the object/ process being reviewed under a single perspective, showing that different conceptual approaches make it possible to build distinct objects/processes, that is, the perspective of the observer depends on the conceptual lenses used, or rather, different conceptual lenses enable distinctive views. Hereinafter, we will present two works which, according to our understanding, are examples of the studies which compose this second scenario. Rolkouski (2006) developed research according to oral history parameters and produced narratives from interviews made with five mathematics teachers in an attempt to understand how these individuals became the mathematics teachers they are (or think they are) throughout their lives. Rolkouski’s interviews with these five teachers generated long narratives, a type of interview known as life story,3 in oral history. Therefore, Rolkouski ended up with five life stories of individuals who are mathematics teachers at the present. Initially, these life stories were examined from a sociological perspective, mobilizing the concepts of Elias (1994, 1995, 2001), Elias and Scotson (2000), Bourdieu (1996, 2003), and Bourdieu and Wacquant (1992). Based on the understandings of these authors and advocating that the social groups to which individuals belong markedly affect what the individuals are or can be, Rolkouski (2006) proposes that “becoming a mathematics teacher” does not depend on the respondents’ free will, and that teachers do not have unrestricted freedom to plan their practices as they wish, since the individual’s will is always restricted by regulating pressures of the configurations set forth among and about individuals. By dialoguing with the perspectives of Bourdieu and Elias, Rolkouski claims that social groups and spaces which teachers join have a logic which involves different variables and depends on configurations inside which individuals operate, according to certain rules and in which, despite the limitations imposed, these teachers have relative freedom of choice of their representations and practices. This allows us to understand why individuals make certain choices and how they interpret their lives. For this reason, Rolkouski (2006) stresses the need to carry out studies about mathematics education—particularly due to the fact that we deal with individuals—while focusing on understanding the different contexts in which they are inserted. However, it is not only the sociological focus which subsidizes the reading that Rolkouski (2006) performed of teachers’ lives. By adopting the theoretical/philosophical perspective of sociology, the author faces two distinct issues. The first one occurs when the researcher realizes that embracing sociological parameters involves using a comparative method of analysis. As Rolkouski’s initial concern was not to carry out a comparative analysis of the lives per presuppositions derived from sociology, the choice of his respondents did not comply with criteria suitable for a sociological approach, such as representativity of individuals in different categories such as gender, age, or social standing. The second problem arose when, still anchored on the sociological referential, the author decided to read the life stories individually, as he realized that, under this perspective, one investigates social trends which somehow explain the individuals’ lives; however, this was not among the objectives of the researcher nor what was deemed to be a study on life stories.  The modalities and procedures typical of oral history methodology are the theme of another chapter in this book. 3

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At this point, the work of Rolkouski (2006) points to the need to approach those lives by mobilizing other theoretical/philosophical perspectives, so that the life stories available would not be read within a comparative or explicative, but singular view. Therefore, having already tried a sociological reading, the author attempted a reading based on key parameters which rely on psychology, more specifically, in this case, on the works of Jerome Bruner (2001), who understands that narratives can be interpreted in a plausible and possible manner. Having made a psychological reading, Rolkouski starts a third reading, which he calls “scientific/literary,” as the life stories of the respondents are read side by side with literary texts, in an attempt to develop meanings for the enigmatic, contradictory individuals’ life stories. After these different analytical exercises, Rolkouski (2006) claims that the process of becoming the mathematics teacher the person is, or in a broader sense, the process of formation of a mathematics teacher, takes place within a complex network of variables, of which we can understand elements which continuously change depending on the theoretical/philosophical perspective used. Silva (2007), in turn, explores the constitution of the Center for Mathematics Education (CEM, Centro de Educação Matemática), a group which operated in the city of São Paulo (Brazil) between 1984 and 1997, providing specialized assistance and consulting services on mathematics education to schools and governmental institutions. Silva’s thesis is comprised of 15 narratives of people who participated in the group in several ways. Silva (2007) set out to study the identity of CEM.  However, this identity was not explored by means of one single design: Silva created four identities for CEM with mobilizations of different theories, of different positions and approaches. None of these identities legitimized or excluded the others. Each and every one of such identities, as they drew closer or farther, intertwined and reinvented the relations which constitute CEM, and created something exterior in which mathematics education, was viewed in the scope of the practices and members of CEM, as recognizable and strange to itself. So, how is the game which constitutes the process of assigning identities and which provisionally enables their existence played? To approach this issue, Silva (2007) carries out five analytical exercises to treat the identity of CEM: the first analytical exercise is based on the perspective of René Descartes; the second is based on the sociology of Émile Durkheim, George Herbert Mead, Peter Berger, Thomas Luckmann, and Norbert Elias; the third treats CEM as a community of practice and mobilizes the works of Etienne Wenger; the fourth conceives CEM as a resistance group according to the perspective of relationships of power and group identity of Michel Foucault; and the fifth and last exercise is a dialogue with the perspective of the Model of Semantic Fields (MCS, Modelo dos Campos Semânticos) which analyzes the identities as discursive constructions. It is under this last perspective of Model of Semantic Fields, based in Lins (1999, 2001, 2006), which the author discusses how the four first analytical exercises show that the idea of “identity” is a relational notion often associated with what is “essential” to something. By means of a dialogue with different theoretical/philosophical perspectives, Silva (2007) claims that the identity is not characterized only by what the identity of something is thought to be, but also by that which legitimizes this affirmation, producing understandings about identity which depend on the means of legitimization used, so that an affirmation about identity can be made. Thus, the researcher claims that treating identity involves dealing with processes of production of meanings, which enable the emergence of multiple senses and understandings about CEM: the identities of CEM. One last consideration about the treatment of the concept of “identity,” made by Silva (2007), which must be included herein, is that research which thematizes the policy of identities, even if based on theoretical/philosophical perspectives which attempt to deal with the idea of an individual detached from a unity, may risk legitimizing a structuring discourse of

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continuity of historic identities. Silva believes that different stances may be attained when presenting the constitutions of identities from the social groups themselves, an exercise in which oral history may play a relevant role. From what has been reported above, we can see how mobilization of different theoretical/philosophical perspectives helps researchers deal with the complexity of the objects/ processes they investigate. On the one hand, Rolkouski (2006), with his multiple fronts of analysis oriented by sociological, psychological, and literary references, demonstrates the difficulties of reading the lives of mathematics teachers in order to understand, by means of such reading, which experiences would justify the ideas, practices, and resistances expressed by them while narrating their paths of formation and practice as teachers. On the other hand, when Silva (2007) tries to outline the identity of a group which performed significant work to prepare mathematics teachers in Brazil, by means of analytical exercises, she treats identity as an incomplete, provisional construction which depends on the life references and on the manner of legitimization set in motion by the person who attributes this identity. These works adopted different theoretical/philosophical references when researchers realize that the objects/processes they set out to investigate are complex and would be understood in quite different ways if approached from distinct perspectives. This way, the works mentioned as examples include exercises of analysis which separately mobilize different authors and concepts, and lead to certain understandings. As a whole, however, these works allow researchers to exercise the limits of the conceptual presuppositions that they mobilize when developing their research. It must be pointed out that, as indicated by the authors of these studies, this type of exercise and the understandings they yield were possible given the nature of the data produced from the investigation methodology adopted, that is to say, oral history. Last but not least, the works promote a discourse of acceptance of different theories, other than those that each one mobilizes. Furthermore, the point is not only to accept diversity; these researches help us to promote diversity. 

Final Considerations

Beyond the scenarios presented, we can discuss how the works which mobilize oral history enable the articulation of different theoretical/philosophical perspectives which are utilized according to the circumstances, requirements, interests, and possibilities both of the project and the researcher. Overall, we can say that the possibility of articulating different theoretical/philosophical perspectives arises from a number of relevancies which are defined and triggered by the narratives mobilized by the researcher during the research. In the work with oral history, by means of production and/or gathering of different sources, we allow our collaborators to help us compose ways of uniquely understanding the objects/processes with which we work, and this uniqueness is reinforced in the theorizations made. However, a set of narratives is never homogeneous. For this reason, narratives may lead to distinct understandings. If some narratives are different from others and lead us to distinct and quite often conflicting understandings, we try to show such differences in dialogue with/among theoretical/philosophical perspectives to demonstrate the heterogeneous character of the objects we wish to understand. Therefore, the choice of a conceptual reference, even if made from the researcher’s life experiences, is also the product of readings which attempt to clarify concepts in the context of their development. Moreover, these scenarios help us realize how the dialogue with/among different theoretical/philosophical perspectives and with different movements of theorization conducted on research which uses oral history in mathematics education has made it possible to produce and advocate our conceptions about methodology, history, and historiography. On the

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one hand, this dialogue confirms the view of methodology as a trajectory. It is necessary to clarify the conceptions underlying the procedures adopted at all times, doubt the epistemological foundations and former life experience filters, and be careful with the ways in which the understandings and also the means to obtain them are publicized, so they can be revisited and reconfigured by any person. On the other hand, the dialogue with/among different perspectives makes it possible to demonstrate the production of historical knowledge which not only attempts to show by which means certain social configurations, objects, or processes manifest themselves in different times and spaces but to produce a problematizing historical knowledge in trying to understand provisionally and locally how certain social configurations, objects, or processes arise, circulate, and affect the present time. Finally, we point out that the methodology of oral history in mathematics education research, when open to several theoretical/philosophical perspectives, allows us to create links with other forms of research, experimenting with different treatments both from a conceptual and procedural point of view. This means that oral history provides not only a conceptual, methodological, or procedural basis for research in mathematics education but also an existential basis for the research in mathematics education which brings ethical, aesthetic, and political implications to the way research is conducted.

References Bertaux, D. (1979). Destinos pessoais e estrutura de classe: para uma crítica da antroponomia política (Destins personnels et structure de classe). Rio de Janeiro, Brazil: Zahar Editores. Bourdieu, P. (1996). Razões práticas: sobre a teoria da ação (Le sens pratique). Campinas, São Paulo, Brazil: Papirus. Bourdieu, P. (2003). O Poder Simbólico (F. Tomaz, Trad.). Rio de Janeiro, Brazil: Bertrand Brasil. Bourdieu, P., & Wacquant, L. J. D. (1992). Réponses: pour une anthropologie réflexive. Paris, France: Éditions du Seuil. Bruner, J. (2001). A cultura da educação (The culture of education) (M. A. G. Domingues, Trad.). Porto Alegre, Rio Grande do Sul, Brazil: Artmed Editora. Deleuze, G., & Guattari, F. (1995). Introdução: Rizoma. In Mil Platôs: capitalismo e esquizofrenia (A Thousand Plateaus: capitalism and schizophrenia) (Vol. 1, pp. 10–39). Rio de Janeiro, Brazil: Editora 34. Elias, N. (1995). Mozart: a sociologia de um gênio (Mozart: portrait of a genius). Rio de Janeiro, Brazil: Jorge Zahar. Elias, N. (1994). O processo civilizador: Uma história dos costumes (R. Jungmann, Trad.). Rio de Janeiro, Brazil: Jorge Zahar. Elias, N. (2001). The society of individuals. New York, NY\London, UK: Continuum. Elias, N., & Scotson, J. L. (2000). Os estabelecidos e os outsiders (The established and the outsiders). Rio de Janeiro, Brazil: Jorge Zahar. Foucault, M. (1999a). As palavras e as coisas: uma arqueologia das ciências sociais (The order of things: An archaeology of the human sciences). São Paulo, Brazil: Editora Martins Fontes. Foucault, M. (1999b). A ordem do discurso (The order of discourse). São Paulo, Brazil: Edições Loyola. Garnica, A. V. M. (2008). A experiência do labirinto: metodologia, história oral e educação matemática. São Paulo, Brazil: UNESP.

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Garnica, A. V. M. (2015). Ceci n'est pas un article: Impressões fragmentadas sobre arte e educação matemática. Zetetiké, 23, 15–39. Guattari, F., & Rolnik, S. (2005) Micropolíticas: cartografias do desejo (Molecular revolution in Brazil). Petrópolis, Rio de Janeiro, Brazil: Vozes. Halbwachs, M. (2006). A Memória Coletiva (The collective memory). São Paulo, Brazil: Centauro. Lins, R. C. (2001). The production of meaning for algebra: a perspective based on a theorical model of semantic fields. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school álgebra (pp. 37–60). London, UK: Kluwer Academic Publishers. Lins, R.  C. (2006). Characterizing the mathematics of the mathematics teacher from the point of view of meaning production. In Proceedings of the 10th International Congress on Mathematical Education (pp. 1–16). Copenhagen, Denmark: International Commission on Mathematical Instruction. Lins, R.  C. (1999). Porque discutir teoria do conhecimento é relevante para a educação matemática. In M. A. V. Bicudo (Ed.), Pesquisa em educação matemática: Concepções e perspectivas (pp. 75–94). São Paulo, Brazil: Editora Unesp. Martins-Salandim, M.  E. (2007). Escolas técnicas agrícolas e Educação Matemática: história, práticas e marginalidade. Dissertação de Mestrado em Educação Matemática. Rio Claro, São Paulo, Brazil: Universidade Estadual Paulista. Massey, D. (2008). For space. London, UK: Sage Publications. Morais, M. B. (2017). Se um viajante... Percursos e Histórias sobre a formação de professores de matemática no Rio Grande do Norte. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Morin, E. (1996). A noção de sujeito. In D. F. Schnitman & J. Schnitman (Orgs.), Novos paradigmas, cultura e subjetividade (New paradigms, culture, and subjectivity) (pp. 45–58). Porto Alegre, Rio Grande do Sul, Brazil: Artes Médicas. Morin, E. (2014). A cabeça bem-feita: repensar a reforma, reformar o pensamento (La tête bien faite - repenser la réforme, réformer la pensée) (21st ed.). Rio de Janeiro, Brazil: Bertrand Brasil. Rolkouski, E. (2006). Vida de professores de matemática: (Im)possibilidades de leitura. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo, Brazil: Universidade Estadual Paulista. Schmitt, J. C. (2006). L’Histoire des marginaux. In J. Le Goff (Ed.), La nouvelle histoire nova (pp. 277–306). Bruxelles, Belgium: Éditions Complexe. Silva, H. (2007). Centro de Educação Matemática (CEM): Fragmentos de identidade. Tese de Doutorado em Educação Matemática. Rio Claro, São Paulo: Universidade Estadual Paulista. Silva, H., & Viola dos Santos, J. R. (2012). Sobre teorização, estética ficcional e algumas aproximações entre o Modelo dos Campos Semânticos e a História Oral. In L.  C. Angelo et al. (Eds.), Modelo dos campos semânticos e educação matemática: 20 anos de história (pp. 110–128). São Paulo, Brazil: Midiograf.

Chapter 4 Oral History in Mathematics Education: Possibilities for Effective Intervention in Teaching Heloisa da Silva Destiny is the life of one man, history is the life of all of us. I want to narrate history so as not to lose sight of the destiny of any man. Svetlana Aleksiévitch

Abstract: By presenting and advocating the potential of oral history as a pedagogical strategy in mathematics education and in teacher formation, this chapter describes the aspects and possibilities of this approach in the treatment and problematization of issues relating to educational cultures and mathematics education. This chapter also presents a panorama of history of mathematics education in Brazil as a line of research and its contributions for training mathematics teachers. Finally, there is a debate of a theoretical and methodological nature about the key issues and strategies to be considered in interventions which use oral history in teaching; some examples are given as reference.

Keywords: Oral history · History of mathematics education · Teacher training · Pedagogical approach · Intervention in teaching Introduction The discussions presented in this chapter are anchored in studies and interventions, conducted since 2011, on the potential of oral history as a pedagogical tool in the processes of teaching and/or training mathematics teachers. The idea of a study linking research and interventions with oral history in teacher education and training emerged from the results and possibilities offered by research based on this methodology within the Group Oral History and Mathematics Education (Grupo História Oral e Educação Matemática—GHOEM). Since 2002, this research group has developed discussions in the field of mathematics education, especially regarding research on history of mathematics education. What contributions can oral history bring to teaching mathematics and training mathematics teachers? What contributions to the history of mathematics education can oral history bring to these processes? What is the potential of oral ­history and mathematics education research for these processes? These are the main issues examined by GHOEM on this topic, and we intend to address them in this chapter.

H. da Silva (*) University of São Paulo State, UNESP, Rio Claro, Brazil © Springer Nature Switzerland AG 2019 A. V. M. Garnica (ed.), Oral History and Mathematics Education, History of Mathematics Education, https://doi.org/10.1007/978-3-030-16311-2_4

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The Group Oral History and Mathematics Education has used oral history as a pedagogical approach in their research/intervention processes based on studies about the participation of history in the teaching of varied subjects and in the pedagogical problematization of school culture, particularly mathematical and mathematical education cultures (Miguel & Miorim, 2002, 2004; de Souza, 2011; Gomes, 2014). It is a type of approach which questions elements of school culture both within space and time, aiming to understand and problematize the present.1 In addition to constituting itself as contemporary or present-day history, historiographical research with oral history has shown that, contrary to common belief, oral sources are not produced to serve as a more concrete illustration of known facts. Nor do they serve as a complement to information obtained from other sources, to ensure a “fuller view of an event.” Oral history research allows us not only to approach the ways through which respondents constitute an event but also to understand the meanings which respondents attribute to experience and constitute these events as happenings. As pointed out by Alessandro Portelli (2009), “one of the things which sets oral history apart is that while more conventional history is mainly interested in what happened— why the massacre occurred that way, whose responsibility it is—oral history also asks another question: what is the meaning?” (p. 22). We believe that the difference between the focus of more traditional history and that of oral history is closely related to the difference which we establish between the interests of mathematics and of mathematics education regarding processes involved in training of mathematics teachers. While the former views such processes from a perspective of mathematics teaching—treating concepts as results of a set of axioms, methods, inferences, procedures, algorithms, and so on—the latter goes further, striving to promote training of students and/or future teachers as human beings and citizens and as professionals, encompassing different values and attitudes. The question “what is the meaning?” is key to understanding why we need oral history in teaching (mathematics) and teacher training processes. We need to focus the interest of young people and future teachers (of mathematics) on issues involving the meaning of (mathematics) education, and its multiple dimensions (epistemological, philosophical, political, social, etc.) in a culture of witnessing, identifying, and approaching people and situations within the scope of their reality and profession. We want students and future teachers, as professionals, to examine not only how school curricula in mathematics have historically been transformed into what they are today, and the relationships between that process and the cultural, social, and political aspects of mathematics education, but also their meaning for teachers, students, and other actors involved in these processes; that is, under which conditions educational processes result in certain curricula and what, if anything, can be applied and why? This way, we can help students engage in discussions such as of Base Nacional Comum Curricular—BNCC (National Core Curriculum) as currently proposed by the Brazilian federal government.2 Future teachers not only need to understand the aspects of the new mathematics movement in Brazil, its political and educational motivations, and the c­ urriculum proposal for mathematics education; but, above all, they must  As Miguel and Miorim (2004, 161) state: “When we dialogue with history ... we create new history, not only because we ask the past new questions, but also, and above all, because we incorporate new sources, new voices to the dialogue; we see new possibilities of establishing relationships between apparently disconnected and immeasurable discourses; because we impart new displacements, foci of discontinuity and new links of continuity.” 2  In Brazil, the so-called BNCC, acronym in Portuguese for National Core Curriculum, is a set of provisions, approved by the federal government, in 2018, which radically changed education in Brazilian schools. This set of provisions was hastily drawn up, and as a result, escaped the scrutiny of academic communities or researchers of education. 1

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have an opportunity to evaluate the meaning of this movement for the actors engaged in the mobilization of “modern mathematics” (Souza & Garnica, 2013) within multiple educational institutions. They might even conclude that curricular changes are complex and, for that reason, cannot be forcefully imposed. Although, according to Santhiago and Magalhães (2015), in Brazil, oral history research has been growing since the 1990s, specialized bibliography on oral history as a pedagogical approach has not followed the pace of the movement in, for example, the United States of America since 1972, England and France since the 1980s, or Argentina since 1996. Most of the references are from historians (Benadiba & Plotinsky, 2001; Ferreira, 2016; Llewellyn, 2016; Santhiago & de Magalhães, 2015; Schwarzstein, 2001), whose work has strongly inspired our research/intervention activities in mathematics education. In order to address the issues raised at the beginning of this text, this chapter will be divided into three sections. The first section consists of discussions about the place, identity, and affiliation of the history of mathematics education, in order to make some considerations about aspects of present-day history when the topic is (mathematics) education, as well as about some of the possibilities of such discussions in spaces devoted to the training of mathematics teachers. The second section aims to consider the potential of history in the teaching and training of mathematics teachers, their specificities and challenges, with the prerogative to discuss the role of memories and narratives of experiences in teaching and teacher education (in particular, through oral history). The third and last section includes some considerations about possibilities and contributions of oral history as a pedagogical approach for mathematics teachers’ training, based on investigations/interventions actually conducted by GHOEM. 

History of Mathematics Education: Place, Affiliation, Identity, and Present Time

Questions about the role of the history of mathematics education in teaching and the training of mathematics teachers inevitably result from the ways in which history has been used in the field of mathematics education.3 They also stem from the way lines of research are being constituted and the affiliation to lines of research which relate history and mathematics education in this field. As highlighted by Gomes (2010), understanding the affiliation and identity of research which relates history and mathematics education has required of some researchers an intense effort of analysis of the works and indicators involving the self-­appointed fields of history of mathematics, history of mathematics education, and relationships between history and mathematics education. Such understanding culminated in more specific approximations and dissociations which, in turn, resulted in what came to be called the history of mathematics education. From the productions in this field, identified both in the annals of the three editions of the National Meeting of Research in History of Mathematics Education and in the books published, as a result of such events (e.g., Garnica, 2016; Valente, 2014), we observe a varied and nonhomogeneous production, both in terms of research objects and the comprehension of theoretical-methodological references. However, as Brito & Miorim (2016) point out, such variety is not completely dissonant. It is a type of literature whose aim is to promote the understanding of various nuances of the Brazilian cultures of mathematics and  According to Mendes (2012), the milestone for the discussion of the relationship between the history, pedagogy, and sociology of mathematics and mathematics education was the creation of the International Group on the relations between the History and Pedagogy of Mathematics (HPM), during the workshop on history of mathematics education, in Toronto, Canada, in 1983. HPM is a group affiliated to the International Committee on Mathematical Instruction (ICMI). 3

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mathematics education. Based on summaries of theses and dissertations on the history of mathematics education produced in Brazil, Brito and Miorim (2016) classified the production and stated that they are part of one or more of five thematic axes. The axes are as follows: (a) history of teacher training in mathematics; (b) history of changes in curriculum, methods, and educational practices in mathematics; (c) history of content and school disciplines at different levels of education; (d) history of pedagogical artifacts related to or meant for mathematics education; (e) history of cultural groups of practice4 or communities involved with mathematics education. This literature mostly consists of in-depth and critical investigations, based on a range of theories and methodologies. Our studies involving oral history as a pedagogical approach in mathematics education are related to the history produced in the field of history of mathematics education, not the history of mathematics.5 This is due both to the fact that we are engaged in research in this field and to the possibility of going beyond the questions concerning the epistemology of “mathematics” and the development of “mathematics” when oral history is mobilized in mathematics education. Expanding this characterization and going beyond the way the history of mathematics education is constructed as a field of research, it is worth questioning what this field has enabled us, mathematics educators, to understand. It is also worth asking what can be done with the understanding that has been gained. It is not necessary to list the contributions of the field or show how fruitful, relevant, and singular research problems have been, in order to legitimize this field as an established scientific discipline. However, the questions are relevant to provide insights onto problematizations and possible approaches to the processes related to mathematics education and the training of mathematics teachers. In regard to the aspects involving the school, this field of research has outlined changes and perpetuities of mathematics in the school curricula, as well as allowed us to appreciate the approaches which are used to compose the Brazilian school curriculum. It has been observed that, as the schooling period was extended, the mathematical content present in school, as well as the ways in which such mathematics has been dealt with have also expanded. On the other hand, there is little evidence of innovative or alternative approaches in the treatment of mathematics (coming mainly from present demands,6 in view of the fact that the mathematics education movement in Brazil has been consolidated since the 1980s), although some changes have been detected due to recent historical changes in the role of schools.7 Even though, over time, we have witnessed a number of changes in legislation regarding the organization of educational levels and length of schooling, there has been little change in the form and conception of what school education actually means, in comparison to a more significant change implemented in Brazilian schools, regarding the demand for universal access to education (school). Regarding the history of teacher training  Community of practice is a concept derived from the work of Etienne Wenger. Briefly, it is a group of individuals who meet periodically, according to certain rules, established by the members of the group themselves, and who share an interest in learning something and practicing what has been learnt. 5  According to Miguel and Miorim (2002, p. 187), in the field of research in history of mathematics education, emphasis is placed on the dimension of mathematical activity “exclusively concerned with investigating the intentional processes of circulation, reception, appropriation and transformation of this activity,” and it includes “any study of a historical nature which investigates, diachronically or synchronically, mathematical activity in history, solely the pedagogical practices of circulation and appropriation of mathematical knowledge and in social investigative practices in mathematical education.” 6  As defined by Ferreira (2016), if we understand the “present time” as the time in which we live and of which we have memories, it is a historical period defined by moving beacons which are dislocated with the progressive disappearance of witnesses. 7  Text such as those of Valente (1999) or Carraher, Carraher, and Schliemann (1982) corroborates this evidence. 4

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in mathematics, research shows a history marked by precariousness and urgency. Associated with that, we identify the perpetuation of a school culture in which mathematics (and, consequently, its teaching) is seen as a fixed, immutable field of knowledge which few can access (Vianna, 2000), as well as a movement toward the acceptance of a meritocratic discourse which throughout history, accentuates mathematics as an objective element to gauge the individual merits of people (da Silva, 2013). These are some of the many themes dealt with in the field of history of mathematics education which can and should be problematized in forums dedicated to mathematics education and to the training of mathematics teachers. This is an opportunity for the future mathematics teacher to understand the history of national educational organizations and programs and their objectives, in a moment of great and continuous reforms, which they will have to put into practice; as well as a good opportunity to examine and reflect on unique aspects which involve the culture of mathematics teaching. Questioning aspects of the various stories of various actors linked to school and mathematics teaching may make future teachers feel part of the history of their profession, and give them clarity about the past, and potentially open to understanding and criticizing current changes. Oral history has proven to be a valuable tool to support this endeavor. Even when dealing with aspects of a remote past, it is possible to establish discussions based on oral history. For instance, showing students how to examine the ways through which people give meaning to aspects of the past, related to mathematical culture in schools, through their experiences related to mathematics, education, mathematics education, and related fields.8 In much the same way as done by oralists, when students conduct exercises with oral history, it generates multiple problematizations derived from the associations between history and memory (whether individual or collective), seen as efforts to reconstitute the past, even though they may be of different natures: history seeks to systematize and record sequences of events and preserve memories, while memory preserves only that which is alive from the past or that which can be revived in the consciousness of an individual or a group (Halbwachs, 2004). Regarding the place of the history of mathematics education in institutions aimed at the training of mathematics teachers, we believe that all possibilities of problematizing the history of mathematics education are valid. However, as researchers, we have often chosen to link the historical perspective to effective strategies aimed at both teaching mathematics and training teachers. The problematizations of the history of mathematics education should not only occur in a specific discipline (existing or to be created)9 in the curriculum of teaching degrees10 but in any and all curricular forums in which possibilities of problematization are detected. This topic will be further discussed below. 

(Oral) History as an Approach to Teaching and Training of Mathematics Teachers

The discussions about the potential of history for mathematics education have raised questions about why, for whom, where, and how to deal with history in mathematics education. What problematizations, history, and themes should be part of the process of teaching mathematics or other processes related to it?

 Halbwachs (2004) states that social groups allow us a great arsenal of memories, so that even when people are no longer materially present one can speak of a “collective memory when we recall an event that took place in the life of the group and which we considered; and we still consider, from the point of view of the group” (pp. 40–41). 9  In Brazil, the need to create a specific discipline, in higher education in mathematics, focused on studies on the history of mathematics education, has been discussed, without consensus. 10  In Brazil, the undergraduate university degrees responsible for teacher training are called licensures. 8

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Since the 1980s, quite a lot of research focused on the possibilities of the use of the history of mathematics for teaching has been developed in Brazil, in line with the approach which uses history in the teaching of mathematics.11 Two main arguments justify the relevance of the history of mathematics to the teaching of the discipline12: (a) the opportunity to show students mathematics as a science which does not deal with eternal, infallible, and immutable truths but a dynamic science, always open to the incorporation of new knowledge, and (b) the advantage for students and teachers of appreciating the obstacles involved in the process of constructing mathematical concepts and the possibility of understanding some aspects of their own learning. These justifications are corroborated by official documents regarding the teaching of mathematics, in Parâmetros Curriculares Nacionais—PCN (National Curriculum Parameters). On the other hand, a pedagogical approach based on oral history adds to the potential of the idea of using history in the teaching of mathematics to deal with aspects related to the history of the relationship between people and mathematics, in their daily lives, through time, and also makes it possible to explore, through research, this potential of oral history in the teaching of mathematics. In da Silva (2013), we present an intervention using the oral history approach in the treatment of some mathematical themes in the discipline “Fundamentals of Elementary Mathematics,” in the undergraduate teaching program in mathematics of a large public university in the state of São Paulo (Brazil). The objective of the course was to relate the topics in mathematics taught in elementary school13 to the contents studied in the various undergraduate courses, showing how the latter form the basis for the former, and to explore different approaches to teaching. Six mathematical topics were examined: calculation strategies (algorithms); algebraic thinking; fractions; proportional reasoning; exponents, whole and real numbers; and logic. Groups were formed to study each of these themes, and each group investigated a specific theme to be presented and discussed with the class. In order to provide a general idea and promote more informed critiques about certain materials, resources were selected, studied, and analyzed during the semester,14 as well as written memoirs by students about their experiences and knowledge on the topic, and oral history interviews with retired or practicing mathematics teachers. The results of this research/ intervention showed that, at the end of the course, future teachers had a more meaningful, explanatory, and multidimensional notion of elements related to the mathematics studied in school. It also enabled students to perceive aspects of mathematics education in the materiality of school history, in the singularities of the students’ own experiences, and in the contact with teachers and their history.  As defined by Mendes (2012), the history of mathematics studies the epistemology of mathematics and the development of mathematics as scientific content. The history of mathematics has been present in undergraduate courses in teacher training courses, in Brazil, since the 1980s, although its presence has become more visible since the 1990s. The majority of current undergraduate courses pay attention to history of mathematics (Balestri & Cyrino, 2010). In addition, it can be said that the history of mathematics has been suggested as an approach to teaching mathematics in basic schooling since the publication of documents governing education in the country (especially the National Curriculum Parameters, in 1997). 12  A reference to the history of mathematics (and not to the history of mathematics education) as this is the favored field in Brazilian official documents about teaching mathematics. When it comes to the use of history in teaching mathematics, references are always made to the history of mathematics and almost never to the history of mathematics education. Changing this scenario should be a top priority for Brazilian researchers in the field of history of mathematics education, due to the sharp growth of this field during the last decade and its consolidation. 13  In Brazil, schooling is divided into stages: basic education is composed of preschool (optional), elementary school (compulsory, 9  years in duration), and secondary education. Secondary education lasts 3  years and is compulsory for those who intend to go to university. 14  The materials used were official documents which regulate teaching, in general, and teaching mathematics, in particular, didactic and paradidactic, as well as academic, materials. 11

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Souza’s doctoral thesis (2006) investigated the history of mathematics teaching at a Grupo Escolar.15 One of the several actions carried out during this investigation was the development of an activity with children between 8 and 10 years of age, in order to explore narratives about that school and study the feasibility of a project which would explore orality in the construction of such narratives with the children. The researcher was inspired mainly by the works of Argentinean researchers Benadiba and Plotinsky (2001), who propose using oral history to work with children, and the experience of Museu da Pessoa16 which, among other projects, uses oral history to work with children in order to recover the history of the place where they live. Souza’s project (2006, p. 122) with children entailed the following phases: Explore the building where they study and search for some objects or spaces from the time when it used to be Grupo Escolar; listen to stories about what was found; instigate the children to elicit stories about the time Grupo Escolar was a school, from their parents, grandparents and neighbors; search for photos, books or notebooks from that time; talk about what had been discovered; organize a joint script to interview a former Grupo Escolar student and watch the video resulting from the interview. Clearly, the project did not only involve aspects related to mathematics and mathematics teaching, but several aspects related to school. However, among the questions raised by the children to their relatives or people who, in the past, had studied in that building, object of the investigation, was the question “was mathematics difficult?”, which was suggested by the children themselves. The answers to this question resulted, for example, in the following excerpt from Souza (2006, p. 130): I believe mathematics was much stricter than today, students had to memorize the multiplication table, the divisions, the four operations and we had to learn very well, because in my time, when we left the 4th grade, it was still required to take the admission examination to go the secondary school. So, the teachers made us work really hard; those who wanted to continue their studies. The entrance examination was taken so that we would be admitted to the secondary school, to enter the first grade of secondary school. It was not the 5th grade, it was 1st grade of secondary school. So, a project like this opens possibilities directly linked to mathematics teachers and their work, attempting to understand teaching programs and methods, pedagogical resources, evaluations, and school exercises at a specific moment in time, pointing to oral history as an interesting approach to the exploration of specific aspects of the history of mathematics education. In addition, such a project promotes young children’s interest in a wide range of topics related to the history of education, allowing more elaborate information to be explored and problematized, making children more expressive, curious, inquisitive, and closer to parents  Grupo Escolar was a primary education institution created in Brazil at the end of the nineteenth century, through one of the first major educational reforms aimed at creating a national education system. Such institution was created in the state of São Paulo, but they rapidly spread throughout the country. Legally, Grupo Escolar was extinguished in the 1970s, and the buildings were occupied by elementary schools operating under the responsibility of municipalities. Thus, even if they no longer exist, it is still the case that in cities they often serve as a reference. To study at Grupo Escolar means to study in the institution which today occupies the physical space of what was once Grupo Escolar. 16  Museu da Pessoa (www.museudapessoa.net) is a virtual museum, founded in 1991 in the city of São Paulo, with the objective of creating a free and broad network to give access to life histories and institutions. The Oral History is one of the main methodologies used in the museum for the recording of stories. 15

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and grandparents based on their interest in the life history of such relatives, as well as allowing children to identify themselves and the spaces they frequent, perceiving themselves as potential storytellers As far as the processes related to the training of mathematics teachers are concerned, it is still necessary to question: which mathematics education stories should be dealt with. From this perspective, Gomes (2010) questions the themes, objects, and approaches with which the research on history of mathematics education ultimately deals with: should the focus be the history of school mathematics, as suggested by Valente (2005)? Or history of teaching mathematics, as suggested by Matos (2006)? Or yet the history of teaching and learning mathematics as suggested by the Topic Study Group at CME—International Congress on Mathematics Education 6 (TSG 29, 2004 edition; TSG 38, 2008 edition)? Or, in a more comprehensive way, should it involve the history of mathematics cultures and mathematical education and their multiple dimensions, as suggested by Miguel e Miorim (2004)? In addition, what are the training forums where this history could be approached and problematized? In undergraduate programs? In which courses? Should there be a new discipline? In Brazil, the curriculum guidelines for teacher training determine that undergraduate courses prepare students to face “pedagogical dynamics that contribute to the professional practice and development of the teaching profession through a broad view of the formative processes, their different rhythms, length and depth, in face of the psychosocial, historical-­ cultural, affective, relational and interactive dimensions that permeate pedagogical action” (BRASIL-MEC/CNE, Ordinance n° 2, of July 1, 2015, p. 6). In the state of São Paulo, one of the main regulating bodies, the State Education Council (Conselho Estadual de Educação—CEE, Portuguese abbreviation), recommends that in all teaching degrees (Indicação CEE n° 127/2014 and Deliberação CEE n° 126/2014) students/future teachers should examine the history of education and the socio-philosophical evolution of pedagogy, as well as the Brazilian educational system and its history, so that students/future teachers can conduct critical and comparative analyses of education in the country, as well as understand the context in which they will conduct their teaching practice. Official documents also draw attention to approximations with history, sociology, and the philosophy of education, as they support pedagogical ideas and practices (SÃO PAULO—CEE, 2014). History of education as a discipline was introduced in Brazil on the second half of the nineteenth century, at Escolas Normais,17 by teachers trained in the most diverse areas, even before its expansion as a line of research in education (Gasparello, 2007). According to Nascimento and Nascimento (2010) the goal at the time was “to understand the principles, experiences, successes and shortcomings of past educators... as an antidote, which would prevent the repetition of past mistakes.” That is to say, the goal was linked to a perspective of history as the master of life.18 Concurrently with changes in the way history itself is viewed and its function as a discipline in the humanities. This perspective on the role of history of education in teacher training has also been revisited or reshaped.

 Following the French model of the Ecoles Normales, Brazilian Escolas Normais were created at the end of the nineteenth century. They were secondary education courses aimed at training elementary school teachers, to teach in early childhood schooling. The creation of Escolas Normais and Grupos Escolares in Brazil took place as part of a national project to restructure primary education. The adjective “normal” in Portuguese (just like “Normales” in French) is justified because function of such schools was to train teachers who would be responsible for educating children according to the “norm,” the current social rules. 18  The concept of Historia Magistra, from Latin, suggests that history serves to record noteworthy past events in order to guide humanity in the present, toward a better future. 17

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The educator António Nóvoa (1999, p. 13) states that “having historical knowledge does not imply acting more effectively, but it stimulates a critical and reflexive attitude.” Similarly, the historian Claudia Alves (2012, p. 208) points out that “a historical perspective repositions the educator, not by giving them action plans but by allowing them to uncover several hurdles which hinder their actions.” It helps them detect routine problems, which otherwise would remain unnoticed, as they are commonplace and habitual, thus invisible. In the field of mathematics education, Antônio Miguel and Maria Ângela Miorim (2004, p. 156), in their studies on the role of history both in the research in mathematics education and in the teaching of mathematics, have argued that history can and should be a reference for the pedagogical problematization of school culture and, more specifically, of mathematical culture and school mathematics education, provided they are duly constituted for explicitly pedagogical purposes and organically linked with other variables which intervene in the process of teaching-learning mathematics in school. These authors characterize such history as pedagogically vectoring. Miguel and Miorim (2004) suggest that these histories, written and/or proposed by mathematics educators, should not conceive historiography, in general, as well as that related to mathematics and mathematics education, in particular, “as moralizing repertoires that have the power to resolve ethical, political and pedagogical conflicts, or of any other nature, that are manifested or may manifest in our pedagogical practice in the present” (p. 161). On the contrary, they suggest that the function of history is not to answer questions such as: “should we act this way?” According to the authors, answers to these types of questions, given in the present, involve many kinds of conditioning factors specific and unique to the institutions and situations in which they presently operate. They suggest, therefore, that historiography can be seen as a source of dialogue, not as a source of answers or formulas to be repeated in the present. This perspective about the possibilities of the participation of the history of mathematics education in pedagogical processes suggests, therefore, a type of approach which problematizes knowledge resulting from vestiges of actions and practices of the past (or elements of the school culture situated in space and time), involving mathematics education. We realize that we are doing it with our feet firmly planted in the present, and in specific everyday situations. Potential examples of questions raised in this type of approach are: what aspects of (mathematics) education were considered relevant to the different memory communities of that time (communities associated with social groups involved and/or promoting different multicultural social practices)? What aspects were not considered relevant? Why were they not relevant? What did education mean to those who lived at different points in the established network of power relations? What aspects of education are considered relevant by the different contemporary memory communities? Why? What is no longer considered relevant? Why and by which communities? What are the characteristics of the answers to the preceding questions, that is, where, for whom, and based on what, do we produce such meanings? How does the problematization of these issues help us think about aspects and issues specific to the institutions and situations with which we are involved at present? These types of questions are very closely related to the possible issues of a pedagogical approach involving oral history. The following interview excerpt helps us demonstrate the potential of an oral history interview to explore these issues. This is an excerpt from the interview of a mathematical educator—Adailton Alves da Silva—given to a researcher in mathematics education whose aim was “to understand how the individual becomes, throughout their life, experiences, their relationships with other individuals, their relation to the

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macrosocial context, this specific mathematics teacher: with their own ideas, practices, resistances” (Rolkouski, 2006, p. 169): I started to work in Porto Alegre do Norte, a city where I lived for 13 years… I stated to work for the city as a draftsman. And from then on, I began to work with people who had a more... I would say, at the time, 1987; a vision of education more open to innovation. I started to get along with these people who thought that education should not be an ordeal. Already at that time we discussed mathematics differently from the traditional way. After two years in the city hall, I was finishing the eighth grade, I was invited to teach. Eighth grade and I had already been invited to teach. It is something that today I tell people, and that some find it strange, find it negative, that is what I feel. I was in eighth grade, eighth grade, when I went to the next grade, I taught in the previous grade because there was no teacher. When I finished my first year [of high school], I went to the second year, and then I taught the first year, because there was no teacher yet. When I got to third grade, I taught second grade. This thing was haunting me. … That was 1992, in the city of Porto Alegre do Norte, a city 1200 kilometers from the capital, 700 kilometers from the nearest university, in Mato Grosso19… Back to my first lessons... My classes were very traditional. I was very insecure, so I clung to the textbook. At the time I wanted to see it differently. I kept trying to do some things… I thought about Biology classes and kept thinking. So, I remember a time to change the class I wanted my students to discuss mathematics and I thought: Ah, I will invent something. I took my students to study the mathematics of the burning fields. In that region there came a time when the fields were set on fire; the smoke is unbearable, people set fire to the fields,20 nowadays people don’t do it anymore. And everyone would complain, about the environment, and so forth, but no one took these issues to the classroom. So, me and my students went to see how that worked in real life. And off I went, I had nothing planned. We just went to Cerrado.21 Cerrado was close to schools; and we dug like, about six inches, a square meter, 20 centimeters deep where the fire had burnt, and a square meter and 20 centimeters deep where there had been no fire, and we saw what was still alive there, to compare. There, we worked on area, volume and living things. There they saw the discrepancy in the life found. I thought of this because I was worried about a formula I was learning in Biology and I decided to do this with my students. I expected this from my teacher, so, in fact, I did for my students what I expected from my own teacher. Research on the history of mathematics education according to the parameters of oral history indicates that oral narratives of teachers such as the one above help bring to light different areas of knowledge, approaches, tones, authors, different perspectives and aspects, and the possibility of understanding what is central and what is subsidiary. They substantially contribute to the analysis of aspects involving the culture of mathematics and mathematics education in different regions and different times, as well as contribute to understanding the varying conditions in which such training occurred, how teachers

 Mato Grosso was a state in Brazilian Midwest. In 1979, it was divided in two new states: Mato Grosso and Mato Grosso do Sul. 20  The purpose of this operation was to burn native vegetation to make room for cattle farming or agriculture. 21  Cerrado is a region of Brazil, a biome with specific fauna and flora. It is one of the largest biomes in Brazil and indisputably one of the most biologically diverse in the planet. 19

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mobilized/mobilize teaching materials, followed/follow or subverted/subvert legislation and other existing provisions (Garnica, 2013). Some questions arise from the excerpt of Adailton’s interview which serve to illustrate how discussions and investigations of the history of mathematics education in Brazil, in the 1980s and 1990s, can be introduced: “What aspects of Adailton’s narrative can be highlighted?” What does Adailton’s life have in common with that of other mathematics teachers at that time, in that place? What does Adailton mean by “education as an ordeal,” “traditional education,” and “a vision of education open to innovation”? Considering the emphasis given by Adailton on the fact that “this was ninety-two, in the city of Porto Alegre do Norte, a city 1200 kilometers from the capital, 700 kilometers from the nearest university in Barra do Garças, Mato Grosso,” how did the communication between cities and people take place in the 1990s, in Mato Grosso? What aspects can be highlighted about the political, educational, and socioeconomic situation of that Brazilian region at that time? Under which conditions did Adailton begin teaching and what strategies did he use? What educational legislation in force in Brazil at the time allowed a loophole such as that experienced by Adailton, who was able to teach while still a student of at same level? Would this be possible today? If not, when did it become impossible? What loopholes does the latest government’s educational restructuring proposal present? What are the differences between the current political educational scenario and that in Adailton’s narrative? Considering Adailton’s account, “I was very insecure, so I clung to the textbook,” what mathematics textbooks were available at that time, in that region? Is it possible to associate the class described by Adailton with any current trend in mathematics education? Which one? What aspects of this trend enable such association? It is worth pointing out that an interesting strategy to elaborate this type of (mathematics) teacher training proposals is to characterize this history as pedagogically vectoring, and to provide a sufficiently broad set of sources, which students may or may not use. This choice might depend on what they consider important to meet their research needs. This example helps demonstrate that, if memory is constituted of past issues situated in space and time, thus creating a place for the individual or collective senses, oral history allows us to problematize aspects of cultures based on the pair, experience/meaning (Larossa, 2016), guided by present issues. The examination of the narrative of experience belongs to this discussion because it allows an association between what affects or happens to individuals (their experience) and the meanings which individuals attribute to what happens or affects them, as “the protagonist of experience is, above all, a space where events take place” (Larossa, 2016 p. 25). The protagonist of experience is therefore different from the “protagonist of information, opinion, work, knowledge, judgment, action, power, will” (p. 25). The knowledge of the protagonist of experience differs from scientific knowledge and information knowledge, and its existence is distinct from that of the protagonist of skill and work. Experience arises from the mediation between knowledge and human life, knowledge which is subjective (not scientific) and human life which is singular (not merely relative to the satisfaction of needs induced by the logic of consumption) (p. 25).22 Knowledge through experience is “what people acquire through the way they respond to what is happening throughout their lives, and how people make sense of what is happening to them. Knowledge through experience is not about the truth of how things are, but to the meaning or senselessness of what happens to people” (Larossa, 2016, p. 32).  “Experience [similar to the word existence] is the way of a being that exists, of a being that has no other being, another essence, besides its own corporeal, finite existence, embodied in time and space, inhabiting the world with others. And existence, as well as life, cannot be conceptualized because it always escapes any determination, because it is in itself an excess, an overflow, because it is in itself possibility, creation, invention, event. Maybe that is why we should keep the experience as a word and not make it into a concept” (Larossa, 2016, p. 43). 22

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Oral history as a pedagogical approach gives the student or future teacher the possibility to access this experience/meaning. It enables the problematization and discussion of issues which would otherwise be difficult or impossible to address. An excerpt from the account of a teacher, given to an undergraduate student in the teacher training course in mathematics, about their history as a mathematics teacher, in a public school, in a poor urban area, in Brazil may perhaps help clarify for the reader what we have tried to explain: The school in the outskirts of the city was the one that worn me out the most, the one that drained me as a human being, I came home crying. I tried to teach math. I do not know if I succeeded. In that school, in the lower-income suburbs, I tried to focus their attention on the job tenders for public servants; on jobs. It was not even for the college entrance exam,23 I never even uttered the word vestibular (college entrance examination), I spoke about jobs, as this was more urgent. The focus was assessing them, gauging if they could do anything with mathematics, not for college, but for work, to help them with that. … I worked in a school where a girl told me that her mother had said that if she continued to study, she would break her teeth, and I said that her mother must have been nervous, with her own problems, that school was a good thing, that she should continue studying... One day the girl came to school, all her teeth were broken, and I paid for her treatment... The mother was an alcoholic and I did not know that at the time. I was in that school… If you think about it, I learned more in that school, than any college could have ever taught me... Because college taught me, through the books I read, that I was going to teach little blonde girls, in pretty little dresses and ponytails, no one ever prepared me, no school prepared me… And in no book had I ever read anything about what I had to face later on. I came home, I was like, "Oh my God, what have I done?" No college, no book, no one could have prepared me for what I found…. (Account of Ana Rita Motta, 2018) Ana Rita’s account reveals aspects of experiences involving the school environment which in general, are not problematized in (mathematics) teacher training courses, even though undergraduate students are afforded experiences in schools, through internships mainly focused on teaching practices. This interaction with Ana Rita enabled students to come in contact with the feelings and meanings she attributes to events in her experience as a teacher. No formal visit to any school would allow this kind of experience. In her testimony, Ana Rita reveals the meaning she attributes to her experiences in a public school located in a low-income housing project on the outskirts of the city, in opposition to the absurdity of the realities presented in the books or considered in the discussions about school while she was working toward her degree. In a sense, this is an example of how teachers face adversity, a record of their dayto-day interactions in institutions, which enables a fruitful analysis of the goals and ideals of teacher training over time. To we researchers, mathematics educators and teacher educators, the excerpt from Ana Rita’s testimony draws attention to how the pair experience/meaning presents the challenge of thinking about education from a different point of view. It leads us to think of situations which other words would not allow us to perceive, would not allow us to say, would not allow us to see, would not allow us to feel. This brings to light other effects of the truth,

 In Brazil, vestibular is a mandatory examination taken by students in order to gain access to higher education. There are public universities (maintained by state and federal governments) and private universities and colleges. Students must pass vestibular in order to enroll in any such higher education institution. 23

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other effects of meaning. At this point we come back to what was initially said about our role as mathematical educators: we must foster the development of the student and/or future teacher as a human being, a citizen, and professional, focusing on a multitude of values and attitudes. We should not ignore the meanings that arise in the present time. In the years working with oral history as a pedagogical tool, we have come to the realization that for we teacher trainers, who seek to problematize histories of mathematical education (of the present time) in educational spaces, the challenges are even greater. We deal with the living memory of contemporary teachers, who influence our work with issues about the legitimacy of schools and the society in which we live. The interviews conducted by our students have been a forum where (mathematics) teachers call for legitimization of their needs and confrontations as professionals in education and ask that we endorse their points of view. Which brings us to the heart of the question about the meaning of oral history as a pedagogical approach in teacher training and teaching situations in general. This work is a strategy which brings out, in such instances, different stories about an event, a place, a person, but also enables the emergence and problematization of multifaceted realities. Using oral history to train teachers is a strategy which seeks to question common sense or naturalized notions, held as static truths. Through the use of this strategy, instead of trying to convince students of our point of view or the “researcher’s point of view,” we seek to empower them and help each one to create arguments which allow them to reject simplified treatments involving aspects of mathematics education and school, to form opinions in a sound and well-­grounded way. This is a work which allows future teachers to see themselves in a place of work which could be theirs, to see themselves in another teacher’s shoes. They may or may not identify with them through the stories which have been told, but it enables them to evaluate and reflect on the various situations and problems which involve working in schools and mathematics classrooms, which are not only situations and problems related solely to teaching and learning specific content. 

Contributions of Oral History to the Training of Mathematics Teachers

Our research-intervention activities with oral history in the senior year of an undergraduate course for training mathematics teachers have been characterized by a process of investigation, elaboration, application, and analysis of teaching strategies in different spaces. Some guiding principles considered in the elaboration of such strategies are the following: (a) the problematization of knowledge produced from traces of actions and practices of the past (school) involving mathematics education; (b) contact with different life situations, educational and political contexts, as well as school and teaching practices; (c) awareness of the differences among people and contexts; (d) the production of plausible24 meanings for mathematics and mathematics education practices of the past, considering that they make sense in the present universe and collaborate with the discussions and decision-making in relation to actions and practices in the present; (e) recognition of those involved as protagonists of history.

 The meaning of something does not exist “in itself,” it is not inherent to things, it is attributed by individuals. Thus, one cannot speak of “true meaning.” The expression “plausible meaning” is intended to highlight this characteristic: teachers in their interviews, as well as other sources, can deal with a particular object. The collation of a variety of sources allows the researcher to assign a plausible meaning to a particular thing, since the meaning “in itself ” cannot be objectively communicated by any source. The attribution of meaning is always a careful movement of approximation. 24

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So far, investigation-intervention actions in processes directed to the training of mathematics teachers have involved disciplines of the course for obtaining a degree in mathematics, such as Fundamentos da Matemática Elementar (Fundamentals of Elementary Mathematics) (da Silva, 2013), Política Educacional Brasileira (Brazilian Educational Policy) (Tizzo & Silva, 2016), História e Sociologia da Educação: questões da Educação Matemática (History and Sociology of Education: Issues in Mathematics Education), a pedagogy25 course, and Conteúdo, Metodologia e Prática do Ensino da Matemática (Content, Methodology and Practice in Mathematics Teaching) (Flugge, 2015; Tizzo, Flugge, & da Silva, 2015), as well as other teacher training venues, such as those created through specific official programs for teaching initiation, which aim to create partnerships between university (undergraduate) courses and schools, in which future teachers will work (da Silva, 2016). All such initiatives contribute for the use and research of oral history as a pedagogical tool. The program for the course History and Sociology of Education: Issues in Mathematics Education was proposed by a group of mathematics educators who work with teacher training courses in mathematics offered by a large public university in the state of São Paulo. The following topics are listed in the course syllabus: basic issues of history and sociology of education and mathematics education; issues in mathematics teaching; human knowledge, in general, and in particular that of mathematics; history of ethnic-racial relations; AfroBrazilian and African history and their relation to mathematics education; different approaches for teaching mathematics and mathematics education from historical and sociological points of view. Generally speaking, the teaching method for this course involves reading, discussion, and elaboration of texts about written, oral, and audiovisual narratives related to the topics of the discipline. In all classes, the students are divided into smaller groups to discuss the texts or videos brought by the teacher. Later on, starting with the presentation of a rapporteur from each group, the larger group will discuss possible issues. In 2018, oral history contributed to the discipline in three different ways. One of them consisted of a group dynamic that aimed to exercise the students’ perception of the aspects to be considered in the historical analysis of school and school disciplines26 (in this case, mathematics). The second way involved fieldwork conducted by the students, in pairs: it included interviews with three different generations of teachers (as suggested, a retired teacher, a teacher who had been working for 5 years, and a teacher who had been working for 15  years). They were asked questions about their training, school, and mathematics classes. And, finally, the third way oral history was used in this discipline was through a request from the students after a class discussion of a text by Gelsa Knijnik (2004): O que os movimentos sociais têm a dizer à Educação Matemática? (What Social Movements have to say to Mathematics Education?). The discussion based on this text instigated the student’s interest in knowing about innovative projects, programs, or proposals in which there was a significant interaction between mathematics education, school, and communities. In order to do that, we organized excerpts from two oral history interviews with teachers who work at Universidade Federal do Paraná Litoral27 and work with innovative projects involving the  In Brazil, the degrees in pedagogy form the teachers who will work in early childhood education, the so-called elementary teachers. Such teachers are responsible for all initial education of the children. This includes literacy in the mother tongue and introduction to elementary mathematics (numbers, counting, etc.). 26  School discipline is used here according to Chervel (1990). 27  Universidade Federal do Paraná (UFPR, abbreviation in Portuguese) is one of the oldest higher education institutions in Brazil. Campus Litoral—located in a city on the coast of the state of Paraná—was recently created and has an innovative pedagogical project, in comparison to other higher education institutions in Brazil. UFPR Litoral was studied by Matucheski (2016). 25

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community. Below, we describe the ways in which oral history was used in this discipline, as well as some developments. The first way oral history was used is through an activity involving the new mathematics movement, from a text by Chervel (1990). First, the teacher revisited with the students the aspects the author considered important to analyze historically a school discipline, which are: (a) the social and political objectives of the school and the discipline at the time; (b) the purposes of the objective (what should school teach?)28; and (c) the actual objectives (why do schools teach what they teach?).29 After that, the teacher provided several materials which addressed the theme of modern mathematics, such as pages from mathematics textbooks of the time, which followed the guidelines of the movement; some excerpts of the text by Souza and Garnica (2013)30 providing information about the educational program of the state of São Paulo during the movement; notes from the thesis of Baraldi (2003) which describes the movement, its propositions, groups based on the movement in Brazil, and the national and international political scenario; and, finally, an excerpt of a textualization of an interview given by a teacher to Baraldi.31 This activity allowed students not only to discuss and understand a kind of historical analysis about a moment related to the teaching of mathematics in our country, but also to understand aspects which differentiated the conceptions about modern mathematics, and consequently, the different ways it was applied, which led us to deal with modern mathematics (in the plural), that is, the many and quite different ways the movement called new mathematics was treated by researchers and teachers. This contact with multiple sources made it possible for students to observe the intrinsic relationship between the knowledge about the past and the sources consulted, thus broadening their experiences about the question of how historical knowledge is constituted and also about “the danger of a single story.”32 The second way in which oral history was used was different from the first one. This time, the idea was not to discuss the results of research, nor interviews conducted for specific research, although, similarly to the first activity, it was also intended to sensitize students to the work with oral history. To do so, a script for a collective interview was developed, which contemplated questions suggested by each pair of students. Different scripts were developed, one for interviewing retired teachers, the other for teachers who were still working. The teacher presented the oral history methodology, to show the class its principles, intentions, and procedures, including providing a sample presentation letter to be given to the interview-

 Sources for obtaining information for this purpose are, for example, the official documents on education and curriculum. 29  Sources for information on this purpose are materials which explain content, teaching methodologies, and motivational practices, such as lessons, exercises, activities, evaluations, student materials (such as notebooks and projects), teaching programs, textbooks, lesson plans, etc. 30  This is the article entitled As matemáticas modernas (no plural): um ensaio sobre os modos de produção significado ao(s) movimento(s) no ensino primário Brasileiro, in which the authors argue that a movement (in this case, the new mathematics movement) is carried out in very different ways; thus there cannot be a single movement, but a multitude of conceptions in relation to the movement which are effected through practice. It so happens that, in Portuguese, some adjectives, like promos, and the noun “mathematics” have a plural form; therefore, the title of the article emphasizes this plural form, which cannot be translated to English (one modern mathematics—singular—and a myriad of modern mathematics—plural). 31  This is the interview with João Linneu do Amaral Prado, available in Baraldi (2003), in which the interviewee talks about his life both as a teacher and a student at the time of the new mathematics movement in Brazil. 32  Chimamanda Adichie’s video “The danger of a single story” was discussed in the first class of this discipline, with the objective of raising students’ awareness in relation to the production and recording of historical knowledge. Freely available with Portuguese subtitles at: https://www.ted.com/talks/chimamanda_adichie_the_danger_ of_a_single_story/transcript?language=pt-br. 28

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ees, as well as a sample of the assignment of rights agreement33 to be signed by each of the interviewees. The students had time to deal with the material and, in due course, each pair presented their results to the rest of the class. The presentations caused a number of issues to arise. In the second phase of the work, each pair furthered the discussions about one of the topics which emerged from interviews and presentations.34 The teacher helped the groups, providing materials which could help them study the topics chosen. As a result of this process, each group produced a final written text which dealt with the theme in question, linking it to the narratives of the teachers interviewed. The final work was also shared by each pair with the whole class. In addition to providing contact and conversations with teachers about aspects of mathematical education, their realities, and meanings, the activity also helped students exercise a type of problematization similar to that used by oral history researchers. After the presentations of the interviews, it also allowed associations between the opinions of the interviewees on the topics. The following excerpt illustrates the spirit of the third way oral history was used in the discipline História e Sociologia da Educação: Questões da Educação Matemática. It is an excerpt of an interview with a university professor (Matucheski, 2016). Matucheski studied the implementation of the pedagogical proposal of UFPR Litoral35 in the course for obtaining a teaching degree in science, and the ways both the teachers and management of the institution have implemented the proposal: I believe that the aim of Universidade Federal do Paraná—Litoral (UFPR/ Litoral) is to challenge and change a paradigm. I believe this because the idea of the institutional political-pedagogical project is to break with the perspective that has steered traditional teacher training provided in courses based on positivist and liberal principles, which separate theory from practice. Theory comes before practice, because practice is the application of theory... What is being proposed here is to break with this dichotomy. There is a desire to abandon this dichotomy and find another training proposition, in which the concrete reality of the place and the people are considered. And this is the reason UFPR/Litoral is located here, on the coast: there was a real need for a university here... There was a need for an educational hub that would dialogue with this place, offer alternatives to this place. So, UFPR/Litoral relies heavily on a concrete reality to reflect upon and then act on that reality. The University suits the location as much as the location suits the University When I work with teacher training, I usually say that the first question we need to ask ourselves is this: “If the school closed tomorrow and you did not come to class, what would happen?” and I add: “If the people don’t do anything... don’t say: ‘What a shame!’… Then, you—the teachers—are not accomplishing anything here... You will not be missed here!”. If something is not  Transcription, textualization, and letters of assignment are steps of the work with oral history. These steps are discussed in a specific chapter in this same book. In summary, letters of assignment give the researcher the right to use the interview for academic purposes. 34  The topics which arose during the activity were the historical aspects of the relations between family and school, technical schools in Brazil, the history of youth and adult education in Brazil, aspects of the history of educational legislation in Brazil since 1930, a history of how problem solving is treated in mathematics teaching in Brazil, historical aspects of educational evaluation systems in Brazil and evaluation in mathematics, a history of mathematics textbooks in Brazil, a history of the training of mathematics teachers in Brazil, a history about teaching of geometry in basic education, and a history of the mathematics curriculum in basic education in the State of São Paulo. 35  According to Matucheski (2016), “the political scenario in the beginning of 2000 favored the creation of UFPR Litoral: the federal government, Paraná state government and municipal governments of the coastal region entered into a partnership for installing a university in a coastal city in Paraná. The project had the support of the then President of the Federal University of Parana” (p. 435).” 33

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missed, it is because it did not make sense in the first place. Here, at UFPR/ Litoral we work with three curricular areas: Learning Projects; Cultural and Humanistic Interactions; and Theoretical—Practical Foundations... These curricular areas are independent, non-linear, and dialectic. They are complementary and have their own movement, suitable to the space and the students. So, each student can have a conceptualization of what each curricular area is. If you talk to 20 students, each one might say different things about the curricular areas. And this is exactly what we aim for! We hope that this will happen, because it is expected that the student will appropriate the scope and dialogue with it, starting from their history in the course, their propositions and desires. I’ll tell you how I see these areas... Learning Projects is the space for the individual to rescue that curiosity or longing for discovering things, for investigating. Starting with the students’ life stories, we asked them, "What do you want? What is your dream? What would you like to do? What do you want to find out? What do you want to study? You know that road sign in front of UFPR Litoral? It is the result of a learning project, in which I was the advisor! It was a learning project for the students of the Tourism and Hospitality course, when I was this course coordinator... And the idea of this learning project came from a trip to Gramado, Rio Grande do Sul. Some students went to Gramado and were in awe, because everything is typical there. As the city was colonized by German immigrants, the houses are built with German architecture and the signposts, too. They saw that. Instead of having these ugly little tin plaques, in Gramado the signs are made of wood, they are more beautiful... So, the students said: “Wow! Matinhos36 lacks proper road signage and there are many craftspeople there who do woodwork… why don’t we do a learning project about that?” From then on, the students began to identify artisans in Matinhos and investigate how they could do something to materialize this idea, so that it would be a reference for the other craftsmen, and the idea of doing the signage for the University eventually came up. But during the learning project, someone noted the following: “If the signage is only inside, it will not get much visibility… People going by on the street cannot see it.” Then, another person said: “The University still does not have a sign that says UFPR/ Litoral is here! Let’s make a sign!” This whole journey to the actual sign took about two years. And what was the meaning of this learning project? They realized that through the Tourism and Hospitality course, they were able to do things that made the city more visually pleasing and at the same time could evidence the work—hitherto invisible—of local artisans. This is social responsibility… Another aspect which we learn from the research interventions which use oral history in institutional training spaces is that the study of teachers’ professional memories enables, on the one hand, an understanding of aspects of the development of teachers’ professional knowledge (how and why) and, on the other hand, brings meaning to the adversities experienced when trying to implement reforms or proposals which radically alter the status quo. As Tyack (1989, p. 419, quoted in Mcculoch, 2012) put it: The historical study of continuity and constancy in instruction may reveal... why attempts to alter its fundamental character can be like writing in the snow. Adding a time dimension to current ethnographies and other classroom studies could raise important issues hitherto neglected in current political debates.

36

 Matinhos is a coastal city in the state of Parana, where UFPR Litoral is located.

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During the activities proposed for the discipline, from the interviews available in Matucheski (2017), it was possible to correlate what the interviewees said to the ideal of a democratic and liberating public school, guided by critical and emancipatory pedagogical actions, such as defined by Paulo Freire.37 The reading of the excerpt above also allowed us to return to the question posed by Knijnik (2004): “Is there any room for ‘misbehaving,’ ‘subversive’ ideas to operate, and materialize within educational systems, public (or even private)?” The examples quoted in the excerpt from the interview provided a reflection on the part of the students about how social projects involving educational institutions and the community are exemplary in the way they oppose hegemonic public policies which produce ruptures in the curricula of most schools and universities. The examples brought up here and in this text as a whole justify some of our conclusions about the work we have done: it must be clear that it is not a question of convincing students of one fact or another but of equipping them so that they can position themselves properly in relation to aspects of mathematical education; tapping into sources such as interviews is important to raise students’ awareness regarding controversial issues and professional confrontations. But it is necessary to go beyond timely sensitization and ensure the continuity of the reflections and discussions brought to light; it is always important not to lose sight of the problematization of truths (in the plural) and from the knowledge of the past bring out in students a disposition to face new and unknown situations.

References Balestri, R.  D., & Cyrino, M.  C. C. (2010). A história da matemática na formação inicial de professores de matemática. ALEXANDRIA Revista de Educação em Ciência e Tecnologia, 3(1), 103–120, maio de. Baraldi, I.  M. (2003). Retraços da educação matemática na região de Bauru (SP): Uma história em construção. 241f. Tese (Doutorado em Educação Matemática) – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. Benadiba, L., & Plotinsky, D. (2001). Historia oral: Construcción del archivo histórico escolar. Una herramienta para la enseñanza de lãs ciencias sociales. Buenos Aires, Argentina: Ediciones Novedades Educativas. Brito, A.  J., & Miorim, M.  A. (2016). A institucionalização da história da educação matemática. In A. V. M. Garnica (Ed.), Pesquisa em história da educação matemática no Brasil – sob o signo da pluralidade. São Paulo, Brazil: Editora Livraria da Física. Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1982). Na vida, dez; na escola, zero: os contextos culturais da aprendizagem da matemática. Cadernos de Pesquisa, São Paulo, 42, 79–86, ago. Chervel, A. (1990). História das disciplinas escolares: Reflexões sobre um campo de pesquisa. Teoria & Educação, Porto Alegre, 2, 177–229. da Silva, H. (2013). Integrando história oral e narrativas a abordagens pedagógicas problematizadoras na formação inicial de professores de matemática. Revista Educação PUC-­Campinas, Campinas, 18(3), 269–285, set./dez.  Internationally renowned for his studies in education, Paulo Freire was secretary of education of the city of São Paulo between 1989 and 1992. At the time, he proposed and put into practice a process of curricular reorientation based on his conceptions on education. According to Saul and Silva (2009), as of 1992, many Brazilian states and municipalities, committed to administration by the people, chose to create curricular policies according to the same assumptions implemented in the city of São Paulo. However, such proposals have since been changed because new management teams often had different ideological biases. 37

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da Silva, H. (2016). A história oral como abordagem em espaços formativos formais de professores de matemática. Histemat, 2, 168–184. de Souza, L. A. (2011). Trilhas na construção de versões históricas sobre um Grupo Escolar. 420f. Tese (Doutorado em Educação Matemática) – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. de Ferreira, M. (2016). História do tempo presente, história oral e ensino de história. In C. S. Rodeghero, L. Grinberg, & M. Frotscher (Eds.), História oral e práticas educacionais (pp. 125–136). Porto Alegre, Brazil: Editora da UFRGS. Flugge, F. C. G. (2015). Potencialidades das narrativas para a formação inicial de professores que ensinam matemática. 255f. Dissertação (Mestrado em Educação Matemática)  – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. Garnica, A. V. M. (2013). Cartografias Contemporâneas: mapa e mapeamento como metáforas para a pesquisa sobre a formação de professores de matemática. Alexandria: Revista de Educação em Ciência e Tecnologia, Florianópolis, 6(1), 35–60, abr. ISSN 1982–5153. Disponível em: https://periodicos.ufsc.br/index.php/alexandria/article/ view/37927. Acesso em: 06 set. 2018 Garnica, A. V. M. (2016). Pesquisa em história da educação matemática no Brasil – Sob o signo da pluralidade. São Paulo, Brazil: Editora Livraria da Física. Gasparello, A. (2007). Encontros de saberes: as disciplinas escolares, o historiador da educação e o professor. In A. M. Monteiro, A. Gasparello, & M. Magalhães (Eds.), Ensino de história: sujeitos, saberes e práticas (pp. 73–89). Rio de Janeiro, Brazil: Mauad X/ FAPERJ. Gomes, M. L. M. (2010). História da Educação Matemática: a propósito da edição temática do BOLEMA, 23(35A). Rio Claro, Brazil: Bolema. Gomes, M.  L. M. (Aug, 2014). História da Educação Matemática, Formação de Professores a Distância e Narrativas Autobiográficas: dos sofrimentos e prazeres da tabuada. Bolema. Rio Claro, 28(49), 820–840. Available from http://www.scielo.br/ scielo.php?script=sci_arttext&pid=S0103-636X2014000200820&lng=en&nrm=iso. Accessed 6 Sept 2018. Halbwachs, M. (2004). A memória coletiva. São Paulo, Brazil: Editora Centauro. Knijnik, G. (2004). O que os movimentos sociais têm a dizer à Educação Matemática? Anais do VIII Encontro Nacional de Educação Matemática. Educação Matemática: um compromisso social. Recife (PE), Junho de. Larossa, J.  B. (2016). Tremores: Escritos sobre a experiência. Belo Horizonte, Brazil: Autêntica Editora. Llewellyn, K. R. (2016). De volta para ao futuro: o poder político da História Oral na educação. In C. S. Rodeghero, L. Grinberg, & M. Frotscher (Eds.), História oral e práticas educacionais. Porto Alegre, Brazil: Editora da UFRGS. Matos, J.  M. (2006). História do ensino de matemática em Portugal: constituição de um campo de investigação. Revista Diálogo Educacional, Curitiba, 6(18), 11–18. Matucheski, S. (2016). Diferenciação e padronização: Um estudo sobre o Setor Litoral da Universidade Federal do Paraná. Tese (Doutorado em Educação Matemática) – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. Disponível em: https://repositorio.unesp.br/bitstream/handle/11449/146715/ matucheski_s_dr_rcla.pdf ?Sequence=3 Mcculoch, G. (2012). História da educação e formação de professores. Tradução: Juliana Passos. Revista Brasileira de Educação, 17(49 jan.-abr). Mendes, I.  A. (2012). Tendências da pesquisa em história da matemática no Brasil: A Propósito das Dissertações e Teses (1990  – 2010). Educ. Matem. Pesq., São Paulo, 14(3), 465–480.

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Miguel, A., & Miorim, M. A. (2002). História da matemática: Uma prática social de investigação em construção. Educação em Revista, Belo Horizonte, (36), 177–203, dez. Miguel, A., & Miorim, M. A. (2004). História na educação matemática: Propostas e desafios (p. 198). Belo Horizonte, Brazil: Autêntica. Nascimento, M. I. M., & Nascimento, M. N. M. (2010). O lugar da história na formação do professor. Revista HISTEDBR. On-line, Campinas,, (38), 186–196. Nóvoa, A. (1999). Apresentação. In F. Cambi (Ed.), História da Pedagogia. Trad. Álvaro Lorencini. São Paulo, Brazil: UNESP. Portelli, A. (2009). What makes oral history different? In L. D. Giudice (Ed.), Oral history, oral culture and Italian Americans. New York, NY: Palgrave Macmillan. Rolkouski, E. (2006). Vidas de professores de matemática – Impossibilidades de leitura. 298 fl. Tese (Doutorado em Educação Matemática) – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. Disponível em: https://repositorio. unesp.br/bitstream/handle/11449/102138/rolkouski_e_dr_rcla.pdf ?sequence=1 Santhiago, R., & de Magalhães, V. B. (2015). História oral na sala de aula (1st ed.). Belo Horizonte, Brazil: Autêntica Editora. (Coleção Práticas Docentes). 206 p. Saul, A. M., & Silva, A. F. G. (2009). O legado de Paulo Freire para as políticas de currículo e para a formação de educadores no Brasil. Revista Brasileira de Estudos Pedagógicos, Brasília, 90(224), 223–244, jan./abr. SÃO PAULO. Conselho Estadual de educação. Deliberação CEE Nº 126/2014. Altera dispositivos da Deliberação 111/2012. São Paulo: CEE, 2014. Schwarzstein, D. (2001). Uma introducción al uso de la história oral en el aula. Buenos Aires, Argentina: Fundo de Cultura Económica. Souza, L. A. (2006). História oral e educação matemática: Um estudo, um grupo, uma compreensão a partir de várias versões. 2006. 314 f. Dissertação (Mestrado em Educação Matemática)  – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro Souza, L. A., & Garnica, A. V. M. (2013). As matemáticas modernas: Um ensaio sobre os modos de produção de significado ao(s) movimentos(s) no ensino primário Brasileiro. Revista Latinoamericana de Investigación en Matemática Educativa, 16(3), 369–393. Tizzo, V. S., & da Silva, H. (2016). A História Oral na Formação Inicial de Professores (de Matemática): uma abordagem possível no estudo de política educacional brasileira (Vol. 8, 15, jul./dez). Campo Mourão, Brazil: Revista NUPEM. Tizzo, V. S., Flugge, F. C. G., & da Silva, H. (2015). Práticas Possíveis com a história oral na formação inicial de professores (de matemática). Bolema, Rio Claro, 29(53), 887–908, dez. Valente, W. R. (1999). Uma história da matemática escolar no Brasil, 1730–1930. São Paulo, Brazil: Annablume/FAPESP. Valente, W. R. (2005). A matemática na escola: um tema para a história da educação. In D. Moreira & J. M. Matos (Eds.), História do ensino da matemática em Portugal, 1(1), 21–32. Lisboa, Portugal: Sociedade Portuguesa de Ciências da Educação. Valente, W.  R. (2014). História da educação matemática no Brasil, 1(1), 331. São Paulo, Brazil: Editora Livraria da Física. Vianna, C. R. (2000) Vidas e circunstâncias na educação matemática. Tese (Doutorado em Educação) – Universidade de São Paulo, São Paulo.

Chapter 5 History of Mathematics Education and Oral History: Possibilities for the Classroom Vinícius Sanches Tizzo and Antonio Vicente Marafioti Garnica

Abstract: Besides presenting some experiences conducted with oral history in mathematics teaching (focusing on teacher formation courses), this text also proposes practical possibilities for classroom practice. Examples of several natures are shown, particularly relating to topics associated with the history of mathematics education. We believe that history, seen as a potent aid in the building of knowledge and because of the richness and diversity of the elements involved in a historiographic operation, enables problematization of some fundamental circumstances which lead us, both teachers and students, to become what we are. Therefore, these historiographic practices, together with oral history, are essential aids in teaching and learning processes (in general and, particularly, in mathematics) when transposed to the educational scenario in a promising and instigating manner. The examples shown must be understood as possibilities for action which trigger perspectives so that each reader, with their own interests, conceptions, and curricular demands, may adapt, transform, and transcend these suggestions, thus creating their own sequences and didactic scripts so as to include history, oral history, and history of mathematics education in teaching and learning processes.

Keywords: Mathematics education · History · Oral history · Classroom · Interventions Introduction Initiatives to use history of mathematics education in teaching are still not very common. This is largely because even though this line of research in mathematics education has grown swiftly in the past decade, it is still new if compared to other research trends, and the work developed often has a more theoretical approach, focused in the study of historical themes, without much concern for the use of such studies in school practice. However, in Brazil, there are some researchers who have developed and proposed historiographical

V. S. Tizzo (*) University of São Paulo State, UNESP, Rio Claro, Brazil A. V. M. Garnica University of São Paulo State, UNESP, Bauru/Rio Claro, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2019 A. V. M. Garnica (ed.), Oral History and Mathematics Education, History of Mathematics Education, https://doi.org/10.1007/978-3-030-16311-2_5

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exercises in the classroom to discuss aspects of teaching and learning mathematics, using oral history or not, or to discuss mathematical content effectively, though to a lesser extent. Examples of such exercises are (a) the use of interviews about school, educational legislation, and mathematics education with former teachers and school administrators; (b) the development of scripts for interviews with students about teaching practices they have experienced; (c) studies focused on specific moments in mathematics teaching, such as the new mathematics movement, and how this movement was distinctly appropriated in different schools and academic levels; (d) interviews with authors of textbooks; and (e) the examination of old school textbooks. Such exercises are mainly conducted during teacher-training courses (particularly teacher-training courses in mathematics). There are also examples of the use of oral history with children in early childhood education. This chapter intends to present and discuss some of such experiences. In general, these proposals are guided by a perspective according to which the diversification of paths toward knowledge is a potent constituent of the initial teacher-training process, as it tends to encourage and value student autonomy. In addition, they can contribute to active, participatory, and collaborative learning. This assumption enables us to understand that students are not destined to be receptacles of data but can be active and creative. As the initiatives using the history of mathematics education and oral history as strategies for teaching are still meager in quantitative terms, academic publications on this topic are rarely divulged. Very little has been published about it, and the papers dealing with this issue, with rare exceptions, treat it in an abstract way, or, as it seems to be more frequent, only encourage appropriation, without necessarily taking the time to show examples, techniques, challenges, references, and the potential of the introduction of the history of mathematics education and oral history in educational practice. This observation gave rise to this text, and we decided to base it on examples of effective implementations in specific pedagogical practices. Objectively, we intend to offer readers some practical possibilities for implementing the history of mathematics education and oral history in the classroom. With this objective in mind, we present an inventory of initiatives which can be incorporated into the creative work of other teachers and adapted to the unique circumstances faced by each professional, in their context, school, and classroom—in short, in their particular work scenarios. For instance, the geographical location of an institution can be a decisive factor in the process of choosing the themes to be worked on. The moment a class is formed can significantly influence the reaction it will have to the proposal of working with the history of mathematics education and/or oral history. Thus, far from being models to be rigidly followed, the examples of pedagogical implementation of history of mathematics education and oral history presented should be seen as triggers of perspectives, so that each reader, faced with their specific purposes and curricular demands, and guided by their conceptions, can adapt, transform, and, perhaps, transcend the examples with their own sequences and activities involving history in teaching and learning processes. Even though the following examples are marked by the different institutional contexts in which they are inserted, it is always worth pointing out that planning a teaching proposal is an undertaking which requires creativity. History can be transposed into the educational setting in a thought-provoking way, both for its significant role in the process of building knowledge, and for the richness of the experience of understanding past circumstances which determine who we are in the present. For instance, oral history can be used in pedagogical practices which enable students and teachers to develop a problematizing approach to learning which does not ignore the feelings of belonging, integration, and historical responsibility which emerge, for example, from interviews brought to the classroom with various educational agents, present and past. It is an approach which is not limited to the stimulation of memory, but also acts in favor of memory.

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By presenting some practical examples of pedagogical actions and their developments, without presuming to prescribe a playbook, as we understand, this is not what pedagogical practice is about. We have attempted to list some guidelines, so that teachers, at various levels of education, can venture in their first incursions into the history of mathematics education and oral history with their students, or simply enrich their own repertoire of pedagogical instruments. 

History in Mathematics Education and Initial Training: Possibilities for Integration

We begin listing examples and possibilities of historiographical exercises in the classroom with the proposal presented by Miguel and Miorim (2004), as it was, to some extent, a groundbreaking study of its kind in Brazil. It not only served as a theoretical reference, but as an inspiration for other initiatives which will be exemplified further on. For this reason, we will dedicate a more generous portion of the text to this example, as we believe that the problematizations presented are mirrored in the other examples mentioned. Miguel and Miorim (2004) proposed a teaching approach for training mathematics teachers which includes the concepts of communities of memory and social practices linked to the history of mathematics and mathematics education in Brazil. In order to problematize the mathematical and educational cultures of teachers during the initial training process, the authors evoke memory as a heterogeneous social construct with the purpose of promoting activities which aim to analyze social practices of different groups, which operate in different geographic spaces, and are organized according to institutionalized work relations. From this scenario, which involves specific social agents and specific social practices, developed through institutional means, stems what Miguel and Miorim (2004) call “communities of memory.” Among examples of materials produced by different communities of memory and which can trigger possibilities for the initial teacher-training process are individual reports of future teachers about a particular mathematical theme and its teaching, textbooks, academic texts, documents and proposals, and also reports from teachers about their teaching experiences. To think about the possibility of organic participation of history in a classroom setting, particularly in the formal processes of teacher training in mathematics, is a strategy which breaks with a mystical perspective, usually preserved in mathematics courses, of going from theorem to theorem as if this sequence of theorems was natural, both as a sequence, and in the idea that the work in mathematics occurs in a well-drawn sequence which allows it to infallibly overcome any obstacle, as it is believed that mathematical contents have always been logically organized and are established beforehand. Moreover, at times, the elegance and objectivity with which the contents are explained limit the possibility of showing the difficult path mathematicians have tread to achieve this structure: linear, effective, powerful, and eternal. For Miguel and Miorim (2004), the mathematical disciplines which integrate the teacher-­training curriculum focus on axiomatic-deductive approaches which limited by formality and logical sequence, ignore significant elements in the development of teachers who will work in schools. They ignore, for instance, the development and constitution of mathematical propositions and concepts in the midst of distinct socially constituted practices throughout history. Thus, Miguel and Miorim (2004) problematize history and mathematics as a stimulus to achieve pedagogical objectives related to mathematics pre-service teachers. Some such objectives, for example, are enabling the perception that mathematics is a human creation and that there are reasons—often quite simple—which prompt people to use mathematics. There are practical, social, economic, and physical demands which stimulate the development of mathematical ideas; there are numerous connections between mathematics and

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other fields of knowledge; and curiosity has the potential to create generalizations and broaden ideas and theories; and the notions that mathematicians possess about mathematical objects can change, transform, and develop over time; and there are conditions for the creation of a structure, a concept, axiomatization, and proof. Miguel and Miorim, in particular, developed this exercise in a specific discipline, focused on discussing methodologies for teaching mathematics, in a training course for mathematics teachers at a large Brazilian university.1 First, the authors suggested that the group be divided into thematic research groups which would remain unchanged until the completion of the course. The first activity was related to what the authors call “individual or subjective memory” about the theme to be investigated, that is to say, they suggested that the students individually write down their memories in relation to that theme. Each student’s text was later shared with the class. In the exercise reported by Miguel and Miorim, four themes were suggested to students: trigonometry in Brazilian school culture; logarithms in Brazilian school culture; geometry in Brazilian school culture; functions in Brazilian school culture. The objective of this first step is to promote a problematization of the individual academic history of future teachers and the pedagogical functions of this academic history— particularly the role that these individual memories play in the way each teacher/student conceives mathematics and mathematics teaching. The texts created during this activity support the recognition of a social memory of that group of students and would be the foundation for the development of the teaching proposal. The teacher responsible for the discipline mediated the sharing of individual memories, which is done orally. For Miguel and Miorim (2004), this collectively problematized sharing is a preliminary form of academic history of mathematics and mathematics education,2 a history which resisted the action of time and was kept in the memory of that particular group of students. In fact, this first form of history of mathematics and mathematics education should not be understood simply as an imaginary and arbitrary construction of the student, based exclusively on their cognition. In a certain sense, it is subjective knowledge, because it is emotional and uniquely constituted, but it is also objective, because it is produced in a culture (academic) which circulates within institutions (school and family, for example). According to Miguel and Miorim (2004), the next step of this pedagogical activity is the elaboration of guiding questions. For the authors, such questions, elaborated by the students themselves (pre-service teachers), effectively trigger the investigation, as the reference of the problems is simultaneously individual prospective judgments (since they are designed for further pedagogic discussions) and retrospective (because their constitution is conditioned by the perspectives and limits of memory, bringing to the present a situation experienced by the student in the past, when they were not yet future teachers). The second step proposed by Miguel and Miorim (2004) is comprised of two stages. In the first, the students must conduct an analysis of the teaching programs and proposals, the educational guidelines for the teaching of mathematics in Brazil, based on official documents and other supplementary texts, from the mid-1900 to the present. During the second stage, students must conduct an analytical exercise with mathematics textbooks which were used throughout the country in the same period. One of the objectives of this stage is to  It is the Universidade Estadual de Campinas (UNICAMP), a public academic institution in the state of São Paulo. 2  It is important to point out that the focus of these activities is always on the study and use of the history of school mathematics, that is, in the ways in which mathematics is revealed in teaching and learning situations, whether in formal instances (such as schools) or not. This is not a study focused on what has been called the history of mathematics. 1

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enable students to reconsider the different ways through which mathematics teaching was viewed over time, and how educational policies determined what should be taught and how, when the students themselves initially had contact with school mathematics. This period (from the mid-1900s to the present) is purposely fixed, given the existence of historical sources. This leads students to dialogue with other manifestations of school mathematics, besides those which are part of their personal experiences. After that, students discuss, in their respective groups, a series of activities developed by the teacher trainer. These activities are elaborated as problems to be solved and are based on the themes initially proposed for the investigation of each group. Miguel and Miorim warn that these activities are elaborated with reference to two criteria, namely, the first is based on the exploration of problems and situations whose objective is to show that the themes under investigation are constituted from different social practices, which are instituted in multiple cultural spaces and at different times; the second criterion is based on the idea that the themes under discussion suffered historical transformations impacted by factors and problems linked to diverse social practices. For this reason, the activities follow a predefined sequence, in order to allow a chronology of the transformations to be retraced. Miguel and Miorim (2004) point out that the last step of the classroom proposal consists in the elaboration and discussion, based on adequate literature, of interview scripts to be carried out with high school3 teachers and students. Subsequently, pre-service teachers go to the field to interview teachers and students. It is worth mentioning that questionnaires can also be used as another way of obtaining information from high school teachers and students, who in turn constitute other communities of memory. Finally, the information obtained is analyzed within each group of students and then shared with the whole class. Based on the investigations completed, students write texts which aim to gauge the complexity of the dialogue between different communities of memory, creating a personalized, meaningful, enlightening, interactive, and multidisciplinary history of the school mathematical culture, related to each of the topics studied. 

History of Mathematics Teaching: An Introduction

Our second example is derived from the work of Gomes (2012). For questioning several aspects of the program of the course in history of mathematics teaching, it has become the reference material for students of the final phase of the extramural teaching degree at Universidade Federal de Minas Gerais—UFMG.4 According to Gomes (2012), both research and curricular proposals have pointed to history as a component of significant relevance to the problems which involve mathematics teaching and learning. For the author, the understanding of the power of history in training  In short, in Brazil, education is divided into several stages of schooling: basic education is composed of early childhood education (optional), elementary school (compulsory), and secondary school (seen as a right of all citizens). Early childhood education is the first stage of basic education, and its main purpose is the development of children up to 6 years of age. Elementary school is compulsory for children between the ages 6 and 14. It lasts 9 years. Secondary school lasts 3 years and is compulsory for those wishing to have university education (higher education). To have access to the higher education, students must pass a selection exam for a specific course. There are public institutions (free and secular) and private institutions which offer all levels of education, from kindergarten to university. Public universities are of two natures: state universities (managed according to the legislation of each state) and the federal (whose responsibility befalls the federal government), both circumscribed minimum requirements as per federal legislation. 4  In Brazil, two majors are offered for university-level courses in mathematics: bachelor of mathematics (aimed at training mathematicians) and the teaching degree in mathematics (aimed at training mathematics teachers). Extramural courses are offered by public and private institutions and follow specific legislation. UFMG is a federal public institution. 3

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settings has fostered the inclusion of disciplines related to history as part of the curriculum of teacher-­training courses. The analysis of several curricular projects of undergraduate courses in ­mathematics, of several institutions, shows that, indeed, disciplines with these characteristics have become a part of the teacher-training process. However, Gomes (2012) notes that the inclusion of the discipline history of mathematics in the initial training of teachers is usually done in reference to the historical development of mathematical concepts, without emphasis to the historical dimensions which involve the teaching of such concepts. Recently, however, this trend has been changing, due to a movement which occurred in several countries, including Brazil, and consists in the creation and consolidation of a field of research known as “history of mathematics education.” Gomes (2012) argues that the historical understanding of aspects related to teacher training and practice is a substantially relevant component in the training of new teachers, as such knowledge, if properly examined, has the potential to foster an expansion of the students’ understanding of their own conceptions regarding the profession and teaching practices which involve mathematics. Moreover, if adequately woven into the training process, such knowledge can help future teachers propose alternative and positive actions, as opposed to what has always occurred in school whose practices are usually limited to the reproduction of past methods without effectively understanding them. While analyzing the circumstances of the creation of the extramural teaching degree in mathematics at UFMG, Gomes (2012) observed that the curricular proposal values historical aspects related to the teaching of mathematics. There is even a specific discipline in the syllabus, which the author proposes and is described here as an example. The course is comprised of 30 classes and aims at fostering the discussion of the evolution of mathematics teaching in Brazilian elementary education, as well as the emergence and main characteristics of movements such as the New School and the New Mathematics, while continuously examining the universalization of basic education in Brazil. As Gomes recalls (2012), it is worth mentioning that as the theme is the history of mathematics teaching in Brazil, the syllabus could accommodate other themes. However, Gomes (2012) argues that a discipline should not have the presumption of exhausting all possible approaches and themes. The course should not merely play an informative role, its main objective should be to create the conditions for students to broaden their understanding of their studies, especially in the future, when they become teachers. Considering these dispositions, the discipline history of mathematics teaching is constituted of three units. Unit number one aims to present a chronological overview of the history of mathematics teaching in Brazil, considering Brazilian education in the colonial, imperial, and republican periods,5 with specific focus to the teaching of mathematics. Considering that the consolidation of an education system in the country occurred only in the twentieth century, a significant portion of the time is devoted to the discussions about the Brazilian Republic. At this stage, there is a more direct focus toward mathematics, especially the first attempt at the organization of the Brazilian education system in 1931, and the new mathematics movement between 1960 and 1970. According to Gomes (2012), the second unit of the discipline history of teaching mathematics is dedicated to the specific task of discussing the transformations of mathematical contents taught at school over time. The example chosen for this discussion involves rational 5  Brazil was “discovered” in the year 1500, as a result of the Portuguese great navigations of the sixteenth century. The “colonial” period lasted from the beginning of the sixteenth century (in 1530, Portugal sent its first colonizing expedition to Brazil after the discovery) until 1822, when the first emperor is crowned (Pedro I of Brazil, also known as Pedro IV of Portugal), who declared the independence from Portugal. Pedro I abdicated in 1831, in favor of his son. The Republic was proclaimed in 1889, when the then emperor Pedro II was deposed after a 58-year reign.

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and irrational numbers and the references are mainly textbooks. In this unit, in particular, three different periods of the twentieth century are examined: the first three decades, from 1931 to the early 1960s, and the period between 1960 and 1970, the time of the introduction and consolidation of the new mathematics movement in Brazil. In order to expand the discussion proposed in the first two units of the program, several texts are studied. To facilitate student access, the texts are available through a digital platform, called Moodle. The third unit has a double function: (a) show students the power of autobiographical sources to understand the past of mathematics teaching in Brazil and (b) lead future teachers to think of themselves as participants in the history of teaching mathematics in Brazil. The work related to autobiographical sources is conducted from excerpts of memoirs of Brazilian authors who lived in the twentieth century and who narrate their school experiences. The effort to make students feel as active participants in history is mainly based on the proposition of a reflexive exercise related to the personal and academic life of the student, who must prepare a memorialistic text using autobiographical texts already studied as references. 

Articulating Oral History and Elementary Education: Possibilities for Work with Children

The third example presented is an activity proposed by Souza (2011). The author sees the work with history not as merely informative, but formative, because it enables the perception that each individual is a historical being. The activity in question was performed by Souza with children between 8 and 9 years of age. At the time, the children were students of a municipal school in a small city in the countryside of the state of São Paulo. This municipal school occupied a building which originally had been built to house an institution called Grupo Escolar,6 from 1920 to 1975. Souza (2011) recalls that for years the so-called Grupo Escolar (TN: low-schools or grade-­schools in English) has been a symbol of cohesion and status and recognized by a significant portion of the population as “real schools.” The operation was as follows: those institutions consolidated the previously isolated schools into a single physical structure, segregating students by level, with one teacher responsible for each of the classes and operating under the supervision of a school principal. In addition, as these were graded schools (as pupils were distributed according to the different “grades” of schooling), Grupos Escolares served as the initial homogeneous standard for early childhood education, imposing regulations on behavior by re-enacting daily rituals aimed at domesticating the body and shaping the soul, according to the socially acceptable norms. The Grupo Escolar of the small town where Souza conducted this activity was inaugurated on April 1920, and until its extinction in 1975, was considered a symbol of modernity, both with regard to education and to its architecture, which were really unique and grandiose for such a small city. To this day, the impressive building clearly stands out from the other constructions in the area.  Grupo Escolar (low-school or graded school) were primary education schools, created in Brazil at the end of the nineteenth century, as a result of a policy aimed at emphasizing the initiatives of the Republic, in detriment of the elitism of the Empire. These were urban schools. New strikingly modern buildings were constructed to house these institutions in order to characterize them as “Temples of Knowledge.” The same structure which created Grupos Escolares also provided for the grouping of students according to educational levels. Thus, each grade was housed in a different room, with a teacher for each room, and a schoolmaster for the whole school (a position created as a result of the educational reform). The time spent in school was strictly controlled, and all school activities were duly logged and assessed. Grupos Escolares were created in the state of São Paulo, but they rapidly spread throughout the country. This type of institutions was extinguished in the 1970s, and the buildings were passed on to municipal governments. Nowadays, early childhood educational institutions occupy such buildings.

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In the beginning, with the objective of exploring notions of space, time, and school material culture, the author presented the students with some historical information about the building they were in, and an exhibition with photographs of furniture and other school objects, and drawings which highlighted traces of the architecture of the building which resisted so well through time since its construction. This initial stage of the activities, according to Souza (2011), served to trigger two exercises: one aimed at the recognition of the building itself and its history and the other involved conversations7 of the students with neighbors, relatives, and friends who had studied in that institution. Such exercises prompted students to look for places, furniture, and utensils in the building, which later served as the basis for a discussion of the objects found and their importance in the present, and to suggest ways in which they might have been used in the past. The aim was to make clear that some practices specific to the pedagogical organization of the school, teaching methods, and educational modernization processes can be understood by examining school materials. That is to say, the objective of the exercise was the problematization of the material culture of the school. Later, in order to promote a discussion with the students about the different situations which occur in the same scenario, the children were encouraged to talk about the physical space of the school and tell the stories they had heard in the conversations with their parents and friends. According to Souza (2011), this new step contributed to reinforce the idea that space is a “place practiced,”8 always “moving,” which reinforces important concepts for history such as of a “movable landscape.” Among the stories presented by the children, many were linked to urban legends9; to the justification of the use of school uniforms; segregation by gender; to the conduct of former students and their relationships with teachers. With the purpose of analyzing the stories as triggers for problematization, the next step consisted of the elaboration of a script for an interview to be conducted with a retired teacher who had studied at the old Grupo Escolar in the 1950s. She had later become the principal of the school and at the time of the interview, she was the Secretary of Education and Culture of the city. The last step consisted in presenting the video of the interview with the retired teacher and an exhibition of photos of the former Grupo Escolar and moments of the project. For Souza (2011), this activity provoked the curiosity of the students and operated as a potent conduit of knowledge about local history which is rarely reported in books. It also fostered the understanding that history is the flow of life and that, therefore, each of the children was a historical being living in the community. In order to conduct all such activities at the school, it was necessary to obtain authorization so that the teachers of three classes could be partially freed from their usual duties and participate in the project. This shows that the partnership with the institution’s pedagogical team is vital. Prior to the beginning of the activities, a detailed survey was carried out in the institution’s former school archive, which supported the articulation between school documents and educational memory, resulting in important information for each of the activities conducted. The interest of the children in the old institution led to the elaboration of questions whose main themes were the behavior of former students, forms of punishment, style and function of school uniforms, teaching (particularly mathematics) and ways topics were  It should be noted that the children involved in this activity were illiterate; therefore the interviews were conducted and shared orally. For the elaboration of the interview script, which will be dealt with next, the children suggested the questions which were registered and organized by the researcher. 8  A practiced place is not merely an existing space but space which is lived, meaningful to those who use it and move around it. 9  As Grupos Escolares were installed in very large buildings, now very old, they are linked to several stories, many of which are fantastic, ghost stories, which children hear and tell, thus creating an urban-legend atmosphere. 7

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taught, learning difficulties, situations and school spaces which no longer exist today (such as the orchard, classrooms in the basement of the building), school meals, and the fact that boys and girls studied in separate classrooms. 

Oral History in the Initial Training of Mathematics Teachers

The fourth example described in this chapter is taken from an activity presented by Silva (2013). In that text, the author deals specifically with work done with oral history, aimed at developing specific topics proposed in the syllabus of a discipline offered to students attempting to obtain a teaching degree in mathematics (future teachers of mathematics) of a large Brazilian university, in the state of São Paulo.10 Silva (2013) does not conceive the discipline “Fundaments of Elementary Mathematics” only as a scenario for the formal demonstration of basic mathematics results but also as a space which allows students to revisit, deepen, and broaden concepts and pedagogical practices related to elementary mathematics. It is according to this perspective that the activities and strategies developed by the author are described. First, seven themes directly related to elementary mathematics were presented to the students, as follows: calculation strategies (algorithms); algebraic thinking; fractions; proportional reasoning; exponents, whole and real numbers; and logic. The activities were developed in groups of three or four members. Each group got involved with a specific investigation whose outcomes and results were presented to the whole class at the end of the semester. In order to present her intentions to the students and to support the studies which each group would conduct, the researcher (also the teacher responsible for the discipline) previously selected and made available to students a series of materials, including the book Mathematics in Elementary Education—Teacher Training and Classroom Application, by Van de Walle (2009)—which discusses mathematical content from the perspective of mathematics education, taking into account some research results in the area, the National Curricular Parameters (NCP), the Curricular guidelines of the state of São Paulo,11 textbooks, supplementary materials, and other academic texts. An interview with an elementary school teacher should be conducted by each group, which would then be included in the set of supporting materials for the proposed activities. Each group analyzed and presented each of the materials to the whole class, except the book by Van de Walle, which was studied and discussed collectively by the class, under the guidance of the researcher. The activities began with the elaboration by the students of an individual memorialistic account about the theme chosen by the group, based on the seven themes initially proposed. This strategy is in line with the work of Miguel and Miorim, already described herein. The objective of this activity is to trigger an investigation in mathematics education and support the elaboration of a script for the interview each of the groups will conduct with an elementary school teacher. Silva (2013) discussed some guidelines for this activity, for example, in relation to “fractions,” the researcher suggested that the students included in their narratives what each knew about the theme, that is, the ideas which they thought were related to it; the contents of elementary school which involve fractions in any way; methods for teaching fractions they knew; the way each one learned this content; memories of experiences relating to fractions; and so on.  “Fundaments of Elementary Mathematics,” a mandatory discipline for obtaining a teaching degree in mathematics at Universidade Estadual Paulista, Campus Rio Claro 11  National Curricular Parameters (Parâmetros Curriculares Nacionais, known as PCN, in the Portuguese abbreviation) are official documents developed by the Ministry of Education (thus at federal level) which contain all national educational legislation for public schools throughout the country. Each state of the federation simultaneously issues Curricular Guidelines, which complement PCN. 10

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Silva discussed with the students the general objectives of the use of each of the materials made available in order to evoke students’ critical thinking about the materials and their uses, as well as enhance their ability to question and establish relationships between the different social practices which involve each of the themes, Subsequently, the students conducted the interviews with elementary school teachers. This sequence of activities resulted in the elaboration of the scripts for the interview, and during the interviews, strictly speaking, made the dialogue between the different memory communities (the researcher-teacher, students, and interviewees) more complex and expressively richer. According to the researcher, the preparation and conduct of the interviews provide students, future teachers, direct and more realistic contact with the difficulties and possibilities which involve teaching and learning mathematical content, as classroom practices are reported by people who are actually teaching. In order to organize this step, the fundaments and procedures of oral history methodology were discussed with the students, and this prepared them for the interviews. At the beginning of the activity, under the guidance of the researcher, the students developed a script with standard questions for the interviews. This script was then adapted by each of the groups, thus incorporating questions arising from the analyses of other materials and individual memories. Prior to the interviews, the interviewees received the script with the questions and a cover letter explaining the objective of the activity. For instance, the letter informed interviewees that the audio of interview would be transcribed12 and would be made available to them who, after reading, would sign a letter of assignment of rights, so that the contents could be used for academic purposes, as well as other pedagogical activities, should the opportunity arise. The students did the recording and transcription, whereas the textualization work was carried out by an undergraduate research intern, whose work consisted in systematizing the data produced during the intervention and research and analyzing some parts of the work. Silva (2013) concluded that the dialogue promoted between the different manifestations of mathematics and mathematics education, observed from the perspective of different communities of memory, enabled future teachers to compare different points of view on the themes examined in the discipline, promoting a critical multidimensional view of the themes investigated. 

Oral History as Support for Problematizing Public Policy in Education and Mathematics Education

The fifth example presented here, similarly to the previous one, refers to an intervention with undergraduate students, pre-service mathematics teachers. The results of this intervention, which involved the use of oral history methodology as a didactic-pedagogical approach in the training of future mathematics teachers, were published as a master’s thesis (Tizzo, 2014). The study, involving research and intervention, was developed in a specific discipline, in the same course in which Silva had previously conducted research. However, there are important differences between the two investigations, two of which are very significant: (a) contrary to what happened with Silva, Tizzo’s intervention was developed in a

 Transcription, textualization, and letter of assignment of rights are steps in the work with oral history. These steps were discussed in a specific chapter in this book. In short, through transcription interviews (conducted orally) are registered in writing; during textualization the written account is refined in order to gain greater fluency and purge the usual oral language idiosyncrasies. Letters of assignment are documents signed by the interviewees, which give the researcher the right to use the textualization. 12

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discipline aimed at studying Brazilian educational policy,13 not specific aspects of teaching and learning mathematical content, and (b) Tizzo was not responsible for teaching the discipline but acted in cooperation with the professor in charge. Initially, the syllabus of the discipline was carefully studied for the elaboration of strategies with oral history as a didactic-pedagogical approach. After that, an action plan for the intervention was prepared in cooperation with the professor in charge of the discipline. The plan listed several themes to be approached by students: Education and politics in Brazil; memory: some aspects of Brazilian educational policy; debate between exclusion and inclusion; educational and the mathematics teacher: evaluation methods; school violence. Six pairs of students were formed, two of which chose the theme “school violence.” Due to the partnership, the professor in charge of the discipline agreed to use the resulting papers as a substitute of a written final exam. The study on these themes was scheduled to take place in nine steps, as follows: 1. Writing an individual memorialistic text about the theme chosen for the investigation. 2. Study of texts made available both by the researcher and the professor responsible for the discipline, in relation to the themes chosen. 3. Presentation and discussion of the fundaments and procedures of oral history. 4. Creation of an interview script, so that each pair of students could later conduct an interview with a practicing teacher. 5. Contact and interview. 6. Interview transcription. 7. Textualization of the transcription. 8. Presentation and discussion of the work of each pair by the whole class. 9. Formal acceptance of textualization by interviewees and issuance of letters of assignment of rights.

While analyzing this intervention, Tizzo (2014) found that there was a significant involvement of the students, future teachers, during all activities, and that this involvement seemed to have positively influenced students’ expectations and perspectives regarding their future performance in real classrooms. According to Tizzo, the intervention proved to be significant in the training process of future mathematics teachers, allowing students to experience a scenario in which personal circumstances, whether linked to school practices or not, related or not to the teaching of specific mathematical content, were problematized and evaluated so that they could be integrated into the future practice of those students, allowing each of them to reflect on decisions they will routinely have to make when they actually start teaching. 

Oral History Approach in the Discipline “Content, Methodology, and Teaching Practice of Mathematics” in a Pedagogy Course

The sixth example given is derived from the work of Flugge (2015) who developed research whose objective was to understand some aspects of how the work with oral history could help future teachers reflect on school routines and broaden the discussion about teaching practices in mathematics. While in the fourth example, Silva dealt with an intervention with future mathematics teachers in a discipline which involved basic mathematical content and the work of Tizzo, for the fifth example, dealt with future mathematics teachers

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 The title of the discipline is “Brazilian Educational Policy.”

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studying public policy (not necessarily focusing on mathematical content), our sixth example, Flugge’s work, investigates an intervention with students at a pedagogy course14 and a discipline focused on discussing content and methods for teaching mathematics.15 The main component for the intervention was the use of oral history, in a similar manner to that of the exercise proposed and analyzed by Tizzo. In order to develop the investigation, Flugge (2015) observed classes (as the researcher was not in charge of the discipline) and used a virtual platform: Moodle. Additionally, the researcher studied extensive documentation which included the pedagogical political project for the major in pedagogy (TN: B.Ed.) and the syllabus of the discipline with its specific bibliography. The activity proposed by Flugge resulted in work which was used for the final assessment of students’ performance in the discipline. Mathematical content examined were as follows: quantities and measurements; geometry (space and shapes); introduction to fractions; addition and subtraction; multiplication and division; and treatment of data. The entire intervention was programmed to be conducted in five stages: 1. Elaboration of a memorialistic account about the previous knowledge of students in relation to the mathematical content chosen. 2. Study of content, based on texts and other materials available, at the end of which a summary on the topics and a questionnaire for interview with a mathematics teacher (or a teacher teaching mathematics16) would be created. 3. Discussion of oral history procedures for the preparation, conduction, and treatment of the interview (transcription and textualization). 4. Preparation of an individual text about the steps taken during the intervention. 5. Presentation of a seminar on the results obtained by each group.

After the intervention, and with the results of each group in hand, Flugge (2015) identified a variety of experiences of the students. These experiences, reported to the researcher by the students themselves, were about the ways they conceive mathematics and mathematics teaching. According to Flugge, the use of oral history enabled students to come in contact with the experiences of several people, which fostered significant reflection and raised questions regarding the practices of students who would soon teach mathematics in the initial schooling years. In addition to enriching the understanding of the topic investigated by each group, the intervention revealed the difficulties of these students in relation to mathematics and mathematics teaching. The anxiety of these teachers-in-training in relation to the mathematics that they would have to teach was quite noticeable.

 Pedagogy courses are university courses for training teachers to work at elementary education level. In elementary schools, during the first five years of the so-called elementary education, a single teacher is in charge of a classroom and must take care of fundaments—such as literacy in the mother tongue, counting, elementary operations, and introduction to history, geography, health, etc. (TN: equivalent to a Bachelor of Education in most countries). 15  This discipline is called “Mathematical Content, Methodology, and Teaching Practice,” and is offered as part of the program for a major in pedagogy at Universidade Estadual Paulista, Campus Rio Claro, a large public university in the state of São Paulo. 16  Literature dealing with mathematics education refers to those teachers who have a degree in pedagogy (as this major also deals with introduction of mathematics to students, in addition to other content) as “teachers who teach mathematics,” whereas the teachers who hold a specific teaching degree in mathematics are called “mathematics teachers.”

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History in Mathematics Education: A Discipline and a Pedagogical-Methodological Contribution for Teaching and Research

The seventh example presented in this chapter is the didactic-pedagogical proposal made by Gonzales and Martins-Salandim (2015). The authors describe an intervention whose objective was to elaborate, apply, and evaluate a didactic, pedagogical, and methodological experience conducted with a group of pre-service teachers. These were students in the program for obtaining a teaching degree in mathematics in a large public university in the countryside of the state of São Paulo.17 The activities were developed in the discipline “History in Mathematics Education.” The work began with a discussion about how each student understood history and mathematics education. Shortly after, in order to revisit some of the ideas which emerged from that discussion, the class conducted a debate about the documentary “The danger of a single story”18 whose central themes are concepts such as versions, truth, plausibility, narrative, history, and memory. After that, readings and discussions were proposed, which, in addition to suggesting investigations and analyses, focused on giving readers an overview of teaching attitudes and possible resources to be used in mathematics classes, problematize the relation between history, teaching, and mathematics. Examples of such texts are D’Ambrósio (1999),19 Miorim and Miguel (2001),20 Gomes and Brito (2009),21 and Garnica (2013).22 A study by Martins-Salandim, Souza, and Fernandes (2010),23 and the movie Narradores de Jave,24 were presented and discussed with the class, in order to provide sup This is Universidade Estadual Paulista, Campus Bauru.  Video (freely available) of a TED talk (technology, entertainment, design)—Ideas Worth Spreading—by Nigerian writer Chimamanda Adichie, in July 2009. The author narrates situations from which she concludes that reducing several versions to a single one annihilates life. In her particular case, she claims that this homogeneous form of storytelling has limited her view of her own country and her own people. She goes on to say that when she moved to the United States, her roommate could only see her through the socially created stereotypes about something called “Africa” and that several times, she yielded to the temptation of those hegemonic versions and stereotypes linked to a “country” called Africa. In this narrative, Chimamanda examines various beliefs about Africa and shows how she also gave into the temptation of hegemonic versions. The version used by Gonzales and Martins-Salandim has Portuguese subtitles (https://www.ted.com/talks/chimamanda_adichie_the_danger_of_a_ single_story/transcript?language=pt-br) 19  In this text, D’Ambrósio discusses aspects of historiography to examine how it is possible to understand and study certain political issues involving mathematics education through a historiographical approach. It is written in Portuguese and has not been translated into English. 20  Miorim and Miguel (2001) discuss the different ways history has been used in studies about mathematics education, establishing three great approaches that the authors call history in mathematics education, history of mathematics education, and history and mathematics education. It is written in Portuguese and has not been translated into English. 21  The text by Gomes and Brito, published in the oldest Brazilian journal of mathematics education, is a study of all the works related to the history of mathematics education presented at the Brazilian Meeting of Graduate Students in Mathematics Education (EBRAPEM). Written in Portuguese not translated 22  Garnica (2013) discusses traditional and alternative conceptions of history and historiographical practice to propose an updated approach to the history of mathematical education. The text, written in Portuguese, has not been translated; however some of its considerations can be found in the chapter the second author contributed for this book. 23  In this text, Martins-Salandim, Souza, and Fernandes discuss methodological aspects of oral history linking such methodology to the research in mathematics education. 24  Narradores de Javé is a Franco-Brazilian film of 2003. It features the small town of Javé which is on the verge of being submerged due to the construction of a dam in the region. The inhabitants of Javé will not be compensated and have not even been notified because they have no proof of ownership of the land they occupy. Aggrieved, the villagers find out that the town could be saved if the historical value of the town could be proven “through scientific documentation.” The villagers then invest all their hopes in the elaboration of a history of the small town, which must be written by the only literate resident—the postman. The memorialistic narrative of each resident is done orally and recorded by the postman. The elaboration process of this “historical work” is slow, as each inhabitant insists on continually altering “some lines” and ensuring that their name is quoted. 17 18

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port for the interviews that students would later conduct with people who experience or have experienced teaching in elementary education to some extent (the next activity proposed). In face of these discussions and considering the question “How do you relate to mathematics inside and outside of school?” as the underlying theme, students began to write interview scripts. The interviews were transcribed and textualized and the results shared during classes. Another exercise proposed to the students was intended to foster the reflection (and use) of the history of mathematics education as didactic-methodological approach, in order to facilitate the understanding of the construction of mathematical concepts and knowledge. To this end, Gonzales and Martins-Salandim (2015) suggested the reading of the text by Mendes (2009),25 as the starting point for the activity. In view of all the readings and discussions which took place in the classroom, under the coordination of the researchers, debates about the potential of the history of mathematics education as a didactic resource were conducted and the concept of a logarithm was chosen to be the central mathematical object of the exercise. Based on an investigative exercise proposed by Miorim and Miguel (2002), initially, the teacher trainers presented an activity to the students in which they should point out, among other topics, those that had any relation with logarithms. The objective of this first activity, according to Gonzales and Martins-Salandim (2015), was to search for evidence of the use of logarithms in social contexts and whether this daily use of logarithms was known to students and/or was part of the studies in their undergraduate course. Before the debates, to contextualize the activity, a video was presented in which the use of logarithms in the calculation of the human development index (HDI) is discussed. Discussions involved arithmetic and algebraic-­functional concepts of logarithms and allowed researchers and students to come to the realization that the dominant conception of students about logarithms was algebraic-­functional,26 which supports the concept of logarithms from potentiation and graphic representation of logarithmic functions. After analyzing several aspects of the topic (e.g., the etymology of the word logarithm and definitions of logarithm presented in textbooks of the early twentieth century), the discussion turned to the so-called “algebraic-functional” conception of logarithms and the distinction between this concept and the arithmetic and geometric conceptions of logarithms, as in mathematics teaching the prevailing conception (algebraic-functional) represents a rupture between preceding conceptions (arithmetic and geometric). Gonzales and Martins-Salandim (2015) believe that this process served to show students that the formulation of the algebraic conception of logarithm was triggered by studies and applications of the properties of logarithms in the fields of mathematical physics, geometry applied to nautical sciences, music theory, and atmospheric theory. In short, in such applications, connections were established between certain natural phenomena and the theory of logarithms which initially, was associated with the field of differential and integral calculus, thus giving rise to the algebraic-functional conception.

 In this text Mendes (2009) points out some similarities and differences between the history of mathematics and the history of school mathematics. The book is written in Portuguese, and has not been translated into English. 26  Under an arithmetical conception, logarithms are numbers, and teaching them is a consequence of teaching of progressions, sequences of numbers with a specific formation law. Under a geometric conception, the logarithms are seen as values of a function (the logarithmic function), and the focus lies on the variation (functional conception), dealt with mainly from graphical representations. The algebraic-functional conception approaches logarithm from the study of exponents and then defines them as values of a specific function (in teaching, according to this conception, logarithms derive from the study of functions). 25

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The experience with history in mathematics education makes it possible for work which involves history and mathematics education to go beyond what has been done in classroom environments. The use of history in teaching environments is usually limited to what can be found in textbooks, which bring fragments of historical information about the concepts without establishing the link between such concepts and teaching them, that is, without relation to school mathematical culture. According to the authors, history displays great methodological potential. Not only social circumstances can be known and understood, but also mathematical contents can be (re)learned and deepened, fostering critical thought in relation to the very development of mathematics and the presence of mathematics in school culture. 

Oral History: Other Exercises Conducted in Mathematics Teaching Courses

The eighth and last example to be presented here was published by Andrade and Sachs (2018). The authors share methodological and analytical notes with reference to the introduction of oral history practices in three different scenarios of pre-service teacher training, in a course for obtaining a mathematics teaching degree, at a university in the state of Paraná, Brazil. The three scenarios are (a) the discipline Methodology and Practice of Mathematics Teaching in Basic Education, (b) the discipline History of Mathematics, and (c) a federal program aimed at exposing future teachers to the reality of Brazilian schools.27 The first activity, proposed to the group of students of the discipline Teaching Methodology and Practice of Mathematics in Basic Education, was conducted concurrently with the other activities provided for in the syllabus of the discipline. It consisted of an exercise to approach oral history for classroom work, such as those proposed, for example, by Santhiago and Magalhães (2015). To do so, initially the students read texts about oral history. The topic “supervised internship”28 was the common thread which drove the activity, conducted by the students, in groups. Each group had to invite three people to be interviewed, as follows: (a) a higher education professor who was involved in supervised internships; (b) a teacher who teaches mathematics in elementary school, who has already had interns in their classroom; (c) a student seeking a degree in teaching mathematics who has experienced supervised internship. The interview scripts were created collectively. After the interviews were conducted, the recordings were transcribed and textualized. After the letters of assignment of rights for the use of the interviews had been issued, the groups conducted the analysis of the material produced, which generated a narrative about supervised internships for obtaining a teaching degree in mathematics. Finally, to wrap up the activity, each of the groups presented their “results” to the others. The second activity presented by Andrade and Sachs (2018) was developed to comply with one of the requirements of the syllabus for the discipline History of Mathematics Education. The first step of this second activity consisted in the study and discussion of texts related to the history of mathematics education, so that each student would later conduct interviews with relatives, friends, and/or neighbors, of different ages, about

 PIBID is an acronym in Portuguese for Institutional Program for Teaching Scholarships Initiatives (Programa Institucional de Bolsas de Iniciação à Docência). This program, implemented by the federal government, provides scholarships for undergraduate students and elementary and high school teachers who work in partnership with university professors in public schools, in specific teaching and research projects. 28  In Brazil, the completion of an internship in public elementary and high schools (under the supervision of a university professor and the teacher in charge of the classroom in which the internship will be carried out) is compulsory for obtaining a teaching degree in mathematics. 27

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mathematics teaching in elementary education, at different times. For such interviews, students developed scripts which addressed issues related to teaching and learning in general, and particularly to teaching and learning mathematics. As in the previous activity— and all activities involving the use of oral history—transcripts and textualizations of the interviews were produced. From the textualizations, students created a narrative based on the stories of each interviewee. Discussions were conducted based on such narratives, which resulted in a list of topics would later be studied in depth. In this case, the students were interested in investigating the punishments administered in schools; the mandatory nature and function of school uniforms; multigrade classrooms29; the different physical structures of schools (especially the facilities of rural schools); the gender issue (especially marked in women’s interviews); teaching manuals; and the different roles played by teachers in the past (teachers in small schools often also worked as janitors and lunch ladies, for instance). The analyses of these themes revealed changes and stabilities regarding school spaces and practices, throughout history. The students concluded that one of the main objectives of a discipline which works with history is to show that, as the time goes by, many things change, but many things remain unchanged. Finally, the third activity presented by Andrade and Sachs (2018) was conducted with students enrolled in the PIBID program. The objective of this activity was investigating how teachers slowly alter their practices over time and become the teachers they are. The students wrote accounts of their own life histories, detailing why they chose to pursue a teaching degree in mathematics. Subsequently, these texts were shared with other students. The sharing of the accounts, meant to be a simple exercise for writing “life histories,”30 acquired other meanings which made the students reflect about the reasons which made them chose to become teachers. For the next step, based on the question “How do teachers become the teachers they are over time?” the students interviewed three elementary and middle school teachers, who were also participants in the PIBID program. As in all projects involving oral history, recordings of the interviews were transcribed and textualized, and each of the interviewees signed a letter of assignment of rights for the use of interviews in the academic environment. With all the material generated by the interventions in the classrooms in hand, Andrade and Sachs (2018) discuss the potential of this approach to train future mathematics teachers. For the authors, it is important to value the history of the people with whom students share a connection, that is, teaching, learning, and the teaching profession. These different points of view show that teaching is not objective work, with completely predetermined steps, and that teachers are not standardized beings, all with the same training and the same behavior. Nevertheless, they share some perspectives which make them mathematics teachers. They also leave their mark in the world as a result of their work as teachers.

 More widespread in the past, but still found in very small elementary schools (in rural areas, for instance), multigrade classes are spaces in which students of different levels of schooling (grades) will be taught by the same teacher. 30  Life history is a type of oral history. While doing a life history, individuals narrate their lives, from childhood to the present. Life history is, in this sense, broader narratives than the thematic oral history (another type of oral history) through which the interviewees create a more restricted narrative, talking about some specific moment in their lives. 29

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Some Considerations

A common denominator of the exercises described herein is the fact that they are pedagogical initiatives carried out with students at different training moments and scenarios. All of them share the strategy of creating situations in which history is used in instances which involve mathematics, teaching, and learning mathematics, in some way. They all share the determination to streamline the boundaries between mathematics, history of mathematics, and history of mathematics education. In all the examples, strategies which would not otherwise be meaningful without problematization and investigation are elaborated, applied, and evaluated. In none of the examples given, history is a mere set of static and immutable truth; on the contrary, in all examples history serves to show that different meanings are produced in different conjunctures and times by different people. Despite all differences and heterogeneities, people constitute (and constitute themselves in) communities of memory which each one subjectively appropriates. The didactic initiatives exemplified in this text eloquently reiterate the impossibility of composing and presenting a single history of mathematics education; however, they show the value and potential of considering the many stories constructed by what creates the so-called school community.

References Andrade, M.  M., & Sachs, L. (2018). “Obrigada por ter apresentado a História Oral”: propostas desenvolvidas e analisadas em um curso de Licenciatura em Matemática. Bolema, Rio Claro, 32(60), 212–230. D’Ambrósio, U. (1999). A História da Matemática: Questões Historiográficas e Políticas e Reflexos na Educação Matemática. In M. A. V. Bicudo (Ed.), Pesquisa em educação matemática: Concepções & perspectivas (pp. 97–115). São Paulo, Brazil: Editora UNESP. da Silva, H. (2013). Integrando história oral e narrativas a abordagens pedagógicas problematizadoras na formação inicial de professores de matemática. Revista Educação PUC-­Campinas, Campinas, 18(3), 269–285. de Souza, L.  A. (2011). Trilhas na construção de versões históricas sobre um grupo escolar. 420f. Tese (Doutorado em Educação Matemática)—Instituto  de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. Flugge, F. C. G. (2015). Potencialidades das narrativas para a formação inicial de professores que ensinam matemática. 255f. Dissertação (Mestrado em Educação Matemática)  – Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. Garnica, A.  V. M. (2013). Sobre historiografia: Fragmentos para compor um discurso. Rematec, (12), 51–65. Gomes, M. L. M. (2012). História do ensino da matemática: Uma introdução. Belo Horizonte, Brazil: CAED-UFMG. Gomes, M. L. M., & de Brito, A. J. (2009). Vertentes da produção acadêmica brasileira em história da educação matemática: as indicações do EBRAPEM. Bolema Rio Claro, 22(34), 105–130. Gonzales, K.  G., & Martins-Salandim, M.  E. (2015). História na educação matemática: Uma disciplina, uma contribuição didática-metodológica para o ensino e a pesquisa. In Seminário Nacional de História da Matemática, 11. Natal. Anais... Natal: UFRN (pp. 1–11). Martins-Salandim, M. E., de Souza, L. A., & Fernandez, D. N. (2010). História oral em educação matemática: Contribuições para um referencial metodológicos. Ciências Humanas e Sociedade, Seropédica, 32(02), 55–66.

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Mendes, I. A. (2009). A história da matemática e o ensino da matemática escolar. In I. A. Mendes (Ed.), Matemática e investigação em sala de Aula: Tecendo redes cognitivas na aprendizagem (pp. 97–115). São Paulo, Brazil: Livraria da Física. Miguel, A., & Miorim, M. A. (2004). História da educação matemática: propostas e desafios. Belo Horizonte, Brazil: Autêntica. (Coleção Tendências em Educação Matemática, 10). Miorim, M.  A., & Miguel, A. (2002). Os logaritmos na cultura escolar brasileira. Natal, Brazil: Editora da SBHMat. Miorim, M. A., & Miguel, A. M. (2001). A constituição de três campos afins de investigação: história da matemática, educação matemática e história e educação matemática. Teoria e Prática da Educação, Maringá, 4(8), 35–62. Santhiago, R., & de Magalhães, V. B. (2015). A história oral na sala de aula. Belo Horizonte, Brazil: Autêntica Editora. Tizzo, S. V. (2014). A história oral como uma abordagem didático-pedagógica na disciplina política educacional Brasileira de um curso de Licenciatura em Matemática. 345f. Dissertação (Mestrado em Educação Matemática)—Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro. Van De Walle, J. (2009). Matemática no ensino fundamental: formação de professores e aplicação em sala de sula. Porto Alegre, Brazil: Artmed.

Chapter 6 Oral History in Mathematics Education: On Non-historiographical Research Maria Ednéia Martins Salandim

Abstract: The purpose of this chapter is to demonstrate how oral history has been mobilized as a research methodology in the area of mathematics education in Brazil, more specifically in research which has no historiographical approach. We describe how oral history became incorporated into the area with the emergence of a research group with a specific interest in problematizing the potential of this methodology for mathematics education. We also describe protocols, procedures, and topics which have been explored in the research. Moreover, we list the contribution of this methodology for the understanding of topics, practices, and scenarios of Brazilian mathematics education through accounts of experiences of people who are or have been involved in it.

Keywords: Research methodology · Research protocols · Non-historiographical research Introduction The first indications of use and discussion of oral history in research in Brazilian mathematics education date back to the late 1990s. Among the initial productions, now considered references, the research of Gilda Lúcia Delgado de Souza (Souza, 1998), developed at the Universidade Estadual Paulista (Unesp), in the Mathematics Education PostGraduation Program; the research of Carlos Roberto Viana (Vianna, 2000), carried out at the Universidade de São Paulo (USP), in the Post-Graduation Education Program; and the article by Antonio Vicente Marafioti Garnica, entitled “Written and Oral: An Initial Discussion about History Methods” (Garnica, 1998) can be highlighted. However, it was in the early 2000 that researchers in this field started to hold more formal meetings, aiming to discuss the potential of oral history for the development of research in mathematics education. These meetings resulted in the creation of the Oral

M. E. M. Salandim (*) Universiy of São Paulo State, UNESP, Bauru, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2019 A. V. M. Garnica (ed.), Oral History and Mathematics Education, History of Mathematics Education, https://doi.org/10.1007/978-3-030-16311-2_6

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History and Mathematics Education Research Group (GHOEM),1 under the leadership of professors Antonio Vicente Marafioti Garnica, Antonio Carlos Carrera de Souza, and Carlos Roberto Vianna, with the participation of students, most of them enrolled in postgraduation programs in mathematics education or education for science, both at the Universidade Estadual Paulista (Unesp). Since 2002, GHOEM has brought together mathematics educators with the objective of examining and implementing oral history as a research methodology in specific research exercises—but not without facing resistance in the field. The confrontation of this resistance was essential for the group to produce theories and strengthen its presence in the field of mathematics education. The academic production of GHOEM is significant and has become recognized by the Brazilian mathematics education community and even outside it, becoming a national reference in this field. In addition, GHOEM has promoted partnerships to create new groups and bring together researchers interested in mobilizing and problematizing oral history. Although the name oral history contains the word history, in mathematics education, it has been viewed as a research methodology used to produce narratives from interview situations or as a didactic-pedagogical approach. In this field, history has been mobilized both in a thematic form and through life histories. When talking about life histories, interviewees talk about their experiences and their lives as a whole, whereas when oral history has a thematic form (in this case called thematic oral history), interviewees talk about experiences related to a certain aspect of their lives, defined by the researcher. However, this does not prevent episodes outside this scope from being narrated. The narratives constituted through oral history2 interviews are recognized as useful sources for both the research for which the narratives are intentionally produced, but also as historical sources in general. However, even though only oral sources can trigger an investigation, they are not always sufficient to conduct research in its entirety (this, as a rule, occurs in historiographical research) and, in such cases, other sources, such as written documents (minutes, legislations, student notebooks, school records, among others), photographs, books, etc., as well as oral narratives integrate the researchers’ set of references. In this scenario, how have we been doing? What contributions by oral history can be highlighted in the field of mathematics education? What has our research problematized? In  this paper, we will revisit the research conducted with non-historiographical bias and ­without intention to intervene.3

 GHOEM was created in 2002. The initial intention was to bring together mathematics education researchers who were interested in the possibility of using oral history as a methodological resource. At present, the main interest of the group is the study of school culture and the role of mathematics in this culture. GHOEM involves researchers from different Brazilian universities. www.ghoem.org 2  Many qualitative research approaches use interviews as a strategy for data collection. What we call “oral history interviews” are the interviews conducted according to specific research parameters whose central methodology is oral history. Such parameters are discussed in more detail in other chapters of this book. 3  Oral history research as a methodology may be historiographical research (or generated by a properly historiographic question, whose objective is to understand the changes and permanence of a given situation over time) and intervention research (when oral history is used as a strategy—or part of a strategy—for teaching and learning) and for studies of other natures. The first two research modalities are discussed further in other chapters of this book. This chapter will specifically address this third modality, that is, which uses oral history without historiographical intentions a priori (although any research can be used as a historical source and the work with oral history always implies generating historical sources) and research in which oral history is not properly, or a priori, a strategy for teaching and learning mathematics (even though any research can be used as a teaching and learning strategy in mathematics education). 1

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About Oral History in Mathematics Education

Oral history in Brazilian mathematics education, in general, has been used to constitute narratives in interview situations. It comprises the research methodology, with which we tackle issues. The methodology involves a constant feedback between a set of procedures and the theoretical framework. In this sense, we consider oral history as a methodology in flux. In each investigation which mobilizes oral history, researchers are prepared to problematize the way oral history was used in other investigations and the results produced as well as by their own actions, frameworks, and analyses. Under this perspective, oral history is described not only as a technique or a set of procedures to establish oral sources, it is also defined as a methodology. Some stable procedures can be identified in the oral history which has been practiced in Brazilian mathematics education. Such procedures constitute a research protocol, which in itself is not sufficient to distinguish research which displays historiographical bias or not, as this protocol only guides the production, editing, and disclosure of oral sources, and says nothing about the nature of what is investigated. The protocol, prepared by researchers associated with GHOEM, deals with the selection and contact with interviewees, preparation of interview scripts, recording or filming of interviews, transcription and textualization of the interviews, revisions of texts generated, signing acceptance and letters of assignment of rights by interviewees, and, finally, the analyses of the narratives produced. So, how has research uses oral history been practiced in GHOEM? Generally, after the objectives of each study are defined, a search for possible interviewees, based on a profile which meets the specificities of the research theme, is launched. The so-called criterion network, according to which an interviewee suggests names of other possible interviewees to the researcher, has been widely used. However, this criterion has been used with care, so that it does not result in a network of interviewees with little diversity, thus limiting the researcher’s perspective. After obtaining a referral, initial contacts are made (in person; by phone, email, mobile apps, or social networks; directly with the person; or with an aid or anyone who can legitimately intermediate the contact with the interviewee). During the first contacts, the aim of the research is explained as clearly as possible, as well as the procedures to be followed during and after the interviews, including reviewing the material and the signing by the interviewee of the letter of assignment of rights over the recordings and textualization, with or without restrictions. Dates, places, conditions, and the length of each interview are also negotiated, always considering the availability and needs of each interviewee. In addition to negotiations with the parties involved, the preparation of the interview demands the elaboration of scripts, flashcards, collection, or production of images and other materials, to which the interviewee has full access before, during, or after the recording and which serve as prompts for the interview itself. This preparation is closely related to the research theme and interviewee’s profile. After the preparations, which may vary for different interviews, and may even occur distinctly for interviews during the same investigation, it is time to record and/or film the interviews. There are several ways to do this, but all of them require sensitivity on the part of the researcher to approach and establish rapport with the interviewee. Experience with interviews has shown that such interlocutions carry multiple biases and serve as opportunities for different interests and actions: an interview can be a time for denunciations, reflection, analyses, nostalgic reminiscence, purging, homage, expression of resentment, to fantasize and to highlight accomplishments. Thus, the interview is not a time of mere descriptive narration of episodes or questions and answers. The role of the interviewer has been discussed by researchers such as Portelli (2010), emphasizing that “rather than merely ‘collecting’ memories and verbal performances, [the interviewer] should trigger them and, literally, contribute to its creation: through his presence, his questions and reactions” (p. 20).

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Having recorded the interviews (in some cases, in addition to the audio, one chooses images) we proceed to transcription. Transcription is the time to transform the oral recording into written text. In this process, interlocution is not fully preserved, though efforts must be made to record, as carefully as possible, what has been said and the general circumstances of the moment of the interview, even indicating silences, gestures, looks, and interruptions, so that the transcribed text will retain very clearly characteristic marks of spoken language. After the transcription, through the editing of the transcription, textualizations are produced. At this stage, the main concern is to maintain the characteristics of an interviewee’s speech, though language idiosyncrasies may be excluded, and the text reordered, thematic or chronologically, depending on the interest of the researcher. Even if direct questions were asked at the time of the interview and the themes have been restructured in the textualizations, we believe that the “vital tone” of the interviewees’ speech, the ways they express themselves, must be preserved in the textualization, so that they are recognizable during reading. Repetitions, specific interjections, and word contractions may be maintained, excluded, emphasized, or minimized, in order to elaborate the narrative, highlight information, “send messages,” call the interviewer’s attention, and organize and reorganize memories and arguments. Even when the interviews are based on scripts or themes, they do not develop along a single thread or axis. Oral history interviews are not conducted by the interviewer alone. The interviewee, their narrative, and experiences interfere directly with the interview proposed by the interviewer. The driving axis is also determined by the interviewee and not only by the interviewer, even though the interviewer established and communicated the topics of interest to the interviewee in advance. The option to preserve such elements and how to do it is limited by the need of the interviewee to recognize themselves in the textual narrative. Although this is neither the narrative transcribed, nor the narrative recorded, it is necessary to recognize that moments are evanescent and that what happened during an interview is never fully retrieved by any recording, no matter how refined the apparatus, records, and devices may be. Therefore, the textualization of the recording is not intended to preserve the organization of the interview, but rather to structure a text in which some of the investigators’ concerns are evidenced, either through text reordering (with the objective of clarifying certain passages narrated by interviewees), or by the inclusion of explanatory footnotes. The inclusion of several footnotes has the objective of complementing or clarifying or contextualizing the information for readers. Therefore, it is worth pointing out that even though the research does not have a historiographical bias, the sources generated for the research, with the use of oral history, are always intentionally constituted as historiographical sources. During the negotiations about the form of the textualizations, some interviewees may decide to change the form significantly, either because they are concerned about making the text even more distinct from spoken language, because they wish to include or exclude information, or because they wish to organize the text differently. All interviewees have full rights to their memories, and even if such interventions disfigure the narratives or make them less interesting for research, they must be fully accepted by the researcher who may, however, negotiate with the interviewee. In general, it is from the textualizations provided by interviewees that the analysis is developed, even if we (researchers) are affected by the whole process which involves the production of such narratives (Martins-Salandim, 2012). When researchers choose oral history, they are concerned with the subjectivity of the other person, about the historicity of the person, about similarities and differences between narratives which talk about experiences in the world, and about the world. We are not only focused on recording the information which an interviewee can give, but also on who the

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person is, on what they are like, and the way that person is constituted as an individual while narrating experiences in the interview. We are interested in the person with whom we are dialoguing, so, as a rule, the actual names of interviewees are kept in the research, as long as they agree. In our experience as researchers/interviewers, we have gradually understood the complexity of that moment and its consequences. All interviewees and/or their legal representatives sign letters of assignment of rights, except in cases when, at the time of recording, the interviewee orally waives rights to the interview and/or textualization, which has been very rare, in the research conducted so far. In some cases—also rare—interviewees may request that the letters of assignment be formally registered before a notary. In all such cases, the interviewee’s wishes must be met. In our experience with oral history procedures, there were few cases in which the interviewee did not authorize the use of their narrative after the interview. More common, however, but also not so frequent, some potential contributors refused to give interviews under different allegations: for lack of time, because they consider that their memories cannot contribute to the research, for fear that their narrative may be used against them by other people, or even for fear that the researcher manipulates their speech. While the steps described above seem, in principle, very technical, they have been constantly problematized. Based on that, we have turned to other authors to broaden our understanding of the use of narratives. When the theoretical framework is explained, we argue that the issues raised are meaningful solely because they are involved and based on a research perspective which contemplates the potential of oral narratives for the understanding of mathematics education. Our research questions make sense considering oral history methodology for data production as reference, enabling and fostering the dialogue with the authors which we believe can assist in the development and registration of our understanding about the research topic. This means that our referential has implications in the way we conceive our questions, and not only how we pursue evidence with the objective of elaborating understandings based on research questions. Analyses, generated in an interpretative process, are entirely linked to our intentions, possibilities, and way of perceiving the world. Therefore, acting ethically does not imply being free to neglect or naively ignore the power relations which take shape in the production and interpretation of data. Acting ethically means that power relations are interwoven by dialogue and, therefore, by the negotiation of meaning, which occurs mainly in the research group and with a group of interlocutors, of which even interviewees¸ who have been included in the research, can participate. On the one hand, if at certain moments, the researcher has a privileged position to attribute meaning; on the other hand, interviewees often use their influence, visibly at the moment of their narrative, thus deciding what they want to narrate, and at the moment they review the textualization, by exercising their power of veto. It also occurs when they comprehend the role and position of the researcher, who needs the narratives which only the interviewees can provide, because they are individual expressions of an experience which, as such, is always unique. Still, in these cases too, the attitude of favoring negotiation makes the process feasible and certainly educational, as education is, after all, a conflictive process, a struggle for the attribution of meaning. Through our experience of conducting research in a group, we gradually understood the complexity of interlocution. This universe, which involves negotiations to produce interviews, timing, reviewing of the data collected, and signing of the assignment letter, is based on protocols followed by all members of the group, even if such protocols are not rigid, and do not follow previously fixed and defined norms and rules, but are all negotiated based on established relationships with each interviewee (Garnica & Martins-Salandim, 2014).

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The basis of the research protocol is the responsibility of the researcher, which obviously includes interpretations taken from previous interviews. However, this protocol is not intended to limit the research. In this sense, when using oral history, each researcher tests the limits of techniques, procedures, and frameworks which have already been used by other researchers. Researchers are thus able to create new ways of acting, always in consonance with their themes, objectives, and interviewees’ profiles. These research practices show evidence of the plasticity of the research protocol. The so-called oral history in mathematics education preserves characteristics similar to oral history practiced in other fields of knowledge, as the procedures and justifications were created in dialogues with different researchers from the most diverse fields. However, by relying on new approaches and questions which arise during the investigations conducted in the field, the oral history that we practice takes a unique form, configuring itself as a specific mode of thinking and doing research which meets the objects of study. So, we could ask if oral history has been perceived as significant in addressing themes, questions, and research objectives in mathematics education. It has proven to be potentially significant to all research whose aim is to understand elements and scenarios of this and other correlated fields, responding to researchers who recognize the legitimacy of knowledge constructed from the narratives of the many agents who have experienced situations that researchers want to understand and share. 

About Themes and Research Intentions

The research issues in mathematics education in Brazil involve school culture, its correlated fields, and the participation of mathematics in this cultural territory. In the case of research which has used oral history as a methodology, we emphasize that it is often linked to a broad project of the Oral History and Mathematics Education Research Group (GHOEM), whose aim is to map the schooling and teachers’ practices for teaching mathematics in Brazil, in different institutions and levels. Other research projects problematize the potential of oral history or the ways in which researchers in mathematics education have mobilized this methodological framework. Although most research which uses oral history in mathematics education has a historiographical bias and is carried out within GHOEM, there are a number of non-historiographical research projects, as well as research, which uses oral history as a didactic-pedagogical strategy. In this chapter, we will consider a set of more than 30 completed master’s and doctoral4 thesis in the field of mathematics education, which mobilized oral history. In general, this research does not have historiographical perspectives per se, nor does it specifically target or study possible interventions; however, in some of the research such perspectives are also featured. This set of research projects, mostly linked to the Oral History and Mathematics Education Research Group (GHOEM), helps problematize the diversity of objectives, themes, and interviewees’ profiles. They are also potential sources for understanding how each research strengthens the notion of a methodology in flux, as already stated at the beginning of this text.

4  The postgraduate system in Brazil is divided into stricto sensu and lato sensu. Stricto sensu postgraduate courses comprise masters and doctoral programs. These courses are presence-based, and their conclusion implies a specific title (master or doctor) in a particular field. Masters courses can also be the professional type, directed to qualify professionals in various fields of knowledge, or academic, aimed at preparing for doctoral study whose objective is training researchers. Lato sensu postgraduation courses encompass distance or presence-based specialization courses. One can think of lato sensu postgraduation as instances of continued education or preparation for stricto sensu postgraduation work. Any post-graduate course in Brazil requires as a prerequisite an undergraduate degree.

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The investigations thematized here are based on narratives produced from interviews involving about 200 people: adults, adolescents, and children; people living in rural, urban, or indigenous areas5; teachers who teach mathematics at different levels of education,6 holding teaching degrees in mathematics or not; students’ parents; and professionals who work in schools, universities, or governmental agencies linked to education. We are interested in the persons who speak, how they speak, where this person speaks from, and how those who speak perceive the situations that each research project seeks to investigate. The themes of this set of investigations are diverse and involve dialoguing with authors from different fields of knowledge such as philosophy, psychology, and sociology. Some researchers have investigated teaching practices and areas of activity. Vianna (2000) interviewed 15 university professors to defend the thesis that professors who choose to work in the field of mathematics education, within the mathematics7 departments of universities, face resistance rooted in the prejudice on the part of their colleagues. Zanardi (2012) investigated the scenario of mandatory transference of professors from one city to another, when a public university was created in the state of São Paulo, through the merger of different institutions, which generated the extinction of some undergraduate courses. To do so, the author interviewed 14 university professors who had experienced this process, in order to understand and attribute meaning to events which resulted in resentment caused by the process. Silva (2004) interviewed 10 teachers with the objective of discussing the cultural identity of the mathematics professor. Rolkouski (2006) interviewed five mathematics teachers with different levels of education, aiming to understand how they became the mathematics teachers they are, with their ideas, practices, and resistances. Bernardes (2003) interviewed four teachers seeking to understand the relationship between teaching and professionalization and to understand the relation between knowledge and power and how it influences mathematics teaching. In line with the previous research, Bernardes (2009) interviewed three mathematics professors who acted as deans in higher education institutions, in order to problematize education as an instrument of distribution of power and knowledge. Aiming to present and analyze narratives to assign meaning to the conceptions and practices related to teaching and learning of mathematics, in early-childhood education, in the Brazilian state of Santa Catarina, Silva (2014), interviewed 16 teachers who were involved in “Mathematics Fairs,”8 establishing the history of an event which has been held for the past 25 years. Galhart (2015) constituted narrative sources about mathematics literacy,9 interviewing five students who studied at different institutions managed by a religious congregation.  Indigenous lands in Brazil are a portion of the national territory, owned by the Union, inhabited by one or more indigenous peoples. Currently, in Brazil, there are 462 legalized indigenous reservations. They represent about 12% of the national territory. More information are at http://www.funai.gov.br 6  There are two levels of education in Brazil: basic education and higher education. Basic education comprises three stages: early childhood education (for children up to 5 years of age), elementary education (for students aged 6–14), and high school (for students aged 15–17). Higher education comprises courses in undergraduate and post-graduate courses stricto sensu and lato sensu. 7  As a rule, in Brazilian universities, departments are the basic units which compose the university, grouping professors with related objectives (usually by specific field of knowledge) who teach, research, and supply extension of services to the community. 8  Santa Catarina is a southern Brazilian State. Mathematics Fairs of Santa Catarina are large educational events where school projects which were developed in classrooms from various regions of the State are presented (Silva, 2014). 9  Current literature defines mathematics literacy as a perspective which extends the role of mathematics in traditional read-write-count triad. Mathematical literacy deals with the teaching and learning of mathematics which goes beyond the decoding of numbers and the realm of the resolution of the four basic operations, to become a tool for reading the world (BRASIL, 2014). 5

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Still on the theme of school spaces, other research projects specifically tackled questions related to daily life in school. Tuchapesk (2004) investigated the interrelationship between school, family, and mathematics, producing 17 interviews which involved adolescent students, parents, mathematics teachers, and other school staff. Silva (2013), recorded the narratives of three mathematics teachers’ daily lives and investigated their everyday usual confrontations, related or not to their school experiences. In this school routine scenario, some researchers investigated inclusive education issues. Rosa (2014) studied school inclusion and the process of teaching and learning mathematics of students with disabilities, pervasive developmental disorder, as well as highly functional and gifted students. For her research, she interviewed nine teachers who teach mathematics. Rosa (2017), aimed to understand how the students and their legal guardians, in their personal lives and their education, perceive the schooling process in the context of special education,10 from the perspective of inclusive education. To this end, the author conducted six interviews which involved four mothers and five visually impaired children. Greca (2015) interviewed five professionals, among them teachers who worked directly with deaf children in a specific school, and three students attending the school. The objective of the study was to investigate what they say about mathematics literacy and what pedagogical practices they implement or are subjected to. The results of this research were shared with a deaf teacher, in order to determine how the dialogue between the school and the experiences lived by hearing-­impaired individuals happened (or could happen). Silveira (2018) produced oral sources from interviews with three hearing impaired individuals, former elementary education students, about their experiences in mathematics classes. Other research projects have investigated how teachers or students perceive certain aspects of specific educational programs related to teaching, learning, and teaching methodologies. In these cases, most of the research was focused on the initial years of the Brazilian educational system. Mindiate (2015) interviewed four university professors in order to review a specific Brazilian public policy, Pacto Nacional pela Alfabetização na Idade Certa (National Pact for Literacy at the Right Age – PNAIC),11 whose focus is education, particularly children’s literacy. Costa (2016) aimed to provide oral sources from interviews with six literacy teachers who participated in National Pact for Literacy at the Right Age (PNAIC) in 2014. Passos (2017) investigated the use of games in the mathematics literacy process of children. In order to do so, six professionals who work in schools and were involved with the National Pact for Literacy at the Right Age (PNAIC) were interviewed. Dombrowski (2013) aimed to build sources from interviews with four teachers enrolled in the Mathematics Pro-Literacy Continuous Education Program.12 Martins (2015) sought to investigate aspects of mathematics literacy, more specifically literacy, conducting interviews with five teachers and four students in the initial years of elementary education. Two of interviewees were indigenous students living in different villages in indigenous lands in the state of Paraná, Brazil. Bagio (2014), in turn, sought to understand the educational processes of teachers in order to facilitate the implementation of Non-Euclidean Geometries

 The expression special education is used in Brazil to refer to school education (regular or informal) of individuals with disabilities (hearing, speech, vision, and intellectually impaired). 11  The National Pact for Literacy at the Right Age—PNAIC (acronym in Portuguese)—is a government initiative to ensure full literacy for all children up to eight years of age. 12  Pro-Literacy Continuous Education Program for Early-childhood Education is a program of the Brazilian Ministry of Education (MEC), in partnership with universities, states, and municipalities. It is a continued teacher training program aimed at improving the quality of learning of the reading, writing, and mathematics in the initial years of schooling. 10

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content in the elementary education13 system. To do so, Bagio (2014) interviewed five teachers who got involved in the development of the mathematics curriculum guidelines14 for the state of Paraná. In addition to mobilizing oral history, Candray Menjívar (2018) also used comparative methodology, interviewing two teachers, one from Brazil and one from El Salvador. Both of the teachers were coordinators of two major continued education projects in their respective countries, and both sought to understand the concepts of mathematics education and continued education in large scale, which served as the basis for the aforementioned programs. Fontes (2018) interviewed five teachers who were involved with remedial classes programs in mathematics,15 in public schools, in the state of São Paulo, during the 2007–2016 decennial, their aim being to investigate, and to understand, how such programs work. Zaqueu (2014) interviewed two teachers who worked as general coordinators of teacher-training programs. Zaqueu also considered (auto)biographical narratives of four former scholarship holders, aiming to bring to the discussion the perspective of former alumni scholarship students from the Programa Institucional de Bolsas de Iniciação à Docência (Institutional Scholarship Program for Teaching Initiation), PIBID.16 Still targeting specific educational programs, though focusing on methodological approaches for mathematics teaching, the objective of Barbosa (2012) was to present the views on mathematical modeling of four mathematics teachers, who had participated in a specific training program—Programa de Desenvolvimento Educacional (Educational Development Program or PDE, Portuguese acronym)17—whose capstone project theme was mathematical modeling. Maron (2017) characterized mathematical modeling practices as language games, according to Wittgenstein’s Philosophical Investigations. The starting point of this characterization was the interview with a researcher who uses mathematical modeling in their teaching practice. Lopes (2013) investigated how teachers suggested the exploration of Multimedia TV18 to approach mathematical contents. The three teachers interviewed for this research project were authors of some didactic-pedagogic material posted on an online environment related to the program. Understanding the structure of Brazilian mathematics education and the movements it has undergone were also the goals of researchers who mobilized narratives constituted through oral history. The objective of Fernandes (2014) was to understand how mathematics education was constituted as a field of research, by conducting interviews with three researchers based on their memories of the examinations they took to become associate professors.19  The study of basic concepts related to presence of non-Euclidean geometries in Brazilian teacher training programs. This can be detected even in older materials, as in the case of textbooks produced at the time of the New Mathematics movement. However, in the state of Paraná, non-Euclidean Geometries have recently been introduced in a more systematic way, as requirements in the basic education syllabus, coordinated by specific public policies. This is the first attempt to do so in Brazil. 14  In Brazil, each state of the federation has autonomy to develop curricular guidelines. Such guidelines establish and guide the curricular organization of elementary education. 15  Remedial classes are strategies, concurrent to regular classes, for students with learning disabilities or who did not score adequately in school performance assessments. In Brazil, there are numerous strategies in effect for projects carried out by and with co-called remedial classes. 16  PIBID—acronym in Portuguese—is a project proposed by the Ministry of Education (MEC) whose objective is to expose teachers-in-training to practical situations involving the daily practices of basic education in public schools and the context in which they are inserted. 17  This is a continuous training program for public-school teachers in Paraná. 18  This is a strategy developed in public schools in the state of Paraná. 19  Associate professorship is one of the last career steps for those who teach in public universities in São Paulo state. Persons applying to become an associate professor must undergo a specific evaluation which involves, among other things, the presentation of a memorandum on their teaching and researching activities. 13

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Santos (2012) conducted seven interviews with mathematicians and mathematics educators with the aim of theorizing about the mathematics education of professors, who work in undergraduate courses which grant teaching degrees in mathematics.20 Souza (2016) sought to understand aspects of the movement of circulation and appropriation of an ideational/ educational movement. In order to do so, Souza (2016) interviewed 13 Brazilian mathematics educators, working at Brazilian universities, who are the main mediators between French Mathematics Didactics (in this research taken as a specific ideology, based on which the study was conducted) and Brazilian mathematics education. For the accomplishment of such objectives, oral history interviews were conducted and later transcribed and textualized. The way some researchers based their investigations, tested the limits of method, or adapted their procedures corroborates the notion of oral history as a methodology in flux in the field of mathematics education. 

On Oral History in flux in Mathematics Education: Some Examples

Research which mobilized oral history as a research methodology in the field of mathematics education, as discussed here, have adopted oral history because they believe that oral history interviews make it possible to create narratives of people who have experienced events or circumstances which were significant to their investigations. The majority of researchers constructed and used the narratives to problematize specific themes, moments, experiences, and circumstances of interviewees’ lives in their studies, even though broader narratives about interviewees’ lives were also important. Therein lies a differentiation in the work with oral history: the researcher may be interested in oral life histories (broader) or in thematic oral histories (restricted to a specific theme). Vianna (2000) and Rolkouski (2006), for example, due to the nature of their research objectives, focused on the life histories of the interviewees. In this chapter, the emphasis will be given to the way which some researchers have enhanced and tested the limits of the procedures adopted in their research while conducting or editing interviews or analyzing the narratives. As oral history has been mobilized and is thought of as still in flux (evolving), procedures and reflections on the methodology have been revisited and replicated. However, the main objective of this chapter is testing its limitations and potential, in order to understand how procedures and frameworks have been adapted and (re)configured to deal with new situations. When preparing for interviews, researchers who use oral history make sure that the theme studied can be problematized during and from the interviews. Thus, scripts with questions are developed; flashcards with words, expressions, or questions are prepared; and games are created or adapted. In order to achieve (and maintain) a more meaningful interlocution, numerous strategies are employed to serve as guides for conducting the interview. When interested in the interrelations between school, family, and mathematics, Tuchapesk (2004) was faced with the challenge of selecting students from a school to be interviewed. The researcher decided to ask students to write an autobiography and selected the interviewees based on their autobiographies, which also served as a guide for the interviews and as the basis for developing the interview script.

 Higher education courses for teachers in Brazil are called “licentiate degrees.” The Licentiate Degree in Mathematics, therefore, qualifies mathematics teachers. 20

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Vianna (2000) conducted interviews in two stages. During the first stage, the researcher prepared an initial presentation, which interviewees read in the beginning of the interview, then proceed to speak freely about their life histories. With this strategy, the researcher, who thematized prejudice and explored the potential of history through the perspective of life histories, intended to unveil the manner through which interviewees constructed their narratives, the choices they made while narrating, and how interviewees viewed their lives. In the second stage, the researcher spread over a desk several sheets of paper with a questionnaire, in which some items had been highlighted, in uppercase, to draw the attention of interviewees to the theme, although they knew they didn’t have to address all the topics. After the completion of the work, Vianna (2000) presented a list with the real names of the interviewees, who had initially been referred to by pseudonyms, in the body of the text. In doing so, the researcher challenged readers to identify the interviewees from the textualization, thus highlighting that life histories really constitute the individual before a community, at a given historical moment, as all the interviewees are easily identified from their stories. Hence, the game proposed by Vianna questions the notions of identity and subjectivity so dear to the Western, modern way of doing research, showing that often the pseudonyms used by researchers to identify their interviewees are extremely artificial and innocuous resources. The whole structure of Vianna’s thesis (2000) tests the limits and potential of oral history in the field of mathematics education. Exploring the potential of listing questions of interest for the research, so that each interviewee would choose what to talk about and the sequence in which they would narrate their perceptions and experiences, Rolkouski (2006) elaborated flashcards in such a way that each card would bring in writing one of the topics that the researcher wanted to discuss. For this investigation, whose intention was to understand how teachers become the teachers they are, the interviews were also carried out in two stages, both mobilizing the cards. General topics related to the life histories of the interviewees were explored in the first stage. During the second stage, more specific themes, connected to educational experiences and professional practices of each interviewee, were explored. When thematizing the daily confrontations of mathematics teachers, Silva (2013), developed interview scripts focused on three distinct moments, each for one interview. Three interviews were then conducted with each interviewee: (a) the first one was based on a script related to the presentation of the teacher, with questions directed toward their education, work, and life; (b) with the second script, the researcher was interested in exploring themes derived from the first interview or which had been studied about the theme; in this case, the types of flashcards used as prompts were similar to those Rolkouski (2006) used; and (c) a third script was composed of different types of clippings, from the two previous interviews, specific bibliographical references, and/or clippings from public documents. The textualizations of different moments of each interview were presented separately, alternated with instances of analysis. Rosa (2017), interviewed visually impaired students and their mothers and also produced scripts with questions and thematic flashcards. Considering the interviewees’ visual impairment, the cards were made with words printed on enlarged fonts as well as in Braille. A digital version of the thesis of Rosa (2017) was made available to be used on a screen reader (specific software) suitable for blind people. With these strategies, deriving from the consideration of the author regarding her interviewees, the researcher widened the possibilities of interlocution with other blind people, who may also be potential readers. After interviewing deaf students, Silveira (2018) developed a pilot project, as he ­realized how complex it would be to practice oral history with hearing impaired people. The author problematizes the technical and ethical issues related to different moments of the research, since it is still unusual to have deaf people as interviewees in studies which use

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oral history methodology. The initial contacts, invitation to interviewees, all the stages of verification of the material produced, and the signatures of the letters of assignment of rights were requested both in sign language21 and in Portuguese. During the recording of the interview, which was supported by an interpreter who interpreted simultaneously (to Portuguese) what was “said” in sign language, the researcher problematized the spatial dynamics for a more appropriate interview recording, as sign language is visual and signs are articulated in a signaling space. Silveira (2018) points out that an average of 20 hours are required translation and for the transcription of 1 hour of sign language recording. In the research, textualizations of what was narrated in sign language by the hearing-impaired interviewees are presented in Portuguese. When developing research with the aim of highlighting relationships between the use of children stories and mathematics learning, in early-childhood education, Murbach (2017) interviewed both teachers and children. In order to get recommendations of children to be interviewed, the researcher faced sensitive negotiations, initially involving school staff and, subsequently, the children’s legal guardians. The author chose to record the interviews and make the audio-recordings available to the families, which after having access to the content of the speech, could authorize, or not, their use in the investigation. To record interviews with children, Murbach (2017) adapted a storytelling game and systematized a script with questions, in case it became necessary. Murbach (2017) then proposed that each child wrote two words in cards previously delivered to them and put those cards in a box. At the time of the interview, two words were drawn to start the game, that is, the interview. The lack of familiarity, the attempt to relate or create narratives, or drawings from the two words were central elements of this proposal. The author points out that some children did not play the game, as these children became more interested in a “regular” interview, done as a conversation. Excerpts of the transcripts of the interviews with children were inserted as text boxes in the textualizations of the interviews with teachers. They function as images whose objective is to disturb and unsettle the reader during the reading experience. These are just a few examples. Many others could help compose this text. However, we believe that the highlights brought here are enough to show how oral history has been practiced, in an adaptation movement, in flux in Brazilian mathematics education. Flexibility in the use of the methodology therefore seems to be fundamental for such adaptations, so that each research project develops its themes with greater effectiveness and vigor, as in qualitative research design, the procedures of the methodological structure are determined by the themes, people, and circumstances. 

Final Considerations

Having presented some themes, paths, and some of the objectives of different research which mobilizes oral history in mathematics education, we conclude this text by examining the contributions of narratives generated from interviews. Orality has allowed us to consider the subjectivity of the actors who, somehow, participate in mathematics education. The narratives constituted are not mere testimonies of something which was actually seen, something which “really” happened, but records of what the researcher perceives at the time of the interview, of what the interviewee decided to talk about, what they experienced, and

 Brazilian Sign Language (LIBRAS, acronym in Portuguese) is a set of manual gestures used by hearing impaired people. 21

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how they experienced it. The constitution of sources through oral history parameters allows researchers to have a vital participation in this recording of subjectivity and share with interviewees the conditions and instances of the production of records; thus, it is possible to explain to potential readers the negotiations, comings and goings, circumstances, proximity, and distance in this attempt to capture in writing aspects of subjectivity narrated orally, on the interviewees’ terms, as interviewees only share what they believe to be appropriate. We believe the contribution to oral history practice in mathematics education of these “non-historiographical” investigations is the possibility of understanding themes, practices, and scenarios in this field, from the experiences of people who effectively have experienced the situations and contexts that the researcher wants to examine. Oral history has allowed numerous scenarios to be created in mathematics education, many of which unprecedented.

References Bagio, V. A. (2014). Da escrita à implementação das DCE/PR de Matemática: um retrato feito a cinco vozes e milhares de mãos. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Barbosa, A. A. de S. B. (2012). Modelagem matemática: relatos de professores. Dissertação (Mestrado em Educação em Ciências e em Matemática)—Setor de Ciências Exatas. Curitiba, Paraná: Universidade Federal do Paraná. Bernardes, M. R. (2003). As várias vozes e seus regimes de verdade: um estudo sobre profissionalização (docente?). Dissertação (Mestrado em Educação para a Ciência). Bauru, São Paulo: Faculdade de Ciências, Universidade Estadual Paulista. Bernardes, M. R. (2009). Educação, relações capitalistas, estratégias e táticas: um ensaio a partir de algumas escolas de ensino superior de Maringa (PR). Tese (Doutorado em Educação para a Ciência). Bauru, São Paulo: Faculdade de Ciências, Universidade Estadual Paulista. Brasil. (2014). Secretaria de Educação Básica. Diretoria de Apoio à Gestão Educacional. Pacto Nacional pela alfabetização na idade certa: Apresentação. Brasília, Brazil: MEC, SEB. Candray Menjívar, J. C. (2018). CFD e GESTAR II: compreensões sobre a formação continuada de professores que ensinam matemática a partir de um estudo comparado. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Costa, E.  X. da C. (2016). Narrativas de professores alfabetizadores sobre o PNAIC de alfabetização matemática. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Dombrowski, A.  E. (2013). Pró-Letramento, prática pedagógica nas aulas de matemática: relatos de professoras de União da Vitória. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Fernandes, F. S. (2014). A quinta história: composições da educação matemática como área de pesquisa. Tese (Doutorado em Educação Matemática). Rio Claro, São Paulo: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Fontes, C.  M. (2018). Projetos de recuperação em matemática no estado de São Paulo: um estudo sobre o decênio 2007–2016. Dissertação (Mestrado em Educação para a Ciência). Bauru, São Paulo: Faculdade de Ciências, Universidade Estadual Paulista.

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Galhart, A.  C. (2015). Alfabetização matemática: contas e contos, em vozes, encontros. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Garnica, A. V. M. (1998). O escrito e o oral: Uma discussão inicial sobre os métodos da história. Ciência e educação, Bauru, São Paulo: UNESP, 5(1), 27–35. Garnica, A. V. M., & Martins-Salandim, M. E. (Org.). (2014). Livros, leis, leituras e leitores: exercícios de interpretação para a história da educação matemática. Curitiba, Paraná: Appris. Greca, L. C. M. (2015). Surdez e alfabetização matemática: o que os profissionais e as crianças surdas da escola têm para contar. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Lopes, R. C. (2013). Produção didática de professores para uso com tecnologias em aulas de Matemática. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Maron, C. do R. de M. (2017). Modelagem matemática como jogo de linguagem. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Martins, I. M. H. (2015). Alfabetização matemática: um ato lúdico. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Martins-Salandim, M. E. (2012). A interiorização dos cursos de matemática no estado de São Paulo: Um exame da década de 1960. Tese (Doutorado em Educação Matemática). Rio Claro, São Paulo: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Mindiate, M. J. (2015). Uma compreensão da alfabetização matemática como política pública no Pacto Nacional pela Alfabetização na idade certa. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Murbach, M.C.G. (2017). Histórias infantis e alfabetização matemática. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Passos, C.M.S.M. dos. (2017). Jogos na alfabetização matemática: reflexões sobre propostas do PNAIC. Dissertação (Mestrado em Educação em Ciências e em Matemática). Curitiba, Paraná: Setor de Ciências Exatas, Universidade Federal do Paraná. Portelli, A. (2010). Ensaios de história oral (de F.  L. Cássio e R.  Santhiago, Trad.). São Paulo, Brazil: Letras e Voz. Rosa, E.  A. C. (2014). Professores que ensinam Matemática e a inclusão escolar: algumas apreensões. Dissertação (Mestrado em Educação Matemática). Rio Claro, São Paulo: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Rosa, F.  M. C. da. (2017). Histórias de vida de alunos com deficiência visual e de suas mães: um estudo em educação matemática inclusiva. Tese (Doutorado em Educação Matemática). Rio Claro, São Paulo: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Rolkouski, E. (2006). Vida de professores de matemática: (Im) possibilidades de leitura. Tese (Doutorado em Educação Matemática). Rio Claro, São Paulo: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Santos, J. R. V. dos. (2012). Legitimidades possíveis para a formação matemática de professores de matemática (Ou: Assim falaram Zaratustras: uma tese para todos e para ninguém). Tese (Doutorado em Educação Matemática). Rio Claro, São Paulo: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista.

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Silva, A. A. da. (2013). Narrativas de professores de matemática sobre seus enfrentamentos cotidianos. Dissertação (Mestrado em Educação Matemática). Rio Claro, São Paulo: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Silva, S. R. V. (2004). Identidade cultural do professor de matemática a partir de depoimentos (1950–2000). Rio Claro, São Paulo, Brazil: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Silva, V. C. (2014). Narrativas de professoras que ensinam matemática na Região de Blumenau (SC): sobre as Feiras Catarinenses de Matemática e as práticas e concepções sobre ensino e aprendizagem de matemática. Tese (Doutorado em Educação para a Ciência). Bauru, São Paulo: Faculdade de Ciências, Universidade Estadual Paulista. Silveira, D. D. (2018). O que os surdos contam sobre a matemática nas suas trajetórias escolares: a produção de fontes em libras a partir da história oral. Dissertação (Mestrado em Ensino de Ciências e Matemática). Pelotas, Rio Grande do Sul: Universidade Federal de Pelotas. Souza, C.  A. (2016). Circulação e apropriação de ideias em educação matemática: Aproximações. Tese (Doutorado em Educação). São Paulo, Brazil: Faculdade de Educação, Universidade de São Paulo. Souza, G. L. D. (1998). Três décadas de educação matemática: Um estudo de caso da Baixada Santista no período de 1953–1980. Dissertação (Mestrado em Educação Matemática), Rio Claro, São Paulo, Brazil: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Tuchapesk, M. (2004). O movimento das tendências na relação escola família matemática. Rio Claro, São Paulo, Brazil: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Vianna, C. R. (2000). Vidas e circunstâncias na educação matemática. Tese (Doutorado em Educação). São Paulo, Brazil: Faculdade de Educação, Universidade de São Paulo. Zanardi, L.  S. de O. (2012). O trânsito de professores durante o processo de criação da Universidade Estadual Paulista Júlio de Mesquita Filho (UNESP): A questão dos ressentimentos. Tese (Doutorado em Educação Matemática). Rio Claro, São Paulo, Brazil: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista. Zaqueu, A.  C. M. (2014). O programa institucional de Bolsas de Iniciação à Docência (PIBID) na formação de professores de matemática: Perspectivas de ex-bolsistas. Dissertação (Mestrado em Educação Matemática). Rio Claro, São Paulo, Brazil: Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista.

Chapter 7 Narratives and Their Power Against Silencing in and by Scientific Research Luzia Aparecida de Souza

Abstract: The key role of oral history is to identify narratives which may be used in several ways. This text deals with the experience of the Oral History and Mathematics Education Research Group (GHOEM), which works with oral history and, as such, uses narratives as the core element of its investigation practices. The objective of this chapter is to debate the concept of “narratives,” as well as the ways they can be created/built and how they can be analyzed. Other topics are also approached such as sources, attentive listening, multiplicity of truths, form, and content (form-as-content), and the need to motivate/promote different sensitivities which are not common within academia when scholars search for new ways of thinking and building knowledge.

Keywords: Narrative · Listening · Analysis of Narratives · Knowledge · Sensitivity Bruner (2014) starts his book by questioning if speaking about the narrative is still required. Answering “yes” to his own question, he points to one of the great dangers to knowledge building: taking things for granted. In order to know, one needs to wonder. This text approaches some of the estrangements which have directed the use of narratives made by the Oral History and Mathematics Education Group—GHOEM. Here, the term “narrative” is interpreted according to Garnica and Souza (2012, p. 78), that is to say a narrative is a way to externalize feelings and representations required for building the self. It is about telling a story, and this can be done either orally, or in writing, via images, tales, and songs, among other forms which enable individuals to devise ways to constitute the world or exist in the world. When narrating to someone, individuals constitute themselves as historical beings before that person. Benjamin (1994) “In a certain sense, the narrative itself... is an artisanal form of communication. The narrative is not concerned

L. A. de Souza (*) Federal University of Mato Grosso do Sul, UFMS, Campo Grande, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2019 A. V. M. Garnica (ed.), Oral History and Mathematics Education, History of Mathematics Education, https://doi.org/10.1007/978-3-030-16311-2_7

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with conveying the ‘purity’ of what is narrated in itself, like a piece of information or a report” (p. 206). There are multiple ways of telling a story. In the context of childhood, for example, a sentence can frequently be a narrative.1 Bruner (2014) stated that It is through narrative that we create and recreate individuality; the “self ” is the product of our story-telling, not essence to be scrutinized in the depths of subjectivity. Nowadays there is evidence that, if we lack the ability to concoct stories about ourselves, individuality cannot exist. (p. 96) Considering the investigations of GHOEM, it is remarkable that one of the first concerns of the group was related to the power of speech not as opposed to silence but as opposed to silencing those who can be identified as infamous (or discredited) by historiography and science. Since 2002, the work which has been developed under this perspective has sought to include in the debate about mathematics education the experiences and contexts of production of teachers, students, and other members of marginal school communities, as compared to the institutions and names commonly targeted by contemporary writings about mathematics education. It is worth noting that this movement was constituted as a historiographic discussion within the group, in the sense of abandoning the notion that there can only be history if there are sources (which implies mobilizing records or great names and renowned institutions). This is in line with the notion, already signaled by the Annales School, that history can only exist if there are questions. Not only does this orientation lead to a search for historical sources, but to their creation, which fosters a decentring effort. Although the historiographic discussion is critical, it is not the only discussion currently taking place within GHOEM. There are several works without historiographical bias, and whose objective is to understand the processes related to teaching and/or learning of disciplinarily organized mathematics and multiple mathematics. Whatever the motivation, it is worth pointing out the political dimension of all these works. The silencing imposed by historiography, and science in general, on the so-called “common” people has happened in Brazil—however not solely in Brazil—as a political project which must be challenged. Silencing consists in imposing onto others the weight of an arrogant existence that elects groups which are worth being heard, but also, and more importantly, it is the perversion of convincing others that they have nothing to say. The drawing of Susano Correia (2018, pp. 76–77), an author who uses the concept of “witchcraft,”2 may serve to illustrate this silencing (see Figure 7.1). The work we have performed with narratives also raises the debate about silence and silencing. There are several ways to navigate this dimension which contemplate a range of topics: from a timid insertion of new sources to foster academic discussion to proposing a confrontation against the creation of centers and margins. Some of these ways may be seen as stages of a process which ontologically empties the terms “center” and “margin.” One way or another, the attempt to break up silencing led researchers to affirm the oral history methodology as an effort to give a voice to those who thought they had no voice. The incursion into other debates would reveal paternalism and also the colonizing perspective present in this

 For example, when her teacher asked her what school is for, Marina, 4 years old, answered: “To go to school.”  These are visual aphorisms which express a thought or a feeling under the view of daily “witchcrafts” or “charms.” 1 2

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Figure 7.1.  Man carrying the weight of silence (in Correia, 2018). attitude. One cannot give voice to a person.3 A group becomes a movement when it regulates its actions. The work with narratives, as we advocate in this text, as represented by the provocation of Susano Correia, seeks to equip speakers with amplifiers, so they can holler as loud as possible. The work with narratives also aims to create environments so that, when speakers shout, they may perhaps be heard. 3  In this sense, we must point out an excerpt from the most recent book published in Brazil by Alessandro Portelli (2016). Writing about his relationship with Dante Bartolini, a member of Italian anti-fascist groups, Portelli stated: “The relationship among intellectuals and members of the working class in places such as Terni, can be very complicated. In the archives of the Communist Party, I found a report from 1927, when the party was being forced underground. This report stated that comrade workers were frustrated because they were being silenced by people with formal education who were joining the party and knew how to speak the political jargon better than they could. In fact, the workers were being silenced by us. …Dante inverted the relation: his comrades and he were in command, and in fact they used me. With Dante, I learned that the idea which we were ´giving a voice´ to someone who had no voice is meaningless. Dante was not a person who had no voice, but I was. I could not sing, I had no stories to tell, and I could write only because people like Dante gave me a voice. I returned the favor by hearing and amplifying their voices.” (pp. 31–32).

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The main debate about this issue did not arise from theorists recognized by academia; the main debate came from the research participants themselves, who, while talking about their memories, reveal the place from where they talk, the “place of speech,” in its own historicity. This perception arises from an effective listening exercise (Larrosa, 2007) which proposes to be mindful in hearing people of different ages, places, genders, and histories, who, while talking about themselves, speak about a perverted process, of which most times they are not aware or is not clear to them, but through which they learned to see themselves as non-historical beings,4 or speak about school as a place where “one must toe the line.”5 When we encounter this type of self-reflection, it is important to remember a passage by Ítalo Calvino (2005), often mobilized in our research along the years: The state of things in the world was still confusing in the remote time when this story takes place. Often one would encounter names, thoughts, forms and institutions that did not correspond to anything that existed. On the other hand, the world thrived with objects, faculties and people that had no names and were not distinguishable from the rest. It was a time when the will and stubbornness to exist, to leave a mark, to make friction with everything, were not used entirely, since many people did not do anything with that—either out of misery or ignorance, or perhaps because everything was going their way just fine—and so certain people were lost in a void. It could even happen that, at a certain point, this will and self-awareness, so diluted, would condensate into a clot, same as the imperceptible water droplet condensates into cloud flakes, and this tangle, either by accident or instinct, stumbled onto a name or strain, as there were many available at the time, onto a certain patent of a military organization, a set of tasks to be carried out and a set of established rules and, most of all, onto an empty trap, since without it, with the passing times, even an existing person would run the risk of disappearing.... I, who tell this story, am sister Theodora, a nun from the religious order of Saint Columban. I write at the convent, deducing things from ancient documents, from talks overheard in the parlatory and from some rare testimonies of people who have been there. We, nuns, have few chances to talk to soldiers: and so, what I do not know, I imagine; otherwise, what would I do? And not everything in the story is clear to me. You will excuse me: we are girls from the countryside. Although our origin is noble, we have always lived in retreat, in lost castles and then in convents, except for religious functions, triduums, novenas, farm work, threshing of cereals, vintages, slave flogging, incest, fires, hangings, invasions of armies, loots, rapes, pests, we have seen nothing. What can a poor nun know about the world? (pp. 31–32) The production of a narrative in research demonstrates the willingness of the interviewee to hear and speak and requires nothing less from the researcher. This way, one can say that, if there is something to be said or heard in an exchange of glances, one has an encounter. It is indeed this encounter which is expected from oral history interviews: the encounter out of which narratives and individuals emerge.

 This happens, for example, when former teachers, now retired, suggest we search information for research in school archives, because they believe that they have nothing to contribute about educational practices they used to perform for decades. 5  This is what a 4-year-old child told the teacher while (subversively) drawing beyond the lines of the composition which had been previously provided. 4

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From Benjamin (1994), we have: Story-telling has always been the art of telling stories again, and this art is lost when the stories are no longer preserved. It is lost when nobody else spins or weaves while listening to the story. The more the listeners forget about themselves, the more deeply what is heard is etched into them. When their pace of work takes possession of them, they listen to the stories in such a way that they spontaneously get the gift to narrate them. (p. 205) If, on one hand, the first movement to confront silencing is to exercise the power of speech, on the other hand, being heard is essential. However, there is much more involved. Larrosa (2007) called “attentive listening” the process of hearing the other person as another person, not as another “me.” When I hear the other person from my position, I take this person for granted and, in this situation, hardly any production of knowledge will take place in (or from) this; only recognition will take place. If we take the other person for granted, we are talking about recognition. Therefore, working with narratives, in our point of view, is not only a theoretical option but, mainly, a political option. 

On “Inappropriate” Questions for the Narrative

According to the dictionary, “inappropriate” relates to an act or procedure which is improper, incorrect, and inopportune and which cannot be recommended. But we wonder: are there any inappropriate questions? Could stating that there are inappropriate questions also be a way to silence others? We do not presume to indicate what should or should not be questioned, what may or may not be asked, but to explain that the narratives do not answer some types of questions by the very foundations of their construction, the nature of their own constitution. Examples of inappropriate questions (which we have often encountered) are the following: (a) what actually happened; (b) how many narratives are required in order to generalize and call any given statement “historical”?; and (c) what side should one take when different narratives describe different truths, whether comparing narratives or comparing them to written documents? The work with narratives, the research which seeks to apprehend and confront the world by listening attentively to the narratives of others, will not be a suitable field for readers to whom these “inappropriate” queries make sense. If such improprieties are considered appropriate, it is because we do not operate—ourselves and the readers—under the same language game.6 The issues we deem inappropriate are those that no narrative will address or answer, precisely because of the lack of meaning which characterizes them.

 A proper way of using language, based on the discussions proposed by Wittgenstein (1999). “A major point in the midst of these provisions is that for this second Wittgenstein, there is not something common and essential to all languages, but only similarities which can vary from one “language game” to another. For this author, every form of life has its own way of using language. It is language use which makes a particular form of life constitute itself as such, for each form of life establishes how words, expressions and gestures are used and how meanings are consequently negotiated for those words, expressions and gestures. Everyday expressions can be easily found to be used in different ways, depending on the environment and the situation. Merely consulting a dictionary will show countless uses of any given word” (Garnica & Pinto, 2010, pp. 207–244). 6

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The phrase “what actually happened” appeals to the “pure existence” or “the (version of) majority, which is therefore valid.” If something happens to a group of people, who can say what happened? “What happened” exists beyond what each individual can say about what happened? If the descriptions of all respondents converge on a given situation, with the exception of one, is the version of the majority truer than the version of the individual whose description diverges? Is it accepted that divergent opinions or conflicting accounts are the expressions of the way through which “what has happened” has been given meaning? Is the individual with the dissimilar account lying? Is “historical truth” the version of the majority? If we consider that each narrative is produced from the singular intersections, the subjective affections of those who produced it, would it make sense to search for a general construct which actually erases the differences which narratives have the power to reveal? One of the greatest powers of the work with narratives resides in the ability to remind us that we are unique in our multiplicity and that any attempt to treat narrative protagonists as representative of any category is perverse. It is necessary to define multiplicity as much as it is necessary to reiterate the importance of differences, but not in the sense of romanticizing the discussion. It is not enough to work toward accepting diversity, it is necessary to problematize the production of differences. “Diversity [can be seen as] the result of a relational, historical and discursive process of constructing difference” (Silva, 2015, p. 101). Moreover, it is not only a matter of following a line of research or interfering in the world in such a way that diversity and multiplicity are tolerated: diversity and multiplicity must be respected. More than that, they should be promoted. In this sense, narrative research must embrace the protection of multiple stories not cancel out differences, challenge all attempts at generalizations, and must not condone preconceptions and intolerance. The truth can only exist in a provisional and plural state: truths (!). Fruits of cultural and often moral intersections, this plural truth is defended through a coherent discourse which supports the legitimacy and richness of contradictions. According to Albuquerque Junior (2007), new history itself is linked to the field of symbolic practices, in which objects and subjects become denaturalized, they are no longer metaphysical, and are therefore thought of as historical fabrications, as the fruit of discursive or non-­discursive practices that institute them, summarize them, name them, classify them, enable us to see and say. (p. 21) Bruner (2014), on the other hand, stated: Through narrative we build, rebuild, and somehow reinvent yesterday and tomorrow. Memory and imagination are merged in this process. Even when we create the possible worlds of fiction, we do not desert the familiar, but subjunctivize it in what could have been or what might be. As much as the human mind has exercised its memory and refined its recording systems, it can never fully and faithfully capture the past. On the other hand, it can never escape the past. (p. 103) Assuming the inventive character of the narrative (thus agreeing with Bruner, 2014), of history and historiography (agreeing with Albuquerque Júnior, 2007) is to understand its construction as derived from the perambulations of the narrator.

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Sources, Formation, and Narrative Aesthetics as Possibilities of Ethical Commitment in Academic Work

Consistent with the views of Veiga Neto (2007) and Albuquerque Junior (2011), we argue that texts are neither the object nor a representation of an object. When text is produced (oral, written, imaginary, etc.), we construct the object of which we speak. The characters of the stories narrated, for example, are “subjects in a state-of-word.” As Albuquerque (2011) stated: The man who tells a story is not the same man who lives it, even when he tells his own life. When recounting memoires or autobiography, the narrator does not coincide with the narrated subject, the subject of the narrative is not the same character in the narrative. Because the subject of the narrative is alive, a being of flesh and blood, it is a subject in whose veins blood flows. The narrated subject is in a state of speech, made of paper, a subject in whose veins ink runs. The historian who presumes to be speaking about the subject of flesh and blood, will in truth speak about the subject of paper and ink, which reaches him through distinct forms of representation, even though such forms of representation allow them to at least ensure that the character was indeed a part of the past. The historian will give the character a new life, in ink and paper, although part of the reading covenant between the reader and the author of historiographical texts is to believe that it has spoken about the person of flesh and bone who once inhabited the past. (p. 254) For Bruner (2014), there is no readily recognizable self, instead, we constantly build and rebuild ourselves, to meet the needs of situations we face, and we do so with the guidance of our memories of the past and our hopes and fears for the future. (p. 74) In this sense, the term “narrative” has been used in our research in at least two distinct movements, both as a source and as a perspective for analysis: (a) oral narratives are sources for teaching and for research; (b) narratives are a type of analysis based on singularities (and not on elaborate categories aimed at generalization). The work with narratives also enables a potential aesthetic turn, evidenced in the escape from the rigid and limiting classical academic guidelines. 

Narratives as Sources

Life histories and the reports resulting from thematic interviews stand out in relation to the first of such movements—oral narratives taken as sources for teaching and research. Both life stories and thematic narratives are structured through textualizations.7 The work  Textualizations are the edited form of the written record of interviews. According to the canonical procedures of oral history, the interviews are recorded and later transformed into written text (at first, the recordings are transcribed). The textual elaboration after transcription is called textualization. Textualizations—not recorded tapes or transcriptions—have been the sources for our oral history research and interventions. Interview scripts can be created in many different ways. Some researchers opt for scripts composed of more targeted queries, others opt for more open questions. The scripts directly focused on a situation or specific moment of the interviewees’ experience (their experience as a teacher, for example, or as an agent of public policies related to education) characterize a modality called “thematics oral history,” while broader scripts with no specific restrictions characterize what has been known as “oral life stories.” Other chapters in this book deal with this differentiation and terminology, specific to oral history work, in more detail. 7

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is developed from a generating question which prompts interviewees to talk about themselves, the way they live, and the constructions and elaborations carried out throughout this process. This type of approach has been shown to be significantly rich, since it is possible to recognize how some issues of special interest for mathematics education are shown to be deeply linked with peculiar experiences and circumstances. Transcribing an interview consists recording, through writing, dialogs, speech, silences, and interruptions. This is intended to be as literal a record of the interview as possible. Because it is the record of something produced to be heard, for the most part, the text is not suitable for reading. Textualization, on the other hand, results from a first analytical exercise which does not seek a literal approach in relation to the moment of the interview: textualization is a text produced for reading and, therefore, without the obstacles, problems, and vices of language recorded in the transcript. Our experience with research exercises which use oral history has shown that interviewees opt for reading the textualization,8 not only because of how the text flows but for the possibility that in this textual production one perceives themselves as whole, not shattered to pieces. This perception is fundamental for an effective negotiation process of the final version of the document. When examining the narrating subject produced in the text, the respondent is more likely to detect distortions arising from the alteration of the order of ideas (when the researcher seeks to approach similar themes while working on text), distortions of meaning (when there is a clear problem of communication between interviewee and researcher), and/or distortions of emphasis (explained by the choice of punctuation). The policy of GHOEM, of creating opportunities for interlocutors to identify themselves or not with the narrative constructed from the textualization, is very important, but insufficient. It is also necessary to create conditions for the interviewees to evaluate the texts as something they would say (since, in effect, textualization is a production of the researchers who obviously consider what was narrated to them by the interviewees), suggesting changes, exclusions, and inclusions of excerpts in order to guarantee the inalienable right of interviewees to their memories. This entire negotiation process between researcher and interviewee results in a letter of assignment of rights that the respondent grants the researcher who, in turn, undertakes to divulge the documents produced from the interview (audio and/or video recordings, transcription, textualization, according to the determinations of the interviewee in the said letter of assignment of rights), as well as its ethical use. In addition to this letter, this process results in a great approximation with the oral sources (respondents) and the resulting writings. This approximation between sources, in turn, is fundamental for researchers to listen to what has been said, resisting the temptation to listen solely to what they expect the answers to their research questions will be. Viveiros de Castro (2010) indicates the need for such care in the following way:. “Let’s see how the natives solve our problems in relation to nature.” And we are disappointed to realize that first the natives have other problems and second, they do not solve ours. ... So we can only know what their problems are and how listing such problems can solve our own impasses, which are not theirs. (p. 19) Under this light, the researchers of GHOEM value the integral insertion of textualizations obtained through research in the body of the reports of their work. This option results from

 Transcription and textualization are made available to interviewees to be verified and, in case of written records, complemented or corrected before being effectively included and analyzed in the research papers. Usually the interviewees choose to check only the textualizations, and it is these texts that researchers have effectively used as their main sources for their work. 8

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the intention of allowing other people to create guidelines, from a multiplicity of stories, to get out of their own impasses and also to problematize obvious surrounding issues. Thus, it can be said that, either as a source or as a strategy of analysis, the resulting narrative is a joint production between researcher and interviewee seeking to share communicative space. According to Bolívar (2002), subjects are constituted in narrative while speaking about themselves to others, and this speech changes naturally with time, with the circumstances, and due to the “other” before them or in their mind. Thus, a narrator is not always the same. The narrating subject, the subject created by the narrative, changes, as Fernando Pessoa (1980) has explained: Everything we see, we should always see it for the first time, as it is really the first time we see it. Each yellow flower is a new yellow flower, although it is called the same as yesterday’s flower. We are not the same and neither is the flower. The colour yellow itself can no longer be the same. (p. 267) Narrative sources constitute not only narrating subjects but also notions about teaching, learning, mathematics, others, teaching, school, etc., which are essential to the understanding of the way individuals face and construct the world and, in a more specific way, create the school and develop teaching as a profession. This way, the narratives constituted by the research that we have conducted have been explored in their potential for research and teacher training. GHOEM has created, from the work of Oliveira (2013), HEMERA,9 a textualization registry, to divulge hundreds of oral testimonies of former teachers, students, and members of the school community in different Brazilian institutions and states. It is interesting to note the ending of the presentation of HEMERA to researchers: “We hope HEMERA will help you produce many narratives.” This phrase expresses two central questions to the group: the potential of narratives and the political need that they be seen in multiplicity, as a way of promoting diversity. The use of these sources for research has been done within a broad project of mapping the training and practice of teachers who teach, or have taught mathematics, in Brazil as well as other parallel projects related to inclusive education, studies on childhood and illiteracy, on teaching mathematics, or about the very constitution of mathematics education as a field of study in Brazil. The power of narratives in education and teacher training has been widely explored by GHOEM, through work carried out with the objective of constructing and problematizing experiences with school, teaching, and mathematics. Such problematization is fundamental for the understanding of the narrator’s own practices and posture, whether it is conducted by the narrators themselves, or by future teachers-in-training who work with the narratives. Considering that teachers live all their school life in a space destined for the profession, it is natural that they acknowledge this space and the actions to be implemented by the professionals in their areas of activity. While talking about narratives in school settings, Bolívar (2002) stated that the narration is the real matter which constitutes teaching and that its exercise is an “epistemological proposition [in which] the knowledge of teachers is expressed in their own terms by narratives and can be better understood this way” (p. 46). Therefore, the work with narratives has a political and pedagogical function: provoking estrangement in order to produce knowledge.

 Available at http://www2.fc.unesp.br/ghoem/index.php?pagina=hemera_filtrar.php. The last search was conducted on October 30, 2018. 9

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Narratives as Possibility for Analysis

In another movement, but still concerned with looking at those who are narratively constituted, we have exercised a narrative analysis, or, in other words, sought to construct a story (a narrative) in which those other stories (narratives) constructed by research are articulated, in order to understand the singularities and the coexistence of different and sometimes opposing perspectives. Within GHOEM this analytical perspective has been potent in the sense of demonstrating the simultaneous occurrence of multiple histories, of different sensitivities in relation to objects and situations. It is a research approach that, by avoiding generalizations from personal histories, promotes diversity and thus becomes politically engaged so as not to act as a settler by imposing anything as true, or any story as unique. Narrative analysis, as the group has tried to implement, articulates sources, narratives, in a story, also singular, elaborated by the researcher from what they read and heard during the investigation process. The result is an analytical text, usually a first-person narrative, which seeks to construct a plot which articulates different histories without assuming that each separately or all combined characterize “history that is more comprehensive or truer” than others. It is an effort to understand and problematize the production of diversity and the multiplicity of world. Under this light, in the works developed by the research group it is common to use words in the plural: there are several mathematics, modern mathematics,10 developments, histories, etc. It is worth noting, however, that in our day-to-day research, we often use categories and classifications to organize arguments and understandings, as a way of producing discourse. Is this a contradiction? Certainly, not only are we unique but we are also plural in our uniqueness. Here, “singular” and “singularity” are not taken as opposites to “plural” and “plurality.” What we say about our interviewees, about history, and our research, naturally applies to us as well. Thus, there is no denying the coexistence with contradictions and paradoxes which make us who we are. Influenced by different perspectives (either theoretical or not), we try to reconcile oppositions, without excluding conflicting points of view, without assuming that any of such viewpoints (the non-excluded) are homogeneous or better than others. This is one of the examples of the ongoing effort to work with narrative analyses. Such effort involves a departure from Cartesian thought that operates under a cause-­ effect perspective, to escape contradiction. However, if the objective is not to hide or circumvent contradictions, neither it is to emphasize or point out contradictions in a judgmental way. It is rather a quest to pacify contradictions, a search for understanding them, considering the world as a place where opposites coexist and point to the multiple meanings which our research seeks to unveil. It must be reiterated that the aim is not to judge, but to heighten awareness of different ways of life, different perceptions and experiences. This requires engendering thinking into thought. It entails an inclusive view of the contradictions, sensibilities, and insecurities which characterize any living being. This kind of thought tends to lead toward an escape from rigid structures and imposed boundaries (e.g., those dictated by classical academic discourse), which steers research toward another possibility of narrative analysis: form as content.  The reader should note that in Portuguese, adjectives as well as nouns have plural. In the specific case of the term matemática, the plural is matemáticas. The plural of matemática moderna (modern mathematics) is matemáticas modernas. This is important because the plural, according to the intent of the group, manifests an idea of diversity and plurality, which the present chapter seeks to highlight. 10

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Form-Content Rational thought has long been associated with sanitized textual elaboration, produced under the deceptive guise of neutrality, of complete distance between the researcher who shows something and that which is shown—the researchers who expose without being exposed themselves. Not infrequently, in academic form which is considered as canonical, quotations of authors viewed as references are catenated as if in forming a shield designed to protect the positions defended in the text. An idea is supported and protected by great authors and recognized theoretical trademarks. Who would dare question statements based on the authority of so many theories and theorists? Surely, it is not a question of denying the importance of citations and the dialog with reference authors: but of defending the need for theorists to mobilize effective and efficient tools themselves. We understand that the production of any given author is theirs, and even though they decide to position themselves behind shields, this still reveals something about those who apparently feel protected. Our production indicates the manner through which we operate from the crossroads presented to us by several texts (written, oral, imagistic). There is no neutrality or distance. Every theory is a toolbox, and as such only makes sense if and when it is actually used. Not using the tools available and those we claim to possess is innocuous. Moreover, it calls into question the legitimacy of any research. In light of this perspective, we state that there is not a single form of textual elaboration—a unique structure for the composition of texts—which demonstrates one’s production and divulges it. Thus, the production of an alternative aesthetic for narratives in the elaboration of research reports, articles and theses has been shown to be a significant exercise in the works developed by the group. In order to reiterate the position that opposites and conflicts can coexist, it should be pointed out that alternative textual elaborations have coexisted, harmoniously, with the classical aesthetics of canonical, traditional, and even hegemonic elaborations in academia. If we consider that when speaking about an object we constitute it before others, we view the aesthetic of the narrative as a way of speaking about oneself, about the research movement, and the process which researcher undergo while becoming and acting as researchers before the scientific community. When we use the perspectives defended here about the work with narratives while researching, the manner of divulging the research is not previously decided or defined. Rather, it is gradually revealed during the investigation, as the emphasis on any aesthetic for textual elaboration will be defined or made possible by the movement which creates the object of investigation. This way of navigating the world through research produces intensity and thereby forms. The aesthetic power of the narrative format seems to reside precisely in the explication of oneself and of all sensibility, effectivity, and affectivity which intersect in any path of human existence, including that of academic research. It is common to find work which argues in favor of the qualitative research approach. Among the several aspects which characterize such qualitative approaches is that which emphasizes process, rather than product. Although citations involving this aspect are very common in academic work, silencing on how to operate with this idea is also very common. How does the investigative process manifest in the production of research? How have the “results” of analyses implemented effectively subsidized the training of researchers? How have authors emphasized the processes, and why or why not are they as important or more important than the products generated? With regard to our work with the narrative format, we see that textual elaborations have mobilized multiple ideas and strategies, combining letters (epistolary literature), fiction, novels, essays, orientation sessions, and multi-volume reports, without any predefined order by the author, and radio programs, among others, which seek to reveal something

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through their form. The form of research reports that we have produced and divulged through articles, books, and theses is now thought to be politically and ethically articulated with the conceptions and manner with which researchers position themselves in and before the world. Moreover, as Portelli (2016) has stated, oral accounts do not specifically generate texts but performances. “In orality we are not dealing with finished discourse, but with the discourse in process (in fact, dialogic discourse in process)” (p. 19). Thus, it is necessary that the form used in textual elaboration for the dissemination of our work be in tune, in order to capture the diverse nature of our sources and practices. Therefore, form has been seen as content, as a way of establishing a relationship with research, looking for means to “speak differently,” that is “to say other things,” “to think in other ways,” “to propose means for engendering thinking into thought.” As Burke (2005) pointed out: “A declaration of love, for instance, is not, or rather not just an expression of feelings. It is a strategy for encouraging, amplifying or even transforming the feelings of the loved one” (p. 142). The work with narratives, as defended in this chapter, is not intended as an alternative to replace one way of doing research with another, or to argue for the use of narratives as a “new” trend to be mobilized: the aim has been to defend an alternative viewpoint, centered on the plurality of ways of thinking, recording, and problematizing investigative processes and researcher training.

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Author Index

A Adichie, Chimamanda., 61, 79 Albuquerque Junior, Durval M. de, 106, 107 Alexievich, Svetlana, 1, 11 Almeida, Shirley P.N. de C. e, 30 Alves, Cláudia., 55 Andrade, Mirian M., 81, 82 B Bagio, Viviane A., 92–93 Balestri, Rodrigo D., 52 Baraldi, Ivete M., 16, 30, 61 Benadiba, Laura, 49, 53 Benjamin, Walter, 101, 105 Berger, Peter, 42 Bernardes, Marisa R., 91 Bolivar, Antonio B., 5, 109 Bourdieu, Pierre, viii, 41 Brito, Arlete de J., 49, 79 Bruner, Jerome, viii, 5, 42, 101, 102, 106, 107 Burke, Peter, 112 C Calvino, Italo, 104 Candray Menjivar, J.C., 93 Carraher, David W., 50 Carraher, Terezinha N., 50 Certeau, Michel de, viii, 25 Chartier, Roger, viii Chervel, André, 60, 61

Correia, Susano, 102, 103 Cury, Fernando G., 25, 30 Cyrino, Márcia C.C., 52 D D´ambrosio, Ubiratan, 79 Deleuze, Gilles, viii, 39 Delgado de Souza, Gilda L., 85 Descartes, René, 42 Durkheim, Emile, 42 E Elias, Norbert, 41, 42 Esquisani, Rosimar S.S., 27 F Faria Filho, Luciano M. de, 23 Fernandes, Déa N., 22, 30 Fernandes, Filipe, 35–44, 93 Ferreira, Marieta de M., 49, 50 Flugge, Flávia C.G., 60, 77, 78 Fontes, Cilene M., 93 Foucault, Michel, viii, 39, 42 G Gadamer, Hans-Georg, viii Galetti, Ivani P, 17 Galhart, Anna C., 91 Galvão, Ana M. de O., 22

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Garnica, Antonio Vicente M., 1–18, 22, 24, 25, 27, 31, 32, 36, 49, 57, 61, 67–83 Gasparello, Arllete, 54 Gomes, Maria Laura M., 21–32 Gonzales, Kátia G., 30, 79, 80 Guattari, Félix, viii, 39 H Hartog, François, viii Herodotus, 1 Hobsbawm, Eric, 22 J Janotti, Maria de Lourdes M., 28 Joutard, Philippe, 2 K Knijnik, Gelsa, 60, 64 L Larrosa, Jorge B., 104, 105 Le Goff, Jacques, viii Lins, Romulo C., 42 Lopes, Elliane M.S.T., 22, 23 Luckmann, Thomas, 42 Lugli, Rosário G., 24 M Maron, C. do R. de M., 93 Martins, I.M.H., 92 Martins, Maria Ednéia, 14–16 Martins-Salandim, Maria Ednéia, 24, 30, 37, 38, 40, 79, 80, 85–97 Massey, Doreen, viii, 39 Matos, José M., 54 Matucheski, Silvana, 60, 62, 64 Mcculoch, Gary, 63 Mead, George H., 42 Mendes, Iran A., 49, 52, 80 Michelet, Jules, 22 Miguel, Antonio, 48, 50, 54, 55, 69–71, 75, 79, 80 Mindiate, Manuel J., 92 Miorim, Maria Ângela, 23, 48–50, 54, 55, 69–71, 75, 79, 80 Morais, Marcelo B. de, 32, 35–44 Morin, Edgar, 39 Murbach, Márcia, 96

Author Index Pires, Rute da C., 17 Plotinsky, Daniel, 49, 53 Portelli, Alessandro, 1, 26, 48, 87, 103, 112 R Ricoeur, Paul, viii, 5, 26, 29 Rolkouski, Emerson, 41–43, 91, 94, 95 Rolnik, Sueli, 39 Romanelli, Otaiza, 23, 24 Rosa, Erica, 92 Rosa, Fernanda M. da, 92, 95 S Sachs, Linlya, 81, 82 Santhiago, Ricardo, 49, 81 Santos, João R.V. dos, 35–44, 93 Sarlo, Beatriz, 26 Saul, Ana M., 64 Saviani, Dermeval, 23 Schmitt, Jean C., viii, 38 Schliemann, Ana L., 50 Schwarzstein, Dora, 49 Scotson, John L., 41 Silva, Alexandre, 55, 60, 92, 95 Silva, Heloisa da, 22, 36, 42, 43, 47–64, 75–77 Silva, Viviane, 91 Silveira, Daniel D., 92, 95, 96 Souza, Carla, 94 Souza, Luzia A. de, 23, 31, 48, 49, 53, 61, 73, 74, 79, 101–112 Souza, Rosa Fátima de, 31 T Tanuri, Leonor, 23 Thompson, John B., viii Thompson, Paul, 5, 6 Tizzo, Vinícius S., 60, 67–83 Tuchapesk, Michela, 92, 94 V Valente, Wagner R., 23, 24, 49, 50, 54 Van der walle, John, 75 Veiga, Cynthia G., 23 Veiga Neto, Alfredo, 107 Vianna, Carlos R., 1–18, 51, 85, 86, 91, 94, 95 Vicentini, Paula P., 24 Viola Dos Santos, João R., 36

N Nietzsche, Frederich, viii Novoa, António, 55

W Wenger, Etienne, 42, 50 Wittgenstein, Ludwig, viii, 93, 105

P Paiva, Paulo H.A. de, 30 Pinto, Thiago P., 105

Z Zanardi, Luciana, 91 Zaqueu, Ana M., 93

Subject Index

A Analysis of Narratives, 6

L Listening, 104, 105

C Classroom, vii, 2, 12, 31, 56, 59, 63, 67–83, 91 Concepts, vi, vii, 1, 4, 29, 37–43, 48, 50, 52, 54, 57, 69, 70, 72, 74, 75, 79–81, 93, 102

M Mathematics education, v–viii, 1–18, 21–32, 35–44, 47–64, 67–83, 85–97, 101–112

H History, v–viii, 1–18, 21–32, 35–44, 47–64, 67–83, 85–97, 101–112 History and historiography of mathematics education, 24–28 History of mathematics education, vi, vii, 3, 13, 14, 18, 21, 23, 40, 47, 49–57, 67–83 History of training and activities of mathematics teachers in Brazil, 3, 22, 25, 26, 29, 62

N Narratives, v–viii, 3–9, 25, 27, 31, 36–43, 49, 53, 56, 57, 60, 62, 75, 79, 81, 86–96, 101–112 Non-historiographical research, viii, 85–97

I Interdisciplinarity, 35 Intervention in teaching, 47–64 Interventions, vi, vii, 5, 8, 10, 17, 47–64, 76–79, 82, 86, 88, 90, 107 K Knowledge, v, 2, 4–6, 10, 11, 15, 18, 22–24, 26–28, 36, 37, 44, 50–52, 55–57, 59–61, 63, 64, 68, 70, 72–74, 78, 80, 90, 91, 101, 105, 109

O Oral history, v–viii, 1–18, 21–32, 35–44, 47–64, 67–83, 85–97, 101, 102, 104, 107, 108 Oral history and mathematics education, vi–viii, 3, 12, 14, 16, 21–32, 35–44, 47–64, 85–97 P Panorama, 22–24 Pedagogical approach, 48–50, 52, 55, 58, 59, 76, 77, 86 Philosophical foundation, 35 Procedures, vi, 3–6, 9, 12, 25–27, 36, 41, 44, 48, 61, 76–78, 87, 89, 90, 94, 96, 105, 107

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128 R Research methodology, 2–4, 86, 87, 94 Research protocols, 87, 90 S Sensitivity, 6, 87, 110

Subject Index T Teacher training, vii, 22–24, 27–30, 40, 50–52, 54, 57–60, 62, 68, 69, 71, 72, 75, 81, 92, 93, 109 Theorization, 36, 37, 40, 43