Following in the steps of the socio-political turn of the discipline, Equity in Mathematics Education: Addressing a Chan

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*Table of contents : CoverSeries pageEquity in Mathematics EducationLibrary of Congress Cataloging-in-Publication DataContentsForewordPrefaceCHAPTER 1: Equity and Social Justice in Mathematics EducationCHAPTER 2: From Equity and Justice to Dignity and ReconciliationCHAPTER 3: Why the (Social) Class You Are in Still CountsCHAPTER 4: Beyond the Binary and at the IntersectionsCHAPTER 5: Right to Learn MathematicsCHAPTER 6: Disability and Equity in MathematicsCHAPTER 7: Identification and Educational Support for Students With Learning Difficulties in Mathematics in Denmark, Finland, and SwedenCHAPTER 8: Parental Involvement and Equity in MathematicsCHAPTER 9: Promoting Equitable Teaching in Mathematics Teacher EducationCHAPTER 10: “Pie π in the Sky”LIST OF CONTRIBUTORS*

Equity in Mathematics Education

A volume in Cognition, Equity, and Society: International Perspectives Bharath Sriraman, Series Editor

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Equity in Mathematics Education Addressing a Changing World

edited by

Constantinos Xenofontos University of Stirling, United Kingdom

INFORMATION AGE PUBLISHING, INC. Charlotte, NC • www.infoagepub.com

Library of Congress Cataloging-in-Publication Data A CIP record for this book is available from the Library of Congress http://www.loc.gov ISBN: 978-1-64113-728-7 (Paperback) 978-1-64113-729-4 (Hardcover) 978-1-64113-730-0 (ebook)

Cover photo by Panayiotis Michael. Special thanks to the graphic designer Savvas Xinaris for the concept of the cover.

Copyright © 2019 Information Age Publishing Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the publisher. Printed in the United States of America

CONTENTS

Foreword............................................................................................... vii Preface.................................................................................................... ix 1 Equity and Social Justice in Mathematics Education: A Brief Introduction............................................................................................ 1 Constantinos Xenofontos 2 From Equity and Justice to Dignity and Reconciliation: Alterglobal Mathematics Education as a Social Movement Directing Curricula, Policies, and Assessment................................... 23 Peter Appelbaum 3 Why the (Social) Class You Are in Still Counts.................................. 41 Peter Gates 4 Beyond the Binary and at the Intersections: Chronicling Contemporary Developments of Gender Equity Research in Mathematics Education................................................................... 65 Luis A. Leyva 5 Right to Learn Mathematics: From Language as Right to Language as Mathematically Relevant Resource.......................... 93 Núria Planas and Mapula Ngoepe 6 Disability and Equity in Mathematics................................................111 Vasiliki Chrysikou, Panayiota Stavroussi, and Charoula Stathopoulou

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7 Identification and Educational Support for Students With Learning Difficulties in Mathematics in Denmark, Finland, and Sweden.......................................................................... 131 Pirjo Aunio, Pernille Ladegaard Pedersen, Inger Ridderlind, Judy Sayers, and Pernille Bødtker Sunde 8 Parental Involvement and Equity in Mathematics........................... 159 Susan Sonnenschein and Brittany Gay 9 Promoting Equitable Teaching in Mathematics Teacher Education............................................................................................ 179 Marta Civil and Roberta Hunter 10 “Pie π in the Sky”: Imaginative Possibilities to Foster Diverse Diversities in Primary School Mathematics...................................... 201 James Biddulph and Luke Rolls List of Contributors............................................................................ 229

FOREWORD

Equity in Mathematics Education: Addressing a Changing World is the 11th volume in the Cognition, Equity, and Society: International Perspectives series, coming 3 years after the release of Critical Mathematics Education: Theory, Praxis, and Reality. The series is highly selective in the volumes that it publishes primarily to avoid replication of topics already covered in the existing volumes. This volume edited by Constantinos Xenofontos situates itself in the sociopolitical realm based on the issues it brings up in relation to the neoliberalized world that institutions and people are situated in. Some chapters call for a grassroots effort to resist the mandates put down by larger institutions. Social class is another issue that is brought up since most curricula are situated in frameworks of institutions that espouse “middle-class” myths situated within a culture of materialistic consumption. Newer issues of identity are explored in some chapters related to gender, disability, multilingualism, and home milieus of students. The book as a whole conveys the scale of research related to equity that pertain to mathematics education, and in a sense convey the sociopolitical nature of the field. —Bharath Sriraman Missoula, Montana

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PREFACE

Mathematics education (or didactics of mathematics, as it is typically called in continental Europe) as a research domain is, essentially, the offspring of two other scientific disciplines—mathematics and psychology—born during the first half of the previous century (Andrews & Rowland, 2014; Kilpatrick, 2014). Until the 1980s, the influences of psychology on the research directions of the newly-established domain (which was still struggling to gain its own identity) were apparent, as the research interest focused on understanding children’s mathematical thinking, through the employment of quantitative methodologies, psychometric methods, and statistical techniques. Around the 1990s, we see mathematics education taking what Lerman (2000) calls a social turn, by examining issues of a sociocultural nature, as for example, the influences of culture on the teaching and learning of mathematics. From 2000 onwards, the research domain takes another important turn, this time of a sociopolitical character (Gutiérrez, 2013), by setting and investigating questions regarding the relationships between sociopolitical concepts (i.e., authority, power relations, the social construction of success/failure) and school mathematics. Following in the steps of the sociopolitical turn of the discipline, Equity in Mathematics Education: Addressing a Changing World emerged as a response to the enormous changes that have in the last years occurred at a global level (e.g., the ongoing war in Syria, the political [in]actions of powerful nations to fight climate change, the rise of far-right parties in many countries around the world, etc.). In recent years, massive migration waves from the Middle East have caused significant demographic changes to many

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European countries, Canada, and the United States, that are reflected in schools and classrooms. These observations have led us to reconsider the concept and/or practice of equity, and its related concept, social justice, and the role of mathematics education research in addressing and promoting a fairer world. Contrary to other, perhaps highly specialized books concerned with similar topics, the book at hand aims to provide a smooth, yet deep introduction to those who are new to this research area. It refers to, and concerns, policy-makers, school teachers, researchers, undergraduate and graduate students in (mathematics) education, and anyone with an interest in the field and beyond it. This edited volume contains 10 chapters, each reconsidering current developments in a specific area (addressing the question of “What is?”) and offering ideas and suggestions of how equity (and social justice) could be promoted, addressing, therefore, the question of “What could be?” The first chapter, “Equity and Social Justice in Mathematics Education: A brief Introduction” functions as a platform providing readers with an understanding of important concepts, like equity and social justice. The chapter discusses how these two notions and practices relate to mathematics education, how they are often used interchangeably, and how they differ from each other. Furthermore, it examines a number of successful examples from the international literature that intend to enable readers to begin considering how equity and social justice can be promoted in the mathematic classroom. Finally, the chapter discusses the importance of mathematics teacher education (both initial teacher education and in-service professional development) in the development and promotion of the two concepts. In Chapter 2, “From Equity and Justice to Dignity and Reconciliation: Alter Global Mathematics Education as a Social Movement Directing Curricula, Policies, and Assessment,” Peter Appelbaum proposes an alternative approach to equity in mathematics education that is not framed by national policies at the local level. On the contrary, such an approach considers the importance of post-colonial experiences, global economic forces, and the spread of consumerism in addressing equity and social justice in (mathematics) education. Appelbaum goes on to discuss how the concepts of equity and social justice could ultimately be replaced by the notions of recognition and dignity, and how individuals can take action in alterglobal social movements towards this direction. From a Marxist perspective, Peter Gates explores in Chapter 3—“Why the (Social) Class You Are in Still Counts”—the links between social class and mathematics attainment. Social structures, as this chapter informs, employ effective mechanisms for hiding discrimination. Gates sheds light on how various factors like curricula, pupil organisation (i.e., ability grouping), and certain pedagogical approaches structure children’s experiences,

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maintaining the status quo, and reproducing inequalities related to social class. In closing, Gates draws on the conclusions of 6 decades of research and proposes a number of suggestions that might reinforce attempts to eliminate the effects of structural inequalities. Luis A. Leyva in Chapter 4, “Beyond the Binary and at the Intersections: Chronicling Contemporary Developments of Gender Equity Research in Mathematics Education,” focuses on the development of how gender has been conceptualized and studied in mathematics education, over 45 years of related research. In doing so, Leyva uses two broad categories, namely achievement and participation, to highlight themes across the research, and to discuss how various conceptualizations of gender have influenced learners’ performance and learning opportunities in mathematics. The author emphasizes the need for intersectional analyses of gender as a social construct, which would enable a better understanding of the changes required for the provision of more equitable educational opportunities in mathematics. In Chapter 5, “Right to Learn Mathematics: From Language as Right to Language as Mathematically Relevant Resource,” drawing on their experiences from Catalonian and South African mathematics lessons, Núria Planas and Mapula Ngoepe challenge the norm of monolingualism in mathematics teaching and learning, by addressing the role of language in general and of the home languages of learners in the mathematics classroom. Switching home and school languages and switching verbal and visual languages enable learners to build a sustainable understanding in mathematics. In closing, Planas and Ngoepe emphasize the role of teachers in the promotion and activation of newer norms, those of multilingualism and multimodality, an alternative to monolingual mathematics instruction. In Chapter 6, “Disability and Equity in Mathematics,” Vasiliki Chrysikou, Panayiota Stavroussi, and Charoula Stathopoulou begin by pointing out the lack of mathematics education research for students with disabilities through a sociocultural perspective. Subsequently, they turn their attention to learners with intellectual disabilities and present evidence from an action-research study with three students, aiming at informing and enriching mathematics teaching and learning in this area. This chapter concludes with the presentation of a model of several important and interrelated factors affecting mathematics education for learners with disabilities. The model includes, inter alia, the immediate educational context, mathematics as the academic subject, the classroom environment, and other contextual factors like family, educational policy, and so on. For Chapter 7, “Identification and Educational Support for Students With Learning Difficulties in Mathematics in Denmark, Finland, and Sweden,” Pirjo Aunio, Pernille Ladegaard Pedersen, Inger Ridderlind, Judy

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Sayers, and Pernille Bødtker Sund provide a Nordic perspective on the topic under scrutiny. While the Nordic countries in general are internationally perceived as having well-developed policies on social inclusion in education, little is known on the identification and provision of support to children with learning difficulties. The authors present and discuss examples from the three countries and conclude that, while in all systems, low attainment is seen as a signal for learning difficulties, each country follows different strategies for identifying and supporting children. In Chapter 8, “Parental Involvement and Equity in Mathematics,” Susan Sonnenschein and Brittany Gay consider how U.S. parents’ socioeconomic status, beliefs, and practices impact children’s academic development in general, and mathematical development in particular. Home-based opportunities, such as parents as role models, children’s engagement in mathematical or mathematics-related activities, and the amount of mathematical talk children hear, are strongly related to the socioeconomic status of their parents and this, consequently, impacts children’s mathematical learning and attainment. As Sonnenschein and Gay argue, the designing of interventions for parents of low socioeconomic status on how to facilitate the mathematical development of their children needs to be in accordance with parents’ beliefs about their role in their children’s learning, to involve schools in the training, and provide enough training for parents to feel confident in their mastery of the relevant skills. In Chapter 9, “Promoting Equitable Teaching in Mathematics Teacher Education,” Marta Civil and Roberta Hunter explore four equity-focused instructional practices, and discuss how these have informed their work in mathematics teacher education in New Zealand and the United States. Such instructional practices, according to the authors, can provide mathematically rich experiences for learners from traditionally marginalized groups in the two countries: Pāsifika students and Mexican-American, respectively. Civil and Hunter provide a number of suggestions concerned with how equitable teaching practices can be promoted in mathematics teacher education. These suggestions include, among others, the engagement of teachers in action-research or design research, teachers’ ethnographic household visits with a focus on learning from the families, and the handling and encouragement of learners’ home language(s) as an important resource for learning mathematics. Finally, in Chapter 10, “‘Pie π in the sky’: Imaginative Possibilities to Foster Diverse Diversities in Primary School Mathematics,” James Biddulph and Luke Rolls explore the complexities entailed in the politics of diversity, and the challenges of fostering and enacting a social justice pedagogy for mathematics education. The authors provide examples of experiences from the University of Cambridge Primary School, on how, as part of the school’s

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ethos, an inclusive mathematics culture and reflection upon practices that hinder inclusion and social justice were actively encouraged and put into action. Biddulph and Rolls stress the importance of critical reflection as a skill that all teachers should develop in consideration to the particularities of different contexts. Equity in Mathematics Education: Addressing a Changing World contributes to the understanding of equity and its complex relations to mathematics education. We hope that it will support individuals in teaching, educational research, policy making and planning, and teacher education, in becoming more aware of the interplay between school mathematics and sociopolitical issues that, ultimately, impacts the lives of learners and their communities, teachers as practitioners and as citizens, the wider society, and the world as a whole. Even though each chapter can be read independently of others, an engagement with all chapters in this volume will provide readers with a solid holistic understanding of the research territory of equity and mathematics education. The publication of this book would not have been possible without the help and support of various individuals, whom I would like to thank. First of all, I would like to express my sincere gratitude to Prof. Paul Andrews (Stockholm University, Sweden), with whom we had started this project together, but, due to personal circumstances, had to withdraw. Our initial discussions about the structure and content of the book were particularly important for the organization and formulation of the overall material. Many thanks to Prof. Bharath Sriraman (University of Montana), who embraced the idea of this project from the beginning and, as the editor of the IAP series Cognition, Equity, & Society, provided the book with a home. In addition, I would like to thank a number of people who helped me with the editing process, by acting as external reviewers and providing thoughtful and thorough comments on previous versions of the chapters. Dr. Angela Hadjipanteli (drama education), Dr. Yiannis Georgiou (digital technologies in education), Dr. Stella Mouroutsou (inclusive education), and Dr. Maria Evagorou (science education), your assistance is greatly appreciated. Special thanks to Dr. James Biddulph, who, besides contributing to this volume, acted as a reviewer for other chapters. Finally, I would like to thank my good friend Maria Petrides (independent writer and editor) for reading all chapters before publication and providing useful linguistic, stylistic, and content suggestions to the authors. —Constantinos Xenofontos Stirling, Scotland December, 2018

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REFERENCES Andrews, P., & Rowland, T. (2014) (Eds). Master class in mathematics education. International perspectives on teaching and learning. London, England: Bloomsbury Academic. Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68. Kilpatrick, J. (2014). History of research in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 267–272). Dordrecht, The Netherlands: Springer. Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport, CT: Ablex.

CHAPTER 1

EQUITY AND SOCIAL JUSTICE IN MATHEMATICS EDUCATION A Brief Introduction Constantinos Xenofontos

Reading the world always precedes reading the word, and reading the word implies continually reading the world. —Paulo Freire (Freire & Macedo, 1987, p. 35)

By virtue of mathematics being political, all mathematics teaching is political. All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not. —Rochelle Gutiérrez (2013b, p. 11)

In mathematics education, both at the research and policy level, measuring and comparing learners’ attainment are very common and highly favored, particularly by policymakers and politicians. For example, pupils’ achievement in mathematics can be compared with that of other school subjects, aiming at finding connections between, let’s say, performance in

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mathematics, science, and language (i.e., Ireson, Hallam, Hack, Clark, & Plewis, 2002). In experimental and interventional designs, pupils’ results before and after an experiment/intervention are compared, as well as examined in relation to those of control groups (that is, groups that have not participated in the intervention), to investigate, for instance, the effectiveness of an instructional program, curriculum, and/or particular teaching materials (i.e., Slavin, Lake, & Groff, 2017). Furthermore, large-scale international studies, like TIMSS and PISA, take measurements from many countries by employing standardized tests, make comparisons across national settings, and often present their results in ranking lists—as typically perceived and promoted by mass media—of inevitable losers and winners (Andrews, 2012). Comparisons of any kind can be very useful and provide deep insight into the topic under scrutiny, especially when seen from the perspective of boundary crossing ( Jablonka, Andrews, Clarke, & Xenofontos, 2018). Boundary crossing is not limited to a documentation of similarities and/ or differences, but takes the act of comparison to a level that serves the benefits and well-being of all comparing groups or entities. However, intranational comparisons, typically in North America and in many European countries, have indicated large discrepancies between the mathematical performances of White male middle-class pupils and pupils from, what is called in the literature, marginalized groups. These so called marginalized groups include, but are not limited to, pupils with racial/ethnic background other than that of the dominant group, girls, children whose sexual orientation is constructed outside heterosexuality, pupils whose home language is not that of school and instruction, children with intellectual, emotional, and kinaesthetic disabilities, and pupils from families of low socioeconomic status (Atweh, Vale, & Walshaw, 2012; Berry & Wickett, 2009; Boaler, Altendorff, & Kent, 2011; Bullock, 2012; Graven, 2014; Stinson, 2013; Strutchens et al., 2012). The observed disparities between the performances of these different marginalized subgroups and the dominant group, typically measured with the use of standardized tests, are labelled, not least in the context of the United States and Britain, as the achievement or attainment gap. This chapter is in line with the voices of scholars who oppose the widespread tendency to measure and remeasure the so-called achievement gap. It is questionable whether such an approach, which Gutiérrez (2008, 2009, 2013a) calls gap gazing, is productive. As she notes, “Deepening our knowledge in this arena is unlikely to advance the cause of marginalized students” (Gutiérrez, 2008, p. 357), since, in reality, most of the research on measuring and comparing the performances of various subgroups retells us what is already known (Lubienski & Gutiérrez, 2008): Disparities exist. The achievement gap is more of a symptom than an underlying

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cause (Flores, 2007). In fact, schools, states Apple (1979), exist through their relations to other more powerful institutions, which are combined in ways that generate, reinforce, and reproduce structural inequalities of power and access to resources. This means that education in most countries of the western world is built around the hegemonic idea of sustaining the current structures of society and the status quo, which put White middle-class male pupils in the very centre of everything, at the same time that the achievement levels of the dominant group are seen as the standards of “excellence” (Gutiérrez, 2008), leaving children from marginalized groups pushed outside the center. Mathematics education is political (Gutiérrez, 2013b; Skovsmose & Valero, 2005; Stathopoulou & Appelbaum, 2016; Swanson, Yu, & Mouroutsou, 2017; Valero, 2004). From the perspective of academic research, this means, among other things, that following the social turn of previous decades (Lerman, 2000), the field has now taken a sociopolitical turn (Gutiérrez, 2013a). In other words, in recent years, more and more mathematics education researchers employ sociopolitical theories and concepts, such as power and identity, to understand and talk about the dynamics between mathematics curricula, policy, politics, and the teaching and learning of the subject. In light of this, Sriraman and Steinthorsdottir (2007) claim that mathematics education is more associated with politics than with mathematics itself, with school mathematics having, traditionally, served as the gatekeeper to many other areas of study. Learners who do not have access to mathematical knowledge or do not acquire it are thought to be disadvantaged and prone to being victims of racism, sexism, and other forms of discrimination and social exclusion (Borba & Skovsmose, 1997; Frankenstein, 1989; Frankenstein & Powell, 1994). Another political dimension of mathematics education has to do with curriculum politics, and questions like, “Who decides what is taught in K–12 mathematics, and how these political forces connect to the implementation of socially just curricula and pedagogy?” (Appelbaum & Davila, 2007, p. 1). Taking all the above into consideration, this chapter examines how equity and social justice are directly related to mathematics education, and how school mathematics and its teaching can contribute to the development of a fairer society. Subsequently, the focus is turned to preservice teacher education and in-service professional development, through an examination of research attempts to introduce equity and social justice to mathematics teachers in order to help them become critical educators who support the needs of all children, further develop a sense of what Freire (1970) calls conscientização (critical consciousness), and later facilitate such a development in their pupils.

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ADDRESSING EQUITY AND SOCIAL JUSTICE IN MATHEMATICS EDUCATION To talk about equity and social justice, and their intimate relation to mathematics education, is not an easy task. A closer look at the relevant literature reveals a variety of reasons why approaching these concepts in an attempt to define them is not an uncomplicated activity. One reason is that in some cases the two terms have been used interchangeably (see, for instance, Esmonde & Caswell, 2010; Jackson & Jong, 2017; Meyer, 1989; Planas & Civil, 2009) while in other written accounts, they appear to be distinct; yet the links between them are explicitly presented and discussed (i.e., Gregson, 2013; Healy & Powell, 2013; Secada, 1989). Furthermore, both equity and social justice are ideologically charged and their pillars are built on values of fairness. Nonetheless, what is fair to one person or in certain contexts is not necessarily fair to another person or in a different context (Bartell, 2013; Gates & Jorgensen, 2009; Gutiérrez, 2009; Strutchens et al., 2012). Equity and social justice have also been associated with other related concepts, as for example, diversity and culturally responsive pedagogy, based on who is perceived as disadvantaged and marginalized each time (Atweh et al., 2012). Finally, there appear to be different concerns about equity and social justice across countries (Lee, Kim, Kim, & Lim, 2018) which also add to the expansion of the gamut of working definitions. For example, in North America, particularly in the United States, ideas and practices of equity and social justice are examined in relation to race; in the United Kingdom, Europe, and Australia they are typically seen through the lens of socioeconomic status; in some European countries issues of ethnicity and home language are also of major concern; while in China, many African countries, and Latin America, research is more focused on rurality, while in countries like South Africa, various health issues, like HIV, are prominent (Graven, 2014). This apparent lack of consensus regarding the dissimilar priorities associated with practices and policies of equity and social justice that each country aims to develop have led authors like Nolan (2009) to think about how we can talk and act effectively about these in relation to mathematics education. The whole discussion on equity and social justice is, or should be, based on the premise that inequalities and injustices exist, and that particular instructional approaches, if we talk on the micro-level of the classroom, not only resist contesting them, but, in fact, reinforce them (Berry, 2005; Boaler et al., 2011; Gutstein, Lipman, Hernandez, & de los Reyes, 1997; Gregson, 2013). In this spirit, Solar (1995) discusses a number of pupil–pupil or pupil–teacher interactions in the mathematics classroom that are discriminatory, such as sexism, ethnocultural discrimination and racism, heterosexism, other forms of discriminations (based on assumptions about religion,

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age, class, or disability), the use of prejudiced language, and discriminatory nonverbal behavior (i.e., gestures and grimaces). Most of the times, teachers are not aware that their own practices reproduce inequalities, and do not easily see connections between how general social behaviors are associated with school mathematics and affect its learning (de Freitas & Zolkower, 2009). Yet, research has documented how teachers’ expectations regarding the attainment of pupils from marginalized groups affect those children’s actual learning (see, e.g., Bartell, Meyer, Knott, & Evitts, 2008; Berry, 2005; Jamar & Pitts, 2005; Jorgensen & Niesche, 2008). Drawing on Pierre Bourdieu’s (1979) notion of habitus,1 Jorgensen and Lowrie (2015) describe how pupils from marginalized groups in the mathematics classroom behave in ways that confirm their teachers’ low expectations, in the form of a selffulfilling prophecy. In the same spirit, Stathopoulou (2017) shares a fable as narrated by a Roma grandparent in Greece, which concludes with the normalized “moral” that the 15-year-old boy (one of the main characters of the fable) was still in the first grade of elementary school due to his “gypsy genes.” It is apparent that such biologically deterministic approaches to learning held by some teachers, not only affect pupils, but also members of their immediate family and extended community, who in turn adopt and internalize the prejudiced representations that the dominant group holds regarding them. Equity and Mathematics Education As stated earlier, in the published mathematics education literature, there are many definitions of equity. Some of them are concerned with pupils’ scores in standardized tests (Boaler, 2006), reducing the whole discussion to outcomes and numbers. Here, I follow Gutiérrez (2002), who defines equity as the “erasure of the ability to predict students’ mathematics achievement and participation based solely on characteristics such as race, class, ethnicity, sex, beliefs and creeds, and proficiency in the dominant language” (p. 153). In subsequent work, Gutiérrez talks about four dimensions of equity (see Gutiérrez, 2008); namely, access (i.e., resources available to engage with quality mathematics), achievement (i.e., standardized test scores, participation rates), identity (maintaining cultural, linguistic, familial connections), and power (agency to affect change in school or society). Gap gazing studies typically focus on the first two dimensions, those of access and achievement, while identity and power are the least examined. In acknowledgement of this, Boaler (2006, 2008) proposes another dimension of equity, which she terms relational equity. Relational equity in mathematics classrooms moves the focus away from school outcomes and achievement in tests; instead, it draws attention to the ways pupils learn to treat peers and the respect they learn

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to have for people from different circumstances to their own. As such, she claims, relational equity has three important outlooks: (a) respect for other people’s ideas, leading to positive intellectual relations; (b) commitment to the learning of others; and (c) learned methods of communication and support. It could be argued that Gutiérrez’s first two dimensions fall under what Lee et al. (2018) call equity as a standard of excellence (measurement of outcomes), while the latter two, along with Boaler’s concept of relational equity, are related to equity as a moral imperative (warranted by the ethical and moral standards of a democratic society, where every individual has the right to learn). Finally, although equity is seemingly similar to that of equality, due to their underlying assumptions of what is fair, the two are, in fact, different, in the sense that equality is concerned with giving all pupils the same amount of a “good” (i.e., resources, support, etc.), while equity assumes and acknowledges that different individuals have different learning needs; therefore, differentiated support is provided so that each pupil can reach a goal (see, e.g., Levitan, 2015; Secada, 1989). Instructional Practices That Promote Equity in the Mathematics Classroom In recent years, discussions on equity in mathematics education have come to the forefront (Meaney, Trinick, & Fairhall, 2009) and have become more and more mainstream (Gutiérrez, 2009), especially after the publication of the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (NCTM, 2000). Although the NCTM is concerned with the U.S. scene, its principles and standards have an impact of an international nature (Hodgson, Rogers, Lerman, & Lim-Teo, 2012). The first of NCTM’s (2000) six principles is the equity principle: “Excellence in mathematics education requires equity—high expectations and strong support for all students” (p. 12). While the inclusion of equity in the document highlights its importance and the fundamental challenges mathematics education encounters in a changing world, many scholars have criticized how the NCTM presents the principle, commenting, inter alia, on how the document does not say much on how to transform inequitable mathematics classrooms into equitable ones (Gutstein, 2003), does not take a stronger position on equity (Berry, 2005) and makes no reference to race, racism, and social justice (Martin, 2003). It is, therefore, not surprising that efforts to associate equity with the field of the mathematics classroom more explicitly are initiated by individual scholars rather than curriculum designers and policymakers. For example, Berry and Wickett (2009) argue that equitable practices on mathematics include strategies that address the needs of other language pupils, gifted pupils and pupils with special needs,

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culturally relevant strategies, multilingual and multicultural, as well as strategies addressing social issues, like gender, poverty, and racism. In this chapter, however, I shall not engage in any detail with these concerns, as readers can refer to other chapters in this volume, which are more specialized. I would rather present more general instructional practices that can be used in the mathematics classroom to address equity. For instance, Bartell and colleagues (2017) summarize nine equitable practices for teaching mathematics, which are described in Table 1.1. Readers please note that Table 1.1 is an adapted version of that which appears in the work of Bartell and her colleagues. The original version of Table 1.1 links each practice to a set of other research articles, which provide evidence concerning the effectiveness of each practice.

TABLE 1.1 Equitable Instructional Practices in Mathematics Equitable Practice

Examples of the Practice

Draw on pupil’s funds of knowledge.

• Build on community, cultural knowledge, and practices. • Recognize pupils’ cultural and linguistic resources. • Have robust knowledge of pupils, validate shared ideas and experiences, and connect instruction to pupils’ experiences and interests.

Establish classroom norms for participation.

• Recognize that pupil voice has implications for power and authority, and builds agency. • Set-up and guide discussions so that pupils from nondominant backgrounds develop strong mathematical identities. • Connect pedagogical practices to pupil participation. • Question whose participation norms are valorized.

Position pupils as capable.

• Construct social structures that enable pupils to develop strategies that help maintain certain positions and reduce others. • Challenge and counteract social stereotypes and inequities to which pupils and communities are subjected. • Attend to how the curriculum may influence perceptions of pupils. • Share power in the classroom by allowing pupils to provide meaningful input in making decisions about classroom practices, curriculum, and assessment.

Monitor how pupils position each other.

• Assign competence to support pupils’ repositioning of one another. • Attend to reification of existing status structures so as to reposition some pupils with their peers. • Position pupils to use one another as mathematical resources. (continued)

8 C. XENOFONTOS TABLE 1.1 Equitable Instructional Practices in Mathematics (continued) Equitable Practice

Examples of the Practice

Attend explicitly to race and culture.

• Make connections to pupils’ mathematical, racial, and cultural identities. • Recognize that certain groups have been positioned as anti-intellectual.

Recognize multiple forms or discourse and language as a resource.

• Facilitate respect among pupils by cultivating culturally responsive relationships among pupils and validating possible differences in their language practices. • Co-construct resources with pupils in moment-tomoment interactions around mathematics. • Consider linguistic choices and acknowledge home language as a valid language of mathematics. • Bridge language practices through affirming pupils’ home languages, modeling code switching, and fostering interactional patterns familiar to pupils.

Press for academic success.

• Assess pupils’ learning, build on pupils’ strengths, explicitly communicate expectations of pupils, and communicate teachers’ responsibility in pupil success. • Have high academic expectations while maintaining pupils’ cultural and psychological well-being rather than accepting deficit views about pupils’ intellectual potential.

Attend to pupils’ mathematical thinking.

• Recognize, understand, and build from children’s understanding of mathematics. • Respond to developmental needs as to not expect a pupil to do mathematics they are not ready for.

Support development of a sociopolitical disposition.

• Incorporate critical texts, discuss controversial topics, serve the community, and allow social issues to drive instruction. • Provide opportunities to explore sociopolitical topics using mathematics. • Engage pupils in conversation about real-world problems and how mathematics can be used to examine them.

Source: Adapted from Bartell et al., 2017.

Similarly, Boaler (2006, 2008) talks about a large four-year study she and her research team conducted in three high schools in California. In one particular urban school, to which Boaler has given the pseudonym Railside, pupil population was highly diverse, both in terms of student’s home languages and their cultural/ethnic backgrounds (approximately 38% Latinas/os, 23% African American, 20% White, 16% Asian or Pacific Islanders, and 3% from other groups). The mathematics teachers at Railside employed various instructional practices (Boaler, 2006), and the research team’s long-term observations, interviews with pupils, and detailed analyses,

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were evidently important in the promotion of what Boaler calls relational equity. Furthermore, these practices, presented in Table 1.2, have been found to promote pupils’ high mathematics achievement, especially when compared to the other two schools which participated in the study, and whose pupil population was less diverse than that of Railside.

TABLE 1.2 Equitable Practices Employed by Mathematics Teachers at Railside High School Equitable Practice

Brief Explanation

Multidimensionality

• Step away from the typical practices of executing procedures correctly and quickly. • Multidimensional classes that value many dimensions of mathematical work. • Use of open-ended problems that illustrate important mathematical concepts, allowing for multiple representations.

Roles

• Pupils placed into groups and given particular roles to play (i.e., facilitator, team captain, recorder/reporter or resource manager). • All pupils have important work to do in groups, without which the group cannot function.

Assigning competence

The teacher raises the status of pupils that may be of lower status in a group, by, for example, praising something they have said or done that has intellectual value and bringing it to the group’s attention; asking a pupil to present an idea; or publicly praising a pupil’s work in a whole class setting.

Teaching pupils to be responsible for each other’s learning

Related to the assessment system: • Teachers occasionally grade the work of a group by rating the quality of the conversations groups have. • Teachers occasionally give group tests, in several formats (i.e., pupils working together but the teacher grades only one of the individual papers, and that grade stands as the grade for all the pupils in the group). • Teacher asks one pupil in a group to answer a follow-up question after a group has worked on something. If the pupil cannot answer the question, the teacher leaves the group to further discussion before returning to ask the same pupil again.

High expectations

The teacher keeps the demand of lessons intellectually high, both by providing complex problems and by following up with highlevel questions.

Effort over ability

The teacher frequently offers strong messages to pupils about the nature of high achievement in mathematics, continually emphasizing that it is a product of hard work and not of innate ability.

Learning Practices

Teachers are very clear about the particular ways of working in which pupils need to engage. For example, teachers can stop the pupils as they are working and talking, and point out valuable ways in which they were working.

Source: Adapted from Boaler, 2006.

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Social Justice and Mathematics Education The role of education in promoting social justice routinely adopted by many politicians (Smith, 2012), who systematically make grandiose statements in public speeches about how it will create or lead to a “better” society; however, a closer look at such assertions reveals how the term social justice is employed to disguise different agendas between politicians. The expression “education for social justice,” should be questioned, since in reality no one seems to explicitly promote education for social injustice (Penteado & Skovsmose, 2009). As mentioned earlier, social justice is driven by ideology and perceived differently, according to the beliefs, priorities, values, and agendas of an individual, a social group, or political representatives (Atweh et al., 2012; Gates & Jorgensen, 2009). Education for social justice (or, critical education, as authors have called it) has its roots in the work of the Brazilian educationist and philosopher Paulo Freire (1970), found mainly in his book Pedagogy of the Oppressed. In his work, Freire, who had spent years supporting illiterate Brazilian adults learn how to read and write, talks about how education needs to move towards decolonization, the breaking of relations between the “oppressor” and the “oppressed,” and lead to liberation. Central to the achievement of these is the idea of conscientização (conscientization or critical consciousness) the process of developing a critical awareness of one’s social reality through reflection and action. Simply put, the main goals of critical education are to help learners understand inequalities, injustices, and oppression, while at the same time, provide them with the necessary tools and skills to act towards changing the world (Erchick & Tyson, 2013; Stinson, Bidwell, & Powell, 2012). In such a process, educators should not just have scholarship in critical education, but be activists themselves (Apple, 2008). Drawing on Freire’s ideas, Frankenstein (1983) opens the field of critical mathematics education, “an attempt to reconceive school mathematics as a site of political power, ethical contestation, and moral outrage,” which “refers to a set of concerns or principles that function as catalysts for reconceiving and redesigning the lived experience of school mathematics” (de Freitas, 2008, p. 48). Other scholars use the term mathematics education for social justice synonymously. Since Frankenstein’s inaugural work, research in critical mathematics education has expanded, with various proponents pursuing its agenda in different ways (de Freitas, 2008): the design of new mathematics curricula that address social justice issues, the examination of the role of mathematics teacher disposition to social justice pedagogy, a deconstruction of the instructional strategies unique to school mathematics that inhibit increased participation, the generation of a sociopolitical ethics of mathematics education, and the expression of visions of alternative teaching practices. As noted earlier, in the mathematics education literature, the

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terms equity and social justice are often used interchangeably. Here, I share the views of authors like Gutstein et al. (1997) and Secada (1989), who perceive social justice pedagogy as the broadening of the value of equity. Equity is about supporting learners’ individual needs. Social justice, which presupposes and includes equity, goes further by explicitly developing and promoting critical awareness regarding the roots of marginalization, structures of inequalities, practices of injustices, and the urgency to tackle these in order to get closer to a fairer world. Instructional Practices That Promote Social Justice in the Mathematics Classroom Teaching mathematics for social justice has two central pedagogical goals: to deepen learners’ sociopolitical understandings of the world and, at the same time, to strengthen their mathematical proficiencies (Bartell, 2013; Gregson, 2013; Gutstein, 2007). The association of mathematics with social phenomena and issues helps pupils develop mathematical knowledge, experience how mathematics can be a tool for understanding and changing the world, as well as develop a positive disposition towards learning the subject. For example, in his 2-year study about teaching and learning mathematics for social justice in an urban, Latina/o classroom, Gutstein (2003, 2007), having the dual role of teacher and researcher, observed that the involvement of his pupils in related activities facilitated the development of their ability to read the world (understand complex issues involving justice and equity) using mathematics, to develop mathematical power, and to change their disposition towards mathematics. Yet, teaching mathematics through the lens of social justice is not an easy thing to do, since such an approach is not a matter of method; on the contrary, it is an ongoing process and a lifelong undertaking (Bartell, 2013; Bateiha & Reeder, 2014), of a cause which teachers need to believe in. Teachers are humans who carry, consciously or not, their personal life experiences and ideologies into the mathematics classroom (Brown, 2009; Foote & Bartell, 2011; Gates & Jorgensen, 2009). Furthermore, in many cases, teachers do not question the content of school mathematics (Gutiérrez, 2013a), or what Gutstein (2007) calls classical knowledge (formal, in-school, abstract knowledge), in order to connect it with other types of mathematical knowledge, such as pupils’ community knowledge (knowledge that resides in individuals and communities and has typically been learned out of school) and critical knowledge (knowledge about the sociopolitical conditions of one’s immediate and broader existence). In some cases, certain sociocultural issues are considered to be socially restrictive, making teachers reluctant to associate them with school mathematics, as for example, the case of Tanko (2014), a male teacher and

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researcher teaching a group of Middle-Eastern Muslim women (ages ranging between 16–36) learning mathematics through social justice pedagogy. Finally, social justice problems are typically integrated in ready-made lesson plans, and most of the times, they are concerned with statistics, graphs, and figures (Nolan, 2009), an approach which is very simplistic and reduces the whole philosophy of the field to “ready-to-make food.” Various authors report a number of successfully applied activities for children of all ages. Nevertheless, readers should be alerted that these are not prescribed methods to be followed, but examples that were used in certain contexts with specific participants (teacher and pupils), of specific social concerns and surrounding issues. One example is that provided by Murphy (2009), a researcher and teacher of a first-grade classroom. Murphy used an allegoric book2 for storytelling, which referred to the colonization of Australia by European colonists. In the book, Australia’s Aborigines were represented as kangaroos, while European colonists who destroyed aborigine traditional culture and natives’ relationship with the land were presented as rabbits. Murphy used cuisenaire rods with the children of his class to represent the rabbits and kangaroos of the story. As he concludes, the children were able to successfully represent the story with the use of the manipulatives and show that they understood issues of power and social justice. The pictures in Figure 1.1 are taken from Murphy (2009), showing the work of his pupils. In a similar vein, Winter (2007) presents various social problems and how these have been linked to a number of mathematical concepts in an undergraduate introductory science, technology, engineering, and mathematics (STEM) course, concerned with algebra, precalculus, and calculus. These are presented in Table 1.3. As readers may observe, the mathematical concepts involved are examined in many secondary mathematics curricula around the world.

Figure 1.1 Student work. Source: Murphy, 2009.

Equity and Social Justice in Mathematics Education 13 TABLE 1.3 Examples of Social Problems and the Mathematical Concepts Involved Social Problem

Mathematical Concepts Involved

Racial and gender imbalance in new HIV infection rates in the United States.

• Comprehension and integration of information presented in numerical, algebraic, geographical, and verbal formats

Political corruption and economic development in Africa, Southeast Asia, Europe, and South America.

• • • • •

Manipulation and analysis of numerical data Graphic data points Finding patterns Linear regression Interpretation of the slope and intercept of a linear function

Climate change in Zimbabwe and the impact on agriculture.

• • • •

Recognizing and creating linear functions Graphing linear functions Finding intersection points Interpreting the practical meaning of intersection points

Water security and native peoples’ rights in Botswana.

• • • •

Domain and range of a function Functions defined in pieces Finding intersection points Interpreting the practical meaning of intersection points

Ancestral lands and petroleum resources: the Uighur minority and redheaded mummies of western China

• Exponential functions • Half-lives and radioactive decay • Setting up and solving exponential functions numerically and with logarithm

Estimating the impact of chemical fertilizers in subSaharan Africa

• • • •

Quadratic functions in standard and vertex form Quadratic regression Completing the square Locating the vertex of a quadratic function and interpreting its meaning

The global disease burden of malaria.

• • • •

Power, exponential, and logarithmic functions Quadratic regression Completing the square Locating the vertex of a quadratic function and interpreting its meaning

The cost of providing ARV medications in Uganda.

• • • • •

Power functions Combining functions to produce new functions Laws of exponents and algebra Setting up and solving power equations Finding intersection points for power graphs

How quickly can polio be eradicated? Medical, political, economic, and social factors.

• • • •

Formulas for polynomial functions Recognizing polynomial functions from graphs Polynomial regression Locating the zeros of a polynomial function

Milestones in the HIV/AIDS pandemic.

• • • • • •

Patterns in data Recognizing linear, exponential, and polynomial functions Regression Graphing and interpreting inverses The horizontal line test Finding formulas for inverses of linear and exponential functions

Source: Winter, 2007

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THE IMPORTANT ROLE OF TEACHERS AND TEACHER EDUCATION Undoubtedly, to transform the mathematics classroom from a space that deals with classical mathematical knowledge to a space where equity and social justice are effectively addressed and exercised—restructuring what Freire (1970) calls systematic education (the educational system, its policies, values, and politics)—is significant. This, however, cannot happen from one day to another, as radical changes and educational transformations typically come from social movements (Apple, 2008). Nonetheless, it is important to bear in mind that “mathematics education in practice is, and always should be, mediated by human teachers” (Bishop, 1988, p. 189), and that between curricula and policies (indented curriculum) and pupils’ actual learning outcomes (attained curriculum) stand teachers, with a certain knowledge, beliefs, ideologies, identities, preferences, experiences, and instructional practices. Many teachers have social and political intuitions; they sense the interplay between school mathematics and political issues, but do not always know how to put these intuitions into practice in ways that help their pupils (Apple, 2008). The majority of teachers, however, are not aware of the cultural and political dimensions of mathematics education (Xenofontos, 2015), and hold opinions like those that claim “I’m just one of those math for math’s sake people,” which “implicitly legitimates an entire set of social practices associated with school mathematics, and thereby serves to reproduce the power relations enacted therein” (de Freitas & Zolkower, 2009, p. 190). To address issues of equity and social justice successfully, teachers need to make conscious decisions regarding instruction (Averill et al., 2009; Rousseau & Tate, 2003), provided that they hold high expectations of all pupils, regardless of background (Jorgensen & Niesche, 2008; Planas & Civil, 2009). The key to this is teacher education (Nolan, 2009; Penteado & Skovsmose, 2009; Vomvoridi-Ivanovic & McLeman, 2015), both at the level of initial teacher education and training (for prospective teachers) and continuing professional development (for in-service teachers). A number of studies report attempts to introduce prospective teachers— both elementary and secondary—to how equity, social justice, and mathematics education are linked, through specifically designed modules, as part of their undergraduate studies and teacher preparation programs. Different types of approaches can be found in the literature, such as the engagement of prospective teachers in activities examining mathematical concepts through the lens of equity and social justice (i.e., Bateiha & Reeder, 2014), an enactment of scenarios and role playing (i.e., Boylan, 2009), and, reading, discussing and reflecting on related research papers (i.e., Jackson & Jong, 2017). In most cases, similar conclusions have been reached: prospective teachers’ engagement in activities of this kind helps them develop both

Equity and Social Justice in Mathematics Education 15

a conceptual understanding of mathematics and a deeper understanding of social problems, as well as learn how these two can be connected in practice. Yet, mathematics teacher educators who work in this area agree that familiarizing trainees with issues of equity and social justice and their relation to school mathematics is a hard process, which can neither take place in one-semester courses nor can a one-size-fits-all approach work for all contexts. On the contrary, prospective teachers need to have multiple and ongoing experiences throughout their studies. Such experiences should be related to trainees’ prior experiences and the particularities of the context in which they are expected to work. From the perspective of in-service teacher education and professional development, addressing these issues is an even more challenging endeavor. Many practicing teachers appear to hold the deep-rooted belief that addressing equity and social justice shifts the focus away from mathematical content, which makes them feel uncomfortable (Erchick & Tyson, 2013). Even teachers who are more inclined towards adopting an equity/social justice approach are concerned with the negotiation of dilemmas, such as finding the balance between covering the classical mathematics curriculum and examining social issues in the mathematics classroom (Gregson, 2013), and, at the same time, addressing the learning needs of all pupils effectively. On the other hand, there is an encouraging number of studies that documents positive shifts of in-service teachers’ beliefs and practices, after their involvement in relevant professional development programs. For example, Bartell (2013) reports a study of secondary mathematics teachers who, as part of a course, were engaged in readings, reflections of sample lessons, and developed and taught lessons in their classrooms. These activities supported teachers in negotiating the goals of mathematics, equity and social justice, as well as the connections between themselves as teachers, pupils, school mathematics, and the social context. In the same spirit, in their study, Planas and Civil (2009) observed that the participation of teachers as co-researchers in a school-based professional development program, led teachers to question their previous perceptions on equity and social justice, as well as reconstruct their teaching practices in ways that improved their pupils’ mathematical performances (many children had physical and/or other learning difficulties). Nevertheless, as concluded in studies with preservice teachers, most studies from the in-service teachers’ perspective highlight that teaching for equity and social justice is a lifelong process. It challenges one’s perceptions, especially if one has taught mathematics in traditional ways for years, and requires time (Erchick & Tyson, 2013) and a commitment that ought to come from within one’s heart and mind (Foote & Bartell, 2011). As teacher educators, we need to provide opportunities and create spaces in which such experiences take place, new ideas develop, evolve and are refined; creative spaces within which teachers

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(both preservice and in-service) feel secure and supported in putting sociopolitical intuitions and knowledge into action in order to address and practice equity and social justice in their mathematics teaching. AN OVERVIEW OF KEY POINTS To end this introductory chapter, I would like to recapitulate key ideas that readers might find useful to take with them as they read on to the next chapters: • Comparisons between the mathematical performances of White, male, middle-class boys (the dominant group) and children from marginalized groups (namely, girls, children whose sexual orientations are constructed outside heterosexuality, pupils whose home language is not that of school and instruction, children with intellectual, emotional, and kinaesthetic disabilities, and pupils from families of low socioeconomic status), have shown large discrepancies. Children from the dominant group appear to outperform the rest. • These discrepancies have been described in the literature as the achievement gap, which has been documented by a number of studies. Yet, if the goal of education, in general, and educational research, in particular, is to help create a fairer world, then we need to move beyond measuring and remeasuring the achievement gap, towards examining what kind of education, instruction, and curricula can support mathematics learning of all children. At the same time, mathematics can be used to help learners’ develop a critical awareness of the world. • Equity and social justice are crucial virtues and practices. Equity can be seen as that stage which once reached, we will no longer rely on variables such as gender, race, socioeconomic status, ability, disabilities to make predictions about children’s (mathematical) performances. This requires a provision of instructional support tailored to individual children’s learning and other needs. Social justice in mathematics education extends the notion of equity and is concerned with the dual goal to help children learn mathematics, and at the same time, become more critical “readers and writers” of the world. In other words, mathematics for social justice (or, critical mathematics education) is concerned with how mathematics learning and critical awareness of social and political issues are linked, and how these can be developed at once, through classroom activities that work with both.

Equity and Social Justice in Mathematics Education 17

• In the relevant literature, a number of instructional practices have addressed equity and social justice, by helping pupils improve their mathematical competences and become more critical of sociopolitical issues. • Teacher education (both initial teacher education and in-service teacher professional development) is the key to change. Research has found that when prospective and practicing teachers engage with relevant courses, combining mathematics learning/teaching and the examination of social, cultural, and political issues, not only do they show an improved conceptual understanding of the mathematical content involved, but they also challenge their own perceptions on equity and social justice, and on their own instructional practices. NOTES 1. Simply put, Bourdieu (1979) talks about habitus as the embodiment of habits, skills, and dispositions held by individuals, as a result of social expectations. Habitus is created and reproduced unconsciously, and is neither a result of free will, nor determined by structures, but created by a kind of interplay between the two over time. 2. The Rabbits, written by John Marsden and Shaun Tan, and illustrated by Shaun Tan.

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Equity and Social Justice in Mathematics Education 19 Esmonde, I., & Caswell, B. (2010). Teaching mathematics for social justice in multicultural, multilingual elementary classrooms. Canadian Journal of Science, Mathematics and Technology Education, 10(3), 244–254. Flores, A. (2007). Examining disparities in mathematics education: Achievement gap or opportunity gap? The High School Journal, 91(1), 29–42. Foote, M. Q., & Bartell, T. G. (2011). Pathways to equity in mathematics education: How life experiences impact researcher positionality. Educational Studies in Mathematics, 78(1), 45–68. Frankenstein, M. (1983). Critical mathematics education: An application of Paulo Freire’s epistemology. Journal of Education, 165(4), 315–339. Frankenstein, M. (1989). Relearning mathematics: A different third R—Radical maths. London, England: Free Association Books. Frankenstein, M., & Powell, A. (1994). Toward liberatory mathematics: Paulo Freire’s epistemology and ethnomathematics. In P. Mclaren & C. Lankshear (Eds.), Politics of liberation: Paths from Freire (pp. 74–99). New York, NY: Routledge. Freire, P., & Macedo, D. (1987). Literacy: Reading the word and the world. Westport, CT: Bergin & Garvey. Freire, P. (1970). Pedagogy of the oppressed. New York, NY: The Seabury Press. Gates, P., & Jorgensen, R. (2009). Foregrounding social justice in mathematics teacher education. Journal for Mathematics Teacher Education, 12(3), 161–170. Graven, M. H. (2014). Poverty, inequality and mathematics performance: The case of South Africa’s post-apartheid context. ZDM, 46(7), 1039–1049. Gregson, S. (2013). Negotiating social justice teaching: One full-time teacher’s practice viewed from the trenches. Journal for Research in Mathematics Education, 44(1), 164–188. Gutiérrez, R. (2002). Enabling the practice of mathematics teachers in context: Towards a new equity research agenda. Mathematical Thinking and Learning, 4(2–3), 145–187. Gutiérrez, R. (2008). A “gap gazing” fetish in mathematics education? Problematizing research on the achievement gap. Journal for Research in Mathematics Education, 39(4), 357–364. Gutiérrez, R. (2009). Embracing the inherent tensions in teaching mathematics from an equity stance. Democracy and Education, 18(3), 9–16. Gutiérrez, R. (2013a). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68. Gutiérrez, R. (2013b). Why (urban) mathematics teachers need political knowledge. Journal of Urban Mathematics Education, 6(2), 7–19. Gutstein, E. (2003). Teaching and learning mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 34(1), 37–73. Gutstein, E. (2007). Connecting community, critical, and classical knowledge in teaching mathematics for social justice. The Montana Mathematics Enthusiast, Monograph 1, 109–118. Gutstein, E., Lipman, P., Hernandez, P., & de los Reyes, R. (1997). Culturally relevant mathematics teaching in a Mexican American context. Journal for Research in Mathematics Education, 28(6), 709–737. Healy, L., & Powell, A. B. (2013). Understanding and overcoming “disadvantage” in learning mathematics. In M. A. K. Clements, A. Bishop, C. Keitel-Kreidt, J.

20 C. XENOFONTOS Kilpatrick, & F. K. S. Leung (Eds.), Third international handbook of mathematics education (pp. 69–100). New York, NY: Springer. Hodgson B. R., Rogers L. F., Lerman S., & Lim-Teo S. K. (2012). International organizations in mathematics education. In M. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Third international handbook of mathematics education (pp. 901–947). New York, NY: Springer. Ireson, J., Hallam, S., Hack, S., Clark, H., & Plewis, I. (2002). Ability grouping in English secondary schools: Effects on attainment in English, mathematics and science. Educational Research and Evaluation, 8(3), 299–318. Jablonka, E., Andrews, P., Clarke, D., & Xenofontos, C. (2018). Comparative studies in mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education: Twenty years of communication, cooperation and collaboration in Europe (pp. 223–238). Oxon, England: Routledge. Jackson, C., & Jong, C. (2017). Reading and reflecting: Elementary preservice teachers’ conceptions about teaching mathematics for equity. Mathematics Teacher Education and Development, 19(1), 66–81. Jamar, I., & Pitts, V. R. (2005). High expectations: A ‘‘how’’ of achieving equitable mathematics classrooms. The Negro Educational Review, 56(2–3), 127–134. Jorgensen, R., & Lowrie, T. (2015). What have we achieved in 50 years of equity in school mathematics? International Journal for Mathematics Teaching and Learning. Retrieved from http://www.cimt.org.uk/journal/index.htm Jorgensen, R., & Niesche, R. (2008). Equity, mathematics and classroom practice: Developing rich mathematical experiences for disadvantaged students. Australian Primary Mathematics Classroom, 13(4), 21–27. Lee, J-E., Kim J., Kim S., & Lim, W. (2018). How to envision equitable mathematics instruction: Views of U. S. and Korean preservice teachers. Teaching and Teacher Education, 69(1), 275–288. Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport, CT: Ablex. Levitan, J. (2015). The difference between educational equality, equity, and justice . . . and why it matters. American Journal of Education Forum. Retrieved from http://www.ajeforum.com/the-difference-between-educational-equality-equity-and-justice-and-why-it-matters-by-joseph-levitan/ Lubienski, S. T., & Gutiérrez, R. (2008). Bridging the “gaps” in perspectives on equity in mathematics education. Journal for Research in Mathematics Education, 39(4), 365–371. Martin, D. B. (2003). Hidden assumptions and unaddressed questions in mathematics for all rhetoric. The Mathematics Educator, 13(2), 7–21. Meaney, T., Trinick, T., & Fairhall, U. (2009). “The conference was awesome”: Social justice and a mathematics teacher conference. Journal of Mathematics Teacher Educatron, 12, 445–462. Meyer, M. R. (1989). Equity: The missing element in recent agendas for mathematics education. Peabody Journal of Education, 66(2), 6–21. Murphy, M. S. (2009). Mathematics and social justice in grade 1: How children understand inequality and represent it. Young Children, 64(3), 12–17.

Equity and Social Justice in Mathematics Education 21 NCTM. (2000). Principles and standards for school mathematics. Reston, VA: Author. Nolan, K. (2009). Mathematics in and through social justice: Another misunderstood marriage? Journal of Mathematics Teacher Education, 12(3), 205–216. Penteado, M. G., & Skovsmose, O. (2009). How to drag with a worn-out mouse? Searching for social justice through collaboration. Journal of Mathematics Teacher Education, 12(3), 217–230. Planas, N., & Civil, M. (2009). Working with mathematics teachers and immigrant students: An empowerment perspective. Journal of Mathematics Teacher Education, 12(6), 391–409. Rousseau, C., & Tate, W. F. (2003). No time like the present: Reflecting on equity in school mathematics. Theory Into Practice, 42(3), 210–216. Secada, W. G. (1989). Agenda setting, enlightened self-interest, and equity in mathematics education. Peabody Journal of Education, 66(2), 22–56. Skovsmose, O., & Valero, P. (2005). Mathematics education and social justice. Utbildning & Demokrati, 14(2), 57–71. Slavin, R. E., Lake, C., & Groff, C. (2017). Effective programs in middle and high school mathematics: A best-evidence synthesis. Review of Educational Research, 79(2), 839–911. Smith, E. (2012). Key issues in education and social justice. London, England: SAGE. Solar, C. (1995). An inclusive pedagogy in mathematics education. Educational Studies in Mathematics, 28(3), 311–333. Sriraman, B., & Steinthorsdottir, O. (2007). Emancipatory and social justice perspectives in mathematics education. Interchange, 38(2), 195–202. Stathopoulou, C., & Appelbaum, P. (2016). Dignity, recognition and reconciliation: Forgiveness, ethnomathematics and mathematics education. International Journal for Research in Mathematics Education, 6(1), 26–44. Stathopoulou C. (2017). Once upon a time . . . the gypsy boy turned 15 while still in the first grade. In M. Rosa, L. Shirley, M. Gavarrete, & W. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education (pp. 97–123). ICME-13 Monographs. New York, NY: Springer. Stinson, D. W. (2013). Negotiating the “white male math myth”: African American male students and success in school mathematics. Journal for Research in Mathematics Education, 44(1), 69–99. Stinson, D. W., Bidwell, C. R., & Powell, G. C. (2012). Critical pedagogy and teaching mathematics for social justice. The International Journal of Critical Pedagogy, 4(1), 76–94. Strutchens, M., Bay-Williams, J., Civil, M., Chval, K., Malloy, C. E., White, D. Y., . . . Berry, R. Q. (2012). Foregrounding equity in mathematics teacher education. Journal of Mathematics Teacher Education, 15(1), 1–7. Swanson, D., Yu, H. L., & Mouroutsou, S. (2017). Inclusion as ethics, equity and/ or human rights? Spotlighting school mathematics practices in Scotland and globally. Social Inclusion, 5(3), 172–182. Tanko, M. G. (2014). Challenges associated with teaching mathematics for social justice: Middle Eastern perspectives. Learning and teaching in higher education: Gulf perspectives, 11(1), 1–14. Valero, P. (2004). Socio-political perspectives on mathematics education. In P. Valero & R. Zevenbergen (Eds.), Researching the socio-political dimensions of mathematics

22 C. XENOFONTOS education: Issues of power in theory and methodology (pp. 5–23). Dordrecht, The Netherlands: Kluwer Academic. Vomvoridi-Ivanovic, E., & McLeman, L. (2015). Mathematics teacher educators focusing on equity: Potential challenges and resolutions. Teacher Education Quarterly, 42(4), 83–100. Winter, D. (2007). Infusing mathematics with culture: Teaching technical subjects for social justice. In M. Kaplan & A. T. Miller (Eds.), The scholarship of multicultural teaching and learning (pp. 97–106). San Francisco, CA: Jossey-Bass. Xenofontos, C. (2015). Immigrant pupils in elementary classrooms of Cyprus: How teachers view them as learners of mathematics. Cambridge Journal of Education, 45(4), 475–488.

CHAPTER 2

FROM EQUITY AND JUSTICE TO DIGNITY AND RECONCILIATION Alterglobal Mathematics Education as a Social Movement Directing Curricula, Policies, and Assessment Peter Appelbaum

WHAT IS A GLOBAL VIEW OF EQUITY IN MATHEMATICS EDUCATION? One common approach is to compare research and practice within various countries with each other, identifying similarities and differences in the kinds of questions about justice and opportunity that occur within nation states. The comparison can lead to generalizable research themes and potentially replicable policy goals. The similarities and differences might occur at the level of individual learners, or in terms of social subgroups of the larger population: What sorts of cultural, ethnic, racial, gendered, and class categories are most relevant to learning experiences, opportunities,

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and potential life trajectories in each country, and why are they important to consider in terms of educational policy pertaining to mathematics education curriculum and pedagogical practices? How do geographic locations impact on the need for policy and professional recommendations depending on neighborhoods throughout urban centers, among rural, suburban, and urban educational institutions, and in combination with the social categories of difference? A common concern, for example, is the differences in school experiences between social groups, and the ways that particular pedagogies impact on minority populations, learners in poverty, learners with special needs, and learners who experience challenges to accessing various specific mathematics education opportunities. The similarities and differences might otherwise be understood in terms of broader, social processes, and the regulation or lack of oversight of popular culture: How are nation states considering the potential interaction and mutual support of various educational institutions for mathematics education, such as schools, family experiences, religious institutions, popular culture and entertainment, consumer experiences, interactions with governmental agencies? Are specific mathematical skill and concept needs being identified for participation in civic dialogue and democratic decision-making, economic trends and workplace readiness, leisure activities, or other purposes? Policy at national or local levels may be designed to harness the potential of internet, radio, television, or shopping for informal, nonschool, educational purpose, and to use interactions with government agencies as an opportunity to teach specific skills. An alternative approach to equity in mathematics education begins with the concern that comparative studies reify the nation state as an assumed category in an increasingly global world where important issues for teaching, learning, curriculum, and policy transcend borders and previously taken-for-granted regional affiliations. Rather than privilege the nation state as a primary category of analysis, such an approach looks for other ways to enter research and practice. In this approach, the questions are no longer framed within the “outmoded” national policy body, since it is considered more important to focus on the ways that postcolonial experiences are more significant than local policies and practices, how global economic forces and the spread of consumer culture have greater impact than school and family, or how school-based policies and practices are simply incapable of addressing pressing concerns for equity and social justice. For example, one can begin with globally significant social trends and social movements as the site of analysis, and ask how mathematics education is complicit in inequitable practices. This is illustrated by the ways that school mathematics maintains ignorance of economic relationships such as wages and relative costs of living, and the costs of global consumerism for local industries and the environment; another trend can be found in the ways that policies

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create advantages for some learners and not for others through the grouping of certain students for particular kinds of mathematics education experiences, grounded in a version of universal “truths” about what constitutes mathematics as a discipline, what is “good” pedagogy, the psychology of learning, and management of learning. Another way to begin analysis of mathematics education from a global perspective independent of nation states is to ask how mathematics education is (or is not) responding internationally to contemporary global crises, such as severe climate and weather catastrophes, refugee and mass migration, or the perpetuation of entrenched legacies of colonialism and imperialism. Questions for practice and policy might include where and how specific mathematics skills and concepts relevant to survival during weather catastrophes are learned and applied; how refugees and other migrants develop skills and concepts via their personal experiences, and how these skills and concepts are exploited or undermined for socially just life chances; how indigenous mathematics is honored or erased from experience, and conversely how mathematics skills and concepts are wielded in practices of postcolonial reconciliation. In this vein, mathematics education looks to transnational social justice movements, and finds opportunities to support them and to interweave efforts. For example, can popular television programming or school curricula highlight and train children in the relevant mathematical skills and concepts for slow food, slow clothing, and other global “slow” movements? In such movements, efforts are made to promote locally created food, clothing, or other fundamental needs of life, independent of a mass consumer culture, celebrating quality of craft over cheapness of product, and avoiding the need to transport goods internationally, which can have deleterious effects on the environment, and on local economics. Or, might there be an effort by a transnational nongovernmental organization (NGO) to enable youth leadership in the education of adults about the politics and mathematics of encryption that are utilized by banks and cryptocurrencies such as Bitcoin, Litecoin, and Ethereum? Whether a comparative, international approach is taken to equity in mathematics education, or a transnational, global approach, the implications are equally ripe for application at almost any level of educational practice, from classroom pedagogy and curricular design, to school policies and social programs, to regional and global partnerships. Both equity of opportunity and equity of outcomes are possible to consider in either case. At the level of classroom practices, a teacher might take it upon themselves to learn about cultural, class, and other sources of differences in opportunities to learn outside and inside of school, and how these can be turned into possibilities for rich learning in the classroom1; one common recommendation at the practice and policy levels is for the teacher to learn as much as possible about the everyday lives of the learners, in order to recognize

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and exploit the funds of knowledge that they bring with them from home and family experiences into the classroom.2 A teacher might also adapt the curriculum to challenge received notions of (Western, European) mathematics as the definition of the discipline, in order to honor indigenous cultures.3 At the same time, a teacher should maintain data on which students are successful, which are less successful, and which are unsuccessful in their classroom, in order to identify what these groups of learners have in common and thus to become aware of and then to rectify implicit biases in their practices. The content of the school curriculum can and should be analyzed in terms of what is included and excluded, and what the backstory about the purposes and possibilities for mathematics is for the learners. What counts as mathematics in this curriculum? Who decides, and why? How does that matter for the teachers, the students, the families, and the community? These questions are often overlooked in favor of expediency, as if their answers are easy to figure out. The equity ramifications are paramount. The global crisis issues are typically ignored in school curricula, connections with transnational social movements are almost universally unheard of, practical survival skills in the case of severe weather disasters are rarely if ever the focus, and students are hardly ever given the opportunity to experience mathematical participation in civic dialogue. Traditional structures of curriculum and organization of topics, strands, and themes are rarely critiqued, while the implications of how students are grouped into classes, schools, and programs, are maintained rather than analyzed for social justice implications. On the level of governmental policy, there is rare attention to the content of the mathematics curriculum, despite a century of assumptions about mathematics and STEM fields in general being good for international economic competition. The rise of ethnomathematics would have seemed to promise significant changes, yet the use of (Western) mathematics as a marker of “goodness” has had little impact on what happens in official school documents. Meanwhile, there appears to be almost no focus on the potential of mass culture, popular entertainment, and community activities for the learning of mathematics, and for the integration of mathematics and community engagement, including cultural life, political action, and social justice. While there have been studies of the kinds of mathematics used and learned in such everyday life practices as shopping, dieting, and participating in sports, it is rare, for example, to find research into how women learn the skills needed to negotiate governmental assistance for their families when they do not have the resources themselves, how advocates for minority groups or refugee rights develop the skills of representing data to themselves and others, or how those working for LGBTQ safety worldwide engage in sharing mathematical

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concepts that can help each other, and how official agencies might support or elaborate upon these efforts.4 THE LIMITATIONS OF THE ASSESSMENT STANCE In general, many mathematics educators and policy makers appear to embrace mathematics as empowering those who learn it. Empowering in what ways? To do what? One critique of school mathematics would argue that those who are successful in school are most qualified by this success to continue to study more mathematics, and little else.5 There is an international standardization of a school mathematics curriculum that, despite minor differences world-wide, essentially introduces youth to arithmetic, simple geometry, and some algebra, elaborates upon this in the middle and secondary years with an introduction to basic calculus and statistics, and which is structured to suggest that each concept or skill is needed in order to progress to the next series of tasks. Whether one takes a demographic set of categories to analyze opportunities and outcomes of individuals, or a global perspective on how such a curriculum enables participants to engage in transnational social justice efforts, the content and methods of delivery of this curriculum are left unchanged, leaving policy and practice to accept it as given, and to attempt tweaks in the delivery of this curriculum in order to improve outcomes. What this enables is an a priori collection of assumptions and practices that evaluate the quality of a program of formal or informal mathematics education in terms of measurement of achievement on the unquestioned outcomes, leaving many other options for judging equity unrealized. This is what can be called the assessment stance: improvement on the measures of achievement would promise quality of program, and relative improvement of differing categories of learners would then be a measure of equity. Working backwards from this rather limited notion of equity would lead us to the challenges of such an accomplishment, including the relative differences of opportunity, the relative differences in funds of knowledge influencing the experience of the program, the ongoing needs of a program to adjust to the relative differences that are created across groups, and so on. Often, the focus of the assessment stance is on the further development of the evaluation tools themselves, because they typically fail to adequately measure the outcomes anyway.6 And a common indirect implication of such practices is that the tools for assessment wind up functioning as part of a system of sorting some learners into categories of success and others into failure, based on the measures of their performance, reproducing inequities and limiting future opportunities within social groups.

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Unfortunately, the entire enterprise built upon the assessment stance leaves most global equity issues invisible and unattended: In which ways are learners empowered and disempowered through their mathematics education experiences? What potential mathematics education experiences have been lost in the process? What were the implications of who made the choices about the content and methods of the curriculum? Would there really be indicators that we could use to judge equity and social justice in terms of how the formal and informal mathematics education experiences of any group of people were contributing over time to their opportunities to participate in global social justice movements, to take action in their communities in projects that improve the quality of life from their own perspective, to appreciate the complexity of opinions about such a quality of life using mathematics, or to accomplish their own goals of migration, safety, and security for themselves and their families? The answers to the questions in this paragraph are heavily influenced by one’s own position visà-vis equity, privilege, opportunity, and self-identification within and across social categories. Skovsmose and Valero (2005) identified two paradoxes of our current historical moment, growing out of the ambiguity first of all of what any one of us or any group of us might even define as “mathematics,” and further in the context of our increasingly global, information-based society. Their first paradox is that of “inclusion”: it seems that any global effort at inclusion is also simultaneously a form of exclusion. For example, at the same time that some people are brought together through global networks—whether processes of consumer production and distribution, or social media—they are also pulled apart: What is considered global and what local has different meanings depending on who is included and who is excluded from groups and affiliations that may be virtual, physical, or some combination of these. On the one hand, inclusion would presume to want to enable the best mathematics education for all; on the other hand, processes of globalization have created new versions of haves and have-nots who are participating in and/or also not accessing: virtual structures, flows of goods and services, appropriation of traditions and epistemologies, and various new modes of marginalization, economic advantage, poaching of consumer goods, and even the trafficking of human beings. Skovsmose and Valero note that a neoliberal perspective on mathematics education might lead one to ignore inequities and social injustices, since the mathematics education of marginalized groups may not appear to promise enough of a profit in return; or one might argue that a minimal level of mathematics skills would be necessary to integrate workers into the workplace; a third argument about the kinds of skills and concepts essential to an educated community would worry about the need to educate an adequately large consumer base primed to participate at various levels in the modes of production and reproduction

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of economic, political, and cultural structures that perpetuate these modes. Equity in this sense is tied to global economic forces and the need to link pedagogies and policies to the ways that they create opportunities to participate in or close off ways to enter such structures. The second paradox articulated by Skovsmose and Valero (2005) is the paradox of citizenship: With this paradox they are addressing the two-sided nature of acting on one’s membership in society. The romantic sense of citizenship that we often think of in educational contexts is the participant who can decide and choose what is relevant and important for their lives; an extension of this for me is that they can be active agents who are defining what is possible to choose among, and who are inventing methods of choice for themselves and with others, using mathematical skills and concepts. The paradox is that such choices and options require that people are simultaneously interpreted and treated as consumers more than as active agents; the citizen-consumers are studied by experts in order to represent their supposed interests, opinions on public issues, relationships to economic indicators or war casualties or other factors, and these representations are also mixed in and intermingled with advertisements and offers from which these consumer citizens might “choose.” Skovsmose and Valero note that a product might be intangible, as in an idea or a service; prices turn into a complexity of conditions for payment, including rates and terms; consumers might make investments or take a loan, while they might also vote, receive services, fulfill obligations, or, in other words, be citizens. For mathematics education and equity, this second paradox raises the confusing two-sided nature of human beings as both consumers and as agents of cultural and political construction, as subjects of the structures of power and as agents of social change. Appelbaum (1995) introduced the ideas of mathematics curriculum, mathematics educators, and mathematics learners as simultaneous commodities and cultural resources. On the one hand, mathematics curriculum materials are commodities bought and sold, and as subjects of policy are implicated in potential networks of power, justice, authority, and opportunity; educators are bought and sold to potential marketers of products; students are potential commodities, and access to them is bought and sold by and to makers of curriculum materials, producers of assessment tools, analyzers of assessment products, and to policy makers who wish to influence these transactions. At the same time, mathematics as an important subject of the curriculum is sold to students as something they should believe is important. The identification of students as unsuccessful fuels an industry of products and testing and teacher training and policy making, all targeting such unsuccessful learners, and thus creating a marketplace that necessitates the ongoing preservation of inequity for its very existence. Meanwhile, mathematical concepts and procedural skills might be used by individuals and groups to influence others, to analyze

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their own situations as political, economic, and cultural actors, to gather evidence for political action, to make choices about what and how they wish to live, and even to define their own identities. On the latter idea, the premier example in Appelbaum (1995) is that of Malcolm X, the legendary civil rights activist in the 1960s United States, who emphasized in his new name the ways that the legacy of slavery had torn from him his very identity: The notion of a variable was used to evoke an important conception of inequity and social injustice in the fact that his very name could be replaced by any other and still would not be able to reclaim the name he might have had, had the global slave trade not raped Africa and trafficked in human beings brought to lives of despair and horror in the so-called “New World” of the 16th through 19th centuries. Malcom’s use of algebra in the taking on of a new name transformed our very conception of social justice. The question remains, then, how we might shift from the narrow views of an assessment stance toward a global perspective on equity and mathematics education that would help us envision and support the kinds of efforts toward agency exemplified by Malcolm’s ingenious use of algebra to transform the civil rights movement. The duality of commodification and cultural resource would need to be combined with the urgency of transnational crises in establishing a sense of justice beyond the simplistic counting of access to educational opportunities and achievement of outcomes. It would also need to recognize categories of social difference, such as race, class, ethnic origin, gender, and so on, when they are important, yet not be reduced to these dimensions and their intersections when other transnational issues and affiliations are critical. These categories are important when they call attention to patterns of inequity and injustice, loss of opportunity of outcome. Yet, they are constraining rather than enabling when they reduce injustice to demographics, or when they lead to a focus on local issues in a way that is inappropriately disconnected from larger, global patterns. Categories of social difference can also hide systemic forms of injustice resulting from the transnational effects of global economic structures, and crises associated with climate change, war, and so on, effectively blaming the victims of such injustice for their own circumstances. RECOGNITION AND DIGNITY MIGHT REPLACE EQUITY AND JUSTICE On the one hand, we might wish to limit our field of vision: It seems insurmountable to take any and all experience as potentially a form of mathematics education, such as school, family, church, mosque, television, streaming video, radio, consumer culture, and leisure travel, and then to begin to imagine policy or practice across these various terrains of experience in

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terms of equity and social justice. So we might want to restrict our discussion to, say, school mathematics, and official government policies, or to a specific NGO program under development, and to our organizational policies on social justice as informing the design of this program. At least this way of narrowing our scope feels doable in the moment, and supports immediate action. Perhaps it is determined that a school policy for mathematics teaching and learning marginalizes indigenous communities on another continent; or it is determined that official policy declares that it knows better than families of certain children what is best for these children in ways that conflict with the families themselves. Battiste (1998) describes such events as the application of an “educational model” of practice or policy, and renames them “cognitive imperialism.” This fragmented accumulation of knowledge of what is “best” builds on Eurocentric strategies that maintain their knowledge as universal, and derivable from standards of “good” that are universally appropriate. The ideas and ideals are so familiar that they are never questions, and do not need to be challenged or questioned; indeed, all decisions and questions can be posed and resolved within the accumulated universal knowledge. Children and families are on the periphery of the curriculum; indigenous cultures are included but in simplistic and tokenistic ways; non-Western traditions of counting, measuring, locating, designing, playing, and explaining are erased into dispossession: the add-and-stir approach to multiculturalism, which avoids disruption of the central Eurocentric assumptions that govern the educational system (Stathopoulou & Appelbaum, 2016). In some global contexts, immigration and refugee migration patterns following in the aftermath of colonialism have led to unique and locally distinct forms of hierarchy and privilege, often related to the land that had been usurped and then eventually redistributed over several centuries of economic and political transformations. Still other places have developed into special creolized “intercultures” that foster their own versions of land ownership and policy practices, while yet others identify relationships to the land through nomadic communities, often overlapping and coexisting with the legacies of colonialism and postcolonial economic–political–cultural processes. In most of these contexts, the definitions of knowledge and knowing, mathematics and school mathematics, skills and concepts, have been reduced to the Western, Eurocentric practices and associated assumptions that they are enacted via policies in the first place. It might be appropriate, nevertheless, to bring together elders of various communities in a discussion of the varieties of ways that the colonialist structures of school and policy might be the sites of dignity for all. Such meetings would be called “reconciliation,” and would be dedicated to a shift in discourse from one of differences across social categories toward one of constructing dignity and recognition. And these conversations would likely lead to practices

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and policies that cross boundaries of school and everyday life, of culture and politics, community and virtual, diasporic affiliations. ALTERGLOBAL MATHEMATICS EDUCATION Dignity and recognition via processes of reconciliation are one strategy of working with categories of difference in postcolonial global contexts while not feeling restricted by the negative consequences of using these categories. More generally, they might be considered processes of “alterglobal social movements,” those transnational efforts to coexist with globalization while not being fully anti-global, taking the best of globalization and exploiting it, while working against the dehumanizing aspects of contemporary globalization (Appelbaum & Gerofsky, 2013; Pleyers, 2010). Alterglobal movements theorize communities in flux that coalesce and take action without requiring fixed identities, clear goals, justifications, or defined structures, while maintaining a strong commitment to ethical principles of inclusion, diversity, human recognition, and dignity (Butler, 1997, 2010). The term alterglobal has come to represent various forms of collective action responding to the negative aspects of globalization—corporate personhood, the dominance of markets over ethics, the increasing need to understand diaspora identities, the changing nature of identity through social media. Alterglobalization seeks a renewal of political citizenship and activism, bypassing traditional ideas about how to make social change. For example, alterglobal movements typically avoid traditional ideas about how to make a revolution, whether by peaceful or violent means, which usually assume that people make change within nation states by toppling regimes and rewriting rules. The idea is to work across boundaries in collaboration for human dignity and social justice. In this sense, mathematics education itself can be “remade” as an explicitly alterglobal social movement (Appelbaum, 2018). To redefine mathematics education in this way is to reorient its purposes. Rather than to “teach” mathematics, mathematics education would educate with mathematics. Mathematics would be the skills and concepts that support dignity and recognition. Mathematics would be defined as those concepts and uses of number, space, pattern, structure, possibility and probability, and decision-making, and the study of these concepts and uses, that support global cooperation and interaction, but also that oppose the negative effects of economic globalization that are harmful for, or not directly enabling, human values such as environmental and climate protection, economic justice, labor protection, recognition and celebration of indigenous cultures, peace, and civil liberties.

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One way to consider mathematics education as an alterglobal social movement is to flip some directions of causation: Rather than imagining mathematics education as the result of curriculum and policy, we can see curriculum and policy as results of the influence of mathematics education. Assessment would be associated with policy, but would not drive mathematics education. In this sense, any policy that leads to a lack of dignity or recognition would be simply inexcusable. Any policy that enabled learners to develop strategies and skills that supported dignity and recognition with and through mathematics would be one that enhanced the mathematics education movement. And we would have new forms of evaluation for policies and practice: Do they support through mathematics teaching, learning, and action global collaborations around human values such as environmental and climate protection, economic justice, and labor protection? Recognition and celebration of indigenous cultures, peace, and civil liberties? Postcolonial reconciliation practices? If so, how? In other words, it is about time that mathematics education stopped serving the interests of “the state,” “the powerful,” “economic needs,” and so on, and about time these entities started serving the needs of alterglobal mathematics education. Some more pragmatic conceptions of mathematics education as an alterglobal social movement follow. First of all, whether a teacher, curriculum designer, policy maker, NGO program director, family member, or other interested participant in the mathematics education community, you would need to conceptualize your efforts as part of a broad, global, diasporic project of collaborators. On the one hand, this could begin with significant participation and contributions in transnational communities of mathematics educators, consciously contributing and supporting efforts at dignity and reconciliation. Further participation in global meetings, virtually or physically, in such communities, would parallel the alterglobal efforts of the yearly World Social Forum (2018), and other alterglobal affiliations. Just as the World Social Forum brought individuals, organizations, and others concerned that “the systems that rule the world have not worked for the people nor the planet,” together March 13–17, 2018, in Salvador, Bahia, Brazil, you would join with others in international meetings to plan specific projects that spread mathematics education as a force for global good. Possible organizations that you might join include: • The International Commission for the Study and Improvement of Mathematics Education (http://cieaem.org): Investigates the actual conditions and the possibilities for the development of mathematics education in order to improve the quality of teaching mathematics. The annual CIEAEM conferences are essential to realize this goal. The conferences are characterized by exchange and discussion of

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the research work and its realization in practice and by the dialogue between researchers and educators in all domains of practice. • The International Congress on Mathematical Education (https:// www.mathunion.org/icmi/conferences/icme-international-congress-mathematical-education): This quadrennial meeting is organized by the International Commission on Mathematical Instruction, which offers a forum to promote reflection, collaboration, and the exchange and dissemination of ideas on the teaching and learning of mathematics from primary to university level. ICMI works to stimulate the creation, improvement, and dissemination of recent research findings and of the available resources for instruction (e.g., curricular materials, pedagogical methods, the appropriate use of technology, etc.). The commission aims to facilitate the spread and understanding of information on all aspects of the theory and practice of contemporary mathematical education from an international perspective. ICMI has the additional objective of providing a link between educational researchers, curriculum designers, educational policy makers, teachers of mathematics, mathematicians, mathematics educators, and others interested in mathematical education around the world. • The Creating Balance in an Unjust World STEM Education and Social Justice Conference (http://creatingbalanceconference.org/): Believes that mathematics literacy is a human right and seeks ways to make mathematics meaningful, relevant, and a tool to analyze and change the world. • The International Study Group for Ethnomathematics (http:// icem6.etnomatematica.org/index.php/icem6/icem6): Whose international conferences take place every 4 years, aiming to bring together ethnomathematicians from different parts of the world, as well as members of other communities (academic or not) who are interested to know, share, debate, or divulgate their reflections and results about research and practice of the ethnomathematics. Another idea is to attend alterglobal meetings, such as the World Social Forum (https://fsm2016.org) as a mathematics educator, and network with people from other areas with whom you might collaborate. The purpose of meeting virtually and in person is to develop collaborations that cross national and regional borders and boundaries, in efforts to promote specifically dignity and recognition with mathematical activities. This necessitates the construction of global process of curriculum development and implementation, which in turn dictate the need for policies that support these actions.

From Equity and Justice to Dignity and Reconciliation 35

You might also network with scholars in ethnomathematics, who have worked tirelessly in recent decades to recognize and value mathematical ways of being that were delegitimized and in some cases erased as “folk culture” by colonialist practices. Efforts in this direction take seriously the realization that (school) mathematics worldwide has been reduced to Western, mostly European, mathematics, and that other traditions and ways of being mathematical are in some cases lost, and in other cases not even understood as mathematics. Mathematics as a discipline is in this sense a tool of cultural imperialism and epistemicide (de Sousa Santos, 2014; Paraskeva, 2016). When local traditions are further labeled as “indigenous,” they are both lowered in status and also implicated in the maintenance of coloniality, that is, the ongoing structural legacies of colonialism. Thus, it is imperative to be cognizant of the potential for mathematics and mathematics education to inadvertently be implicated in the construction of indigeneity itself, a most insidious and deceitful form of coloniality (Barajas-López & Bang, 2018). TAKING ACTION IN THE MOVEMENT Toscano (2012) identified three spheres of action in alterglobal social movements that can help us consider what the collaborative projects could or should look like: a resistance to domination, the importance of elaborating alternative worldviews, and taking practical steps in the pursuit of these alternatives. Each sphere implies a branch of mathematics education as an alterglobal social movement: directly confronting the ways that groups are dominated or marginalized through their mathematics education experiences, conceptual development of outside-of-the-box ideas for mathematics education, and practical programming using the prior two branches. I suggest that each of these spheres shares an awareness that equity and justice (in mathematics education) is somehow more than providing access to quality forms of educational experience, and somehow more than refusing to follow those practices that you can see are causing some inequity; they share a need to rethink and challenge the notion that mathematics skills and concepts are “gifts” sifted out of the historical legacy of the human condition that each person has a right to have, use, and, in turn, give to others. They shift the mathematics from something that can be more equitably distributed to all, who as humans would be granted the right to access, toward the use of mathematics to achieve dignity and recognition across borders and boundaries. Furthermore, these spheres of action using mathematics avoid the presumption that recognition is a matter of cultural adjustments that would bestow status or privilege upon those who currently do not enjoy such positions; to do so would turn recognition into yet

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another form of redistribution of some greater social good, named recognition. These spheres of action avoid an economics of recognition as much as an economics of knowledge, because they seek to celebrate and acknowledge differences instead of burying them as irrelevant in the redistribution mechanisms. Christine Sleeter’s (2018) historical perspective on the evolution of multicultural education offers some practical lessons to be applied by mathematics educators. First of all, it is critical to be prepared for severe push-back by the neoliberal forces that maintain interests in education as currently practiced. The allure of this global paradigm driving economic and social restructuring presses schools away from democracy for diverse politics and toward competition and privatization for personal gain. Equally compelling is the industry that has emerged to promote a greater need for teacher training in response to unequal educational outcomes, that is, the how one teaches, keeping the what one teaches, and the toward what purposes one teaches untouched. For example, a culturally responsive pedagogy intended to voice and elaborate the needs and urgencies of a minority population is reduced to steps to follow as “best practices” for all learners, while consideration of culture and curriculum is reduced to cultural celebration as an extracurricular activity. In particular, for mathematics education, we often hear the “serious concern” that what we are doing is no longer mathematics per se—it might be labeled “fuzzy math,” “new math,” or heavily criticized as “watered down” and not serving a population that requires greater rigor. Hence the need, articulated by Toscano above, for a program that includes plans for resistance to this kind of push-back, and for parallel efforts that elaborate alternative views with enough detail to legitimate them. Contributing to already-existing alterglobal social movements as the mathematics educator is an important recommendation. Here one might propose, as does Christine Sleeter (2018), to collaborate with bilingual educators. My experience with mathematics education in Northeastern U.S. urban schools leads me to understand the importance of place and location in such recommendations, since, for example, the schools with which I am familiar are rarely if ever bilingual and mostly multilingual, with up to 20 different first languages in any given classroom. Thus, a bilingual approach to mathematics education is received by teachers in these contexts as naïve and ill-informed, whereas the contexts in the Southwest of the United States might work very well with promoting a serious bilingualism and biculturalism through and with mathematics. The general idea in either context is that we can contribute as mathematics educators to “border pedagogy” projects (Appelbaum, 2011): Such educational efforts work in the liminal locations along borders of any kind. One version is to collaborate across national borders of tension, to create events and programs that share the concerns on either side of the border, and in the process, constructing an

From Equity and Justice to Dignity and Reconciliation 37

identity in the border region that transcends any one identity on either side. For example, Jaime Romo (2005), Bernardo Estrada (2006), and Elizabeth Garza (2007) exploit the ways that a border, which would be assumed to split things apart, paradoxically binds the two sides together (like a “zipper”—Appelbaum, 2002), so that the border region can be united through the challenges of a quality education for all border region residents. The strategies that we discover and develop together create a vibrant energy within each of us, an excitement to go back and try these in our classrooms. Through understanding that there are no recipes or formulas, no checklists or advice that describe the one best way to educate all children, Border Pedagogy participants are discovering new ways of relating and engaging students, caminos, that allow students to become critical thinkers and creative problems solvers. (Estrada, 2006, p. 36)

Border pedagogies enable all participants to dwell in “the borderlands,” and in this respect no one is superior or dominant as the authority on borderland experience. Instead of the all-too-common uncanny, haunting gaze of “the other,” a spontaneous regrouping often takes place, based on a sense of a shared identity and experience as borderland inhabitants. Participants are positioned to explore educational topics from that common ground. For example, when discussing the issues of cultural and linguistic diversity in education, Garza’s (2007) students, future teachers, drew on their experience as inhabitants of the border region; they found many common threads, all of which took on new importance for these students because they saw each thread as woven into a type of “alterglobal” preparation for teaching in the border region. Note that border pedagogies are fundamentally different from traditional notions of multiculturalism, which would hope for learning to take place through the bringing together of people who are “different” from each other. Such experiences tend to reify biases and increase distances, rather than building the common ground of a border pedagogy. Border pedagogies are in this way examples of Toscano’s alternative worldviews, and his practical applications of them, at one and the same time. A contrasting alternative worldview presumes mathematics teaching and learning can and perhaps should happen outside of schools, led by the youth themselves, and can be found in the Youth Mathematician Laureate Project (http://yomap.org). This educational effort identifies youth who are already using mathematics as the art that builds communities, and recognizes them for this creative work by supporting new and more ambitious projects that they dream up. The “laureates” are provided administrative support and a minimal budget to help them with mathematically-grounded community-building initiatives, and names these youth leaders as community laureates, officially honored by a city, state, region, or NGO.

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Focusing our gaze from the pedagogies of a border region through the localities of the mathematician laureates into the microlevel of classrooms, we can consider mathematics education as obligated to provide experiences for young people to practice taking action in the community themselves (Appelbaum, 2009). In the middle of their investigation work, students are required (because of the social obligation) to reconsider what they have done, and to identify key aspects of their experience. Based on this reflection, the students are facilitated in designing an interaction with people outside of their class. The designed “impact” can be one of two types: either they must identify a person or group of people upon whom they believe they can have an influence, or they must identify a person or group of people who they believe can help them significantly with their own mathematical work. SUMMARY The key point about mathematics education as an alterglobal social movement is that mathematics is used for activities not particularly defined in mathematical terms, boldly contributing with mathematics teaching and learning to already-existing social movements which share compatible goals of human dignity and reconciliation. This social movement conception of mathematics education is not specifically in opposition to neoliberal education reform nor to traditional school mathematics practices, but instead would coexist with them, in effect co-opting them in the service of the grander, alterglobal project. In this way, alterglobal mathematics education might appear irrelevant to those busy within the educational politics of redistribution that is grounded in standardized tests and other assessment measures of opportunities and outcomes. It might further seem uninterested in parallel activities in the cultural politics of recognition, which also ground themselves within the discourses of opportunities and outcomes, or which lock themselves inside of fixed labels of social difference such as class, race, ethnicity, gender, and so on, preferring instead to form new categories of affiliation and action across borders and boundaries. The most potential is with those actions that ignore presumptions of routine practice, and instead elaborate alternative worldviews, such as collaborative educational projects in border regions that are not based within schools on either side of a border, or community mathematics projects led by youth in community contexts. Such alternative worldviews shift recognition away from the redistribution paradigm toward processes of reconciliation. The closest mathematics education has come to a postcolonial version of reconciliation is within those ethnomathematics projects that incorporate local activism with the exploitation of indigenous mathematical knowledges in collaboration with researchers who seek patterns that are

From Equity and Justice to Dignity and Reconciliation 39

alterglobal rather than fixated on the uniquely local and non-generalizable “lessons to be learned.” NOTES 1. One example of a classroom effort to create a bicultural mathematics education: Lipka, J., et al. (2007). Creating a third space for authentic biculturalism: Examples from math in a cultural context. Journal of American Indian Education. 46(3), 94–115. 2. Two good examples of research grounded in valuing the strengths of minority learners: Leonard, J., & Martin, D. (2013). The brilliance of Black children in mathematics. Charlotte, NC: Information Age; Stathopoulou, C., & Kalabasis, F. (2007). Language and culture in mathematics education: Reflections on observing a Romany class in a Greek school. Educational Studies in Mathematics, 64(2), 231–238. 3. Two versions of research that offer alternative views of the content and practices of mathematics education: Gutiérrez, R. (2012). Embracing “Nepantla”: Rethinking knowledge and its use in teaching. REDIMAT-Journal of Research in Mathematics Education, 1(1), 29–56; Stathopoulou, C., & Appelbaum, P. (2016). Dignity, recognition, and reconciliation: Forgiveness, ethnomathematics and mathematics education. RIPEM, 6(1), 26–44. 4. A rare research study of the interwoven dynamics of honoring indigenous mathematics, political action, and mathematics education for social justice is Knijnik, G. (2002). Curriculum, culture, and ethnomathematics: The practices of “cubagem of wood” in the Brazilian landless movement. Journal of Intercultural Studies, 23(2), 149–165. 5. Recent critiques: Singh, S. (2017). Pi of life: The hidden happiness of mathematics. New York, NY: Rowman & Littlefield; Meyer, D. (2010). Math class needs a makeover. TEDTalk, TEDxNYED. Retrieved from https://www.ted.com/talks/ dan_meyer_math_curriculum_makeover; Lockhart, P. (2009). A mathematician’s lament: How school cheats us out of our most fascinating and imaginative art form. Bellevue Literary Press. 6. Jahnke, T., & Meyerhöfer, W. (2007). PISA & CO. Kritik eines Programms. Hildesheim: Franzbecker; Strauss, V. (2017, April 19). 34 problems with standardized tests. The Washington Post. Retrieved from https://www.washingtonpost.com/news/answer-sheet/wp/2017/04/19/34-problems-with-standardized-tests/?utm_term=.8d49a09db78c

REFERENCES Appelbaum, P. (1995). Popular culture, educational discourse, and mathematics. Albany, NY: State University of New York Press. Appelbaum, P. (2002). Multicultural and diversity education: A reference handbook. Santa Barbara, CA: ABC-CLIO.

40 P. APPELBAUM Appelbaum, P. (2009). Taking action: Mathematics curricular organization for effective teaching and learning. For the Learning of Mathematics, 29(2), 39–44. Appelbaum, P. (2011). Carnival of the uncanny. In E. Malewski & N. Jaramillo (Eds.), Epistemologies of ignorance and studies of limits in education (pp. 221–239). Charlotte, NC: Information Age. Appelbaum, P. (2018). How to be a political social change mathematics education activist. In M. Jurdak & R. Vithal (Eds.), Sociopolitical dimensions of mathematics education (pp. 53–73). Cham, Switzerland: Springer. Appelbaum P., & Gerofsky, S. (2013, July). Performing alterglobalization in mathematics education. Plenary in the form of a jazz standard. Opening Plenary Address, CIEAEM (International commission for the Study and Improvement of Mathematics Education), Turin, Italy. Barajas-López, F., & Bang, M. (2018). Indigenous making and sharing: Claywork in an indigenous STEAM program. Equity & Excellence in Education, 51(1), 7–20. Battiste, M. (1998). Enabling the autumn seed: Toward a decolonized approach to aboriginal knowledge, language, and education. Canadian Journal of native Education, 22(1), 16–27. Butler, J. (1997). Excitable speech: A politics of the performative. New York, NY: Routledge. Butler, J. (2010). Performance agency. Journal of Cultural Economy, 3(2), 147–161. de Sousa Santos, B. (2014). Epistemologies of the South: Justice against epistemicide. Abingdon, England: Routledge. Estrada, B. (2006). The border pedagogy initiative: An opportunity for growth and transformation. Hispanic Outlook, January, 34–36. Garza, E. (2007). Becoming a border pedagogy educator. Multicultural Education, 15(1), 2–7. Paraskeva, J. (2016). Curriculum epistemicide: Toward an itinerant curriculum theory. Abingdon, England: Routledge. Pleyers, G. (2010). Alter-globalization: Becoming actors in a global age. Cambridge, England: Polity Press. Romo, J. (2005). Border pedagogy from the inside out: An autoethnographic study. Journal of Latinos and Education, 4(3), 193–210. Skovsmose, O., & Valero, P. (2005). Mathematics education and social justice: Facing the paradoxes of the informational society. Utbildning & Demokrati, 14(2), 57–71. Sleeter, C. (2018). Multicultural education past, present, and future: Struggles for dialog and power-sharing. International Journal of Multicultural Education, 20(1), 5–20. Stathopoulou, C., & Appelbaum, P. (2016). Dignity, recognition, and reconciliation: Forgiveness, ethnomathematics and mathematics education. RIPEM, 6(1), 26–44. Toscano, E. (2012). The sphere of action of the alterglobal movement: A key of interpretation. Social Movement Studies, 11(1), 79–96. World Social Forum. (2018). Towards World Social Forum 2018, São Paulo, Brazil. Retrieved from https://fsm2016.org

CHAPTER 3

WHY THE (SOCIAL) CLASS YOU ARE IN STILL COUNTS Peter Gates

If you knew that something in young people’s diets caused them to experience anxiety, made them feel bad about themselves, reduced their academic achievement, limited their concentration span, then there would surely be a widespread call to restrict it. It might be sugar, hydrogenated trans fats, salt, food additives. So why is there not an outcry against one everyday school practice, which causes children to experience each of those responses? Ability segregation, setting, tracking, call it what you will, causes symbolic violence and abuse to children up and down the country—and around the world—on a daily basis. Yet, we turn a blind eye; why is that? Well I am not usually one for conspiracy theories, but it is a conspiracy. Why does the educational establishment keep relatively quiet about it? Because we are part of that conspiracy, and we do relatively well out of it—well-paid jobs, pensions, international travel, kudos . . . A recent analysis by Archer et al. (2018) provides us with a clear and detailed account of the processes by which the educational segregation that discriminates against working-class young people becomes represented (or rather “misrecognized”) as both natural and normal but also . . . How can it be other? The data and thus the

Equity in Mathematics Education, pages 41–64 Copyright © 2019 by Information Age Publishing All rights of reproduction in any form reserved.

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analysis, comes from the United Kingdom, yet, we are not alone. Our education system is riven with notions of social class and in this chapter, I take a look at class and its influence on the mathematics classroom, principally in the United Kingdom—and possibly the United States and Australia. So how does, what on the surface appears to be an administrative and pedagogic decision about the organization of teaching groups, turn into a process of social class reproduction? Those of us whose politics and professional ideology sit comfortably within a social justice framework (and especially those of us on the Marxist left), have no problem with the notion of class. It is after all the most defining characteristic that influences attainment, achievement, and engagement in schooling, at least in the United Kingdom—and I suspect also in other similarly capitalistic societies. The specific definition of social class may have changed since the 1840s due to changes in the structural basis of the economy, but what hasn’t changed is the nature of social inequity, and domination by a social, political, and economic elite. Postmodern approaches have attempted to silence the politics of class, yet, like Marx, it refuses to go away. Rightfully, as the late Geoff Whitty reminds us: The mere fact that class does not explain all can be used as an excuse to deny its power. This would be a serious error. Class is of course an analytic construct as well as a set of relations, that have an existence outside of our minds . . . It would be wrong to assume that, since many people do not identify with or act on what we might expect from theories that link, say, identity and ideology with one’s class position, this means class has gone away. (Whitty, 2002, p. 69)

One feature of the failure of many learners to achieve after years of schooling, is the observation that school achievement is not equitably spread throughout society; children from less affluent homes do disproportionately worse than those brought up in relative affluence (Whitty & Anders, 2014). Such children are at risk of sustaining a weak conceptual grasp of mathematical and scientific concepts and in numerical procedures, which hold them back from developing a more sophisticated understanding of STEM subjects (Hoadley, 2007; Oakes, 1985, 1990). This in turn closes off pathways to many careers and professions, but worse, develops into anxiety and rejection of mathematics in particular, contributing instead to an identity of “I just can’t do maths” (Gates, 2001). Whilst much research has attempted to articulate this relationship, much research has simply ignored it, either through denial, or in the belief that by providing good research all will benefit through the “trickle down” principle. The denial may even be part of the conspiracy, but it is a political act. Indeed, an early finding from the ICAAMS study (http://iccams-maths.org/) is that over 30 years

Why the (Social) Class You Are in Still Counts 43 attainment has not changed very much [. . .] The general trend is for results to be somewhat lower than in the 1970s, although there are some exceptions to this. (Hodgen, Brown, Küchemann, & Coe, 2010, p. 8)

By ignoring this critical feature of learners, we appear to have made little improvement in learning—according to ICAAMS at least. There is also some doubt that the improvements in levels of achievement in mathematics and science in the United Kingdom, trumpeted by successive governments have in reality been that real (Dickinson, Eade, Gough, & Hough, 2010) and undoubtedly the same holds true in other countries. So, instead of trying to do the same old thing better, maybe it is time to think anew. My own contact with schools and teachers over 40 years suggests that each academic year, the prospect of yet again teaching fractions to a class of low achieving challenging adolescents strikes abject frustration into mathematics teachers throughout the world. Yet, the reality is that many young people fail to understand even basic mathematics after a decade of schooling. How we get to this position where, after 9–10 years of compulsory education, we are still trying to convince some children that 1/4 = 2/8 is nothing short of an international scandal. Worryingly, this is after decades of curriculum reviews, policy changes, and millions spent on research. Children do not start school—or life—on an equal footing. It is well known in an extensive literature that there is a significant difference in the levels of achievement of children from different social backgrounds (or social classes), and these early differences expand during the course of compulsory schooling (Clifton & Cook, 2012). So, whilst children start school with differential levels of achievement, these gaps increase, rather than decrease during schooling, suggesting schooling does not mitigate against these social advantages, but contribute to their deepening. More specifically, the influence of family and parental wealth upon educational attainment and post-school employment is particularly strong (Jerrim & Macmillan, 2015). So, how does class play out? I will argue that it does so by successfully hiding the discrimination and thereby creating a very effective mechanism. Its greatest effectiveness lies in its invisibility and the presentation of normality and of necessity. Alexandre Pais argues very strongly that the equity and “mathematics for all” discourse is a sham; we live in a society where in order for some to succeed, many others have to fail: In the example of “mathematics for all,” this official claim conceals the obscenity of a school system that year after year throws thousands of people into the garbage bin of society under the official discourse of an inclusionary and democratic school [ . . . ] The antagonistic character of social reality—the crude reality that in order for some to succeed others have to fail—is the nec-

44 P. GATES essary real which needs to be concealed so that the illusion of social cohesion can be kept. (Pais, 2012, p. 58)

Those who stand to gain by this, cannot allow serendipity to organize failure, or, well, it might end up being random and equitably spread throughout society and that won’t do. So, those in power activate several class weapons in this process—curriculum, pupil organization, pedagogy, to mention just three of the most significant factors in structuring pupil experience. In addition, the mode of communication, specifically the use of verbal and literal rather than visual forms of instruction, serve to further exacerbate class divisions (Gates, 2015, 2018). CURRICULUM The first strategy is to create alienation through the curriculum. The curriculum becomes an object that is “alien” to, or outside of, the pupil. Marx sees alienation as a key component in an exploitative economy meaning it becomes an object, an external existence. But that it exists outside of him, independently as something alien and that it becomes a power on its own confronting him. (Marx, 1844/1975, p. 272)

Whilst Marx is referring to economic labor, exactly the same process relates to educational labor; young people engaging in tasks in the classroom. Whereas some have a personal investment in the certification the curriculum brings for future advancement, others sit outside that—the losers as Pais calls them—to ensure the winners win. Government-controlled curricula in the United Kingdom and elsewhere, presents largely decontextualized (alienated) skills, devoid of any particular rationale; why do you complete the square? This is as true today as it was nearly 40 years ago: Mathematics lessons in secondary schools are very often not about anything. You collect like terms, or learn the laws of indices, with no perception of why anyone needs to do such things. There is excessive preoccupation with a sequence of skills and quite inadequate opportunity to see the skills emerging from the solution of problems. As a consequence of this approach, school mathematics contains very little incidental information. A French lesson might well contain incidental information about France—so on across the curriculum; but in mathematics the incidental information which one might expect (current exchange and interest rates; general knowledge on climate, communications and geography; the rules and scoring systems of games; social statistics) is rarely there, because most teachers in no way see this as part of their responsibility when teaching mathematics. (Cockcroft, 1982, p. 141)

Why the (Social) Class You Are in Still Counts 45

Alternative approaches to the mainstream “about nothing at all” remain just that—alternative. Using mathematics in a creative and critical way rarely gains official legitimacy, not because it is explicitly banned, but because the system operates to marginalize such approaches placing them outside the accountability mechanisms set up to maintain control. Eric Gutstein provides one such approach where mathematics can be used to help students investigate, critique, and subsequently oppose injustice and oppression (Gutstein, 2006), yet the prospect of such ideas influencing mathematics teaching more widely seems remote. Children from disadvantaged backgrounds have forms of knowledge that do not allow them to fit so well into the expectations of schools as do those from more affluent or middle-class homes (Zevenbergen, 2000). Whilst this seems to be true generally, there seems to be specific differences in learning of mathematics (Case, Griffin, & Kelly, 1999) where the most significant and consistent predictor of academic achievement in school seems to be the parental income, which has an effect stronger even than parental educational background. Where ethnicity and gender are factors, they are usually confounded with socioeconomic status (SES; Jordan, Huttenlocher, & Levine, 1992, p. 652); in the first 2 years of formal education, school makes little difference to this (Stipek & Ryan, 1997, p. 721). Yet, one major impediment to the amelioration of mathematics teaching and learning around the world is that much work in mathematics education is so politically focused as to ignore the social class basis of mathematics learning. Whilst this is lamentable, it is not surprising; indeed, it would be surprising if the field of mathematics education were quarantined from the left–right/radical–conservative dispositions that exist everywhere else. Marilyn Frankenstein from the United States makes a quite radical suggestion: Traditional mathematics education supports the hegemonic ideologies of society [ . . . ] Even trivial math applications like totalling grocery bills carry the ideological message that paying for food is natural and that society can only be organized in such a way that people buy food from grocery stores. (Frankenstein, 1983, p. 328)

Making mathematics relevant, is also taken to mean making it real. So, let us explore the real world for a moment—best buys. In a national supermarket chain when I was writing this chapter I found the following: Kellogg’s Cornflakes 790g – £2.52 (32p/100g) Kellogg’s Cornflakes 450g – £1.80 (40p/100g) So, who pays more for their food, and why? At least one supermarket knows the answer to this:

46 P. GATES Today, research reveals that the UK’s lowest income homes are being forced to spend a disproportionate amount of their weekly expenditure on food shopping. The average household in the UK spends 11 per cent of its weekly expenditure on food. However, 20 per cent of households (those on lower incomes) are actually forced to spend proportionately at least 30 per cent more of their current weekly food spend than the national average. (Morrissons, 2012)

So those who can only afford to buy less, pay more; surely a much more important issue than “which is cheapest”? Is this fair? Certainly not, and the increasing levels of food poverty in the United Kingdom under the rightwing conservative government between 2010 and 2018 have been shocking: Whilst the level of food poverty is worrying enough, what is of greater concern is the exponential growth in the numbers of people across the UK who are experiencing real hunger and hardship. Perhaps the most extreme manifestation of food poverty is the rising number of people who depend on emergency food aid. (Cooper & Dumpleton, 2013)

There is data out there to allow mathematics lessons to be about something important. However, the politicization of the curriculum makes such a critical stance alien to mainstream mathematics education in schools. Oh, you cannot raise that, it is political. Ideological. Brainwashing. Yet as Ole Skovsmose argues, mathematics teachers work within a larger political framework: Whatever the mathematics teachers try to do, it will be done within the overall socio-economic and political formation of society. As the function of mathematics education is determined by this formation, any claimed educational improvements at the micro level will be illusory. As a consequence, it does not make sense to talk about improvements in the mathematical classroom unless one changes society. Conditioned essentialism is an axiomatic element of the classic Marxist outlook. The capitalist order of things is a determining structure, which conditions what is possible and what is not possible to do. (Skovsmose, in press)

So, what we teach and how we teach it is critical to the maintenance of the dominant social forces, but that is not sufficient on its own, we need to structure the experience so that the alienation can be best directed through structured grouping practices. PUPIL ORGANIZATION I opened this chapter by presenting “ability segregation” as a key mechanism, which operates very effectively and secretly by constructing a notion

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of the “naturalness” of elitist educational segregation which goes on to “play a key role in maintaining the status quo in England with regard to the pervasiveness of setting” (Archer et al., 2018, p. 121; see also Francis et al., 2016). Whether one should use “ability” or “attainment” here is a moot point. The very usage of “ability” has a tendency to concretize pupils’ achievement; whereas “attainment” pushes our attention onto a more fluid claim. Yet, while researchers and the sceptics might insist on using “attainment,” many schools and the teaching profession hang on to “ability” for very purposeful reasons; pupils need to be selected, labeled, and segregated. Whatever term we use, the research on pupil grouping is fairly clear: the educational effect of grouping by attainment is insignificant, but has a negative influence on those pupils placed in lower groups (Archer et al., 2018; Francis et al., 2016). I will not explore the arguments here, it is not needed as it is all so well laid out in an extensive literature. The most critical issue for me here is that such grouping of pupils has in the United Kingdom become hegemonic—the ruling and dominant ideology. Yet as with all forms of hegemony, it is a system of ideas and associated practices which serve the need to achieve a specific form of domination. It is not necessary for mathematics teachers to realize this, or to cynically operate a system in order to segregate poorer pupils. Indeed, it is better they do not know so it can be more easily hidden. So it becomes perpetrated as a universal; how can it be otherwise? Well it can be, and indeed is, otherwise: In Sweden ability grouping is illegal because it is known to produce inequities. In the USA parents have brought law-suits against school districts that have denied high level curricula to students at high school age; the idea that such selectivity in “opportunity to learn” (Porter, 1994) could happen at elementary school is inconceivable for most Americans. In Japan (Yiu, 2001) students are believed to have equal potential and the aim of schools is to encourage students to attain at equally high levels. Japanese educators are bemused by the Western goal of sorting students into high and low “abilities.” (Boaler, 2005, p. 136)

This is not to suggest that each of these countries are free of the segregation by social class we see in the United Kingdom, it just plays out differently. However, it is not just played out through the forms of organization but also through the pedagogy adopted. PEDAGOGY Sarah Lubienski studied the mathematical experiences of pupils with an eye to looking at pupils’ backgrounds (Lubienski, 2000a, 2000b, 2007). Whilst she naturally expected to find SES differences, what she actually found

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were very specific differences in two main areas—whole class discussion and open-ended problem solving. These are two well-researched pedagogical strategies and classroom practices which at least in professional discourse are held in some esteem. Discussion based activities were perceived differently by pupils from different social backgrounds. High SES pupils thought discussion activities were for them to analyze different ideas whilst low SES pupils thought it was about getting right answers. The two groups had different levels of confidence in their own type of contributions with the low SES pupils wanting more teacher direction. Higher SES pupils felt they could sort things out for themselves—as their parents do in life presumably. I suspect this is not an uncommon feature of many schools but where does it emanate? Here then, social class is a key determining characteristic largely absent from much literature on discussion-based mathematics. A second area where Lubienski noted differences was that of open-ended problem solving. The high level of ambiguity in such problems caused frustration in low SES pupils which in turn caused them to give up. High SES pupils just thought harder and engaged more deeply. It is well-known that middle class pupils come to school armed with a set of dispositions and forms of language which gives them an advantage because these dispositions and language use are exactly the behaviors that schools and teachers are expecting and prioritize (Zevenbergen, 2000). High SES pupils have a level of self-confidence very common in middle-class discourses whilst working class discourses tend to be located in more subservient dependency modes, accepting conformity and obedience ( Jorgensen, Gates, & Roper, 2014). Middle-class pupils after all tend to live in families where there is more independence, more autonomy, and creativity (Kohn, 1983). Studies of parenting suggest different strategies are used in different class background. Low SES, working-class parents are more directive, requiring more obedience. Middle-class parents tend to be more suggestive and accommodating reason and discussion (Lareau, 2003). The middle classes grow up to expect and feel superior with more control over their lives. Crucial to understanding the influence of class of learning though is specifying the types of mathematical knowledge on which the discrepancy is present (Siegler & Ramani, 2009). On nonverbal numerical tasks, preschoolers’ performance does not vary significantly with economic background (Ginsburg & Russell, 1981; Jordan et al., 1992; Jordan, Levine, & Huttenlocher, 1994). However, on tasks with verbally stated or written numerals, the knowledge of preschoolers and kindergartners from lowincome families lags far behind that of peers from more affluent families. The differences are seen on a wide range of tasks: recognizing written numerals, reciting the counting string, counting sets of objects, counting up or down from a given number other than one, adding and subtracting, and

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comparing numerical magnitudes (see Alexander & Entwisle, 1988; Geary, 1994, 2006; Ginsburg & Russell, 1981; Griffin, Case, & Siegler, 1994; Jordan et al., 1992; Jordan, Kaplan, Olah, & Locuniak, 2006; Siegler & Ramani, 2009; Starkey, Klein, & Wakeley, 2004; Stipek & Ryan, 1997). Significant is the argument that the problem lies deep within the way in which schools divorce children from the informal intuitive forms of understanding they had experienced before formalized education. Ginsburg and Russell (1981) investigated the associations of social class and race with early mathematical thinking arguing that early mathematical thought develops in a robust fashion regardless of social class and race, and that school failure, specifically in mathematics cannot be explained by initial cognitive deficits (p. 56) a finding in conflict with many early years teachers’ beliefs. However, Ginsburg and Russell (1981) argue that it was cognitive competence not a cognitive deficiency that might be in existence. Specifically, low-income children seemed to have a less developed set of what Case and Griffin call “central conceptual structures” (Case & Griffin, 1990a, 1990b) that went on to underpin future cognitive development specifically of mathematical and numerical processes (Griffin et al., 1994, p. 36), that without these detailed structures early on, children would go on to develop a “rote” approach to learning which would limit the scope of their level of achievement (p. 47). However, through a taught “RightStart” program focusing on conceptual bridging, multiple representation, and affective engagement, Griffin et al. (1994) were able to demonstrate elimination of differences. The importance of looking at the competencies of children very early on is the more significant neurological influence of the developing brain, since children’s early mathematical capacities show a considerable degree of differentiation by social class during the years when the neurological circuitry on which they depend is showing its most rapid development. (Case et al., 1999, p. 148)

Case et al. (1999) go on to argue that whilst SES differences are not observable at birth they do begin to appear around 3 years old (see also Ginsburg & Russell, 1981), but by kindergarten this had become a year and a half difference in capabilities (p. 131). These early differences in mathematical knowledge have a lasting effect as preschoolers’ performance on tests of mathematics is predictive of mathematical achievement at age 8, 10, and 14 and even later in upper secondary school (Duncan et al., 2007; Stevenson & Newman, 1986). This stability of individual differences in mathematical knowledge reflects to some extent the usual positive relationship between early and later knowledge, but the stability of individual differences in mathematics is unusually great. This might be because mathematics

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is something of a secret garden, avoided by low SES parents (Siegler & Ramani, 2009): Observations of homes and preschools, as well as the self-reports of teachers and parents, suggest that the home and preschool environments provide children with relatively little experience where their attention is focused on mathematics, far less than literacy-oriented experience. (Siegler & Ramani, 2009, p. 558)

It must be no surprise then that the U.K. conservative government from 2010 destroyed “Surestart” children’s centers that had been a model policy for social inclusion of the 1997–2010 Labour Party governments. VERBAL AND VISUAL For most pupils, skill at visualization is not instinctive but “one learns to ‘see’” (Whiteley, 2000, p. 4) though we might observe in many mathematics classrooms pupils learning to repeat or learning to say. However, there is some evidence, that social class effects upon the development of mathematical skills is more marked for verbal than nonverbal forms (Jordan et al., 1992). Children from middle-income families do better when the mode of representation is verbal. Yet, where the mode of representation is visual or nonverbal, the social class gap is much reduced possibly because verbal and written forms of communication are less prioritized in workingclass families. Alternatively, for working-class families “knowledge that has been constructed directly from their own actions on objects as well as their observations of the world” applies equally to development of visual and nonverbal modes (Jordan et al., 1992, p. 651). A study by Mayer (1997) suggested that learners with low prior knowledge (or “low domain knowledge”) might be particularly supported by visual models and these are likely to be those very pupils from less affluent backgrounds: Students who possess high levels of prior knowledge will be more likely than low prior knowledge learners to create their own mental images as the verbal explanation is presented and thus to build connections between verbal and visual representations. In contrast, students who lack prior knowledge will be less likely than high prior knowledge learners to independently create useful mental images solely from the verbal materials. Thus, low prior knowledge learners are more likely than high prior knowledge learners to benefit from the contiguous presentation of verbal and visual explanations. (Mayer, 1997, p. 15)

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Mayer takes this further by looking at those learners identified as “poor readers,” who may be so because of an imbalance in text vs visual processing, and whether this would benefit from a visual approach: Previous research on children’s processing of narrative texts has shown that the poor readers profit generally more from text illustrations with regard to comprehension and learning than good readers (Cooney & Swanson, 1987; Levie & Lentz, 1982; Mastropieri & Scruggs, 1989; Rusted & Coltheart, 1979). This suggests that poor readers are able to construct a mental model from a text with pictures, whereas they would fail on the basis of a text alone. Similar results have been found for adult learners’ processing of expository texts. Learners with low prior knowledge benefit from pictures in a text, whereas learners with higher prior knowledge seem to be able to construct a mental model of the described content also only from the text. (Mayer, 1997)

Consequently, school pedagogies may privilege certain learners—those confident and at ease with literal forms (Winn, 1987). In many, but not all cases, “graphics have done more to improve the performance of low-ability students than those of high ability” (Winn, 1987, p. 169), particularly in science (Holliday, Brunner, & Donais, 1977) and mathematics—where it is claimed that visuals reduced “the reading-related working memory overload” in poor readers (Moyer, Sowder, Threadgill-Sowder, & Moyer, 1984, p. 343). Though there is a claim that “low-ability learners” have particular difficulty with materials that are informationally rich and with redundancy (Allen, 1975), those very learners are labelled and placed in lower achievement groups. However, further evidence indicates that there is a lack of explicit instruction in dealing with graphics—that unsuccessful learners would benefit from support and guidance in mapping between graphic and text information and the resulting mental models (Schnotz, Picard, & Hron, 1993). A U.S. study of 13 randomized control trials (RCT) on learning difficulties in mathematics (see Gersten et al., 2009, p. 30 for a full bibliography) reported empirical support for using visual representations with learners who were achieving poorly in mathematics even if this was cited in some studies as providing only “moderate evidence” (Gersten et al., 2009, p. 30). They placed visuals explicitly within a framework consistent with Bruner’s enactive, iconic, symbolic representation situated specifically between physical manipulatives and abstract symbolic representations. In this way, diagrams and visual representations should be used specifically to support learners’ reasoning through transitions between physical models and symbolic representations. It is further argued that student understanding of these transitions can be strengthened through the use of visual representations of mathematical concepts (Hecht, Vagi, & Torgesen, 2007):

52 P. GATES A major problem for students who struggle with mathematics is weak understanding of the relationships between the abstract symbols of mathematics and the various visual representations. (Gersten et al., 2009, p. 30)

They go on to argue that materials specifically for pupils with difficulties, “provide very few examples of the use of visual representations” (Gersten et al., 2009, p. 36). We can see the same reluctance to place visual reasoning in reteach studies examining instructional strategies—for example Darch, Carnine, and Gersten (1984) who offer “explicit instruction” with no attempt to consider any visual forms between word problems and solution. A conclusion for mathematics educators is to foster an approach with teachers to recognize and respect the visual and diagrammatical form as a pedagogical tool to represent and work on mathematics. Low attainers seem to have greater difficulty seeing the salience in a problem than can be represented in multiple ways—particularly the visual—or even to have a disposition to do so. They conclude “reasoning with a diagram is a difficult process that students may need more time and experience to develop” (Garderen, Scheuermann, & Poch, 2014, p. 147). In addition, we do not have an understanding of the way in which diagrammatic competence develops over time, maybe because we have little idea of what we mean by diagrammatic competence and have rarely used it as a legitimate pedagogical device within mathematics. For it is only once we recognize the difficulty and “lack of transparency . . . Can we begin to identify and adopt strategies to support students” (Rubenstein & Thompson, 2013, p. 550). Siegler and Ramani (2009) take this need for privileging of the nonverbal further but argue that whilst preschool children from more affluent backgrounds perform better on some numerical tasks than disadvantaged children, this differential performance can be partially alleviated by regular playing of linear board games—consistent with the hypothesis that playing board games contributes to differences in numerical knowledge among children from different backgrounds, children from middle-income families reported playing far more board games (though fewer video games) than their lowincome peers, indicating part of the gap between low-income and middleincome children’s mathematical knowledge when they enter school is due to differing play experiences (Ramani & Siegler, 2008; Siegler & Ramani, 2009, p. 557). Given that these same disadvantaged children report playing board games at home less than the affluent children, Siegler and Ramani conjectured that this might be partially influential in not providing the cognitive experience that would move them forward (see also Dehaene, 2011): Board games provide a physical realization of the mental number line, hypothesized to be the central conceptual structure for understanding numerical operations in general and numerical magnitudes in particular. (Siegler & Ramani, 2009, p. 546)

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Allocation of blame to working-class parents is common amongst politicians and some researchers, yet, interviews with parents in low-income families indicate that many believe the primary responsibility for teaching mathematics lies with the professionals in schools (Holloway, Rambaud, Fuller, & Eggers-Pkirola, 1995; Tudge & Doucet, 2004) a perhaps not surprising position given the self-importance with which the teaching profession surrounds itself. Indeed, Tudge and Doucet (2004) studied children’s exposure to explicitly mathematical activities in their own homes, other people’s homes, and child care centres, supporting this assertion: A majority of children from working-class backgrounds were observed engaging in mathematical play or mathematical lessons in 0 of 180 observations. If it is indeed correct that working-class parents look to preschool settings to provide children with mathematics experiences . . . our data suggest that they are mistaken—we found no evidence that children are more likely to be engaged in mathematical activities . . . in formal childcare centers than at home. (Tudge & Doucet, 2004, p. 36)

SES affects behavior through its impact on an individual’s aspirations, sense of self-efficacy, personal standards, and emotional states. A strong sense of self-efficacy can help strengthen resilience to adversity often found in the environment of the low SES student. Low SES students often live in chaotic and unstructured environments. They live day to day. They may be unable to manage their emotions, have poor role models, and feel they have no choice or control over their destiny. Students with low SES may also be depressed, have a fear of failure due to past experiences or have acquired failure expectations from their parents. They may be truly capable children who, as a result of previous demoralizing experiences or self-imposed mindsets, have come to believe that they cannot learn. If they doubt their academic ability, chances are they envision low grades before they even complete an assignment or take a test. This has an effect on goal setting in that these individuals also tend to set lower goals for themselves. They may have no real personal goals or vision, but only fantasies of what they hope for. If they do have goals, these children need to learn how they can achieve the goals and develop awareness of the possible self. Goals need to be difficult but attainable in order for significant achievement to be recognized. We need to assign challenging tasks and meaningful activities that can be mastered (Pajares, 1996). One U.S. study looks at how mathematics is organized in effective schools that serve the poor (Kitchen, 2003; Kitchen, DePree, Celedón-Pattichis, & Brinkerhoff, 2007). Looking to “get real” about reform for high poverty communities, Kitchen (2003) suggests three challenging policy changes. The first is over whose interests mathematics education serves:

54 P. GATES Transforming the mathematics education culture to value the mathematical preparation of the majority over the achievements of a select few requires mathematics educators to connect with movements that promote mathematical literacy for those who have been excluded in mathematics. (p. 21)

The challenge is the idea that we should value the interests of the majority over those of a select few—this runs counter to our system. But it surely raises the question—can we both value the achievements of the few (who do well) as well as the many (who do less well)? Kitchen seems to believe we can’t and I agree. There is much in the maths education literature that claims to be socially just because it improves learning for everyone. But unless one explicitly strives to reduce the gap between rich and poor, then one cannot claim to be socially just; rather you are merely giving everyone a chair to stand on. This raises a second, political, challenge: Acknowledging that mathematical education is a political endeavor requires the mathematics education community to recognize the reform movement should be situated in the context of the larger movement for social and political justice (Kitchen, 2003). This too runs counter to our culture and not everyone wants to situate maths education reform in a movement for social and political justice. Kitchen’s (2003) final claim is a little less controversial and possibly achievable: Proponents of reform need to question the role of an education in mathematics, particularly at schools that serve high-poverty communities. (p. 21)

So here is a realistic empirical question. What is the role of mathematics education in such communities? To even ask that question is to take a political stance though—where do those pupils struggling with maths, come from? What backgrounds do they have? What needs do they have? Karen Pellino (2007) argues “the social world of school operates by different rules or norms than the social world these children live in” (para. 5) and summarizes much of the literature on the effects of poverty by drawing our attention to some of the characteristics of children in poverty. They experience: high-mobility, hunger, repeated failure, low expectations, undeveloped language, clinical depression, poor health, emotional insecurity, low self-esteem, poor relationships, difficult home environment, a focus on survival. Kati Haycock (2001) concludes, “We take the students who have less to begin with and then systematically give them less in school” (para. 13), something noticed by Bart Simpson: “Let me get this straight. We are behind the rest of our class and we are going to catch up to them by going slower than they are? Cuckoo!!” (Groening et al., 1996).

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CONCLUSIONS1 For many, the claim that economically disadvantaged children do less well at school will be hardly controversial, or new. Yet, the next stage of that argument often escapes some. This is the “so what” question. A damaging stance is to take a deficit perspective, that “these children” need remediation, that they miss out of stimulation in the home, that both children and parents “lack aspiration,” and even worse, they need a more practical curriculum for a practical future, focusing on “the basics” reinforced through repetition. In a study of 262 U.S. preschool children, Stipek and Ryan (1997) argue that economically disadvantaged preschool children very quickly developed a more negative view of their own competencies and negative attitudes to school, both which lead to a decline in motivation leading to potential future depression of achievement (p. 722): Disadvantaged children are every bit as eager to learn as their more economically advantaged peers. They do however have much further to go in terms of their intellectual skills and, as schools are presently organized, they do not catch up. (Stipek & Ryan, 1997, p. 722)

In a society—and school system—that extols only the virtues of the rich, famous and successful, this is perhaps quite iniquitous but not surprising. Stipek and Ryan (1997) suggest an alternative is to develop instructional methods that will decrease the gap in cognitive competencies specifically targeting the self-esteem and interest of disadvantaged children. This is not an easy policy to enforce, especially how narrowing the gap acts against the social and economic interests of those who benefit from being at the head of the gap, but as Wilkinson and Pickett (2009) point out, when inequity is reduced, the whole society benefits; this is indeed at the heart of poetical struggle, the creation of “el hombre nuevo” (Guevara, 1965). Many studies have indicated ways in which parents might support children in seeing and thinking more mathematically, yet, the practices being advanced might be more readily seen in middle-class families: taking advantage of opportunities to practice spatial thinking (Joh, Jaswal, & Keen, 2011; Newcombe, 2010; Pruden, Levine, & Huttenlocher, 2011); playing construction games that challenge children to recreate a design from a sample or design (Ferrara, Golinkoff, Hirsh-Pasek, Lam, & Newcombe, 2011), encouraging children to gesture when they think about spatial problems (Cook & Goldin-Meadow, 2006; Goldin-Meadow, Nusbaum, Kelly, & Wagner, 2001), playing with tangrams and jigsaw puzzles (Levine, Ratliff, Huttenlocher, & Cannon, 2012), creating and explaining maps (Kastens & Liben, 2007), practicing mental rotation skills including through computer games (Terlecki, Newcombe, & Little, 2008; Wright, Thompson, Ganis,

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Newcombe, & Kosslyn, 2008). Studies of early cognition do suggest potentially useful strategies which might benefit learners of mathematics from more disadvantaged backgrounds. Whilst much work on the links between disadvantage and achievement look to generic social structures, a look toward studies of cognitive development point toward more specific aspects of how that process is operationalized. As a consequence, there is sufficient evidence to consider a greater examination of the mode of communication and representation as playing a significant role. There is another element to this. It is also well-known that young people from disadvantaged backgrounds find it harder to succeed at school mathematics than those young people who have experienced relative economic privilege. Schools don’t make this any easier for them by placing all such pupils together in the same mathematics groups and restricting their curriculum and linguistic opportunities, but also restricting the development of alternative forms of representation. Research has consistently shown that young people specifically from low SES backgrounds do less well (Noble, Norman, & Farah, 2005), and on spatial tasks the SES difference is confounded with gender (Levine, Vasilyeva, Lourenco, Newcombe, & Huttenlocker, 2005). This may be due to their experiences as young children, the toys they have (or don’t have) the use they make of maps, and so on. Hence, there is a need to explore the use of visualization in teaching and learning mathematics and how teachers and pupils can be supported to develop imagery and mental manipulation as a natural part of mathematics—which after all gets increasingly abstract the further you go. This forces us to ask, what use is made of visualization in teaching mathematics and do groups at different levels get the same experience, particularly those in lower-attaining groups populated by pupils with low SES backgrounds? Low teacher expectations can influence the methods that teachers use and can limit the access of these groups to higher order skills. Children from low SES backgrounds may well have an impoverished mathematical experience before school and their progress may be restricted further if teaching methods do not allow them access to the appropriate opportunities for development. If visualization is potentially beneficial to mathematical development, then how and when is this taught in schools, but more importantly, how might it reduce the SES gap in achievement? In conclusion, there is a need to examine central issues in mathematics teaching which are too often kept apart by looking at the experiences of pupils from low SES or less affluent backgrounds. We probably can’t do much about improving the social and economic backgrounds of our pupils; we might however be able to do something about enhancing some of the key skills which they have not previously been required to focus on—visual acuity, visual reasoning, and mental representations.

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How does class work in mathematics classrooms? In a process that sociologists call “symbolic violence” —that is not actual violence—pupils, parents, and schools participate in a stages process of recognition, misrecognition, and exclusion; it becomes such a natural process it is not even noticed, which is why so many teachers support and encourage setting. Being integrated into the system it becomes too complex and onerous to overcome it. So, what can we do? Here I will offer just four suggestions; small but significant steps that might diminish the effect of structural inequalities in schools and which are derived from 60 years of research in schools (Gates, 2012): 1. Engender positive, respectful social and pedagogic relationships with low SES pupils, to explicitly foster self-esteem and resilience in working with mathematics. The system of organizing pupils along some imagined construct of “ability” or “attainment” damages pupils, and robs them of their self-respect, self-esteem and their potential engagement in mathematics. However, this is not always something that is desired by the individual teachers who have little power over school or national policies of exclusion. However, we can do something subversive by countering very explicitly the effects through our own personal relationships with young people in classrooms. 2. Treat low SES students to the same high expectations, with a demanding and rigorous mathematics curriculum that expects all pupils to succeed and understand. Curriculum, pedagogy, and organization each combine to create a very diluted and shallow experience for many young people with a paucity of opportunities to experience deep mathematical ideas. The self-fulfilling prophecy forces teachers to believe these pupils are not capable of sophisticated thinking. This creates a feeling of “how could it be other?” Well it will only be “other” by making it so; expect more, challenge more, and demand more. 3. Recognize and embrace the diversity in the student body, valuing the talents and abilities of low SES learners, encompassing a respect for different life worlds and their contributions to mathematics. Get to know the families and provide differentiated support. By playing their part in the sociology of the school, teachers develop a concept of the “ideal pupil” (Becker, 1952; HempelJorgensen, 2009) as a stereotype of a young person designed to perfectly fit the school system. Usually not too far removed from that of the teachers themselves. However, few pupils fit that mold and diversity is often emotionally felt as deviance by teachers. It then becomes no wonder these young people reject our life-world, we reject theirs. Mathematics is really not a White, male middle-

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class set of rules and theorems. Parents and families need to be embraced rather than excluded from school and educational policies by recognizing the “funds of knowledge” they hold (Moll, 1992). 4. Create and use meaningful tasks involving inquiry and cooperative learning, where low SES learners have some control and responsibility. Mathematics to most working-class pupils is about nothing at all; it is a purposeless sequence of mystifying techniques: expand the brackets, turn upside down and multiply, and so many more. No one enthusiastically engages in purposeless activity for long. Indeed, when incarcerated, purposeless activity became a punishment. One mathematics lesson after another! Furthermore, schools do their utmost to remove any responsibility and control by pupils, even often (at least in the United Kingdom) determining what color socks they wear. Giving young people a purpose, and some responsibility is surely a major step in drawing them into the world of mathematics (for examples of how this might be achieved see Gutstein, 2006). Yet there is political work to do amongst teachers themselves if we are to seek a resolution to the current damaging segregation in our system: Teachers need to be tuned in to the culture of poverty and be sensitive to the vast array of needs that children of poverty bring to the classroom. Social contexts have a significant impact on the development of children. The social world of school operates by different rules or norms than the social world these children live in. Focus should be placed on finding a harmonious relationship between the cultural values of students and values emphasized in school. (Pellino, 2007, p. 5)

Too often we overlook this elephant in our classrooms and try to pretend we are all in it together and striving for the same things. I have argued how maths teaching is a political act and how teacher beliefs are themselves political (Gates, 2001, 2006, 2010). So, what you teach and how you teach it are intimately linked to your own political beliefs. I am encouraged by the argument of Wilkinson and Picket in The Spirit Level (Wilkinson & Pickett, 2009) that inequality is bad for all of us. Treating people unfairly will reduce social cohesion, leading to social unrest and conflict. A challenge for all of us is to fight the demons that cause us to expect little from learners from less affluent backgrounds and, more specifically, to recognize the influence that poverty has on all aspect of teaching and learning mathematics. Engaging explicitly with class and social differences in learning has been shown to have the potential to open up greater opportunities for higher order thinking (Jorgensen et al., 2011), and for raising the intellectual quality of pupil cognition (Kitchen et al., 2007). Class, in some guise or another, is always a latent variable whose invisibility obscures

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possibilities for action. However, this remains not merely an epistemic or empirical question, but a political and an ideological one and your response to this chapter, will be similarly political. NOTE 1. This section is adapted from Gates, 2018.

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60 P. GATES Cooney, J., & Swanson, H. (1987). Memory and learning disabilities: An overview. In H. Swanson (Ed.), Memory and learning disabilities: Advances in learning and behavioral disabilities (pp. 1–40). Greenwich, CT: JAI. Cooper, N., & Dumpleton, S. (2013). Walking the breadline. The scandal of food poverty in 21st century britain. Retrieved from https://policy-practice.oxfam.org.uk/ publications/walking-the-breadline-the-scandal-of-food-poverty-in-21st-century-britain-292978 Darch, C., Carnine, D., & Gersten, R. (1984). Explicit instruction in mathematics problem solving. Journal of Educational Research, 77(6), 351–359. Dehaene, S. (2011). The number sense: How the mind creates mathematics. Oxford, England: Oxford University Press. Dickinson, P., Eade, F., Gough, S., & Hough, S. (2010). Using realistic mathematics education with low to middle attaining pupils in secondary schools. In M. Joubert & P. Andrews (Eds.), Proceedings of the Britishcongressfor mathematics education. Manchester: MMU Institute of Education. Retrieved from http:// www.bsrlm.org.uk/IPs/ip30-1/BSRLM-IP-30-1-10.pdf Duncan, G., Dowsett, C., Claessens, A., Magnuson, K., Huston, A., Klebanov, P., . . . Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–1446. https://doi.org/10.1037/0012-1649.43.6.1428 Ferrara, K., Golinkoff, R., Hirsh-Pasek, K., Lam, W., & Newcombe, N. (2011). Block talk: Spatial language during block play. Mind, Brain, and Education, 5(3), 143–151. Francis, B., Archer, L., Hodgen, J., Pepper, D., Taylor, B., & Travers, M. (2016). Exploring the relative lack of impact of research on ‘ability grouping’ in England: A discourse analytic account. Cambridge Journal of Education, 47(1), 1–17. https://doi.org/10.1080/0305764X.2015.1093095 Frankenstein, M. (1983). Critical mathematics education: An application of Paulo Freire’s epistemology. Journal of Education, 165(4), 315–339. Garderen, D. V., Scheuermann, A., & Poch, A. (2014). Challenges students identified with a learning disability and as high-achieving experience when using diagrams as a visualization tool to solve mathematics word problems. ZDM Mathematics Education, 46(1), 135–149. https://doi.org/10.1007/ s11858-013-0519-1 Gates, P. (2001, July). Mathematics teacher belief systems: Exploring the social foundations. Paper presented at the 25th International Conference of Psychology of Mathematics Education, Utrecht, The Netherlands. Gates, P. (2006). Going beyond belief systems: Exploring a model for the social influence on mathematics teacher beliefs. Educational Studies in Mathematics, 63(5), 347–369. Gates, P. (2010). Beyond belief: Understanding the mathematics teacher at work: Why beliefs are not enough to understand the mathematics teacher. A sociological study of mathematics teaching. Saarbrücken, Germany: Lambert Academic. Gates, P. (2012, July). Why mathematics educators should be bothered about poverty. Paper presented at the PME36, Taiwan. Gates, P. (2015). Social class and the visual in mathematics. In S. Mukhopadhyay & B. Greer (Eds.), Proceeding of the 8th International Mathematics Education and Society Conference (Vol. 2, pp. 369–378). Portland, OR: Portland State University.

Why the (Social) Class You Are in Still Counts 61 Gates, P. (2018). The importance of diagrams, graphics and other visual representations in STEM teaching. In R. Jorgensen & K. Larkin (Eds.), STEM Education in the Junior Secondary: The State of Play. Singapore: Springer. Geary, D. (1994). Children’s mathematics development: Research and practical applications. Washington, DC: American Psychological Association. Geary, D. (2006). Development of mathematical understanding. In D. Kuhn & R. Siegler (Eds.), Handbook of child psychology: Volume 2: Cognition, perception, and language (6th ed.; pp. 777–810). Hoboken, NJ: Wiley. Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009–4060) Washington, DC: National Center for Education Evaluation and Regional Assistance. Ginsburg, H., & Russell, R. (1981). Social class and racial influences in early mathematical thinking. Monographs of the Society for Research in Child Development, 46(6), 1–69. Goldin-Meadow, S., Nusbaum, H., Kelly, S., & Wagner, S. (2001). Explaining math. Gesturing lightens the load. Psychological Science, 112(6), 516–522. Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 25–49). Cambridge, MA: MIT Press. Guevara, C. (1965, March 12). El socialism y el hombre en Cuba. [Socialism and man in Cuba]. Havana, Cuba: Editora Política. Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York, NY: Routledge. Haycock, K. (2001). Closing the achievement gap. Educational Leadership, 58(6), 6–11. Hecht, S., Vagi, K., & Torgesen, J. (2007). Fraction skills and proportional reasoning. In D. Berch & M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 121–132). Baltimore, MD: Paul H. Brookes. Hempel-Jorgensen, A. (2009). The construction of the ‘ideal pupil’ and pupils’ perceptions of ‘misbehaviour’ and discipline: Contrasting experiences from a low-socio-economic and a high-socio-economic primary school. British Journal of Sociology of Education, 30(4), 435–448. https://doi.org/10 .1080/01425690902954612 Hoadley, U. (2007). The reproduction of social class inequalities through mathematics pedagogies in South African primary schools. Journal of Curriculum Studies, 39(6), 679–706. https://doi.org/10.1080/00220270701261169 Hodgen, J., Brown, M., Küchemann, D., & Coe, R. (2010). Mathematical attainment of English secondary school students: A 30-year comparison. Paper presented at the British Educational Research Association (BERA) Annual Conference, University of Warwick. Holliday, W., Brunner, L., & Donais, E. (1977). Differential cognitive and affective responses to flow diagrams in science. Journal of Research in Science Teaching, 14(2), 129–138.

62 P. GATES Holloway, S., Rambaud, M., Fuller, B., & Eggers-Pkirola, C. (1995). What is “appropriate practice” at home and in child care?: Low-income mothers’ views on preparing their children for school. Early Childhood Research Quarterly, 10(4), 451–473. Jerrim, J., & Macmillan, L. (2015). Income inequality, intergenerational mobility, and the Great Gatsby Curve: Is education the key? Social Forces, 94(2), 505– 533. https://doi.org/10.1093/sf/sov075 Joh, A., Jaswal, V., & Keen, R. (2011). Imagining a way out of the gravity bias: Preschoolers can visualize the solution to a spatial problem. Child Development, 82(3), 744–750. Jordan, N., Huttenlocher, J., & Levine, S. (1992). Differential calculation abilities in young children from middle-and low-income families. Developmental Psychology, 28(4), 644–653. Jordan, N., Kaplan, D., Olah, L., & Locuniak, M. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77(1), 153–175. Jordan, N., Levine, S., & Huttenlocher, J. (1994). Development of calculation abilities in middle- and low-income children after formal instruction in school. Journal of Applied Developmental Psychology, 15(2), 223–240. Jorgensen, R., Gates, P., & Roper, V. (2014). Structural exclusion through school mathematics: Using Bourdieu to understand mathematics a social practice. Educational Studies in Mathematics, 87(2), 221–239. https://doi.org/10.1007/ s10649-013-9468-4 Jorgensen, R., Sullivan, P., Grootenboer, P., Neische, R., Lerman, S., & Boaler, J. (2011). Maths in the Kimberley. Reforming mathematics education in remote indigenous communities. Brisbane, Australia: Griffith University. Kastens, K., & Liben, L. (2007). Eliciting self-explanations improves children’s performance on a field-based map skills task. Cognition and Instruction, 25(1), 45–74. Kitchen, R. (2003). Getting real about mathematics education in reform in high poverty communities. For the Learning of Mathematics, 23(3), 16–22. Kitchen, R., DePree, J., Celedón-Pattichis, S., & Brinkerhoff, J. (2007). Mathematics education at highly effective schools that serve the poor: Strategies for change. Mahwah, NJ: Erlbaum. Kohn, M. (1983). On the transmission of values in the family: A preliminary foundation. Research in the Sociology of Education and Socialisation, 4(1), 1–12. Lareau, A. (2003). Unequal childhoods. Class race and family Life. Berkeley: University of California Press. Levie, W., & Lentz, R. (1982). Effects of text illustrations: A review of research. Educational Communication and Technology Journal, 30(4), 195–232. Levine, S., Ratliff, K., Huttenlocher, J., & Cannon, J. (2012). Early puzzle play: A predictor of preschoolers’ spatial transformation skill. Developmental Psychology, 48(2), 530–542. https://doi.org/10.1037/a0025913 Levine, S., Vasilyeva, M., Lourenco, S., Newcombe, N., & Huttenlocker, J. (2005). Socioeconomic status modifies the sex difference in spatial skill. Psychological Science, 16(11), 841–845. Lubienski, S. (2000a). A clash of cultures? Students’ experiences in a discussionintensive seventh grade mathematics classroom. Elementary School Journal, 100(4), 377–403.

Why the (Social) Class You Are in Still Counts 63 Lubienski, S. (2000b). Problem solving as a means towards mathematics for all: An exploratory look through a class lens. Journal for Research in Mathematics Education, 31(4), 454–482. Lubienski, S. (2007). Research, reform and equity in US mathematics education. In N. Nasir & P. Cobb (Eds.), Improving access to education. Diversity and equity in the classroom (pp. 10–23). New York, NY: Teachers College Press. Marx, K. (1975). Economic and philosophic manuscripts of 1844. In Karl Marx and Frederick Engels: Collected works, Volume 3 (pp. 229–346). London, England: Lawrence and Wishart. (Originally published in 1844) Mastropieri, M., & Scruggs, T. (1989). Constructing more meaningful relationships: Mnemonic instruction for special populations. Educational Psychology Review, 1(2), 83–111. Mayer, R. (1997). Multimedia learning: Are we asking the right questions? Educational Psychologist, 32(1), 1–19. Moll, L. (1992). Funds of knowledge for teaching: Using a qualitative approach to connect homes and school. Theory Into Practice, 31(1), 132–141. Morrissons. (2012). Poorest budgets eaten up by food. Retrieved from https:// w w w. m o r r i s o n s - c o r p o r a t e . c o m / m e d i a - c e n t r e / c o r p o r a t e - n e w s / poorest-budgets-eaten-up-by-food/ Moyer, J., Sowder, L., Threadgill-Sowder, J., & Moyer, M. (1984). Story problem formats: Drawn versus verbal versus telegraphic. Journal for Research in Mathematics Education, 15(5), 342–351 Newcombe, N. (2010). Picture this: Increasing math and science learning by improving spatial thinking. American Educator, 34(2), 29–43. Noble, K. G., Norman, F., & Farah, M. J. (2005). Neurocognitive correlates of socioeconomic status in kindergarten children. Developmental Science, 8(1), 74–87. Oakes, J. (1985). Keeping track: How schools structure inequality. New Haven, CT: Yale University Press. Oakes, J. (1990). Opportunities, achievement and choice: Women and minority students in science and mathematics. In C. Cazden (Ed.), Review of Research in Education, 16(1), 153–222. Pais, A. (2012). A critical approach to equity in mathematics education. In O. Skovsmose & B. Greer (Eds.), Opening the cage: Critique and politics of mathematics education (pp. 49–91). Rotterdam, The Netherlands: Sense. Pajares, F. (1996). Self-efficacy beliefs in academic settings. Review of Educational Research, 66(4), 543–578. Pellino, K. (2007). The effects of poverty on teaching and learning. Retrieved from http://www.teach-nology.com/tutorials/teaching/poverty/ Pruden, S., Levine, S., & Huttenlocher, J. (2011). Children’s spatial thinking: Does talk about the spatial world matter? Developmental Science, 14(6), 1417–1430. https://doi.org/10.1111/j.1467-7687.2011.01088.x Ramani, G., & Siegler, R. (2008). Promoting broad and stable improvements in low-income children’s numerical knowledge through playing number board games. Child Development, 79(2), 375–394. Rubenstein, R., & Thompson, A. (2013). Reading visual representations. Mathematics Teaching in the Middle Years, 17(9), 544–550. Rusted, J., & Coltheart, M. (1979). Facilitation of children’s prose recall by the presence of pictures. Memory and Cognition, 7(5), 354–359.

64 P. GATES Schnotz, W., Picard, E., & Hron, A. (1993). How do successful and unsuccessful learners use texts and graphics? Learning and Instruction, 3(3), 181–199. Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games-but not circular ones-improves low-income preschoolers’ numerical understanding. Journal of Educational Psychology, 101(3), 545–560. Groening, M., Brooks, J. L., Simon, S., Swartzwelder, J., & Greaney, D. (Writers), & Anderson, M. (Director). (1996). You only move twice [Television series episode]. In James L. Brooks (Executive producer), The Simpsons. New York, NY: Fox Broadcasting. Retrieved from https://www.youtube.com/ watch?v=XQcBI5SKJgU Skovsmose, O. (in press). Three narratives about mathematics education. For the Learning of Mathematics. Starkey, P., Klein, A., & Wakeley, A. (2004). Enhancing young children’s mathematical knowledge through a pre-kindergarten mathematics intervention. Early Childhood Research Quarterly, 19(1), 99–120. Stevenson, H., & Newman, R. (1986). Long-term prediction of achievement and attitudes in mathematics and reading. Child Development, 57(3), 646–659. Stipek, D., & Ryan, R. (1997). Economically disadvantaged preschoolers: Ready to learn but further to go. Developmental Psychology, 33(4), 711–723. Terlecki, M., Newcombe, N., & Little, M. (2008). Durable and generalized effects of spatial experience on mental rotation: Gender differences in growth patterns. Applied Cognitive Psychology, 22(7), 996–1013. Tudge, J., & Doucet, F. (2004). Early mathematical experiences: Observing young Black and White children’s everyday activities. Early Childhood Research Quarterly, 19(1), 21–39. Whiteley, W. (2000, July–August). Dynamic geometry programs and the practice of geometry. Paper presented at the ICME 9, Tokyo, Japan. Retrieved from http:// www.math.yorku.ca/Who/Faculty/Whiteley/Dynamic.pdf Whitty, G. (2002). Structuring the Postmodern in education policy. In D. Hill, P. McLaren, M. Cole, & G. Rikowski (Eds.), Marxism against Postmodernism in educational theory (pp. 67–87). Lanham, Maryland: Lexington Books. Whitty, G., & Anders, J. (2014). (How) did New Labour narrow the achievement and participation gap? London, England: Centre for Learning and Life Chances in Knowledge Economies and Societies. Retrieved from https://www.llakes. ac.uk/sites/default/files/46.%20Whitty%20and%20Anders%20-%20final.pdf Wilkinson, R., & Pickett, K. (2009). The spirit level. Why more equal societies almost always do better. London, England: Allen Lane. Winn, W. (1987). Charts, graphs and diagrams in educational materials. In D. Willows & H. Houghton (Eds.), The psychology of illustration (pp. 152–193). New York, NY: Springer-Verlag. Wright, R., Thompson, W., Ganis, G., Newcombe, N., & Kosslyn, S. (2008). Training generalized spatial skills. Psychonomic Bulletin & Review, 15(4), 763–771. https://doi.org/10.3758/PBR.15.4.763 Zevenbergen, R. (2000). “Cracking the code” of mathematics classrooms: School success as a function of linguistic, social and cultural background. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 201–223). Westport, CT: Ablex.

CHAPTER 4

BEYOND THE BINARY AND AT THE INTERSECTIONS Chronicling Contemporary Developments of Gender Equity Research in Mathematics Education Luis A. Leyva

TWO CATEGORIES OF GENDER EQUITY RESEARCH IN MATHEMATICS EDUCATION Problematic conflation of the terms gender and sex in mathematics education research has been documented (Damarin & Erchick, 2010). This includes use of the term gender to describe differences in mathematics achievement and participation according to students’ biological sex (namely, being female or male). Such interchangeable use of binary gender terms and theorizations of gender do not take into consideration intersex and gender-nonconforming individuals, thus reifying the idea that there are two distinct biological groups of people (Rubel, 2016). In this chapter, gender is conceptualized as a social construct varyingly produced across individuals and subject to change in different contexts (Butler, 2004; Wilchins, 2004).

Equity in Mathematics Education, pages 65–91 Copyright © 2019 by Information Age Publishing All rights of reproduction in any form reserved.

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Elsewhere (Leyva, 2017), I completed a review of 45 years of peer-reviewed publications on empirical research studies about gender in mathematics education. Publications in this review met search and selection criteria (e.g., journal ranking, citation count) designed to filter research that made notable contributions to the field’s study of gender. I organized reviewed publications using two broad categories based on their approaches to studying gender: achievement and participation (see Leyva, 2017 for additional details on the review methodology). Achievement studies compared females’ and males’ outcomes on mathematics assessments and their relationships with influences on opportunities to learn or succeed, either internal (e.g., problem-solving strategy use) or external (e.g., teacher expectations). Participation research examined patterns of student persistence, classroom interactions and engagement, and students’ negotiations of identity with educational experiences in mathematics. The two categories of gender research arguably address dimensions that constitute Gutiérrez’s (2009) two axes of equity in mathematics education: the dominant axis (access and achievement) and the critical axis (identity and power). Test score comparisons and connections to gendered opportunity differences in achievement research map onto the achievement and access dimensions of the dominant axis, whereas analytical attention to the gendering of norms in mathematics and its impact on identity corresponds to the power and identity dimensions of the critical axis. The two categories of publications, therefore, provide an organizational structure for reviewing conceptual and methodological developments of gender equity research in mathematics education. To contextualize progress in gender research, I first share insights from my review of representative studies for each category published before the year 2000. This is followed by a review of more recent studies representative of each category to capture current developments in gender equity research from each analytical perspective. ACHIEVEMENT RESEARCH In this section, I present two strands of achievement studies. One strand quantitatively examined achievement differences between sexes in relation to internal influences, including cognitive processes (e.g., spatial reasoning, problem solving strategies) and affective factors (e.g., mathematics attitudes and confidence). The second strand used various assessment instruments, such as standardized tests as well as international and national mathematics assessments, to complete quantitative analyses of female–male differences in achievement related to external influences (e.g., classroom instruction, teacher perceptions, sociocultural factors). Participants’ sex or “gender” across both strands was treated as

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a binary variable in statistical models to document “sex(-related) differences” (e.g., Fennema & Sherman, 1977, 1978) or “gender differences” (e.g., Birenbaum & Nasser, 2006). Internal Influences Although research examining achievement differences between sexes in relation to spatial reasoning and strategy use has historically shown negligible sex differences among students in lower grades, increasingly widening disparities in mathematics achievement and interest in upper grade levels have been documented (Bielinski & Davison, 1998; Fennema & Sherman, 1977, 1978). These findings challenged the myth of male superiority in mathematics assessments that largely motivated the detailing of sex-based differences across achievement studies (Fennema, 1979). Fennema, Carpenter, Jacobs, Franke, and Levi’s (1998) longitudinal study of “gender differences” in problem solving strategy use was a turning point in gender equity research through considerations of mathematics task performance and learning. Working with a predominantly White sample of 132 first-grade students across three schools, Fennema et al. (1998) used a problem-solving assessment aligned with participants’ school mathematics curricula as well as cognitive interviews to examine students’ strategy use. The researchers found that males used abstract strategies more frequently than females, which led to their greater success with more complex problem-solving tasks. Extending prior achievement studies’ findings, the Fennema et al. (1998) study showed that “gender differences” in more advanced mathematics achievement can be explained by distinct problem solving approaches between sexes. Other researchers (Boaler, 2002c; Hyde & Jaffee, 1998; Sowder, 1998), however, critiqued this claim of strategy use differences according to sex since the Fennema et al. (1998) study lacked insight into participants’ mathematics learning environments (e.g., classrooms). The use of observations avoids “position[ing] gender as a characteristic of groups of people rather than as a situated response” (Boaler, 2002c, p. 139). Despite minimal insight on how social contexts shaped “gender differences” in mathematics problem solving, the Fennema et al. (1998) study raised initial considerations of gendered performances in doing mathematics, particularly through strategy use. Contemporary Research Developments More recently, achievement research on affective factors problematized earlier studies’ explanations of sex differences that produce deficit perspectives about females as innately less mathematically able than males. Carr,

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Steiner, Kyser, and Biddlecomb (2008), for example, quantitatively analyzed five cognitive and affective factors (accuracy, confidence, fluency, spatial ability, and strategy use) as mathematics achievement predictors among 227 racially diverse second-grade students across seven schools. Their findings challenged two previously proposed explanations for males’ stronger performance on more difficult mathematics test items than females, and vice versa for easier items: (a) more variability of males’ mathematical ability and (b) a shift in assessed abilities from easier to harder test items with the latter related to mathematical reasoning and spatial visualization (Bielinski & Davison, 1998). Carr and colleagues (2008) found minimal evidence to support “gender differences in predictor values or competency [or achievement] are due to boys’ higher variability” (p. 73). Further, sex differences in spatial ability were observed, but there was no statistically significant relationship between achievement and spatial ability. Carr and colleagues (2008) noted the possibility that spatial ability is central and, thus, significantly predictive of achievement in more advanced problem solving beyond second grade. Achievement studies’ lack of consistent findings of males outperforming females in mathematics, both across grade levels and with particular content topics, raised considerations for how schools, classrooms, and other external factors mediated gendered achievement trends (Fennema & Sherman, 1978). Fennema (2000), for example, recently acknowledged limitations of studies on “gender differences” that often overlooked variation in individual learning experiences, including the gendered socialization of mathematics classrooms. Researchers, thus, turned to exploring gendered influences of educational contexts (e.g., instruction, teacher perceptions) on achievement (e.g., Davis & Carr, 2001; Robinson & Lubienski, 2011). Furthermore, lack of consideration for variation at intersections of sex and race or ethnicity, class, and other social identities across achievement studies examining internal influences, left implicit the extent to which their findings generalized (Leyva, 2017). External Influences This section illustrates how gender equity research in the achievement strand exploring external influences addressed limitations of the other strand in two ways: (a) supplementing quantitative analyses of sex differences with student interviews and teacher questionnaires and (b) exploring intersectional variation in gendered patterns of mathematics achievement related to race and class. Despite continued use of binary gender comparisons, these studies shed light on social influences on achievement related to students’ mathematical problem solving and learning. Findings from these

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studies on external influences, thus, contextualized “gender differences” in achievement rather than attributing them to innate ability based on sex. Instruction and Psychosocial Influences Early studies exploring relationships between achievement, instruction, and psychosocial factors used self-report data sources (e.g., interviews, questionnaires) to capture variation in teacher and family influences (Battista, 1990; Carr, Jessup, & Fuller, 1999; Ethington, 1992). Emphasis on instruction of certain mathematical skills (e.g., spatial visualization in Battista, 1990) and abstract problem-solving strategies (Carr et al., 1999) was found to be associated with gendered trends in course grades and task performance, respectively. Further, this strand of research found gendered distinctions in the influence on achievement related to parental encouragement (Ethington, 1992) as well as parents’ and teachers’ perceptions of strategy use indicative of mathematical ability (Carr et al., 1999). Analytical insights about contextual influences on achievement across these studies further problematized the deficit positioning of females through the myth of male superiority in mathematics assessments. Contemporary Research Developments More recent achievement research continued detailing school and family influences on gendered trends in mathematics assessment outcomes and task performance, either attending to variation among girls and among boys (Davis & Carr, 2001) or keeping prior studies’ binary gender comparisons (Ai, 2002; Lubienski, Robinson, Crane, & Ganley, 2013; Penner & Paret, 2008; Robinson & Lubienski, 2011). Davis and Carr (2001) documented variation in strategy use among 84 racially diverse, suburban first-grade students as related to five classroom teachers’ perceptions of their temperament. Findings showed that teachers’ interpretations of student behavior as impulsive or inhibited was associated with differential use of mathematics strategies among boys and girls. The researchers also argued how teacher perceptions of student temperament can produce gendered learning opportunities for strategy development through mathematics instruction. For instance, Davis and Carr (2001) discussed how teacher perceptions of boys’ impulsivity or “difficult” temperament, indicative of low mathematics achievement, may result in instruction of lower cognitive demand and a focus on concrete strategy use as reflected in their problem solving behaviors. The researchers also similarly proposed that girls with lower impulsivity or a “slow to warm up” temperament may have limited access to teacher instruction that encourages development of more advanced strategies of retrieval. Davis and Carr’s (2001) study, thus, is a representative analysis of more recent gender research that not only corroborated earlier findings about gendered socialization influences on mathematics performance, but

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also extended these findings by detailing variation among girls and boys that was left largely implicit in prior studies. Recent longitudinal analyses of “gender differences” and “gender gaps” in students’ mathematics scores on national assessments also continued prior studies of female–male achievement comparisons, but with considerations for school and teacher influences across grade levels (Ai, 2002; Lubienski et al., 2013; Penner & Paret, 2008; Robinson & Lubienski, 2011). For Grades 7–10, Ai (2002) noted differences across schools with varying resources in girls’ average growth rate of achievement as well as achievement effects related to teacher encouragement among boys. These recent longitudinal studies also captured “subtle gendering processes” (Lubienski et al., 2013, p. 644) of teacher encouragement that varyingly impacted students’ mathematics attitudes and achievement. Robinson and Lubienski (2011), for instance, observed that middle school teachers rated girls more favorably than boys for their mathematical proficiency despite boys’ higher achievement scores. Mathematics attitudes had a stronger influence on achievement for boys than girls and was unrelated to any form of encouragement, whereas parent and teacher encouragement shaped the mathematics attitude effect on achievement for girls. With achievement influences in school and classroom contexts documented in this strand of studies, researchers argued for the methodological promise of observations in future research (Carr et al., 1999; Lubienski et al., 2013). Observations would not only provide situated insights into how these predictive factors of achievement operate as gendered mechanisms, but also illuminate variation in their impact across contexts and students. Sociocultural Influences Achievement studies exploring external influences also considered sociocultural factors besides gender, thus nuancing claims about female– male achievement differences through noted variation at intersections of gender, racial, ethnic, and/or class identities (Birenbaum & Nasser, 2006; Brandon, Newton, & Hammon, 1987; Hanna, 1989; Lubienski et al., 2013; Penner & Paret, 2008; Robinson & Lubienski, 2011). In a cross-cultural analysis of 70,000 eighth-grade students’ mathematics assessment score differences by sex across 20 countries, Hanna (1989) found that factoring country and country-by-sex variables in a two-way multivariate analysis resulted in statistically significant differences. Intersectional analyses involving culture and ethnicity were also adopted in more recent achievement studies. For instance, Birenbaum and Nasser’s (2006) study of “ethnic and gender differences” among Arab and Jewish students in their mathematics dispositions and test performance documented an “achievement-enhancing pattern” (Birenbaum & Nasser, 2006, p. 36) among Arab girls. This pattern, related to strong mathematics attitudes and self-confidence, was

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reflected in Arab girls’ omission of less test items, reports of receiving less help from others, and beliefs of parents’ higher academic expectations of them compared to Arab boys. Birenbaum and Nasser (2006) interpreted this finding of Arab girls’ strong achievement as a reflection of gendered cultural expectations—namely, Arab girls experiencing more pressure than Arab boys to academically succeed in Israel. Early and recent intersectional analyses, therefore, illustrate how culture and ethnicity played a role in making sense of variation in gendered trends of achievement across different national contexts. Contemporary Research Developments Other recent intersectional analyses of achievement based on mathematics assessment outcomes from U.S. national datasets (e.g., Early Childhood Longitudinal Study—Kindergarten (ECLS-K), National Assessment of Educational Progress (NAEP)) detailed variation in relation to gender, race, and class (Lubienski et al., 2013; McGraw, Lubienski, & Strutchens, 2006; Penner & Paret, 2008). McGraw and colleagues’ (2006) study of NAEP mathematics achievement and affect data in Grades 4, 8, and 12 revealed statistically significant “gender gaps” favoring males only among White and Hispanic students in the upper grade levels. An opposite and significant gap with Black females outperforming Black males in geometry and data analysis was also observed, in contrast to the Lubienski et al.’s (2013) study that documented a “gender gap” favoring Black males in Grades 3, 5, and 8. Similarly, Penner and Paret (2008) found that the “gender difference” in the top distribution of test scores, particularly males outperforming females, was more pronounced among Asians and students whose parents held a college or advanced degree. The nature of the “gender difference” was not the same for Hispanic students, with Hispanic females having higher scores than Hispanic males. Although Lubienski and colleagues (2013) did not find that home experience to be a statistically significant predictor of “gender differences,” they noted that students experienced more stereotypically gendered upbringing in upper-class families. These intersectional differences in mathematics achievement, thus, problematize the notion of male superiority in mathematics assessments as well as illustrate how multiple systems of oppression shape distinct trends of access and achievement. Despite binary conceptualizations of gender and minimal qualitative insights into students’ experiences, intersectional analyses in achievement research illuminated variation that would have been overlooked by solely attending to gender. Detailing intersectional differences in achievement without losing analytical sight of the interplay between classism, racism, and sexism that produce and perpetuate them is crucial:

72 L. A. LEYVA Educational inequalities have a tendency to both reflect and reinforce structures of inequality, whether they are related to gender, race, or class divisions. In the case of math achievement these three axes of difference interact to produce gender inequalities among boys and girls. . .It is important to focus on the ways in which processes of academic achievement are embedded in and shaped by hierarchical structures of difference and inequality. (Penner & Paret, 2008, p. 251)

This statement captures the analytical promise of intersectionality (Crenshaw, 1991), a theoretical perspective that addresses the unique forms of marginalization and empowerment which individuals experience at the intersections of racism, sexism, classism, and other systems of oppression. To illustrate, Penner and Paret’s (2008) implicit considerations of intersectionality are evident in their argument of class and race shaping varyingly gendered stereotypes that students must manage, as well as differential access to family support that enhances mathematics achievement. Such intersectional considerations can nuance prior findings about school, classroom, and parental influences on mathematics achievement differences according to sex and, thus, further inform institutional change for more equity in mathematics education. Summary Overall, findings from earlier research studies in the achievement category were interpreted according to sex or a binary conceptualization of gender that limited researchers to reporting between-group differences (female/male, women/men) in mathematics outcomes and task performance (Leyva, 2017). Quantitative analyses of large samples may have constrained researchers from exploring gender as a binary variable in their statistical models, but this left unexplored the varying ways that individuals’ gender was socially constructed and expressed across contexts of mathematics education. More recent achievement studies adopted methodologies that accounted for the role of contextual influences in shaping gendered disparities of mathematics achievement, such as classroom instruction, teacher perceptions, and family encouragement. These achievement studies, for example, supplemented assessment outcomes with other data sources (e.g., cognitive interviews, questionnaires) to explore statistical relationships between achievement and dimensions of students’ educational experiences that, in turn, contextualize gendered inequities in mathematics achievement. Considerations for such external factors related to achievement were largely based on self-report data sources, leaving the achievement category void of situated insights about how these influences play out in

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classrooms and other contexts of mathematics education. Thus, implications from achievement research findings of females’ weaker mathematics performance are unfairly suggestive of how females must change to be on par with higher-achieving males as opposed to challenging the gendered (namely, masculinized) norms and expectations of mathematics success across educational spaces (Boaler, 1997). Achievement studies, thus, point to connections between achievement and masculinized values of engagement in mathematics that require more localized analyses of gender to better inform gender-equitable institutional practices. The need for situated insights on gendered influences in mathematics education informed the units of analysis and methodologies adopted in the participation category of research explored in the next section. Furthermore, only a subset of achievement studies attended to the interplay of students’ gender with other social identities for making sense of variation in mathematics achievement. At the turn of the century, Fennema (2000) described how mathematics education was in need of more intersectional analyses of gender and achievement: “The U.S., as many other countries, is a highly heterogeneous society, made up of many layers, divisions, and cultures. The pattern of female differences in mathematics varies across these layers and must be considered” (Research from 1970–1990 section, para. 13). McGraw and colleagues (2006) similarly acknowledged the importance of exploring intersectional differences to avoid reverting back to the oversimplified discourse of male superiority in mathematics assessments and perpetuating the “‘gap-gazing’ fetish” (Gutiérrez, 2008) in mathematics education. While achievement studies in general pointed to the need for disrupting institutional forms of gender marginalization in mathematics, intersectional analyses in particular captured the promise of intersectionality as an analytical lens to generate nuanced understandings of catalyzing such institutional change. Participation Research This section reviews gender equity research explored from a participation perspective using two subcategories based on studies’ conceptualizations of gender: sex-based and gender-based (Leyva, 2017). Sex-based participation studies interpreted findings with a binary theorization of gender and reported group comparisons of participation. Gender-based participation studies theorized gender in nonbinary ways to document variation in students’ perspectives and educational experiences in mathematics. Below I present two strands of sex-based participation studies: one with analyses void of context and the other considering context.

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Sex-Based Participation Research Analyses Without Contextual Considerations One strand of sex-based participation studies analyzed female–male participation differences in relation to mathematics attitudes and course enrollment, perceptions of mathematical ability, and views of gendered stereotyping in mathematics. Because early studies (e.g., Benbow & Stanley, 1982; Sherman & Fennema, 1977) documented few differences in attitudes contributing to mathematics persistence, researchers began to explore students’ experiences and socialization influences as sources of insight into gendered differences in mathematics participation (Pallas & Alexander, 1983). Subsequent studies, for example, explored: (a) the extent to which access to mathematics coursework impacted achievement differences between female and male students (Armstrong, 1981; Pallas & Alexander, 1983); (b) students’ self-perceptions of mathematical ability and the gendered ways by which they are positioned in classrooms (Bornholt, Goodnow, & Cooney, 1994; Casey, Nuttal, & Pezaris, 2001; Pallas & Alexander, 1983); and (c) relationships between mathematical confidence, attributions of mathematics success, and persistence (Casey et al., 2001; Pedro, Wolleat, Fennema, & Becker, 1981; Seegers & Boekaerts, 1996). Contemporary Research Developments More contemporary sex-based participation research has examined how different socialization factors shaped individuals’ perceptions of mathematics as a gendered domain that could impact participation (Bornholt et al., 1994; Brandell & Staberg, 2008; Forgasz, Leder, & Barkatsas, 1998; Forgasz, Leder, & Kloosterman, 2004). Forgasz and colleagues (2004), for example, analyzed female–male differences in responses from Australian and U.S. students (Grades 7–10) for items in two instruments that measure perceptions of mathematics as a gendered domain. Overall, the two countries’ gendered trends in responses were similar to a stronger pattern of gendered beliefs about mathematics among Australian respondents. A majority of Australian and U.S. respondents, especially U.S. females, perceived mathematics as not being a female domain and gender-neutral. While this perception of gender neutrality does not explicitly perpetuate notions of male superiority in mathematics, it complies with notions of gender-blindness that, analogous to color-blindness as a racist ideology in U.S. race relations (Battey & Leyva, 2016; Bonilla-Silva, 2003), does not disrupt the patriarchy and misogyny that produce gendered inequities of mathematics participation (Leyva, 2017). Forgasz and colleagues (2004) also highlighted noteworthy distinctions between the two countries’ sets of responses, including how Australian respondents departed from U.S. narratives of females innately lacking

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mathematical ability. Australian females and males, for example, reported females being more likely to find mathematics easy and enjoyable. Statistically significant differences between Australian females’ and males’ responses were noted for two items: (a) males being more readily than females to find mathematics interesting and (b) mathematics being more useful to females than males in the future. Similar to interpretations of findings from cross-cultural analyses of gendered achievement trends (e.g., Birenbaum & Nasser, 2006; Hanna, 1989), Forgasz and colleagues’ (2004) findings point to how different cultural constructions of gender and related forms of socialization in the United States and Australia can be used to explain each country’s pattern of female–male differences in item responses. Sex-based participation studies in this strand, thus, illustrate how exploring social and cultural contexts can further explain variation in gendered trends of participation in mathematics. Analyses With Contextual Considerations The second strand of sex-based participation studies examined participation as negotiations of meaning that shape individuals’ identities and practices in mathematics (Boaler & Greeno, 2000). Mathematics classrooms, therefore, operated as figured worlds (Holland, Lachiotte, Skinner, & Cain, 1998), or contexts in which students’ mathematics participation was negotiated with peers’ and teachers’ gendered perceptions and interpretations of classroom events (Esmonde & Langer-Osuna, 2013). Studies examined several mathematics classroom influences on female–male participation differences, including teacher–student interactions (Becker, 1981; Hart, 1989), teacher beliefs of students’ mathematical ability (Fennema, Peterson, Carpenter, & Lubinski, 1990; Tiedemann, 2000, 2002), and instruction (Boaler, 1997; Peterson & Fennema, 1985). Early sex-based participation studies examined how teacher influences, such as classroom interactions and beliefs of students’ mathematical ability, contributed to gendered participation trends. In an analysis of teacher– student interactions across 10 predominantly White geometry classrooms (Grades 9 and 10) in urban-suburban and rural high schools, Becker (1981) documented two gendered classroom patterns: (a) teachers more frequently initiated contact with and extended academic encouragement to male than female students and (b) female students kept extremely quiet and passive whereas male students frequently asked questions and called out answers. Teachers’ differential treatment was interpreted as possibly reflecting teachers’ implicit gender biases aligned with stereotypes of females as less able and interested in mathematics than males. Becker (1981) made sense of classroom behavior differences, according to student sex, as responses to a learning environment that “sex-typed mathematics as male” (p. 50) through teachers’ gendered interaction patterns. Thus, Becker’s

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(1981) analysis illustrates how students potentially made meaning of teachers’ differential treatment in ways that affected their classroom participation—namely, females afforded less opportunities to be acknowledged and actively engaged than male peers. Other early findings in sex-based participation research captured teachers’ inaccurate perceptions of students’ mathematics success and differential attributions for it across sexes (Fennema et al., 1990), motivating researchers to further examine the gendered impact of teacher beliefs on mathematics participation. Tiedemann’s (2002) more recent study, for instance, explored teachers’ gendered biases and their potential impact on students’ sense of their mathematical ability and effort through a quantitative analysis of teacher questionnaire responses about student perceptions. The study involved a sample of 288 predominantly White, middleclass students across 48 elementary classrooms in Germany. Tiedemann’s (2002) findings, extending those of prior studies about teachers’ gendered beliefs of mathematical ability, showed how teachers’ inaccurate assessments of student competence and effort was a gendered dynamic mainly observed among low- and average-achieving students. Mathematics achievement, therefore, mediated the influence of teachers’ gendered biases. A statistically significant “gender x stereotype” interaction was also found, as reflected in teachers’ more positive perceptions of mathematical competence and effort for same-sex students than students of the opposite sex. Tiedemann’s (2002) analysis, therefore, nuanced understandings of the influence of teacher beliefs on mathematics participation by identifying conditions, such as prior achievement and teacher–student sex match, that may vary the extent to which beliefs are activated and impact students. Sex-based participation studies reviewed thus far contributed to gender equity research by highlighting how teacher influences operate as gendered mechanisms that limit access to participation opportunities and can be used to explain achievement differences according to sex. What remains largely missing across these analyses are situated insights across mathematics education spaces to capture variation in the manifestation of these contextual influences and how students engage with them in their educational experiences. Contemporary Research Developments Addressing this need for details from students’ everyday contexts as mathematics learners, Boaler (1997, 2002b) completed a 3-year ethnographic study involving interviews and observations that compared female and male students’ engagement with classroom instruction. The study sample consisted of predominantly White, working-class students from two nonselective, comprehensive schools in England (Amber Hill and Phoenix Park) differing in their approaches to teaching mathematics and the

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social class of surrounding areas. Namely, Amber Hill, in an affluent area with greater job accessibility, used traditional mathematics teaching methods that focused on learning procedures. Phoenix Park, in a working-class neighborhood with students’ families living in public housing, adopted reform-oriented and discussion-based instructional practices, including projects. Boaler (1997, 2002b) found that female students experienced less academic struggle and enjoyed mathematics more in Phoenix Park, whereas male students had similarly positive mathematics experiences in Amber Hill. Similar to early achievement research (cf. female–male differences in strategy use, Fennema et al., 1998), Boaler (1997) attributed differences in students’ experiences to alignment (or lack thereof) between schools’ instructional methods and students’ preferred mathematics learning styles according to sex. Boaler (2002b) also detailed female–male differences in students’ strategic ways of negotiating their preferred mathematics learning practices and the two schools’ mathematics teaching practices. In particular, Phoenix Park males “play[ed] the game” (Boaler, 1997, p. 298) through a disregard for building meaning from school mathematics and focus on quickly getting correct answers, despite their school’s value of learning for understanding. Amber Hill females dwelled on their inability to build meanings of the mathematics, lagged behind male peers academically, and felt powerless in changing their school’s mathematics teaching approaches. Despite binary interpretations of gender-related findings and leaving intersectional considerations implicit, Boaler’s (1997, 2002b) analysis advanced gender equity research by adopting qualitative methodologies of interviews and observations to offer situated accounts of how institutional factors, including curricula and instruction, shaped gendered forms of mathematics learning experiences. Furthermore, the detailing of variation among females and males across schools, particularly in negotiating their identities in mathematics with institutional practices of instruction, addressed an underexplored area of gender equity research. Summary Studies in the strand of sex-based participation research with contextual considerations address the other strand’s void of insights about students’ everyday educational experiences in mathematics. More specifically, these studies illustrate mathematics participation as a gendered form of experience through students’ negotiations of masculinized norms for being mathematically successful that are maintained by contextual influences (e.g., classroom interactions, teacher beliefs, instruction). Elsewhere (Leyva, 2017), I argued how these influences contribute to the social construction of gendered hierarchies of mathematical ability in classrooms that position notions of femininity toward the bottom and notions of masculinity

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toward the top (Ernest, 1991). This hierarchy serves as an interpretative lens of how sociomathematical norms1 are masculinized and, thus, produce gender-inequitable opportunities for mathematics participation related to learning and identity constructions. Sex-based participation studies, therefore, shifted the unit of analysis in gender equity research away from being solely about mathematics achievement comparisons and toward “coproductions” (Boaler, 2002c, p. 128) of mathematical success between students and gendered learning environments. At the same time, sex-based participation studies’ binary gender comparisons left implicit considerations of ways in which female and male students’ participation varied, including at intersections with race or ethnicity, class, and other social identities. Intersectional analyses in Boaler’s (1997, 2002b) study, for example, could have examined the extent to which the social class demographics of Amber Hill’s and Phoenix Park’s surrounding areas shaped instructional practices and, thus, students’ gendered patterns of engagement with them. In this body of research, the lack of conceptualization of race and ethnicity, including Whiteness,2 also left implicit how these social dimensions of students’ identities intersected with gender to shape their negotiations of participation with contextual influences. For instance, there was room for consideration of teachers’ ethnic backgrounds in Tiedemann (2002) which leaves unexplored any ethnic variation in the trend of more positive perceptions of competence for same-sex students. Boaler (2002a), in looking back on her study’s findings, raised the question of how future research could better attend to social constructions of identity at intersections of gender, other social identities, and mathematics: “How do identities of race, class, and gender intersect with those of mathematics?” (p. 47). Gender-Based Participation Research Gender-based participation studies conceptualized gender as a social construct (Butler, 2004; Wilchins, 2004) and mathematics success as a gendered source of power (Damarin, 2000; Walshaw, 2001). Such conceptualizations allowed researchers to document variation in individuals’ negotiations of their mathematical pursuits with masculinized discourses and practices in mathematics. Qualitative data sources, such as interviews and observations, were used to detail individual students’ behaviors and narratives of experience for managing gendered discourses and negotiating their positions along the gendered hierarchy of mathematical ability. “Gender differences,” therefore, were examined as dynamic, relational, and situational rather than as static or binary like in achievement and sex-based participation studies (Mendick, 2006).

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Departing from early gender equity studies’ analytical premise of the male superiority myth in mathematics assessments, more recent research from a gender-based participation perspective focused on exploring the masculinization of mathematics through observation and self-report accounts of students’ educational experiences (e.g., Barnes, 2000; Forgasz, 1998; Solomon, 2012). Observations allowed for detailing institutional and relational constructions of gender across spaces of mathematics education that shaped variation in students’ participation. Questionnaires and interviews provided self-reported insights from students on the extent to which gender played a role in making meaning of their mathematics experiences and shaping their identities. Below I review contemporary developments across two strands of gender-based participation research: one focused on classroom and curricular experiences and the other on constructions of student identity. Classroom and Curricular Experiences One strand of gender-based participation studies documented differences in students’ gendered learning experiences in classrooms and with curricular opportunities (Barnes, 2000; Shapka & Keating, 2003; Vale & Leder, 2004). For example, Barnes’s (2000) ethnographic study of an advanced high school calculus classroom in Australia examined how student subgroups engaged varying discourses of masculinity during collaborative learning. Barnes (2000) detailed the discursive production of two subgroups of boys (Mates and Technophiles) who performed differently patterned forms of masculinity for being mathematically successful. With mathematical ability as a gendered source of power, Barnes’s (2000) analysis sheds light on how these competing constructions of masculinity differentially positioned Mates and Technophiles along a gendered hierarchy of mathematical ability in the classroom. The more dominant masculinity among Mates leveraged their social capital of recognized athleticism and extracurricular involvement to approach mathematics with a sense of coolness. Technophiles’ “rational form of masculinity” (Barnes, 2000, p. 163) was premised on their intellectual capital and subordinate to Mates’ masculinity, which was more aligned with gender role expectations for men in and out of the classroom. Barnes’s (2000) study, therefore, extended sexbased participation studies’ binary analyses of gender by attending to variation in the social construction of gender among boys in a single mathematics classroom. Observations in Barnes’s (2000) study also provided situated insight into the mathematics classroom as a figured world where students are constantly engaged in “forms of authoring” (Boaler & Greeno, 2000, p. 173) themselves through discursive negotiations of mathematical success with their gender identities.

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Shapka and Keating’s (2003) more recent analysis explored the effect of an all-girls mathematics and science curriculum on 85 high school girls’ mathematics achievement, persistence, and engagement over 2 years. This longitudinal study statistically compared these outcome measures with those of 319 girls and 382 boys enrolled in a coeducational program at the same high school in Canada. Shapka and Keating (2003) found that the allgirls curriculum had a significantly positive effect on enrolled girls’ mathematics achievement and persistence, but no effect on engagement except for expended effort. In interpreting these findings, Shapka and Keating (20003) argued that the all-girls curriculum seemed to support girls’ ability of not falling victim to gendered stereotypes of mathematical ability in a coeducational learning environment. Shapka and Keating (2003), thus, departed from earlier studies’ binary gender comparisons of mathematics achievement and participation by using comparisons to make interpretative claims about curricular program design rather than the students. The researchers acknowledged several limitations of their study, including: (a) the short length of 2 years for observing change in mathematics attitudes or engagement, (b) the lack of classroom observations and student interviews for qualitative insight on curricular impact, and (c) a homogeneous sample of high-achieving and predominantly White middle- or upper-middle-class girls that limited insight on variation across social differences. With qualitative data sources and a more heterogeneous sampling of participants, similarly designed future studies on curriculum can explore how classroom factors contributed to achievement and participation outcomes as well as capture intersectional differences to inform curriculum design with a broad sense of gender inclusivity. Constructions of Student Identity The other strand of gender-based participation studies used narrative inquiry to examine variation in how students made meaning of their mathematics experiences in and out of the classroom in relation to their gender identities (Mendick, 2003, 2005a; Solomon, 2012). Mendick’s (2003, 2005a) multisite ethnographic study, involving 43 student interviews and 3 weeks of classroom observations at demographically diverse colleges, explored British students’ choices in pursuing mathematics coursework to better understand men’s disproportionate representation in mathematics. Extending prior analyses of discursive productions of gender in mathematics (cf. constructions of masculinity in Barnes, 2000), Mendick (2003, 2005a) used narrative analysis of interview data to detail variation among both women and men in their negotiations of masculinized discourses as mathematics students. Mendick (2005b) referred to these negotiations as gender identity projects, characterized as individuals varyingly constructing their identities in negotiations with a gendered dualism of either being

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“good at maths”3 or “not good at maths.” To illustrate, Mendick (2005a) discussed how mathematical ability is viewed as a masculine attribute through “gendered discourses of rationality” (p. 203) that contributed to women participants’ tensions in identifying as “good at maths.” Further, Mendick (2003, 2005a, 2005b, 2006) documented variation in constructions of masculinity among men in mathematics, similar to Barnes (2000), with hard work for career development positioned as a more dominant attribute of masculinity than natural talent in mathematics. Mathematically successful men, thus, often engaged in gender identity projects that protected their perceived masculinity by separating themselves from nonmathematicians yet describing themselves as hard-working rather than naturally talented. The lack of observations in Mendick’s (2003, 2005a, 2005b, 2006) analysis, however, limited insight into how masculinized discourses in mathematics shaped institutional and interpersonal dimensions of participants’ experiences, including behavioral ways of engaging in their unique gender identity projects. More recently, Solomon (2012) completed a narrative analysis of interviews with two women (Joanne and Roz) to further explore gender-related identity constructions in undergraduate mathematics education. Solomon (2012) used dialogism, a perspective that conceptualizes discursive identity productions as responses to past and future discourses (Bakhtin, 1981; Holquist, 2002), to detail the role of discourses in shaping Joanne’s and Roz’s identities. In addition, Solomon’s (2012) study captured how Joanne and Roz engaged in reflective processes of negotiating their identities with the gendered status quo in mathematics—namely, a refiguring rather than passive acceptance of the masculinized figured world of mathematics. Despite these analytical similarities to prior gender-based participation studies (cf. Barnes, 2000; Mendick, 2003, 2005a), Solomon (2012) extended these studies’ findings by detailing how the two women’s identity constructions, or self-authoring of “new identity spaces” (Solomon, 2012, p. 171), were influenced by another social dimension of experience, particularly social class. Solomon (2012), for example, argued how Joanne’s identity reflected her unique way of negotiating discourses from her past (e.g., coming from a financially privileged family) and anticipated future in mathematics (e.g., class-based divisions between students) to manage being at the juncture of gendered oppression and class privilege as an upper-class woman in undergraduate mathematics. The coupling of such intersectional considerations with contextual insights of experience (e.g., observations, teacher perceptions), however, is an analytical possibility for further uptake in gender-based participation research.

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Summary Gender-based participation studies expanded on binary analyses of gendered “co-productions” to detail students’ unique negotiations of their participation with the masculinization of mathematics. While the first strand of studies captured variation among boys (e.g., Barnes, 2000) and among girls (e.g., Shapka & Keating, 2003) in their engagement with classroom and curricular learning opportunities, the second strand documented variation in students’ negotiations of their identities with gendered discourses in mathematics as a field (e.g., Mendick, 2003, 2005a; Solomon, 2012). Through the use of observations and self-report data sources (e.g., interviews, questionnaires), gender-based participation studies methodologically centered students’ mathematics experiences to capture different social constructions of gender across contexts as well as their relationships with classroom learning, persistence, and identities. The perspectives of teachers, family members, and other key figures in students’ mathematics education, however, were largely underexplored compared to the achievement and sex-based participation categories of gender equity research. Future research, thus, can look across students’ and other individuals’ perspectives on mathematics classroom experiences to better inform gender-inclusive forms of instruction and student support practices. While case studies of classrooms and individual students in genderbased participation research allowed for more analytical depth, there remains room in gender equity research for more longitudinal, case study analyses that engage in extended explorations of students’ engagement and identities in mathematics. Limited consideration was given to intersections of gender with other social identities (e.g., race or ethnicity, class) for explaining variation in participation trends, including that among White students. Sex-based and gender-based participation studies completed in different countries, for example, engaged in minimal interpretations of findings with regard to international variation in cultural constructions of gender, race or ethnicity, and other social identities (Leyva, 2017). Thus, an area for future research is exploring how social and cultural norms for different geographic areas shape gendered participation in mathematics, as taken up in some cross-cultural analyses of achievement (e.g., Birenbaum & Nasser, 2006). CONCLUSION: THE FUTURE OF GENDER EQUITY RESEARCH With the goal of chronicling current developments in gender equity research in mathematics education, this chapter examined how more recent empirical studies published since 2000 have conceptualized and studied

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gender. I drew on studies in my review of gender research in mathematics education (Leyva, 2017), including those that had not been discussed in detail yet were representative of the achievement and participation categories of gender equity research. Below I synthesize insights from this chapter on advancements and underexplored areas of study in gender equity research. This is followed by a discussion of how studies published since I completed my review have begun addressing underexplored areas of gender research. I conclude with a proposed vision for future gender equity research that engages intersectional analyses of gender through methodologies that capture its varying social construction across contexts of mathematics education. Achievement studies largely adopted a binary conceptualization of gender in statistical models exploring internal and external influences on female–male differences in mathematics achievement. Findings challenged the myth of male superiority in mathematics assessments and, thus, highlighted the need to explore the social contexts of mathematics education that contribute to gendered disparities. Some recent achievement research (Ai, 2002; Davis & Carr, 2001; Lubiesnki et al., 2013; Robinson & Lubienski, 2011) attended to how psychosocial influences in students’ relationships with parents and teachers shaped female–male differences in mathematics achievement and task performance, in addition to tracing the extent to which such gendered trends were consistent across grade levels (versus, at one time). While recent research in the achievement category captured potentially gendered interpersonal influences on mathematics assessment outcomes and experiences, there remains a void in gender equity research that foregrounds the perspectives and behaviors of teachers, family members, and other influential figures to complement those of students in mathematics. Mathematics as a gendered experience was examined in the participation literature that theorized gender as either binary (sex-based) or socially constructed (gender-based). Sex-based participation studies used interviews and observations to detail gendered socialization through institutional and interpersonal influences (e.g., classroom instruction, teacher perceptions) that shaped different mathematics learning experiences between girls and boys. Findings highlighted the masculinization of participation norms and standards of success that may have a varying impact among individuals within and across gender identity groups (e.g., gender-nonconforming, women, men) that cannot be captured using a conceptualization of gender as binary. Gender-based participation studies provided insight into within-group differences, particularly among women and among men, in negotiations of their persistence and identities with their mathematics educational experiences. While participation studies provided insight into gendered trends of mathematics participation across geographic borders (e.g., Barnes, 2000; Forgasz et al., 2004; Solomon, 2012; Tiedemann, 2002),

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variation in the cultural construction of gender across these countries as well as how they shaped individuals’ mathematics experiences and identities remained implicit in gender equity research. In addition to explicit considerations of other key stakeholders’ perspectives and international constructions of gender, I have argued that gender equity research in mathematics education can benefit from intersectional analyses that attend to how race, ethnicity, class, and other socially constructed identities shape gendered trends of achievement and participation (Leyva, 2017). A majority of gender equity research, including recent studies reviewed in this chapter (e.g., Shapka & Keating, 2003; Tiedemann, 2002), explored their respective units of analysis using samples of predominantly White students and teachers, leaving unexplored how participants’ White identities intersected with gender to capture possible racial or ethnic variation in the findings. Such analytical bypassing of race and ethnicity, as a result, leaves room for future research to problematize the rendering of Whiteness as invisible and neutral for more intersectional insights on gender equity (Battey & Leyva, 2016). While some recent research studies engaged intersectional analyses with more diverse samples in terms of race and class (e.g., Carr et al., 2008; McGraw et al., 2006; Solomon, 2012), gender largely remained conceptualized as a binary and shaped by intersections of race, class, and gender. The dynamic interplay between gender and other social identities, such as ability and sexuality, thus, was underexplored and can be taken up more explicitly in future gender equity research (Esmonde, 2011; Lewis & Fisher, 2016; Leyva, 2017). With my review’s focus on studies published through January 2016, it is important to acknowledge more recent contributions that have begun to address underexplored areas of inquiry, thus pointing to what could be the future of gender equity research in mathematics education. Recent achievement studies, while still adopting binary conceptualizations of gender, considered contextual influences, such as family support and resources, gender role socialization, and gender-stereotyped beliefs on achievement outcomes across grade levels (Ehrtmann & Wolter, 2018; Moè, 2018; Moon & Hofferth, 2016). Teacher influences on gendered disparities in mathematics participation, including perceptions of student ability and engagement as well as instructional approaches, have been explored using interviews, observations, and questionnaires (Baroody, Rimm-Kaufman, Larsen, & Curby, 2016; Mogari, 2017; Sarouphim & Chartouny, 2017). Some of these recent participation studies also explicitly considered cultural constructions of gender within national contexts in interpreting their findings. For example, Baroody and colleagues’ (2016) quantitative analysis of gender differences in mathematics achievement and attitudes challenged discourses of girls from traditional Lebanese cultural backgrounds being less skilled and interested in mathematics than their boy counterparts. Findings from

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Mogari’s (2017) study of a culturally relevant mathematics lesson activity involving students from a rural, patriarchal South African community showed how boys marginalized girls through classroom behaviors premised on gendered cultural stereotypes. While recent gender-based participation studies continued examining within-group variation in participants’ negotiations of their identities with gendered mathematics experiences (e.g., Foyn, Solomon, & Braathe, 2018; Wolfe, 2017), there remains a significant void of intersectional analyses that illuminate differences at intersections of gender and race or ethnicity, class, and other identity dimensions. Thus, I envision future advancement of gender equity through mathematics education research premised on exploring gender as socially constructed and with explicit attention to the intersectionality of experience for members of different social groups. I have previously argued for the promise of intersectional analyses through a review of studies that either did or did not adopt Crenshaw’s (1991) theory of intersectionality to examine intersectional variation in mathematics achievement and participation (Leyva, 2017). Intersectional analyses in these and more recent studies further problematized the myth of male superiority in mathematics that reinscribes deficit constructions of femininity vis-à-vis attempts to close “gender gaps” in achievement and participation (McGraw et al., 2006; Young, Young, & Capraro, 2018). Furthermore, studies included in my review of intersectional analyses pointed to the potential of complementing quantitative with qualitative methodologies (e.g., interviews, observations) to shed light on how gendered contexts of mathematics education varyingly shape the intersectionality of students’ experiences (e.g., Riegle-Crumb & Humphries, 2012). Conceptual and methodological contributions from a growing body of intersectional studies in mathematics education about members of various social groups—for example, Black women (e.g., Gholson & Martin, 2014), Latinxs (e.g., Leyva, 2016, Oppland-Cordell & Martin, 2015), and queer people of color (e.g., Esmonde, Brodie, Dookie, & Takeuchi, 2009)—can inform future gender equity research aimed at broadening inclusive opportunities in mathematics achievement and participation. NOTES 1. Sociomathematical norms are “normative understandings of what counts as mathematically different, mathematically sophisticated, mathematically efficient, and mathematically elegant in a classroom” (Yackel & Cobb, 1996, p. 461). These norms are relationally produced and negotiated through interpretations of being “intellectually autonomous in mathematics” (p. 458). 2. Whiteness is a fictive ideology from which racism is established and White supremacy is maintained to privilege Whites over other racial groups (Kivel, 2011; Leonardo, 2004).

86 L. A. LEYVA 3. In British English, the term maths is interchangeable with math or mathematics in North American English.

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Beyond the Binary and at the Intersections 87 Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171–200). Westport, CT: Ablex. Bonilla-Silva, E. (2003). Racism without racists: Color-blind racism and the persistence of racial inequality in the United States. Lanham, MD: Rowman and Littlefield. Bornholt, L. J., Goodnow, J. J., & Cooney, G. H. (1994). Influences of gender stereotypes on adolescents’ perceptions of their own achievement. American Educational Research Journal, 31(3), 675–692. Brandell, G., & Staberg, E.-M. (2008). Mathematics: A female, male or gender-neutral domain? A study of attitudes among students at secondary level. Gender and Education, 20(5), 495–509. Brandon, P. R., Newton, B. J., & Hammond, O. W. (1987). Children’s mathematics achievement in Hawaii: Sex differences favoring girls. American Educational Research Journal, 24(3), 437–461. Butler, J. (2004). Undoing gender. New York, NY: Routledge. Carr, M., Jessup, D. L., & Fuller, D. (1999). Gender differences in first-grade mathematics strategy use: Parent and teacher contributions. Journal for Research in Mathematics Education, 30(1), 20–46. Carr, M., Steiner, H. H., Kyser, B., & Biddlecomb, B. (2008). A comparison of predictors of early emerging gender differences in mathematics competency. Learning and Individual Differences, 18(1), 61–75. Casey, M. B., Nuttal, R. L., & Pezaris, E. (2001). Spatial-mechanical reasoning skills versus mathematics self-confidence as mediators of gender differences on mathematics subtests using cross-national gender-based items. Journal for Research in Mathematics Education, 32(1), 28–57. Crenshaw, K. (1991). Mapping the margins: Intersectionality, identity politics, and violence against women of color. Stanford Law Review, 43(6), 1241–1299. Damarin, S. K. (2000). The mathematically able as a marked category. Gender and Education, 12(1), 69–85. Damarin, S., & Erchick, D. B. (2010). Toward clarifying the meanings of gender in mathematics education research. Journal for Research in Mathematics Education, 41(4), 310–323. Davis, H., & Carr, M. (2001). Gender differences in mathematics strategy use: The influence of temperament. Learning and Individual Differences, 13(1), 83–95. Ehrtmann, L., & Wolter, I. (2018). The impact of students’ gender-role orientation on competence development in mathematics and reading in secondary school. Learning and Individual Differences, 61, 256–264. Ernest, P. (1991). The philosophy of mathematics education. London, England: Routledge Falmer. Esmonde, I. (2011). Snips and snails and puppy dogs’ tails: Genderism and mathematics education. For the Learning of Mathematics, 31(2), 27–31. Esmonde, I., Brodie, K., Dookie, L., & Takeuchi, M. (2009). Social identities and opportunities to learn: Student perspectives on group work in an urban mathematics classroom. Journal of Urban Mathematics Education, 2(2), 18–45. Esmonde, I., & Langer-Osuna, J. M. (2013). Power in numbers: Student participation in mathematical discussions in heterogeneous spaces. Journal for Research in Mathematics Education, 44(1), 288–315.

88 L. A. LEYVA Ethington, C. A. (1992). Gender differences in a psychological model of mathematics achievement. Journal for Research in Mathematics Education, 23(2), 166–181. Fennema, E. (1979). Women and girls in mathematics—Equity in mathematics education. Educational Studies in Mathematics, 10(4), 389–401. Fennema, E. (2000, May). Gender and mathematics: What is known and what do I wish was known? Paper presented at Fifth Annual Forum of the National Institute for Science Education, Detroit, MI. Retrieved from http://archive.wceruw. org/nise/News_Activities/Forums/Fennemapaper.htm Fennema, E., Carpenter, T. P., Jacobs, V. R., Franke, M. L., & Levi, L. W. (1998). A longitudinal study of gender differences in young children’s mathematical thinking. Educational Researcher, 27(5), 6–11. Fennema, E., Peterson, P. L., Carpenter, T. P., & Lubinski, C. A. (1990). Teachers’ attributions and beliefs about girls, boys, and mathematics. Educational Studies in Mathematics, 21(1), 55–69. Fennema, E., & Sherman, J. (1977). Sex-related differences in mathematics achievement, spatial visualization, and affective factors. American Educational Research Journal, 14(1), 51–71. Fennema, E. H., & Sherman, J. A. (1978). Sex-related differences in mathematics achievement and related factors: A further study. Journal for Research in Mathematics Education, 9(3), 189–203. Forgasz, H. (1998, November–December). Why are they studying mathematics? Tertiary mathematics students tell all! Paper presented at the annual conference of the Australian Association for Research in Education, Adelaide, Australia. Retrieved from http://www.aare.edu.au/data/publications/1998/for98049.pdf Forgasz, H. J., Leder, G. C., & Barkatsas, T. (1998). Mathematics—For boys? For girls? Vinculum, 35(3), 15–19. Forgasz, H. J., Leder, G. C., & Kloosterman, P. (2004). New perspectives on the gender stereotyping of mathematics. Mathematical Thinking and Learning, 6(4), 389–420. Foyn, T., Solomon, Y., & Braathe, H. J. (2018). Clever girls’ stories: The girl they call a nerd. Educational Studies in Mathematics, 98, 77–93. Gholson, M., & Martin, D. B. (2014). Smart girls, Black girls, mean girls, and bullies: At the intersection of identities and the mediating role of young girls’ social network in mathematical communities of practice. Journal of Education, 194(1), 19–33. Gutiérrez, R. (2008). A “gap-gazing” fetish in mathematics education? Problematizing research on the achievement gap. Journal for Research in Mathematics Education, 39(4), 357–364. Gutiérrez, R. (2009). Framing equity: Helping students “play the game” and “change the game.” Teaching for Excellence and Equity in Mathematics, 1(1), 4–8. Hanna, G. (1989). Mathematics achievement of girls and boys in grade eight: Results from twenty countries. Educational Studies in Mathematics, 20(2), 225–232. Hart, L. E. (1989). Classroom processes, sex of student, and confidence in learning mathematics. Journal for Research in Mathematics Education, 20(3), 242–260. Holland, D., Lachiotte, W., Jr., Skinner, D., & Cain, C. (1998). Identity and agency in cultural worlds. Cambridge, MA: Harvard University Press.

Beyond the Binary and at the Intersections 89 Holquist, M. (2002). Dialogism: Bakhtin and his world (2nd ed.). London, England: Routledge. Hyde, J. S., & Jaffee, S. (1998). Perspectives from social and feminist psychology. Educational Researcher, 27(5), 14–16. Kivel, P. (2011). Uprooting racism: How White people can work for racial justice (3rd ed.). Gabriola Island, Canada: New Society. Leonardo, Z. (2004). The color of supremacy: Beyond the discourse of ‘White privilege.’ Educational Philosophy and Theory, 36(2), 137–152. Lewis, K. E., & Fisher, M. B. (2016). Taking stock of 40 years of research on mathematical learning disability: Methodological issues and future directions. Journal for Research in Mathematics Education, 47(4), 338–371. Leyva, L. A. (2016). An intersectional analysis of Latin@ college women’s counter-stories in mathematics. Journal of Urban Mathematics Education, 9(2), 81–121. Leyva, L. A. (2017). Unpacking the male superiority myth and masculinization of mathematics at the intersections: A review of research on gender in mathematics education. Journal for Research in Mathematics Education, 48(4), 397–452. Lubienski, S. T., Robinson, J. P., Crane, C. C., & Ganley, C. M. (2013). Girls’ and boys’ mathematics achievement, affect, and experiences: Findings from ECLS-K. Journal for Research in Mathematics Education, 44(4), 634–645. McGraw, R., Lubienski, S. T., & Strutchens, M. E. (2006). A closer look at gender in NAEP mathematics achievement and affect data: Intersections with achievement, race/ethnicity, and socioeconomic status. Journal for Research in Mathematics Education, 37(2), 129–150. Mendick, H. (2003). Choosing maths/doing gender: A look at why there are more boys than girls in advanced mathematics classes in England. In L. Burton (Ed.), Which way social justice in mathematics education? (pp. 169–187). Westport, CT: Praeger. Mendick, H. (2005a). A beautiful myth? The gendering of being/doing “good at maths.” Gender and Education, 17(2), 203–219. Mendick, H. (2005b). Mathematical stories: Why do more boys than girls choose to study mathematics at AS-level in England? British Journal of Sociology of Education, 26(2), 235–251. Mendick, H. (2006). Masculinities in mathematics. New York, NY: Open University Press. Moè, A. (2018). Mental rotation and mathematics: Gender-stereotyped beliefs and relationships in primary school children. Learning and Individual Differences, 61, 182–180. Mogari, D. (2017). Using culturally relevant teaching in a co-educational mathematics class of a patriarchal community. Educational Studies in Mathematics, 94(3), 293–307. Moon, U. J., & Hofferth, S. L. (2016). Parental involvement, child effort, and the development of immigrant boys’ and girls’ reading and mathematics skills: A latent difference score growth model. Learning and Individual Differences, 47, 136–144.

90 L. A. LEYVA Oppland-Cordell, S., & Martin, D. B. (2015). Identity, power, and shifting participation in a mathematics workshop: Latin@ students’ negotiation of self and success. Mathematics Education Research Journal, 27(1), 21–49. Pallas, A. M., & Alexander, K. L. (1983). Sex differences in quantitative SAT performance: New evidence on the differential coursework hypothesis. American Educational Research Journal, 20(2), 165–182. Pedro, J. D., Wolleat, P., Fennema, E., & Becker, A. D. (1981). Election of high school mathematics by females and males: Attributions and attitudes. American Educational Research Journal, 18(2), 207–218. Penner, A. M., & Paret, M. (2008). Gender differences in mathematics achievement: Exploring the early grades and the extremes. Social Science Research, 37(1), 239–253. Peterson, P. L., & Fennema, E. (1985). Effective teaching, student engagement in classroom activities, and sex-related differences in learning mathematics. American Educational Research Journal, 22(3), 309–335. Riegle-Crumb, C., & Humphries, M. (2012). Exploring bias in math teachers’ perceptions of students’ ability by gender and race/ethnicity. Gender & Society, 26(2), 290–322. Robinson, J. P., & Lubienski, S. T. (2011). The development of gender achievement gaps in mathematics and reading during elementary and middle school: Examining direct cognitive assessments and teacher ratings. American Educational Research Journal, 48(2), 268–302. Rubel, L. H. (2016). Speaking up and speaking out about gender in mathematics. The Mathematics Teacher, 109(6), 434–439. Sarouphim, K. M., & Chartouny, M. (2017). Mathematics education in Lebanon: Gender differences in attitudes and achievement. Educational Studies in Mathematics, 94(1), 55–68. Seegers, G., & Boekaerts, M. (1996). Gender-related differences in self-referenced cognitions in relation to mathematics. Journal for Research in Mathematics Education, 27(2), 215–240. Shapka, J. D., & Keating, D. P. (2003). Effects of a girls-only curriculum during adolescence: Performance, persistence, and engagement in mathematics and science. American Educational Research Journal, 40(4), 929–960. Sherman, J., & Fennema, E. (1977). The study of mathematics by high school girls and boys: Related variables. American Educational Research Journal, 14(2), 159–168. Solomon, Y. (2012). Finding a voice? Narrating the female self in mathematics. Educational Studies in Mathematics, 80(1–2), 171–183. Sowder, J. T. (1998). Perspectives from mathematics education. Educational Researcher, 27(5), 12–13. Tiedemann, J. (2000). Gender-related beliefs of teachers in elementary school mathematics. Educational Studies in Mathematics, 41(2), 191–207. Tiedemann, J. (2002). Teachers’ gender stereotypes as determinants of teacher perceptions in elementary school mathematics. Educational Studies in Mathematics, 50(1), 49–62.

Beyond the Binary and at the Intersections 91 Vale, C. M., & Leder, G. C. (2004). Student views of computer-based mathematics in the middle years: Does gender make a difference? Educational Studies in Mathematics, 56(2), 287–312. Walshaw, M. (2001). A Foucauldian gaze on gender research: What do you do when confronted with the tunnel at the end of the light? Journal for Research in Mathematics Education, 32(5), 471–492. Wilchins, R. (2004). Queer theory, gender theory: An instant primer. Los Angeles, CA: Alyson Books. Wolfe, M. J. (2017). Smart girls traversing assemblages of gender and class in Australian secondary mathematics classrooms. Gender and Education, 31(2), 205–221. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. Young, J. L., Young, J. R., & Capraro, R. M. (2018). Gazing past the gaps: A growthbased assessment of the mathematics achievement of Black girls. The Urban Review, 50(1), 156–176.

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CHAPTER 5

RIGHT TO LEARN MATHEMATICS From Language as Right to Language as Mathematically Relevant Resource Núria Planas and Mapula Ngoepe

Today, the richness of languages, cultures, and communities produces a complex heterogeneous picture of what it means to teach, learn, and think school mathematics. This picture is one of the reasons for terms like superdiversity entering classroom research in mathematics education (Barwell, 2016) to describe the plethora of intersecting types of diversity in the midst of discourses of uniformity and homogeneity. Despite the evidence of superdiversity in our world, pedagogies based on the belief that monolingualism is achievable and preferable are strongly rooted in history and tradition. In mathematics teaching and learning in particular, multilingualism is still considered exceptional and monolingualism remains the norm. In order to contribute to the body of works that challenge the monolingual norm, in this chapter we address two questions: What is the role of language in the mathematics classroom? What is the role of the languages of the learners? Central to our argument is the

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understanding of the language of the mathematics classroom as conformed by all the languages in the classroom and their uses. Learners and people in general have the right to use their languages, and importantly the right to learn in settings that are linguistically, culturally, and pedagogically responsible. When only the language of instruction or only some languages of only some learners are valued as tools for thinking, teaching, and learning: “What students are afforded the right to learn mathematics robustly, actively, and with understanding? What students are obligated to learn mathematics in less productive ways?” (Langer-Osuna & McKinney de Royston, 2017, p. 647). Our experiences of research and teaching in mathematical lessons of Catalonia and Gauteng, where participants are able to draw on more than one language as they teach, learn, and interact, frame our approach to the issues of power and equity captured in the quote above. In our two contexts, basic language and educational rights similarly exist in principle, with bi/multilingual policies and culturally responsive discourses on paper. Nonetheless, Catalan in Catalonia and English in Gauteng are the languages of teaching and learning privileged in practice, and the forms of knowledge of the mainstream groups are those primarily valued over the course of school mathematics. Throughout this chapter, we discuss the potential of the languages of the learners alongside the language of the teacher and of mathematics in the production of a language of the mathematics classroom that supports mathematics learning. After this introduction, we focus on findings from research to argue in favor of varied uses of diverse languages in mathematics teaching, learning, and thinking. We claim that any instance of language use, regardless of the politics attached to it (Chronaki & Planas, 2018; Ricento, 2014; Ruíz, 1984), is a potential resource to be realized for learners in their mathematics learning (Adler, 2000; Planas, 2018). In the examination of the role of language, we take pieces of data to consider the impact of practices of switching languages that incorporate the home languages, and of practices of diversifying modes of communication on the creation and distribution of mathematics learning. The illustration of data collected in Catalonian schools is utilized to reflect on findings from field observations, conversations with teachers and learners, and lessons in Gauteng as well. We conclude with some implications regarding the adoption of the multilingual norm and the use of related multilingual practices in mathematics education. THE LANGUAGES IN THE LANGUAGE OF THE MATHEMATICS CLASSROOM Following the publication of Speaking Mathematically: Communication in Mathematics Classrooms (Pimm, 1987), the study of the teaching and learning

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of the language of mathematics became strengthened in the field of mathematics education. Since then, this line of research has gained complexity through the articulation of the languages of the learners, the teacher and the mathematics (see the review of the early domain by Austin and Howson, 1979) to imply the plural language of the mathematics classroom (Planas, Morgan, & Schütte, 2018). All these languages are connected, each has a relationship to another, although it is not necessarily obvious how they connect over the course of a lesson. The reference to a triad of inseparable classroom types of language is currently of much importance and helpful in many respects. Attention to this triad has been decisive in the structuring of research on mathematics education and language diversity as visible in the volume edited by Barwell and colleagues (2016). Specifically, the focus on the languages of learners has contributed to unveiling the social and educational disadvantages created for learners who do not speak the language of the teacher at home. While language is a potential tool of communication and of facilitation of mathematical activity, this role of language is especially complex in the mathematics classrooms where the language of the teacher and of instruction is not a home language for all learners. The broad conceptualization of the language of the mathematics classroom as a pedagogic resource (Planas, 2014, 2018; Planas & Civil, 2013; Planas & Setati-Phakeng, 2014), thus, does not imply that mathematics learning is always achieved or encouraged in any language use. Language use does not necessarily translate into mathematics learning and, indeed, such learning may be hindered when classroom practice is primarily oriented to teaching, assessing, and developing proficiency in the language of instruction. The realization of language as a resource for the learning of mathematics requires practices that are mathematically relevant, that is, oriented to teaching, assessing, and developing the language of mathematics and its related meanings. Hence, we need to understand the classroom practices that play a role in making use of language to enhance mathematics learning, compared to the practices that subsume the learning of mathematics under the learning of the language of instruction. Focusing on practices responsible for differences in the effects of language on mathematics learning, several chapters in Hunter, Civil, Herbel-Eisenmann, Planas, and Wagner (2018) describe practices in multilingual lessons that helped to overcome tensions between the languages present in the discussion and communication of mathematically relevant meanings. Some of the practices of flexibly using the languages of the learners appear combined with reasoning algebraically, building models, telling examples, predicting patterns, and drawing representations. Walsh (2011) also provides a variety of classroom practices where participants interact through their diverse languages and, significantly, with the aid of drawings, diagrams, and other forms of visual languages introduced by

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learners in their explanations to the group. These are works in which language diversity has two main interpretations regarding diversity of verbal languages and diversity of modes and abilities of communication. The practice of working in groups with learners using their home languages is largely addressed in the discussion of the diversity of spoken languages in the mathematics classroom. A strong relationship is indicated between the use of home languages and the mathematical participation of learners. In low-income Latino communities of the United States, Civil (2012) reports the contrast between learners’ participation in English and Spanish: “When presenting to the whole class in English, their communication was tentative and stilted. . .When presenting in Spanish or talking in their small groups (where students turned automatically to Spanish), it was a completely different story” (pp. 50–51). This is also consistent with what Planas and Setati (2009) found regarding Latin American migrant learners in Barcelona, the main city of Catalonia, who switched to the home language as soon as the conceptual level of mathematical explanations increased. The mathematical ideas brought up in the home language during group work, on the contrary, did not arise as long as this language remained unused by the teacher and the learners in whole class interaction. This happened with bilingual teachers who shared the language of Latin American learners and who occasionally replied in their language, but also with Arabic-speaking learners and teachers who did not share a main language with the majority of learners. Migrant learners who did not own the language of instruction hardly participated in the whole class and their reasoning was visible at the level of group work only, where activity was relatively free of instruction and direct assessment. Interestingly, in small groups in which learners did not share the home language with all their peers, the communication included unusual drawings and mathematically meaningful sketches. In this way, learners found modes of communicating their reasoning other than speaking either the home language or the language of instruction. Those drawings were as much a part of the set of language resources used by the learners as were their spoken languages. In her work with children of working class immigrant families, Civil (2012) reminds us of the need to look at the politics of the situation. While learners in classrooms from her research share a home language, this is a consequence of policies of segregation with Latino children mostly concentrated in certain neighborhoods and school districts. In spite of the socially ambitious educational policies developed for Gauteng and Catalonia, in our regions the politics of the situation also reveals geographic concentration of language groups. Together with the challenge of meeting the needs of schools with learners who do not own the language of instruction, many other challenges therefore appear in relation to how the minority languages spoken by the majority of learners are to be considered. In historically

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disadvantaged Black African schools, for example, teachers are often fluent in the home languages of their learners but prefer to teach mathematics in English, which is a language that some of them are not proficient in. Setati and Adler (2000) provide findings from mathematics classrooms in these schools with teachers who experimented with work in linguistically homogeneous groups. The discussions in group work were in the home languages, simultaneously or not with English, while interaction with the whole group was through speaking and writing on the board in English. Still in South Africa and similarly to what was found in the work with learners in Barcelona, Mparutsa (2011) reports practices of alternating modes of communication in the absence of a common language. In her exploration of access to school mathematics when the teacher does not share a home language with learners, Mparutsa found that learners move between their languages in peer work and between verbal and visual modes of communication to be able to interact with and understand what is mathematically going on in the lesson. The examples in the next two sections respectively focus on two classroom practices: (a) group work with learners using their home languages and (b) varied modes and abilities of communication in the absence of a common language. We combine pieces of data that reflect stories of participation and realizations of the languages of learners as a resource for mathematics learning. Despite the underlying force of the monolingual norm, the learners in the examples flexibly use their home languages and communication abilities. Although differences at many levels can be argued between Catalonia and Gauteng, the effects of these practices raise substantial commonalities. The relevance of the language of instruction in the lessons, the tensions around the views of the languages and of the speakers, and the ways in which learners nevertheless draw on their home languages and communication abilities are similar. The deliberate promotion of these practices might be a solution to the disadvantage experienced by learners of minority language groups in settings of school mathematics and, more generally, to the loss of learning opportunities experienced by all learners that see their possibilities of interaction and exchange reduced. We believe that the observations in Catalonia and Gauteng might illuminate the interpretation of other lesson contexts that at first sight might appear as different. LEARNERS USING THEIR HOME LANGUAGES The moves between the languages of the learners and the languages of mathematics, and of the teacher operating together within the language of the mathematics classroom provide the context for the possibilities of using

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the home languages. Language switching, languaging, translanguaging, or codemeshing are some of the terms for these moves to mean what learners do with languages, registers, words, pronunciations, grammars, and so on. Regardless of the term used to name the practice, we are interested in reflecting on its pedagogic value for mathematics teaching, learning, and thinking. The home languages and the language of instruction may not share the same space, at the same time, as a presence visible to participants. However, an important issue with respect to the use of the home languages is the use of the language of instruction. All these languages are interconnected in the activity of the learner in the classroom, and they all need to be viewed from the perspective of their contribution to mathematics thinking and learning. Even for learners who are in the early process of learning the language of instruction, this language is never absent or small. It develops in the general school practices of watching, listening, reproducing, and imitating the oral and written texts of the teacher and the classroom. The same applies to learners whose home language is not the language of instruction and, based on their interactions, apparently draw on this language only. The pieces of data in this section are published in Planas and Setati (2009). They illustrate how the home languages function alongside the language of instruction at selected times of mathematics teaching, learning, and thinking. In many of the lessons observed over these years, learners switched to their home languages when discussing in small groups. The home languages were less common in situations of whole group and of writing, in which the language of instruction mostly prevailed. In our research, the flexibility in the use of the languages in the classroom is thus particular to oral interaction in group work. In reflecting back on the data in Excerpts 1 and 2 below, we can think of many other examples in lessons of Catalonia and Gauteng, where group work is one of the methods targeted by the local curricula to promote participation and engagement. One important conclusion from our observations is that learners engage meaningfully with mathematical content when talking in their home languages. Moreover, in the rare cases in which learners write in their home languages or use them in whole class discussions, they provide substantial details of the mathematical reasoning developed in the small group. These results provide arguments for the deliberate promotion of different languages working together so that learners can harness all their languages in their learning. Learners Engaging Meaningfully With Mathematical Content in Group Work This first piece of data serves us to argue that learners use their home languages in group work for engagement with conceptual reasoning that is

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difficult to them. Excerpt 1 shows an instance of group work on the mathematical representation of a tornado in a lesson in Barcelona. The utterances are first given in the languages in which they were made. Italics and non-italics in the English version indicate the moves between Catalan—the language of instruction—and Spanish—the home language of these learners. Language is diverse at several other levels, if we consider the combination of drawings of arrows with gestures to imply the spiral motion. While these two sides of language diversity—spoken languages and modes of communication—are necessary to capture the role of language in mathematics learning, we leave the discussion of the latter for the next section. Excerpt 1 Máximo: Hem de decidir les fletxes que dibuixem i ja està. [We need to decide the arrows that we draw and that’s all.] Eliseo: Primer pensem les fletxes, després les dibuixem i després en parlem. [First we think about the arrows, then we draw them and then we talk about it.] Máximo: Esta idea de las flechas no es fácil. Tenemos que imaginar los diferentes movimientos que existen dentro del tornado. [This idea of the arrows is not easy. We have to imagine the different movements that exist within the tornado.] Eliseo: Una flecha tiene que ser una línea recta para que el tornado baje. Tenemos la t para la traslación. [An arrow needs to be a straight line for the tornado to go down. We have the t for the translation.] Luna: La pregunta pide representar un tornado, ¿no? [The question asks to represent a tornado, doesn’t it?] Nicolás: Sí, diu que s’ha de representar matemàticament un tornado. [Yes, it says that we need to mathematically represent a tornado.] Luna: No és parlar d’un tornado, és representar-lo matemàticament. [It is not to talk about a tornado, it is to mathematically represent it.] Eliseo: Nos puede ser útil representar un tornado antes de dibujarlo. [The drawing of a tornado can be useful before its representation.] Nicolás: Está claro que con una sola flecha no basta, porque un tornado es más que una traslación. [It is clear that only one arrow is not enough, because a tornado is more than a translation.] Eliseo: Hay que pensar en cómo dibujaríamos una espiral. Dibujaríamos curvas. [We need to think about the drawing of a spiral. We would draw curves.]

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Máximo is a second-generation Colombian boy, Luna is a girl born in Peru, and Nicolás and Eliseo are two boys born in Colombia. They all testified that they spoke Spanish at home and that they most often spoke Catalan and Spanish at school. During group work, they used Spanish and Catalan in the discussion of mathematically relevant meanings related to the task. The home language, however, dominated the processes of reasoning and, specifically, the processes of making sense of the task and the procedure to be followed in the resolution. They used what they knew to make sense of what they were asked to do, and the home language is certainly part of what all learners know. In the discussion of the dynamics of a tornado and how to represent it mathematically, they developed a spoken description of the cylindrical spiral shape, which they refined into the description with gestures of a helical spiral motion. They were then challenged by the representation of the helical case as a rotation composed with a translation in the two dimensions of the plane. They started by proposing a composition of arrows that showed the direction of linear motions, and they soon realized that they had to solve the problem of the directions needed for the composed rotation, a transformation for which the two dimensions of the plane were not enough. The discussion continued for some minutes with references to angles, parallelism, and perpendicularity in the planar representation of spatial motions. These learners were not engaged the same in the final discussion guided by the teacher in Catalan. We illustrate this point with the next excerpt. Learners Briefly Reporting Their Mathematical Discussion in Whole Group Regarding participation and reasoning, this second piece of data allows us to argue the contrast between the engagement through the home language in group work and the engagement through the language of instruction in the whole group feedback. In our observations, the active role that the languages of the learners play during the small group discussions tend to decrease when the learning environment is the whole class. Excerpt 2 shows part of the later oral exchange of the group in Excerpt 1 with the teacher, during which only the language of instruction was used. The teacher asked a representative of the group to put up their answers on the board and explain it to the rest of the class. Eliseo hardly shared a brief part of the reasoning. He disregarded the discussion of the cylindrical and helical spirals produced in the group and did not explain how they had addressed the challenge of representing the tornado with the two dimensions of a plane only.

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Excerpt 2 Teacher: Eliseo, per què no dius res? Sé que heu estat treballant en el vostre grup. [Eliseo, why don’t you say anything? I know you’ve been working in your group.] Eliseo: Hem dibuixat algunes fletxes. [We’ve drawn some arrows.] Teacher: I quina heu triat finalment? Vols dir-ho en castellà? [And which have you finally chosen? Do you want to say it in Spanish?] Eliseo: Sabem que han de ser almenys dues fletxes i una és vertical perquè el tornado va cap avall. [We know that there are at least two arrows and one is vertical because the tornado goes down.] Teacher: Una translació vertical? [A vertical translation?] Eliseo: Una vertical. [Vertical.] Teacher: Els altres possibles moviments? [The other possible motions?] Eliseo: El tornado gira; creiem que una fletxa ha de ser la de la rotació. [The tornado turns around; we think that one of the arrows has to be the rotation.] Teacher: I què més? Abans parlàveu molt. [And what else? You were talking a lot.] Eliseo: Res més. [Nothing else.] Although the teacher used Catalan, she prompted the learners to choose the language for communication. These prompts of the teacher to use languages other than Catalan did not lead to use the home language in the whole group feedback. Compared to peer work, Luna, Máximo, Nicolás, and Eliseo were less engaged in the presentation involving the whole class, for which Eliseo was the representative, not without resistance. As a consequence, the other learners missed the opportunity to listen to, for example, the descriptions of the two spiral curves, the helical and the cylindrical, considered in that group. In fact, the term “spiral” and the variety of meanings for it, or the challenges involved in representing the curve of the tornado on a plane did not appear in the whole class feedback. At the end of the lesson, in a conversation with the teacher, she showed concern about the knowledge of the language of instruction and interpreted that this knowledge determined the participation of learners in the whole group. Teacher: Necessiten més confiança amb la llengua. Tot arribarà. . .i se’n sortiran amb els estudis. No participen més per la llengua. [They need more confidence with the language. All in good time. . .and they will get on with their education. They do not participate more because of the language.] Núria: Què passa amb la llengua? [What happens with the language?]

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Teacher: A les posades en comú, els altres parlen català més que ells. El saben parlar una mica. [In whole class discussion, others speak Catalan more than they do. They know how to speak a little.] This conversation shows most of the issues observed repeatedly with other teachers in Catalonia and Gauteng. The teacher in Barcelona did not question the mathematical activity and performance of her learners. Rather, she claimed that some of them had precarious knowledge of the language of instruction and needed more knowledge of this type for participation as well as for continuity with their studies. In our research, various teachers relate the use of the home languages in group work to poor knowledge of the language of instruction. The dismissal of some languages as useless “for getting on with an education” and the elevation of one language over the others for learning reasons are also common findings. The meanings that, either intentionally or not, were communicated by the teacher in the conversation of that day were even more complex. In the case of her learners, it was unclear how much knowledge of the language of instruction was enough and how that knowledge should be gained. Despite there being room for a diversity of opinions as to how much knowledge is enough according to a designated standard form of a language, we can agree that learners need some knowledge of the language of instruction. However, those four learners could speak Catalan well, though this language did not play a part in their lives outside the school. Máximo entered the local school system when he was a child, and his peers, Luna, Nicolás, and Eliseo, learned Catalan in the lessons for “late arrivals” at the school and passed the prescriptive local tests of proficiency in this language. Thus, it may not be precise to argue that the home languages played a remedial role in the group work of these learners. Learners may draw on their languages for different aspects of their lives, and may combine these languages for different purposes in their learning of mathematics. Those of us who speak two or more languages know about the pervasive socially differentiated use of languages. LEARNERS USING THEIR COMMUNICATIONAL ABILITIES In the study of language diversity in mathematics education, research has tended to consider diversity in relation to spoken and written languages rather than uses of language expressed through other modes of communication. In the previous section, the pedagogic value of using the learners’ spoken home languages has served to question the ubiquity of the monolingual norm. Now we expand the problematization of the monolingual norm with the focus on the interaction between verbal and nonverbal modes in

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lessons in which learners may not share a home language. We particularly reflect on the pedagogic value, for mathematics teaching, learning and thinking, of using modes beyond orality and writtenness. In any classroom, communication uses a variety of modes. There is no verbal text in gaze and gesture or in the body languages of signs of the deaf people. Cartesian graphs and geometric designs are also examples of nonverbal tools, culturally encoded to convey mathematical meanings. Indeed, the language of mathematics has been characterized as one in which verbal modes are accompanied by visual and symbolic modes, stimulated in recent years by the growth of digital technologies (Morgan & Kynigos, 2014). Much has been said about the visual and the symbolic in mathematics thinking and learning, but little is still known about the construction of learning spaces oriented to develop abilities other than speaking and writing. The creative use of language in interaction has been less studied, as well as the relationship between the verbal and the visual. The mathematical abilities of learners are mostly considered in their verbal dimension, and learning is primarily assessed through verbal and written performance. In what comes we provide a piece of unpublished data from one of the lessons in Planas (2014). We use it to illustrate how original personal images function alongside explanations in one or more verbal languages to support mathematical reasoning in group work. The drawings generated by the learners in the prior examples with the representation of the tornado would also serve as such a support. Language diversity in terms of modes of communication that include idiosyncratic configurations of individual learners is not a very frequent result in the lessons observed in Catalonia and Gauteng. In our contexts, verbal and verbal-symbolic texts dominate the language of the mathematics classroom except for the occasions in which the teacher prompts learners to work with diagrams of number arrays, bar graphs, tables with organized data or prototypical drawings of concepts in school mathematics, like quadrilaterals with specific orientations. Overall, our example with the production of unusual diagrams in multilingual group work suggests the role of the communicational abilities of learners in mathematics thinking and learning. It is sound to argue that the role of language in mathematics learning continues in the influence and juxtaposition of the verbal languages, the prototypical images and the creative configurations of learners. The effects of idiosyncratic drawings of learners in the mathematical discussion below reveal the strength of the creative use of language. Once more, although this use of language may become particularly pertinent in a learning environment where the learners do not share the spoken language at home, its promotion may be important for all learners in their struggle with the language of mathematics.

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Learners Supporting Mathematical Reasoning With Idiosyncratic Images in Group Work Similar to Excerpts 1 and 2, the teacher of the classroom, in this example, regularly proposed lesson dynamics that made it possible for learners to communicate their thinking with others. Almost at every lesson, she asked the class to develop and listen to different ways of solving the tasks, first in small group work and then in whole group sharing. Not all learners shared in front of the whole class. As noted earlier in the chapter, for the group with Eliseo, most learners whose home language was not the language of instruction, hardly volunteered to present the work done. The piece of data below shows three learners in Barcelona involved in the task of converting algebraic expressions into word texts. Roberto, a learner born in Ecuador, is discussing with Joana and Miquel an alternative to “any odd number” for 2x + 1. Roberto is one of the learners that never shared the group work with the whole class, although he was never absent and he meaningfully engaged in the discussions with his home language regardless of the language used by his peers. Although this is an instance with learners that spontaneously moved between two spoken languages in the interpretation of technical vocabulary (“regular”) and more generally in the discussion of answers to the task, we focus on how the interacting modes of communication strengthened the reasoning that was difficult to them. The drawings of Roberto in Figure 5.1 served to initiate a geometrical interpretation that was later included in the final written performance. This learner first drew a rectangle with two sides of length x and 2 and adjacent to a square with Side 1, and then marked the two areas. During the discussion in Excerpt 3, he drew a second rectangle for a value of x higher than the prior considered. At the end, Joana made the drawing with colors and shapes in Figure 5.2 to represent “The area of either a rectangle or an L form” as one possibility for the conversion of 2x + 1. Importantly, the thinking through drawings introduced the discussion of the numerical set for the values of x, which was not specified in the given task. The more sophisticated the understanding was of x, the more complex was the drawing.

Figure 5.1 Roberto’s drawings.

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Figure 5.2 Joana’s drawings.

Excerpt 3 Joana: Però nombre senar vol dir per tots els nombres senars i el teu dibuix és només per uns casos. No sé si vale. [But odd number mean all odd numbers and your drawing is for some cases only. I don’t know if it works.] Roberto: El cuadrado de costat u tiene que ser siempre así. [The square with side one has to be always like this.] Miquel: Però el rectangle no. [But the rectangle has not.] Joana: No vale. Ha de ser general. [It doesn’t work. It has to be general.] Roberto: Pues necesitamos dos dibujos . . . A lo mejor tres si el rectángulo es perfecto. [So we need two drawings. . .Maybe three if the rectangle is perfect.] Miquel: Perfecte? [Perfect?] Roberto: Si es dos y dos, es perfecto. [If it’s two and two, it is perfect.] Joana: Sí, regular, un quadrat. Però encara no sé això què voldria dir. Com diferents grups de nombres senars? [Yes, regular, a square. But I still don’t what that would mean? Like different groups of odd numbers?] Roberto: ¿Por qué? Yo hablo de áreas regulars. [Why? I speak about regular areas.] Joana: Sí, però quin dibuix tens pel número u? I el número tres? [Yes but what is your drawing for number one? And number three?] Roberto: Pues quatre dibuixos. No quiero un dibujo para el número u, lo quiero para equis cero. Lo que cambio es un lado. Yo tengo muchas más equis. [So four drawings. I don’t want a drawing for number one, I want it for x zero. What I change is one side. I have many more x’s.]

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Joana: Tinc una idea. Ho podem fer general. Puc agafar-te la llibreta? [I have an idea. You can make it general. Can I take your notebook?] This example clearly shows the copresence of mathematical symbolism, visual display, and verbal language in mathematics learning. Moreover, it reminds us that multimodality is not synonymous with the digital world (computers, podcasts, virtual media, and other types of digital literacy). We are aware of the multiple possibilities of the geometry dynamic software and graphic calculators at present, which would easily create representations of 2x + 1 on the screen that learners could quickly modify and visualize. Roberto and Joana introduced rather rudimentary sketches in their mathematical talk, but these sketches proved to be a significant start in their geometrical visual thinking. Without the aid of the software or the calculator, they also found a dynamic way to manipulate and change the length of one side of the rectangle. Any understanding of how they made meaning of the task in that lesson would be incomplete without attending to the modes and languages at play. Taken alone, none of these modes and languages would be sufficient in the development of mathematically relevant conceptual explanation and arguing. The use of only the images or only the languages of the learners might have been relatively limited in service of the reasoning developed. The reproduction of the drawing in Figure 5.2 in the written report for the teacher indicates that, for the learners in this group, the image created was just as important as the texts with the answers. As with the learners in the prior section, the drawings here reveal mathematical reasoning that some learners may have difficulty writing or speaking about in any of their languages. In conversations with other learners regarding images created for the discussion of mathematical tasks in group work, we found that some of them related the utility of some of their drawings in moments of difficulty with the understanding of the spoken languages and the mathematics. These visual representations replaced verbal descriptions. One of the learners from Latin America said that she often made drawings that were “understood by them [the learners in the group] to avoid the mess with the words.” Another learner of the Catalan school system whose home language was Amazigh brought up questions of creativity and normativity in relation to the production of images unintended by the teacher. In a conversation about her low participation in the mathematics classroom, she responded that she was active in the group work with peers. She added, “Sometimes teachers say, hey, draw a picture, and I am good at drawing. . .But sometimes they do not mean any picture. Then I need to know exactly, and I do my best.” This quote raises the question of the limits of creativity in language use in a given classroom culture.

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TOWARDS A MULTILINGUAL NORM THAT IS MATHEMATICALLY RELEVANT We started with two questions: What is the role of language in the mathematics classroom? What is the role of the languages of the learners? We have argued the potential role of the language of the mathematics classroom as a resource for mathematics teaching, learning, and thinking. Specifically, for the languages of the learners, we have presented two multilingual practices that can serve to the realization of the language of the classroom as a resource. These practices are grounded in two major approaches to the understanding of language diversity in mathematics education. Regarding the verbal dimension of language diversity, we have discussed some benefits of the flexible use of home languages. Regarding the wider communicational dimension of language diversity, we have reflected on some benefits of the creative use of modes of communication other than speaking and writing mathematics (in the language of instruction). Since some of the fundamental properties of language include its flexible and creative expression and the fact that it is multimodal (Halliday, 1978), these are possible and achievable practices. A number of reasons, however, indicate that the monolingual norm still prevails in mathematics education. When we refer to the term “multilingual norm” we are indicating an approach to language that does not stay at the surface level of linguistic symbols and grammatical rules. The research community and the school context are in the early stages of producing newer norms aligned with flexibility and creativity in language use during mathematics teaching, learning, and thinking. It is a widespread idea that learners who are not “native” speakers of the language of instruction are deficient communicators of school mathematics. Moreover, it is common to find school practices of slow and simplified language use, along with curricular remedial arrangements, in lessons with learners whose home language is not the language of instruction. The primary goal is often to make these learners monolingual as they work with academic written texts, rather than to support them in their learning of mathematics. The message sent is that the language of instruction is the appropriate language for the learning of mathematics. In this chapter, we have shown multilingual learners in interaction who are proficient communicators of the language of mathematics through their mathematically relevant uses of languages and abilities. An important point in our arguments is that multilingualism cannot be separated from multimodality. Learners naturally switch between languages and modes in their learning processes. Our research indicates that the learners use their home languages as well as idiosyncratic drawings for engagement with a mathematical reasoning

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that is difficult to them, but also that the presence of these resources is lower in the whole group. Following our observations, many multilingual practices are spontaneous initiatives of learners and develop with minimal pedagogical intervention from teachers. A quick interpretation might give the impression that these practices do not have to be taught by the mathematics teacher. Far from this, progress towards the multilingual norm strongly requires coordination of mathematics teachers, educators, and researchers for the design and implementation of mathematical curricula that explicitly refer to the flexible and creative use of languages and modes of communication. Otherwise, multilingual practices can be confined to interaction in the small group, and implicitly censored in the whole class. This cannot be done in practice without awareness of all the languages that pedagogically matter in the classroom: the languages of the learners, of the teacher, and of mathematics. Importantly, the learners’ texts cannot be assessed only or mainly according to orthography, grammar, and lexical errors in the use of technical terms. Assessment of texts needs to inform the achievement of mathematical content goals and the use of meaningrelated phrases (e.g., “What I change is one side”). In the data presented, for example, “different groups of odd numbers” is not the right expression in the academic language of mathematics, but this is an effective meaning-related phrase for explaining the numerical subsets involved in the interpretation of the task. In our two contexts, there is a developmental gap to be filled regarding professional knowledge of language demands that are mathematically relevant in teaching and learning. Mathematics teachers are not trained to think about language and language diversity in specific relation to mathematics teaching and learning. The processes of learning to teach mathematics are mapped to the processes of becoming a teacher of mathematical contents and the idea of language demands is considered on the surface level of vocabulary and grammar. A similar monolingual bias can be found in other countries where the language of instruction is produced as the privileged tool for teaching, learning, and thinking mathematics (Barwell et al., 2016; Essien, Chitera, & Planas, 2016). We thus have a long way to go in developing professional knowledge for mathematics teaching that is language responsible for the benefit of mathematics learning. More could be said about the role of language and specifically of the languages of learners in mathematics teaching and learning. It is our hope to have motivated views that are socially, linguistically, and pedagogically responsible with all learners. There are good reasons why teaching practices that address mathematically relevant language demands should be developed from the strategies learners themselves use.

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ACKNOWLEDGMENTS This collaboration is funded by the Catalan Institute of Advanced Studies and Research—ICREA; and the Spanish Government, EDU2015-65378-P, MINECO/FEDER. REFERENCES Adler, J. (2000). Widening the lens—changing the focus: Researching and describing language practices in multilingual classrooms in South Africa. In H. Fujita, Y. Hashimoto, & T. Ikeda (Eds.), Abstracts of plenary lectures and regular lectures, ICME9 (pp. 20–21). Makuhari, Japan: ICMI. Austin, J. L., & Howson, A. G. (1979). Language and mathematical education. Educational Studies in Mathematics, 10(2), 161–197. Barwell, R. (2016). Mathematics education, language and superdiversity. In A. Halai & P. Clarkson (Eds.), Teaching and learning mathematics in multilingual classrooms (pp. 25–39). Rotterdam, The Netherlands: Sense. Barwell, R., Clarkson, P., Halai, A., Kazima, M., Moschkovich, J., Phakeng, M., . . . Villavicencio, M. (Eds.) (2016). Mathematics education and language diversity. The 21st ICMI Study. New York, NY: Springer. Chronaki, A., & Planas, N. (2018). Language diversity in mathematics education research: From language as representation to the politics of representation. ZDM, 50(6), 1101–1111. Civil, M. (2012). Opportunities to learn in mathematics education: Insights from research with “non-dominant” communities. In T. Y. Tso (Ed.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education (pp. 43–59). Taipei, Taiwan: PME. Essien, A., Chitera, N., & Planas, N. (2016). Language diversity in mathematics teacher education: Challenges across three countries. In R. Barwell et al. (Eds.), Mathematics education and language diversity. The 21st ICMI Study (pp. 103–119). New York, NY: Springer. Halliday, M. A. K. (Ed.) (1978). The social interpretation of language and meaning. London, England: Edward Arnold. Hunter, R., Civil, M., Herbel-Eisenmann, B., Planas, N., & Wagner, D. (Eds.). (2018). Mathematical discourse that breaks barriers and creates space for marginalised learners. Rotterdam, The Netherlands: Sense. Langer-Osuna, J. M., & McKinney de Royston, M. (2017). Understanding relations of power in the mathematics classroom: Explorations in positioning theory. In A. Chronaki (Ed.), Proceedings of the 9th International Mathematics Education and Society Conference (pp. 645–653). Volos, Greece: University of Thessaly. Morgan, C., & Kynigos, C. (2014). Digital artefacts as representations: Forging connections between a constructionist and a social semiotic perspective. Educational Studies in Mathematics, 85(3), 357–379.

110 N. PLANAS and M. NGOEPE Mparutsa, S. T. (2011). Towards the development of instructional strategies for teaching algebra in multilingual classrooms in South Africa. PhD Manuscript. Johannesburg, South Africa: The University of the Witswatersrand. Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. London, England: Routledge Kegan & Paul. Planas, N. (2014). One speaker, two languages: Learning opportunities in the mathematics classroom. Educational Studies in Mathematics, 87(1), 51–66. Planas, N. (2018). Language as resource: A key notion for the understanding of the complexity of mathematics learning. Educational Studies in Mathematics, 98(3), 215–229. Planas, N., & Civil, M. (2013). Language-as-resource and language-as-political: Tensions in the bilingual mathematics classroom. Mathematics Education Research Journal, 25(3), 361–378. Planas, N., Morgan, C., & Schütte, M. (2018). Mathematics education and language. Lessons from two decades of research. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education. Twenty years of communication, cooperation and collaboration in Europe (pp. 196– 210). London, England: Routledge. Planas, N., & Setati, M. (2009). Bilingual students using their languages in their learning of mathematics. Mathematics Education Research Journal, 21(3), 36–59. Planas, N., & Setati-Phakeng, M. (2014). On the process of gaining language as a resource in mathematics education. ZDM, 46(6), 883–893. Ricento, T. (2014). Thinking about language: What political theorists need to know about language in the real world. Language Policy, 13(4), 351–369. Ruíz, R. (1984). Orientations in language planning. National Association for Bilingual Education Journal, 8(2), 15–34. Setati, M., & Adler, J. (2000). Between languages and discourses: Language practices in primary multilingual mathematics classrooms in South Africa. Educational Studies in Mathematics, 43(3), 243–269. Walsh, M. (2011). Multimodal literacy: Researching classroom practice. Sydney, Australia: Primary Teachers Association of Australia.

CHAPTER 6

DISABILITY AND EQUITY IN MATHEMATICS Vasiliki Chrysikou, Panayiota Stavroussi, and Charoula Stathopoulou

Teaching and learning of mathematics is at the heart of research interest in the mathematics education community. In the last decades, the interest of researchers has shifted from focusing on the teaching triangle (student– teacher–mathematics) to focusing on broader sociocultural (Lerman, 2000; Pais & Valero, 2014) and political (Gutiérrez, 2013) factors that affect mathematics teaching and learning. The development of approaches of the last few decades, such as ethnomathematics, critical mathematical education, and post-structural theories has led to revisions of perceptions that mathematics is a neutral object. At the turn of the century, Lerman (2000) called this shift in research interests, a “social turn.” Recently, Gutiérrez (2013) introduced the term “sociopolitical turn” to describe a growing body of research, but also applications that mainly pay attention to a political dimension, other than the social and cultural dimensions involved. Gutiérrez, while referring to equity, summarizes three theoretical perspectives that focus on knowledge, power, and identity as “interwoven and arising from (and constituted within) social discourses” (Gutiérrez, 2013, p. 40).

Equity in Mathematics Education, pages 111–130 Copyright © 2019 by Information Age Publishing All rights of reproduction in any form reserved.

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She has noted that works based on these theories seek not only a better understanding of mathematics education, but by working in such a way, they contribute directly to endorsing justice in their practices. On the basis of a sociocultural and sociopolitical perspective, difficulties faced by students in school mathematics are not only considered the result of a profile of their strengths and weaknesses, but are also related to broader social, cultural, economic, and political factors (Morgan, 2014). What does such an approach to mathematics teaching and learning implies if we consider students with disabilities? A sociocultural perspective of mathematics education for students with disabilities emphasizes several important issues: the social model of disability, the implementation of inclusion practices that offer opportunities for the teaching of mathematics concepts besides procedures, and the dynamic connection between mathematics learning and context. However, research evidence of mathematics education for students with disabilities through a sociocultural perspective is very limited, especially as regards to students with intellectual disability (Lambert & Tan, 2017). The National Council of Teachers of Mathematics (NCTM) outlines six principles for school mathematics that are fundamental to high-quality mathematics education, the first of which is the equity principle. This principle calls for “high expectations and strong support for all students regardless of their personal characteristics, backgrounds, or physical challenges” (NCTM, 2000, p. 12). Equity does not imply that instructional practices should be the same for every student, but “it demands that reasonable and appropriate accommodations be made and appropriately challenging content be included to promote access and attainment for all students” (NCTM, 2000, p. 12) including students with intellectual disability. Teaching mathematics is an important process, which, among other things, contributes to enhancing independence and inclusion of students in the community. This view has gained ground leading to a greater emphasis on the design or adaptation of curricula that include instruction of mathematics concepts, such as algebra, rather than just arithmetic, to students with intellectual disability in order to facilitate and support learning of life skills (Jimenez, Courtade, & Browder, 2008; Martinez, 1998; Rodriguez, 2016). MATHEMATICS EDUCATION FOR STUDENTS WITH INTELLECTUAL DISABILITY People with intellectual disability comprise quite a heterogeneous group. According to the latest definition of the American Association on Intellectual and Developmental Disabilities (AAIDD) “Intellectual disability is a disability characterized by significant limitations in both intellectual functioning

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and in adaptive behavior, which covers many everyday social and practical skills. This disability originates before the age of 18” (AAIDD, 2018). Considering the diagnostic practices and criteria, significant changes have occurred over the years pertaining to the sufficiency of IQ tests for a diagnosis of intellectual disability (McKenzie, Milton, Smith, & Ouellette-Kuntz, 2016). Specifically, adaptive functioning criteria to identify intellectual disability have been added in diagnostic manuals, such as the Diagnostic and Statistical Manual of Mental Disorders—DSM 5th edition (American Psychiatric Association [APA], 2013), while the role of the construct of adaptive behavior is further pointed out in the DSM-5, which relies on the levels of adaptive functioning to specify the severity of intellectual disability. For many years, the diagnostic process was primarily based on measurements of intelligence (IQ tests), although recent diagnostic practices highlight the combination of assessment procedures with an emphasis on the assessment of adaptive behavior, the results of which are interpreted in relation to people of the same age, gender, and sociocultural context (APA, 2013). The construct of adaptive behavior is conceptualized as including conceptual (e.g., money, time, and number concepts), social and practical skills (e.g., use of money; AAIDD, 2018; Tassé, Luckasson, & Schalock, 2016). Such skills have a central role in educational programs for students with intellectual disability, especially in those that provide a combination of academic and life skills to support the needs and enhance the strengths of the learner (Bouck, 2017; Goldman, Hasselbring, & Cognition and Technology Group at Vanderbilt, 1997). Specifically, several educational programs emphasize the instruction of functional academic skills, including functional mathematics, which refer to basic mathematics skills that are useful to students’ daily lives, such as money and time management (Algozzine & Ysseldyke, 2006). Although the teaching of functional mathematics has been studied more systematically, the teaching of mathematics concepts to students with disabilities is a less studied domain that poses challenges as regards instructional content and techniques, especially in the context of promoting access to general education—in the area of mathematics— for students with intellectual disability (Browder, Spooner, & Trela, 2011; Göransson, Hellblom-Thibblin, & Axdorph, 2016). In the last decades, there is a growing research interest in the instruction of mathematics for students with intellectual disability. In particular, researchers have focused mainly on studying teaching of basic/functional mathematics, such as number recognition, money and time management, performance of simple arithmetic operations, and speed and accuracy of problem solving (Browder, Spooner, Ahlgrim-Delzell, Harris, & Wakeman, 2008). In the majority of studies, direct instruction with various strategies, such as task analysis, constant time delay, simultaneous prompting, or flashcards was used (Dihoff, Brosvic, Epstein, & Cook, 2005; Rao & Kane, 2009)

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while the participants attending most of them were primary school students (Butler, Miller, Lee, & Pierce, 2001). However, given that an increasing number of students with intellectual disability enter secondary education, there is a need for more empirical evidence of teaching mathematics to secondary school students and young adults (Bouck, Bassette, Taber-Doughty, Flanagan, & Szwed, 2009; Hord & Bouck, 2012; Hua, Woods-Groves, Kaldenberg, Lucas, & Therrien, 2015; Jimenez, Browder, & Courtade, 2008). Recent research evidence of mathematics education from the perspective of inclusion does not seem to be reflected in studies involving students with intellectual disability (Göransson et al., 2016). Literature reviews reveal an emphasis on the use of behavioral approaches to mathematics education for students with intellectual disability (Browder & Grasso, 1999; Butler et al., 2001; Hord & Bouck, 2012). The development of sociocultural approaches in mathematics education has had little influence on the field of special education and, in particular, education of students with intellectual disability (Boyd & Bargerhuff, 2009; Woodward & Montague, 2002), mainly due to the influence of medically-oriented perspectives that focus on individual factors (Ainscow, 2005). In particular, parameters such as the impact of the sociocultural context of the students do not appear to be a prevalent issue in studies on mathematics education of students with intellectual disability. Therefore, the way that various factors of the social context may impact the processes of teaching and learning mathematics of students with intellectual disability remains unclear. On the one hand, existing research on teaching mathematics to students with intellectual disability focuses mainly on a procedural approach to mathematical knowledge, taking into account only the cognitive characteristics of the students. On the other hand, in the area of mathematics education for students without disabilities, researchers focus increasingly on the role of sociocultural backgrounds, thus understanding mathematics as a network of social practices. Woodward and Montague (2002) describe the situation by stating that research data on teaching mathematics to students with intellectual disability, which are in line with mathematics education reforms, are at a very early stage. Göransson et al. (2016, p. 196) argue that the absence of research data on teaching mathematics to students with intellectual disability in accordance with the international research community of mathematics education may lead to the separation of educational practices for students with intellectual disability from those of general education. Considering the above-mentioned perspectives and issues, the process and the results of an action research study grounded in a sociocultural perspective of mathematics learning are described in the following subsection.

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AN ACTION RESEARCH ON TEACHING MATHEMATICS TO STUDENTS WITH SEVERE INTELLECTUAL DISABILITY Our work draws on sociocultural perspectives of mathematics learning, where emphasis is placed on the understanding not only of the cognitive characteristics of students with intellectual disability, but also of the influence of the school and the broader educational context on their learning process. Through an action research we aim to explore how a sociocultural view can inform and enrich mathematics teaching and the learning of students with intellectual disability. In this approach, an active participation of students in mathematics activities is considered a key notion (Civil & Planas, 2004). At this point we would like to share some examples of the engagement of students with intellectual disability in mathematics activities during the action research study. In doing so, we also provide examples of their mothers’ and the teacher-researcher’s1 practices that reinforce or hinder students’ participation. Consider three students with severe intellectual disability (Katerina, 16; Vangelis, 23; and Anastasia, 28 years old) attending a class of a secondary special school in Greece, who can count, recognize, and write numbers up to 10. At the beginning of the action research project, both teacherresearcher and the mothers approached mathematics in a more “traditional” or “academic” way. Here are some examples depicting the mothers’ practices at home and their overprotection over students’ engagement in mathematics activities outside the home, mostly because of these students’ severity of intellectual disability: We count numbers [ . . . ] with spoons, with pencils, with whatever we find. (Katerina’s mother, Interview) At home we have a pencil board on which we write. Kostas [her brother] tells her: 1 + 1 equals 2, 2 + 2 equals 4. (Katerina’s mother, Interview) He likes to count. For example, on the way, we try to measure how many trucks pass by on the street. (Vangelis’ mother, Interview)

M: In shops where she buys clothes . . . she [Anastasia] walked in, chose something and said “I want this red shirt, I want these shoes.” T-R: And then? Did she pay? M: No, I haven’t done that . . . allow her to manage money. (Anastasia’s mother, Teacher-Researcher, Interview) T-R: Nice, and then what happens at the cashier? M: She helps me put the products in shopping bags. T-R: Does she also help you with money? M: No, she doesn’t know what “money” is. (Katerina’s mother, Teacher-Researcher, Interview)

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In the following excerpt, Anastasia is trying to use the calculator for the addition “3 + 2 =” which is written in her notebook:

T-R: Which number is written first? A: Three. T-R: Excellent! Press “three” [Anastasia presses the button]. Then? A: Two. T-R: Oh. First there is . . . [teacher-researcher points at “three”] A: Three. T-R: Good! After, there is . . . [teacher-researcher points at “plus”] A: Plus. T-R: Excellent! Press “plus” [Anastasia presses the button]. Then . . . [teacher-researcher points at “two”] A: Two [Anastasia presses the button]. T-R: And last is . . . [teacher-researcher points at “equal”] A: Equal [Anastasia presses the button]. T-R: Well done! What is the result? A: Five [Anastasia writes down the result]. (Teacher-Researcher, Anastasia, Classroom)

Both teacher-researcher and mothers seem to use activities, which are not age-appropriate to the students. Moreover, they try to teach students in a very procedural way by presenting mathematics decontextualized and isolated from their everyday activities. This becomes obvious from the answer of Katerina in the following question:

T-R: When you listen to the word “mathematics” what do you think of? K: Subtractions, additions. T-R: How do you feel? K: I say to myself “when will I finish.” (Teacher-Researcher, Katerina, Interview)

Among multiple contextual and background factors that influence the decision-making and teaching practices of the teacher-researcher, a prominent position is held by the teacher-researcher’s academic training, previous experience, and system of beliefs (Ashton, 2015), that may make it difficult to uncover mathematics in everyday contexts. During the action research, the teacher-researcher managed to reflect on her beliefs about mathematics education of students with intellectual disability, as well as on her collaboration with parents: Students [with ID] are not like primary education students. They are adolescents or even adults. Their physical needs are different. Why didn’t I consider their age? (Teacher-Researcher, Journal)

Disability and Equity in Mathematics 117 Why should they [students] show interest in counting plastic glasses or wooden cubes? (Teacher-Researcher, Journal) Why do I care so much about students’ mathematical performance? I should care more about students’ participation in mathematics activities. (TeacherResearcher, Journal) It is unfair that my students don’t learn mathematics that they really need. (Teacher-Researcher, Journal) Students will graduate from school in two or three years. When will they learn the type of mathematics they need in their everyday lives? (Teacher-Researcher, Journal) Why did I allow fear to overwhelm me and not try to collaborate with mothers earlier on? Besides providing me with information, they could help with the implementation of the educational program outside school. We both want what is best for students. (Teacher-Researcher, Journal) Let’s think about it. Is the way you teach mathematics the only significant factor of influence? . . . What about the role of the school? (Critical friend, Discussion)

Three key elements promoted shifts in the decision-making of the teacher-researcher: critical self-reflection, a sense of social justice, and critical friends. Critical self-reflection and a sense of social justice allowed her to question her teaching practices, which she had taken for granted. As Mezirow (1997) notes, “Becoming critically reflective of one’s own assumptions is the key to transforming one’s taken-for-granted frame of reference, an indispensable dimension of learning for adapting to change” (p. 9). Since one’s beliefs develop socially through interaction with others, the critical friends played an important role in helping the teacher-researcher change her perspective framework (Mezirow, 2003). The teacher-researcher in collaboration with the mothers decided to engage students in everyday social activities that contribute to the development of mathematical knowledge. Learning and teaching mathematics were approached not as processes carried out with socially “sterile” material but as sociocultural processes related directly to students’ daily lives. Thus, contexts with a practical application to students’ daily routine were used in a functional mathematics approach. Mothers provided information to the teacherresearcher concerning social contexts that are meaningful to their children, such as the purchase of products for school events, the purchase of ingredients for baking cake, and so on. The teacher-researcher designed and implemented various authentic communication situations with realistic scenarios, giving students the opportunity to actively participate in social activities that enhance both mathematics learning and social inclusion.

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As DiPipi-Hoy and Jitendra (2004) mention, community-based instruction plays a very important role in enhancing social inclusion of people with disabilities. During the action research multiple learning spaces outside the school framework were used (i.e., different stores, bakeries, etc.) helping to build a bridge between everyday life and school knowledge. Community-based instruction offered opportunities to students for generalizing knowledge gained in the school environment while encouraging their socialization and autonomy. During visits to different stores students were supported in becoming active participants in everyday activities that promote both their mathematics learning and social inclusion: [Katerina asks the employee for a brittle] K: I want a brittle. E: There, at the top, is a brittle. Pick one. K: [Katerina chooses a brittle and places it in front of her]. (Pause for 10 seconds) T-R: What would you ask? K: How much does it cost? E: 90 cents. K: Here! [She gives one euro] T-R: Katerina, what did you give? K: 90 cents. E: One euro. K: One euro. T-R: Well done, Katerina! (Katerina, employee, teacher-researcher, visit to a kiosk)

Students faced communication difficulties and restrictions in making money transactions mostly because of their limited engagement during previous experiences of shopping at grocery shops. This, however, was overcome by means of particular adaptations, such as using the one-more-than strategy2 (Denny & Test, 1995) to pay for groceries, and students were given the opportunity to fully participate in a variety of activities. However, because some individual skills, such as making a financial transaction, are difficult to be repeated in real-life conditions, a combination of educational experiences in a realistic context and in a simulation context (i.e., role playing in the classroom; Xin, Grasso, DiPipi-Hoy, & Jitendra, 2005) was used. Students were trained in the classroom using the one-more-than strategy with real money and empty packages of products (i.e., milk, cookies, detergent, etc.):

T-R: Corn flakes cost 3 euros and 89 cents. How many euros are there? A: Three euros. [She gives coins one-by-one without counting them.] T-R: We should count. A: One, two, three, four . . .

Disability and Equity in Mathematics 119 T-R: You said three. Let’s count again . . . one . . . A: Oh yes. One, two, three. T-R: Ok! For the rest, how much will you give me? A: One euro. T-R: Well done, Anastasia! [Teacher-researcher, Anastasia, classroom]

Taking into consideration that students with severe intellectual disability have difficulty initiating things independently and need motivation and reward for their efforts, it was considered crucial to collaborate with the students’ mothers. For this reason, mothers were informed about students’ educational program at school so that they could continue to encourage their children to actively participate in similar activities outside the school framework. Thus, through the creation of DVDs showing students’ participation during visits to different stores, the teacher-researcher provided an opportunity for the students’ families to observe successful attempts to integrate their children into organized money-trading training activities. As a result, mothers and important others in each family recognized students’ efforts, rewarded them, stimulated their self-confidence, and contributed to continual practicing outside the school context: Yesterday we bought some chocolates and they had a sticker with the price on. And she went and got a piece of paper and wrote down the price she saw. Then she asked me, “There, in the kitchen, there is a juice. How much did it cost?” I said, “How much? 0,35€.” She wrote 0 and asked, “Now?” I said, “comma thirty-five euros.” She asked, “3 and 5?” and I said, “Congratulations.” (Anastasia’s mother, Interview)

Although mothers were encouraged to collaborate with the teacher-researcher and allow students to actively participate in similar activities that take place outside the school context, their actual practices varied greatly. During the action research a variety of factors were identified which limited opportunities for students’ active participation in social activities that enhance mathematical thinking. Their mothers’ free time, their low expectations, and their past experiences with regard to the influence of the social context on the social inclusion of students with severe disabilities, appear to have restricted students’ participation: Thinking about others’ opinions is also an issue. (Vangelis’ mother, interview) I’m a bit ashamed [ . . . ] when people see her like this. (Katerina’s mother, interview)

All three mothers strive for the best of their children but the influence of the wider social environment that they live in has led to the formation of specific beliefs, thus restricting social inclusion and active participation

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of students in social activities. Families of students with intellectual disability often experience discrimination because of society’s dominant attitudes towards the social integration of people with disabilities, resulting in the forming of specific attitudes and beliefs about the social adaptation of their children. As Seligman and Darling (2007) mention “in order to prevent stigma-producing encounters, then, families may have to structure their lives to avoid social situations that would require their children to speak or perform roles that would otherwise call attention to their disabilities” (p. 139). Perhaps collaboration with a psychologist in the school context would be useful for encouraging mothers to overcome their second thoughts and help them “build” new pleasant experiences that could enhance the students’ active participation in everyday activities promoting mathematics learning and social inclusion. Unfortunately, due to the economic crisis this special school was not provided with a psychologist during that school year. The absence of this specialty depicts a weakness of the educational system to ensure the necessary educational benefits for all students and their families (Pais, Stentoft, & Valero, 2010). Along with the students’, their parents’, and teachers’ efforts, schools also need to change in order to respond to the education of all students. Thus, highlighting the political parameter, it is obvious that aspects of the wider social context could hinder the learning process and social inclusion of students with severe disabilities. In summary, during the process of this action research guided by a sociocultural perspective, the teacher-researcher managed to reflect on her own practices and beliefs with an aim to change her educational choices concerning mathematics education for students with intellectual disability. Moreover, the development of a collaboration between school and family promoted students’ active participation in contextualized and real-life mathematics activities. During this procedure, broader factors emerged that limited students’ actions, such as the stigma experienced by the families, as well as school parameters, which hinder equity in education for all students (a political parameter). IMPLICATIONS FOR TEACHING MATHEMATICS TO STUDENTS WITH INTELLECTUAL DISABILITY From a sociocultural point of view, which was adopted in the previously presented action research, mathematics education for students with intellectual disability could be enriched by taking into consideration broader social, cultural, and political factors that promote or hinder students’ educational opportunities. Nonetheless, more research data in the field of mathematics education for students with intellectual disability is required for the integration of sociocultural perspectives in educational processes.

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In the process of promoting equity in mathematics education for all children, including students with severe intellectual disability, attention should be paid to sociocultural parameters (such as school culture or parents’ beliefs) that interact dynamically with each other, creating specific conditions during mathematics learning. Regarding school practice, adopting a sociocultural approach to mathematics education for students with intellectual disability calls for greater emphasis on students’ participation and factors affecting their active engagement in the educational process. There is an increasing recognition in the relevant literature of the need to promote the active participation of students with intellectual disability in the learning process on the basis of his/her strengths rather than weaknesses (Shogren, Wehmeyer, Schalock, & Thompson, 2017; Wehmeyer, Shogren, Singh, & Uyanik, 2017). This perspective emphasizes the critical role of teachers in creating interactive and supportive environments and providing students with necessary time and an appropriate guidance and instruction to cultivate mathematical knowledge and skills that contribute to their social inclusion and autonomous living in the community. These activities may be of the simplest, such as counting the number of students absent from the school, to more complex, such as students’ participation in grocery shopping of ingredients for a recipe and implementation at the school kitchen. Nonetheless, students with intellectual disability can benefit from active participation in contextualized and real-life mathematics activities. In line with this perspective, the educational programs for students with intellectual disability usually consist of a combination of academic skills and functional life skills. However, there is an increasing interest in the instruction of mathematics concepts to these students, besides simple arithmetic, in the context of promoting their access to general education in the area of mathematics. As regards the development of math-related instructional objectives and activities, it has been suggested that it should be based on the criterion of “ultimate functioning” (i.e., whether the skills to be taught are useful in everyday life and chronologically age appropriate; Browder & CooperDuffy, 2003). Within this context several issues emerge that consequently affect the curriculum content. On the one hand, the usefulness of a skill to students’ daily life and not just the acquisition of more information is recognized as an important element of the educational process. On the other hand, the objectives and activities selected by the teacher should be ageappropriate and not have, for example, an adolescent who keeps counting spoons (Browder et al., 2004; Storey & Miner, 2011). Overall, the instructional objectives and activities in the mathematics area should aim at positive social outcomes through the implementation of evidence-based strategies in the context of preparing students with disabilities for their transition

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to adulthood (Patton, Polloway, & Smith, 2000; Stavroussi, Papalexopoulos, & Vavougios, 2010). In addition, the students’ sociocultural context is of critical importance in mathematics education for all students, including those with intellectual disability. From this perspective, the student is not viewed merely as an individual within a classroom who passively acquires information but as a person participating in the school’s learning and social processes, and also as a person who at once, has differing roles, as a school member, a family member, a neighbor, a member of a community and society, in general. As regards teacher’s practices, understanding the roles that a student has besides being a member of the class and having access to information of the student’s daily routines and important contexts initiates a pathway for creating valuable resources which can be used in mathematics education. As Porter and Lacey (2005) mention “if we want to understand what is happening in a particular environment then we need to see it through the varied perspectives of those who are part of it” (p. 86). Students with intellectual disability can provide accurate and reliable information about their feelings, thoughts, preferences, and needs, as long as the questions are presented in a structured and supportive form. In addition, families of students could provide useful information about the environments in which students acquire informal mathematical knowledge, about their interests and preferences, and so on, ensuring that school activities students are engaged in will make sense to them and have a practical application to their everyday lives. Civil (2007) supports that parents can be important sources of knowledge for the teacher by providing information that can be used in the design of students’ educational program. As Epstein (1995) mentions: If educators view children simply as students, they are likely to see the family as separate from school. That is, the family is expected to do its job and leave the education of children to the schools. If educators view children as children, they are likely to see both the family and community as partners with the school in children’s education and development. Partners recognize their shared interests in and responsibilities for children, and they work together to create better programs and opportunities for students. (p. 701)

Apart from giving information, parents can collaborate with schools to create appropriate conditions for the development of mathematical knowledge inside and outside the school environment (DiPipi-Hoy & Jitendra, 2004; Tekin-Iftar, 2008). It is important that these conditions are of a supportive and interactive environment to be expanded and shaped within the family context as well, in order for teachers in collaboration with students themselves and their parents, approach the educational process holistically and form appropriate, personalized, and meaningful learning objectives. In addition to the significant contribution of parents to educational practice,

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through collaboration, they will be able to perceive their children’s abilities as well as their needs for independent living and social inclusion (DiPipiHoy & Jitendra, 2004). Especially in the case of adult students, parents may have acquired certain habits that often hinder the autonomous living of their children (Seligman & Darling, 2007). School and family collaboration can contribute to social inclusion and a smooth transitioning into adulthood of secondary school students with intellectual disability, which can be promoted through mathematics education. From a sociocultural perspective, the learning and teaching of mathematics to students with intellectual disability could be considered a continuity of social practices that take place in specific educational settings. During design, implementation, and evaluation of the educational process, teachers play a very important role through choices and decisions they make. On behalf of the teachers, an important issue in perceiving and implementing mathematics education as a process that exceeds the narrow boundaries of school and individual students is to critically reflect on their own personal practices and beliefs, which define their, sometimes unconscious, educational choices. Beliefs that mathematics is a set of absolute and predetermined knowledge leads teachers to restricting the education of students with intellectual disability within the narrow limits of the classroom. On the contrary, perceiving mathematics learning as a process that includes participation in social practices is related to the emergence of the importance of student and teacher interactions, as well as students’ active engagement in everyday social activities. However, it should be kept in mind that teachers’ beliefs may be enacted in given moments of practice while personal factors interact with various external factors situated within social context, such as classroom-context, school-context, and national education policy factors (Ashton, 2015; Buehl & Beck, 2015). Taking into account recent theoretical and research literature as well as the results of our action research study presented above, we suggest that adopting a sociocultural perspective in teaching and learning mathematics would have beneficial outcomes for students with intellectual disability in terms of social inclusion and independence (Göransson et al., 2016; Kroesbergen & Van Luit, 2005). It should be mentioned, however, that there are several key influential and interrelated factors, which could facilitate or impede teaching and learning of mathematics in the context of the aforementioned perspective as regards to students with disabilities. These factors are delineated by features of the direct educational context—such as the student, the teacher, the academic subject—in this case mathematics, and the classroom, as well as other features related to the broader educational and social context—such as school environment, family, school–family collaboration, significant others, community, available research evidence, educational policy, and the social environment (Figure 6.1; Chrysikou, 2016).

124 V. CHRYSIKOU, P. STAVROUSSI, and C. STATHOPOULOU Country (economic situation, beliefs about mathematics, etc.) School context (culture, staff, interdisciplinary collaboration, infrastructure, etc.) Classroom (culture, physical organization, etc.)

Teacher (beliefs, previous experiences, expectations, education, etc.)

Student (age, gender, cognitive characteristics interests, etc.)

Family (composition, practices, expectations, beliefs, etc.)

Learning

Family–School collaboration (practices, beliefs, etc.)

Mathematics (epistemological characteristics, curriculum, etc.)

Important others (critical friends, family, friends, etc.)

Educational policy (curriculum, laws, staff, etc.)

Teaching

Research data (on mathematics education, on intellectual disability, etc.)

Community (beliefs, practices, etc.)

Figure 6.1 Factors affecting mathematics education for students with disabilities.

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In this sense, the effects of the dynamic interplay of the abovementioned factors along with the basic principles of a sociocultural approach should be considered in the process of decision-making about mathematics education for students with disabilities. CONCLUSION Nowadays, it is widely accepted in the research and educational community that we cannot focus on mathematics education without reference to social, political, and cultural parameters (Appelbaum & Stathopoulou, 2016). However, regarding mathematics education for students with disabilities, it seems that this is not the case yet (Tan & Kastberg, 2017). Namely, the rhetoric of education for all students through the prism of equity seems not to be fully translated in policies and practices pertaining to students with disabilities, especially students with intellectual disabilities. These students remain marginalized in terms of participation in general schools’ mathematics classes. Nonetheless, the development of inclusive schools (Ainscow et al., 2006) is still a challenging issue for the educational systems all around the world. In the realm of mathematics education, a fundamental question may arise: why do we need to teach mathematics to students with disabilities and, in particular, with intellectual disability? One could possibly argue that it is vital for them to learn that “two plus one equals three” and rely on the academic skill of arithmetic. However, such an answer raises other questions on the long-term outcomes and objectives of teaching them arithmetic and on why to teach them only simple arithmetic. Answering these questions is, among others, an issue of perspective, for example, do we focus on the individual with disability as being a social participant in an inclusive context or not? Through a sociocultural approach lens, it is important to offer these students a “tool” that will promote their social inclusion and help them become equal members of this society. Equity in mathematics education implies that we should give all students, including students with severe disabilities, opportunities to become active participants in mathematics activities (Civil, 2006). In this chapter, we addressed aspects of the issue of mathematics teaching to students with disabilities, with an emphasis on students with intellectual disability, from the perspective of equity in mathematics education. Considering equity within a sociocultural framework that supports a merge-based approach to social and individual perspectives (Civil & Planas, 2004) we discussed the need of integrating sociocultural approaches into mathematics

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teaching in developing quality practices that empower students with disability to become more independent in both school and out of school contexts, contributing thus to their social inclusion. In support of this perception pertaining to the contributions of a sociocultural perspective in mathematics teaching, we referred briefly to the outcomes of an action research study (Chrysikou, 2016) which pointed out the role of specific strategies—such as, the exploitation of students’ funds of knowledge, collaboration with students’ families, offering learning opportunities in authentic situations, and encouraging students to participate and act in authentic and meaningful for them situations—in helping students to experience recognition of their efforts by important others and to enhance their self-confidence. Moreover, we highlighted the need for instruction in realistic contexts, which could enhance both mathematics learning and social inclusion of students’ with disabilities. In conclusion, we suggest that the current trends in mathematics education which are based on a sociocultural perspective should be more actively present in educational practices concerning students with disabilities since their implementation could contribute to promoting these students’ educational and social inclusion, given—on the grounds of equity—the adoption of an education for all attitude. However, there is a need for sound empirical evidence regarding the implementation of interventions or programs based on these trends in order to effectively link theory to school practice and inform decision-making processes in an educational policies context. NOTES 1. The first author is the teacher-researcher, while the next two authors are critical friends (in the context of the action research procedure). 2. With the “one-more-than strategy” students are asked to pay one more euro than requested. For example, if the price of an item is 2,56€ the student would provide three one-euro coins.

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128 V. CHRYSIKOU, P. STAVROUSSI, and C. STATHOPOULOU Civil, M. (2006). Working towards equity in mathematics education: A focus on learners, teachers, and parents. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the Twenty Eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 30–50). Mérida, Mexico: Universidad Pedagógica Nacional. Civil, M. (2007). Building on community knowledge: An avenue to equity in mathematics education. In N. Nasir & P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 105–117). New York, NY: Teachers College Press. Civil, M., & Planas, N. (2004). Participation in the mathematics classroom: Does every student have a voice? For the learning of mathematics, 24(1), 7–12. Denny, P. J., & Test, D. W. (1995). Using the one-more-than technique to teach money counting to individuals with moderate mental retardation: A systematic replication. Education & Treatment of Children, 18(4), 422–432. Dihoff, R. E., Brosvic, G. M., Epstein, M. L., & Cook, M. J. (2005). Adjunctive role for immediate feedback in the acquisition and retention of mathematical fact series by elementary school students classified with mild mental retardation. The Psychological Record, 55(1), 39–66. DiPipi-Hoy, C., & Jitendra, A. (2004). A parent-delivered intervention to teach purchasing skills to young adults with disabilities. The Journal of Special Education, 38(3), 144–157. Epstein, J. L. (1995). School/family/community partnerships: Caring for the children we share. Phi Delta Kappan, 79(9), 701–712. Goldman, S. R., Hasselbring, T. S., & Cognition and Technology Group at Vanderbilt. (1997). Achieving meaningful mathematics literacy for students with learning disabilities. Journal of Learning Disabilities, 30(2), 198–208. Göransson, K., Hellblom-Thibblin, T., & Axdorph, E. (2016). A conceptual approach to teaching mathematics to students with intellectual disability. Scandinavian Journal of Educational Research, 60(2), 182–200. Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68. Hord, C., & Bouck, E. C. (2012). Review of academic mathematics instruction for students with mild intellectual disability. Education and Training in Autism and Developmental Disabilities, 47(3), 389–400. Hua, Y., Woods-Groves, S., Kaldenberg, E. R., Lucas, K. G., & Therrien, W. J. (2015). Effects of the TIP strategy on problem solving skills of young adults with intellectual disability. Education and Training in Autism and Developmental Disabilities, 50(1), 31–42 Jimenez, B. A., Courtade, G. R., & Browder, D. M. (2008). Teaching an algebraic equation to high students with moderate developmental disabilities. Education and Training in Developmental Disabilities, 43(2), 266–274. Kroesbergen, E. H., & Van Luit, J. E. H. (2005). Constructivist mathematics education for students with mild mental retardation. European Journal of Special Needs Education, 20(1), 107–116. Lambert, R., & Tan, P. (2017). Conceptualizations of students with and without disabilities as mathematical problem solvers in educational research: A critical review. Education Sciences, 7(2), 51. https://doi.org/10.3390/educsci7020051

Disability and Equity in Mathematics 129 Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport, CT: Ablex. Martinez, E. M. (1998). Teenagers with Down syndrome study algebra in high school. Down syndrome research and practice, 5(1), 34–38. McKenzie, K., Milton, M., Smith, G., & Ouellette-Kuntz, H. (2016). Systematic review of the prevalence and incidence of intellectual disabilities: Current trends and issues. Current Developmental Disorders Reports, 3(2), 104–115. Mezirow, J. (1997). Transformative learning: Theory to practice. In P. Cranton (Ed.), Transformative learning in action: Insights from practice—New directions for adult and continuing education, No. 74 (pp. 5–12). San Francisco, CA: Jossey-Bass. Mezirow, J. (2003). Transformative learning as discourse. Journal of Transformative Education, 1(1), 58–63. Morgan, C. (2014). Social theory in mathematics education: Guest editorial. Educational Studies in Mathematics, 87(2), 123–128. NCTM. (2000). Principles and standards for school mathematics. Reston, VA: Author. Pais, A., Stentoft, D., & Valero, P. (2010). From questions of how to questions of why in mathematics education research. In U. Gellert, E. Jablonka, & C. Morgan, (Eds.), Proceedings of the 6th International Mathematics Education and Society Conference (MES6) (vol. 2, pp. 398–407). Berlin, Germany: Freie Universität Berlin. Pais, A., & Valero, P. (2014). Whither social theory? Educational Studies in Mathematics, 87(2), 241–248. Patton, J. R., Polloway, E. A., & Smith, T. E. C. (2000). Educating students with mental retardation. Focus on Autism and other Developmental Disabilities, 15(2), 80–89. Porter, J., & Lacey, P. (2005). Researching learning difficulties. London, England: SAGE. Rao, S., & Kane, M. T. (2009). Teaching students with cognitive impairment chained mathematical task of decimal subtraction using simultaneous prompting. Education and Training in Developmental Disabilities, 44(2), 244–256. Rodriguez, A. M. (2016). Learning to apply algebra in the community for adults with intellectual developmental disabilities. Intellectual and Developmental Disabilities, 54(1), 19–31. Seligman, M., & Darling, R. B. (2007). Ordinary families, special children: A systems approach to childhood disability (3rd ed.). New York, NY: The Guilford Press. Shogren, K. A., Wehmeyer, M. L., Schalock, R. L., & Thompson, J. R. (2017). Reframing educational supports for students with intellectual disability through strengths-based approaches. In M. L. Wehmeyer & K. A. Shogren (Eds.), Handbook of research-based practices for educating students with intellectual disability (pp. 17–30). New York, NY: Routledge. Stavroussi, P., Papalexopoulos, P. F., & Vavougios, D. (2010). Science education and students with intellectual disability: Teaching approaches and implications. Problems of Education in the 21st Century, 19, 103–112. Storey, K., & Miner, C. (2011). Systematic instruction of functional skills for students and adults with disabilities. Springfield, IL: Charles C. Thomas.

130 V. CHRYSIKOU, P. STAVROUSSI, and C. STATHOPOULOU Tan, P., & Kastberg, S. (2017). Calling for research collaborations and the use of dis/ability studies in mathematics education. Journal of Urban Mathematics Education, 10(2), 25–38. Tassé, M. J., Luckasson, R., & Schalock, R. L. (2016). The relation between intellectual functioning and adaptive behavior in the diagnosis of intellectual disability. Intellectual Developmental Disabilities, 54(6), 381–390. https://doi. org/10.1352/1934-9556-54.6.381 Tekin-Iftar, E. (2008). Parent-delivered community-based instruction with simultaneous prompting for teaching community skills to children with developmental disabilities. Education and Training in Developmental Disabilities, 43(2), 249–265. Wehmeyer, M. L., Shogren, K. A., Singh, N. N., & Uyanik, H. (2017). Strengths-based approaches to intellectual and developmental disabilities. In K. A. Shogren, M. L. Wehmeyer, & N. N. Singh (Eds.), Handbook of positive psychology in intellectual and developmental disabilities: Translating research into practice (pp. 13–22). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-59066-0 Woodward, J., & Montague, M. (2002). Meeting the challenge of mathematics reform for students with LD. The Journal of Special Education, 36(2), 89–101. Xin, Y. P., Grasso, E., DiPipi-Hoy, C. M., & Jitendra, A. (2005). The effects of purchasing skill instruction for individuals with developmental disabilities: A meta-analysis. Council for Exceptional Children, 71(4), 379–400.

CHAPTER 7

IDENTIFICATION AND EDUCATIONAL SUPPORT FOR STUDENTS WITH LEARNING DIFFICULTIES IN MATHEMATICS IN DENMARK, FINLAND, AND SWEDEN Pirjo Aunio, Pernille Ladegaard Pedersen, Inger Ridderlind, Judy Sayers, and Pernille Bødtker Sunde

The Nordic countries are admired internationally for their well-developed policies on social inclusion within equitable social and educational systems. However, we know very little about the differences between the countries when it comes to identifying children who have difficulty learning mathematics, and the ways in which they are supported at school. The aim of this chapter is to describe and compare the identification and support processes related to mathematical learning difficulties in three Nordic countries: Equity in Mathematics Education, pages 131–158 Copyright © 2019 by Information Age Publishing All rights of reproduction in any form reserved.

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Finland, Denmark, and Sweden. With this knowledge, we will learn more about how equally the Nordic education systems are treating children who have problems in mathematics learning. In addition, we will perhaps identify new and valuable scientific research aims and questions for future development in education systems and practice. LEARNING DIFFICULTIES IN MATHEMATICS AND (SPECIAL) EDUCATIONAL INTERVENTIONS In the literature, there are several different terms used in relation to learning difficulties in mathematics, such as low performance in mathematics, learning difficulties in mathematics, mathematical learning disability, dyscalculia, and mathematics disorder. These various terms refer to different definitions (e.g., in terms of various cut-off scores) and different origins of the problems ranging from neurological dysfunctions to inappropriate opportunities to learn and practice mathematical skills (e.g., low socioeconomic status of the child’s family; Ansari, 2015; Mazzocco, 2009). Geary (2013), suggests that children who score at or below the 10th percentile on standardized mathematics achievement tests for at least 2 consecutive academic years are categorized as having mathematical learning disability (MLD). He further suggests that all children scoring between the 11th and 25th percentiles, inclusive, across 2 consecutive years are classed as low achievers (LA). Related to young children just starting their school career, it seems to be more appropriate to use the terms “low performing” or “mathematical learning difficulties,” as there is not yet enough information about severity and persistence of their difficulties. The identification of children with learning difficulties, or at risk for learning difficulties, relies on good information related to the learning. In this chapter, we use the concept tests, which includes a set of tasks designed to measure ability, features, or suitability. Screeners are designed to find those who are potentially at risk in their learning or development. Usually, screeners are less precise than tests are, as often there is less items than in tests. They aim to catch all children potentially at risk. Those children who are identified with the screeners will be directed to further investigations. Test and screeners are most often norm based or criteria based, thus relying on data collected from the population the test and screeners are designed to be used. They are easy to use, as manuals are conducted according to the strict criteria, such as The American Educational Research Association (AERA), the American Psychological Association (APA), and the National Council on Measurement in Education (NCME, 2014). At the moment, the concept “intervention” is a popular term and used with various meanings in education. We define an educational intervention

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as a research-based and planned modification of the learning environment made for the purpose of altering behavior or development in a prespecified way (Riley-Tillman & Burns, 2009). Intervention research design can be used to investigate the effects of a particular intervention program, which can then be published and used by educators. The most important way to measure the effectiveness of the educational intervention programs is to study the increase in learning (i.e., achievement) of the children as a result of extra practice, hence intervention (Jimerson, Burns, & VanDerHeyden, 2007). Consequently, we present an overview of each country context highlighting the process of identifying children; the intervention tools and teaching materials; other relevant support students with difficulties receive. In addition, we will describe what future teachers learn about mathematical learning difficulties in teacher education and the role of national educational assessments related to identification of children with mathematical learning difficulties. We will then discuss and highlight some differences and similarities between the contexts, and suggest further areas for research. DENMARK The Process of Identifying Children With Mathematical Learning Difficulties In Denmark, students begin 10 years of compulsory schooling (Grade 0 to 9) in the calendar year when a child turns six. The majority of children (98%) at the age of 3 to 5 years attend kindergarten before starting school. Danish kindergarten has an obligation to provide a description of their learning goals (“læreplaner”) to both authorities and parents, but before the children start school their language competences are screened on suspicion of the child having difficulties (LBK, 2016). This kind of screening is obligatory in Grade 0 for all students (BEK nr 855 of 01/07/2014). There are no national requirements of screening for mathematics performance before entering school or before the national test (mathematics) in third grade. The governing principles of the Danish school (“Folkeskolen”) builds on equality, equal opportunities and inclusion (LBK, 2017; Undervisningsministeriet, 2017a). Students with special educational needs should be included in regular classes through differentiated teaching, small groups, or extra teacher support. If this is not sufficient, the student attends special classes or special schools. In the process of identifying students with special needs, and decisions to what actions to take, the school cooperates closely with the student and family.

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In 2016–2017, 95.2% of students were included in the regular classes (Undervisningsministeriet, 2017b). To date, there has not been a comprehensive investigation of the process of identifying children with learning difficulties in mathematics and the available tests and screeners in the Danish context. Thus, the existing overviews of the different tests and assessment tools (e.g., Hansen, Jess, Pedersen, & Rønn, 2006; Larsen & Bengtsson, 2013) are far from complete in relation to validity and reliability. In Denmark, there is a high level of local self-government, besides the mandatory national tests, a decision can be made at an institutional level (school or municipality) as to what kind of tests and assessment tools should be used across different grades. However, it can also be left to the individual teachers to decide their own praxis. To describe the process of and tools to identifying children with learning difficulties in mathematics in Denmark, we need to address it from a national as well as a school level. In 2010, National Tests were implemented in the Danish school. All students are tested in mathematics in 3rd and 6th grade, and since 2018, in eighth grade too (BEK, 2017). It is a computerized adaptive test, where the difficulty of the next task depends on the correctness of the students’ answer from the former tasks. The students are tested in three mathematics curricular topics: “numbers and algebra,” “geometry and measurement,” and “statistics and probability” (Undervisningsministeriet, 2017c). The tests have been criticized for inconsistencies in the scores on an individual level and it has been suggested that the results are only relevant to classroom, school, and higher organizational levels (Bundsgaard & Puck, 2016; Ministeriet for Børn, Unge og Ligestilling, 2016). (For a comprehensive description of the national tests in English, see Beuchert and Nandrup, 2014.) According to the ministry of education (Undervisningsministeriet, 2017c), the national test cannot stand alone and the teachers also need to evaluate the student in other ways. Therefore, the student’s national test results must be compared with other tests or assessments concerning the learning of the student. In addition to the national test results, the Danish curriculum outlines eight fundamental objectives for mathematics learning (“opmærksomhedspunkter”), three in Grades 1–3, three in Grades 4–6, and two in Grades 7–9 (Table 7.1). These objectives describe the expected level of skill and competence the student should fulfill, to be able to participate with positive outcome in the following school years. It is up to the individual teacher, school or municipality to decide how to assess these objectives and what actions to take if not fulfilled. In 2015, the ministry of education set up a committee of national experts in the field of mathematical learning to develop a test which can

Identification and Educational Support for Students With Learning Difficulties 135 TABLE 7.1 Fundamental Objectives for Mathematical Learning in the Danish Curriculum (“Opmærksomhedspunkter”) Describing the Expected Level of Skill and Competence Needed to Participate With Positive Outcome in the Following School Years Objective After Year 3

The student is able to use 3-digit numbers to describe quantity and sequence. The student can add and subtract simple natural numbers by mental calculation and with the use of a calculator. The student can estimate and measure length, time, and weight in simple everyday situations.

After Year 6

The student can choose and adequate operation for solving simple everyday problems and write a simple math expression. The student can carry out arithmetic procedures in all four operations with the use of estimation and calculator. The student can extract relevant information from simple texts with mathematics information.

After Year 9

The student can carry out simple calculations of percentages with the use of estimation and calculator. The student can substitute variables with numbers in simple formula.

Source: Translated from Undervisningsministeriet, 2014.

identify children with dyscalculia and be used by teachers (Lindenskov & Lindhardt, 2015). The test will consist of two parts: (a) a computer-based screener that will identify children with possible dyscalculia and (b) a oneto-one diagnostic interview for the final diagnosing. In Denmark, tests and assessment tools are predominantly produced and distributed by private publishers. It is therefore not possible to access information on which tools are frequently used. They represent a variety of approaches and test within very different topics. Furthermore, they are not evaluated or authorized by governmental bodies. It is therefore up to the individual teacher to judge which assessment tool would be most suitable in a given situation. An analysis of some of these assessment tools has shown differences in how they identify students with learning difficulties (Sunde & Pind, 2016). This can be partly due to the fact that tests measure various mathematical skills and knowledge (van den Heuvel-Panhuizen & Becker, 2003) and because of tests’ psychometric properties. Intervention Tools and Teaching Materials Most of the intervention materials in Denmark are, like the tests and assessment tools, produced and distributed by private publishers. The

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ministry of education provides general information and inspiration materials through the website www.emu.dk, but does not provide complete intervention programs or teaching materials for students with mathematics difficulties. The ministry of education also provides a guide on how to use the results from the national tests (Undervisningsministeriet, 2017c). Furthermore, the ministry of education funds research and developmental programs that produces tools for the teachers. Most of these programs focus on the final years in school. Two examples of recent and ongoing projects are a developmental project for low-performing schools and a research project on the effect of “learning camps.” One attempt to increase the performance level of students is the national developmental program for the lowest performing schools (approximately 121) measured as the final exam scores in Danish and mathematics. Eightyeight schools were invited to participate in the project and consultants from the ministry of education will visit the schools and facilitate a development project (Undervisningsministeriet, 2017d). The project is developed by Rambøll, Metropolitan University College and VIA University College and evaluated by TrygFonden’s Centre for Child Research. The participating schools were those with the largest percentage of students with exam results under Grade 4 (Grade D). The aim is that the school will have fewer lowperforming students during the final exams in Danish and mathematics over a period of 3 years. If the school accomplishes this, they are given an economic bonus (approximately 1.3–1.5 million Danish kroner) after each year of success. This means that the school has to consistently get better results than the prior year in order to receive the bonus (Undervisningsministeriet, 2017d). Another attempt to increase the level of students’ mathematics performance began in 2016, when the government initiated a 5-year longitudinal RCT (randomized control trial) research project on the effect of intensive “learning camps” (Turboforløb for fagligt udfordrede elever i 8. klasse). The project was based on reviews of the effect of intensive learning camps (e.g., Dyssegaard, Bondebjerg, Sommersel, & Vestergaard, 2015). The research group consists of researchers from TrygFonden’s Centre for Child Research (Aarhus University), Rambøll (consultancy company), VIA University College, and Metropolitan University College (University College Copenhagen). The intervention consists of a 2-week intensive course in mathematics and Danish (“turbo-forløb”) with two intervention groups, one attending a learning camp focusing on learning strategies in mathematics and Danish (“Strategier til læring”) and the other group attending a learning camp focusing on Danish, mathematics, and social skills and competencies (“Dit liv, din læring”). After a 2-week intensive course, the students attend two lessons a week for a period of 8 weeks. The project is for eighth grade students who have been assessed as “not ready to attend

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further education” because they have difficulties in mathematics or Danish, or are otherwise immature. After 2 years, the study featured a total of 2,832 students (1,256 eighth grade students participated in 2017 and 1,576 new eighth grade students participated in 2018). The effect of the intervention is measured in pre- and post-tests in reading and mathematics, pre- and post-surveys on students’ motivation and self-esteem, an analysis of students’ test results of the national tests, and their results of the final exam. Furthermore, the effect on the students’ further educational level will be evaluated, as this is the overall aim of the project: to ensure students succeed in further education (Trygfondens Børneforskningscenter, 2017; Undervisningsministeriet, 2017e). The final results from the project will be published in 2020. In 2016 the ministry of education also funded a national research project for the middle grades: Quality in Danish and mathematics (KiDM—Kvalitet i dansk og matematik). One of the aims is to develop a more inquiry-based learning in mathematics for fourth and fifth grade students (Michelsen et al., 2017). It is expected that through working with raising the quality of mathematics teaching in general, a side effect would be that it also benefits the students in difficulties (Dreyøe et al., 2017). Private foundations also contribute to the research and developmental work, but few projects focus on students with learning difficulties. The largest, privately donated, recent RCT project is the Early Mathematics Intervention Programme for Marginal Groups in Denmark (TMTM Tidlig Matematikindsats Til Marginalgrupper), addressing effects of intervention on learning outcomes of the highest and lowest achieving 20% of second graders. In total, 281 students from 39 schools (28 municipalities) participated in the intervention. The project concluded in finding a statistically significant positive effect of intervention on mathematics performance for the high achievers but not for low-achieving students (effect size not reported) and improved the attitude to mathematics in all students (Schmidt et al., 2016). In 2017, the original project was upscaled to include 1,100 second and eighth graders from 80 schools of other geographical areas in Denmark. This upscaled project called Matematikindsats 2017, built on the principles and materials from TMTM. The group consists of researchers from Metropolitan University College (University College Copenhagen), Danish School of Education (Aarhus University), TrygFonden’s Centre for Child Research (Aarhus University), and VIA University College. Furthermore, it aims to investigate the effect of small-group intervention compared to one-to-one intervention. The assessment tools and intervention materials are available through a private publishing company (Lindenskov & Weng, 2013; Lindenskov, Tonnesen, & Weng, 2016). The results from this project await publication.

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The majority of the private published intervention tools and teaching materials in Denmark are for young children. Some of these materials are connected to a specific assessment tool. They vary in length, progression, approach, and systematics, as well as the level of guidance of the teacher. As with the tests and assessment tools, this put quite a lot of demand on the teachers’ competencies, and it is time-consuming both in choosing the relevant tool as well as in performing the actual assessment. Other Relevant Support for Students With Learning Difficulties The ministry of education has a team of learning consultants in mathematics. Schools and municipalities can consult this team for different developmental or consulting projects. In light of the high level of local autonomy, the organization and level of support differs between schools as well as municipalities. The municipalities have a team of different specialists to support the schools (“Pædagogisk Psykologisk Rådgivning”), but not all municipalities have a mathematics specialist. Likewise, some schools have a pedagogical learning center where teaching consultants with different expertise, guide colleagues on the teaching of students with different kinds of difficulties, but not all schools have access to these specialized mathematics consultants. In 2014, 50% of all Danish schools had a consultant in mathematics (Mogensen, Rask, Lindhardt, Østergaard, & Rostgaard, 2014). Mathematics consultants can participate in a national network of mathematics consultants (Danmarks Matematikvejleder Netværk). They meet once or twice a year in each of the six regions and share knowledge of teaching in mathematics, as well as of the difficulties in mathematics (Danmarks Matematikvejleder Netværk, 2017). Furthermore, all students have access to services of a welfare team of a school dentist, school nurse, and school doctor. Teacher Education and Learning Difficulties in Mathematics The Danish teacher education is a 4-year (240 ECTS) bachelor’s degree on university colleges. This includes four 10 ECTS mandatory courses in mathematics and mathematics didactics for teacher students. In addition, they have a mandatory 10 ECTS general course about students with special needs. There is no mandatory course focusing only on students with learning difficulties in mathematics, but each teacher education has to offer one or two courses in mathematics where differentiated teaching, students with

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special needs and assessment are part of the syllabus (BEK, 2015). Furthermore, the teacher colleges can offer supplement courses. VIA University College, for example, offers a 10 ECTS course on students with special needs in mathematics. The content of the course is assessment and planning of targeted teaching for students with special needs in mathematics (LiA Studieordningen, 2016). Postgraduates have the opportunity to study different competency qualifications at the university colleges (masters/half masters), for instance, mathematics consultants and reading consultants. Usually, the tuition fee is paid by the teacher’s school or municipality. One of the 10 ECTS mandatory courses on the mathematics consultants’ education is “Students with Special Needs in Mathematics.” The overall content in the course is inclusion, assessment, interventions, and differentiated instruction (Professionshøjskolerne, 2017). FINLAND In Finland, comprehensive education begins in the year children turn 7, and ends after 9 years of education (Grades 1–9). Pre-primary education (i.e., kindergarten year) is provided as part of early childhood education in the year preceding the start of compulsory education. Pre-primary education is organized in day care centers and schools, and became compulsory in 2015. The National Board of Education provides the curriculum guidelines and local curriculums can be made at community or school level (National core curriculum for basic education, 2014). The Finnish education system emphasizes early identification and learning support in general classroom settings. The National Board of Education (2011) suggested that the educational support system should have three-tiers: general, intensified, and special support. The first tier, general support, is for all students in basic education and should be part of everyday teaching. General support is mainly offered by general or subject teachers through educational differentiation in adjusting, for instance, the content or instruction. The following support level is the second tier, intensified support. Intensified support should be given for a limited period of time and should be evaluated on a regularly basis to determine whether the support is enough. Special support is given to ensure that students will have the support necessary to complete the compulsory education. With special support, students can follow a general or individualized curriculum. The Finnish tier model is comparable to the “Response to Intervention” used in several countries ( Jimerson, Burns, & VanDerHeyden, 2007). At the school level, the student’s welfare group is important in supporting students and teachers. The main aim of the group is to organize a system

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and screening of students who need support in their learning, and observe that sufficient and well-fitting support is provided. Nationally, attempts in the field of early mathematics, funded by the National Ministry of Education and Culture, has been focused on producing evidence-based knowledge for educators, providing them with assessment tools and intervention programs for children struggling with learning (www.lukimat.fi, http://blogs.helsinki.fi/thinkmath/). The design process of the LukiMat started at the University of Jyväskylä and Niilo Mäki Institute in 2006. LukiMat (www.lukimat.fi) contains three parts: reading, mathematics, and assessment of learning. All parts focus on basic skills for children aged 5–10 years. The mathematics part has been developed to serve both Finnish-speaking and Swedish-speaking educators and children in Finland. It features a knowledge base for parents and educators about mathematical skills development, mathematical learning difficulties, assessment, and educational interventions. The Process of Identifying Children With Mathematical Learning Difficulties Screening of Students With Mathematical Learning Difficulties Based on Aunio and Räsänen’s (2016) core factor model of the mathematical skills that develop in children aged 5–8 years, screeners were designed for the identification of children with learning difficulties in mathematics during kindergarten, first, and second grade. These assessment materials focus on core skill factors: a symbolic and nonsymbolic number sense, understanding mathematical relations; counting skills and basic skills in arithmetic. There are three scales to be completed three times a year with children: at the beginning of the school year (August–September), in the middle of the school year (November–December), and at the end of the school year (April–May). Teachers can decide what scale to use and this is based on teachers’ own activity and decision whether they want to use the materials or not. The study with Swedishspeaking Finnish children featured a total of 1,139 children (587 girls and 552 boys) and reported good internal consistency and developmentally relevant factor structure (e.g., Hellstrand, Korhonen, Räsänen, Linnanmäki, & Aunio, 2019). In addition to LukiMat screeners, the Finnish teachers can use other educational psychology-oriented measurements to identify students with learning difficulties in mathematics (Table 7.2). All of these measurements are based on research. Teachers need to buy the materials to be able to use them.

Identification and Educational Support for Students With Learning Difficulties 141 TABLE 7.2 Finnish Screeners for Mathematical Learning Difficulties Test

Age

Skills Measured

Subskills

Early Numeracy Test (Van Luit et al., 2006)

4.5–7.5 years

Mathematical relational skills, counting skills

The concepts of comparison, classification, one-to-one correspondence, seriation, the use of number words, structured counting, resultative counting, and general understanding of numbers

Banuca (Räsänen, 2005)

Grades 1–3

Number skills and basic arithmetic skills

Dot comparison, addition, correspondence, subtraction, numberline, number comparison, spoken numbers, calculations, arithmetic reasoning

RMAT (Räsänen, 2004)

9–12 years

Calculation skills

Addition skills, subtraction skills, multiplication and division, measurement units, understanding rational numbers

KTLT (Räsänen, 2005)

Grades 7–9

The basic mathematical skills

Basic arithmetic, applied problem solving and algebra

LukiMat-scales (http://www. lukimat.fi/lukimatoppimisen-arviointi)

Grades K–2

Early numeracy skills

Symbolic and non-symbolic number sense, understanding mathematical relations, counting skills, and basic skills in arithmetic

Matte—word problem solving and calculation scale (Kajamies et al., 2003)

Grades 4–5

Mathematical word problem solving skills, calculation skills

Basic arithmetic skills in word problem tasks.

Intervention Tools and Teaching Materials In the LukiMat web service, there are two evidence-based computer games in which low-performing children can practice early mathematical skills. The first, Number Race (Wilson, Dehaene, Pinel, Revkin, & Cohen, 2006; Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006) is based on the idea that number skills develop from approximate representations of magnitudes, and these representations are connected to numbers with the help of counting. The second game, GraphoGame Math, was originally designed as part of the GraphoGame project at the University of Jyväskylä, Finland

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(Richardson & Lyytinen, 2014). The foundations of GraphoGame Math stem from the notion that learning the correspondence between small sets of objects and numbers help children discover the numerical relationships in arithmetic. Research has demonstrated that both games have positive effects on children with learning difficulties in mathematics (Räsänen, Salminen, Wilson, Aunio, & Dehaene, 2009; Salminen, Koponen, Leskinen, Poikkeus, & Aro, 2015; Salminen, Koponen, Räsänen, & Aro, 2015). ThinkMath web service (http://blogs.helsinki.fi/thinkmath/in-english/) development started at the University of Helsinki in 2011 (see also https://thinkmathglobal.com/). It provides educators with evidence-based hands-on intervention materials to be used with children, aged 5–8 years, who have problems with learning early mathematical skills. In a way, it is an additional part of LukiMat, as it focuses on providing hands-on intervention materials to be used with children instead of computerized practice material. There is a knowledge-base with evidence-based information concerning (a) mathematical skills development and learning difficulties, (b) thinking skills development, (c) motivational issues related to learning, (d) executive functions relevance to learning, and (e) (special) educational interventions. In the knowledge-base, there are short videos to explain the main ideas to educators as clearly and quickly as possible. For instance, the material section offers group-based intervention materials for practicing mathematical skills with children in small groups. The design related to pedagogical characteristics followed the newest findings in the research literature (Mononen, Aunio, Koponen, & Aro, 2014). In the ThinkMath mathematical skills intervention programs, explicit teaching was one of the main guidelines along with several ways to practice the skills in focus (e.g., Gersten et al., 2008, 2009). In line with these recommendations, each lesson consists of a teacher-guided activity to model a new mathematical learning concept or strategy as well as guided and peer activities (e.g., hands-on activities with manipulatives, or card and board games based on the current topic). At the end of the lesson, there is a short individual, paper-and-pencil activity. Another general feature is that mathematical ideas are represented following the concrete, representational and abstract levels, thus giving meaning to abstract concepts by using visual representations (e.g., cubes, bundles of sticks, dot cards structured in tens and hundreds; Mononen, 2014). The teacher manual includes 12–15 lesson plans of 35–45 minutes each. The lesson plans include specific instructions for teachers to follow in each activity. The manipulatives are made of low-cost, everyday materials found in every classroom, combined with printable materials (e.g., dot and place value cards) included in the manual.

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Research With ThinkMath Intervention Programs During the development of the intervention materials, the effects of these intervention programs on children with learning difficulties were studied, through quasi-experimental, pre–post measurement with intervention and control groups in different age groups (Mononen, Aunio, & Leijo, 2019; Mononen & Aunio, 2014; Mononen & Aunio, 2016). In a kindergarten intervention study (Mononen, Aunio, & Leijo, in revision) children were divided into two groups: children whose math performances were below the 10th percentile (i.e., very low-performing, n = 20), and children whose mathematical performances lay between the 11th and 25th percentiles (i.e., low-performing, n = 18). The results showed that the number of children who reached an average level of performance at the post-test stage was higher among the group of children with low performance (67%) versus those with very low performance (35%). Mononen and Aunio (2016) investigated the impact of ThinkMath intervention on the performance of Finnish first graders (N = 151, M age = 7 y. 2 m.) with mathematical learning difficulties. This intervention program focused on increasing the counting skills knowledge in the number range 1–20 (Mononen & Aunio, 2012a). It was provided to small groups 12 times during 8 weeks, one session lasted about 45 minutes. The development of intervention children (n = 11) was compared to the development of children with learning difficulties (n = 26) and children with average performance (n = 114). The results showed significant effects of intervention, as the children in the intervention group made significantly greater gains in their mathematical performance from Time 1 to Time 2, compared with the children with learning difficulties and children with average performance. The main conclusion is that a relatively short counting skills supplementary intervention that applied explicit teaching showed promising effects in improving the performance of children with mathematical learning difficulties. The second grade intervention study (Mononen & Aunio, 2014) was done with 88 children (M age 8 y. 2 m.). The intervention program aims to practice counting and conceptual place value knowledge in the 1–1,000 range (Mononen and Aunio, 2012b). The intervention program lasted 6 weeks, and there were two 45-minute intervention sessions per week. The results demonstrated that the learning difficulties intervention group made significant improvements in mathematic scale, and especially in tasks measuring addition and subtraction facts, but did not show significantly better scores compared to the mathematical learning difficulties control group. In addition, neither the intervention children nor the control children were able to perform at the same level of their average peers following the intervention. Although there were not many scientifically significant results, there was an observable trend indicating that when children with mathematical

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learning difficulties received extra support, their skills developed, but when the intensified instruction ended so did the development of their skills, especially in arithmetical fluency skills. National Mathematics Assessment The National Board of Education and the Finnish Education Evaluation Centre (FINEEC) have been responsible for making assessment of learning outcomes for comprehensive education. The national assessments in mathematics have been mostly cross-sectional using randomized samples, and even though there is presently an attempt to follow-up the same students in several years, including students with learning difficulties (Hirvonen, 2012; Niemi & Metsämuuronen, 2010; Metsämuuronen, 2013; Rautapuro, 2013) students with learning difficulties have not always been included (Metsämuuronen & Salonen, 2017). The aim of the assessment is to investigate the learning outcomes at a certain grade level in Finland. Even though individual schools will have their own results, the school level results are not public, hence, no ranking of schools is possible based on these assessments. Identifying children with mathematical learning difficulties by means of a national assessment is not possible as the test focus is the curriculum, which is not conducted every year, and data is collected randomly. Teacher Education Classroom teacher education has no separate course for mathematical learning difficulties. There is one general course of learning difficulties. Students who desire to be special educators or who have special education as a main subject, have one 5 ECT course on mathematical learning difficulties where they learn about the development of fundamental mathematics skills, about assessment of skills, and interventions. SWEDEN The background to the Swedish context has long drawn on the philosophy of equality and learning for all, where documents state that all students have equal rights to personal learning and development, according to the Swedish National Agency for Education (Skolverket, 2015a). Thus, teaching will promote the development of every child and adult. All children in Sweden attend comprehensive school at the age of 7 to the age of 16 (Grades 1–9). All municipalities are obliged to offer a preschool class for

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6 year olds. Although they were initially voluntary, since the autumn of 2018 preschool classes have been compulsory. The majority of 6 year olds in Sweden have traditionally attended such classes. According to the Swedish National Agency for Education (Skolverket, 2011, 2014a), nearly half a million children attend preschool for education, care, play, and preparation for school whilst their parents work or study. The figure for Sweden is reported to be well above the European average (Drabble, 2013) recording 95% of all children between the ages of 3 and 6 to have attended formal childcare. Preschool now constitutes the first educational step in the Swedish education system, with its own national curriculum to reinforce the educational mission of preschools (Skolverket, 2011). At preschool, all children are given opportunities to learn the language and communication, science, technology, and mathematics. What those opportunities consist of is perhaps more difficult to answer; however, Sweden’s revised preschool curriculum in 2016 and 2011 of the curriculum of 1998, clarified the goals in these areas for teachers, with a continuous evaluation and development of the responsibility of the head of the preschool. Although outcomes are not specified, there is a deliberate emphasis on what measures need to be taken by each teacher to improve the conditions for each child to learn, develop, feel secure, and have fun in preschool (Taguma, Litjens, Makowiecki, & Early, 2013). Identifying Children With Learning Difficulties Identifying communication and language difficulties has long been a tradition in Sweden at preschool age; however, in 2017, the government testing agency began a piloting to screen for number sense (early numeracy) with 6-year-old children. It will be compulsory to use it as of January 2019 (https://www.skolverket.se/undervisning/forskoleklassen/kartlaggning-i-forskoleklassen). The purpose of this material is to provide support for classroom teachers to identify children who might have special needs in mathematics, and children who may have progressed further in their mathematical development. It is based on curriculum guidelines but there is no research done, for instance, to publish norms so that teachers would know how well each individual child is performing in relation to the age group. Currently, Grade 1 sees the first compulsory assessment (number sense and geometry) in schools. The test is produced at three levels, (low, middle, and high) and conducted in both the autumn and spring terms, to both identify children’s progress and highlight individual needs. Teachers can use the test again in Grades 2 and 3, but this is optional. There is little assessment or detailed testing of mathematical knowledge or competences officially in the schools, however, there are compulsory national assessment

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tests in Grades 3, 6, and 9. These tests are summative (oral and written), to inform the teachers of their students’ progress. They are administered and assessed by the teachers, who use their prior understanding of the student’s knowledge in conjunction with the test scores to provide an overall grading. Results are collected nationally by the statistics office, and examples of children’s test papers are gathered by the agency (PRIM group), which collects all papers of children born on a particular date in the year. The aim of government guidelines throughout the schooling system (primary to secondary) is to promote good assessment strategies by teachers to identify all students who struggle with mathematics knowledge. Thus, it could be argued, that the Swedish authorities trust the teachers to identify, organize, and develop appropriate strategies to support all children in learning mathematics. If teacher assessments and further school investigations show that special assistance is needed for individuals, the school principal has to establish a program of measures, that is, an action program (Skollagen, 2010), together with the teachers and parents, for all subjects. This can be in the classroom through the adjustment of teaching, or by providing individual support with reading material, writing on the computer, smartphones, increased instruction time, extra lessons or tasks, and so on, as well as parental involvement (Skolverket, 2014b). Special assistance (särskilt stöd) means actions of a more interventional nature. It is the extent or duration of the efforts, or both, which separate support from additional adjustments (Skolverket, 2014b). Thus, there is strong emphasis on the school responsibility to identify and provide support for all learners. If the school cannot provide such additional adjustments, special needs teachers are employed if funding is available, to support learners appropriately. Schools are required to provide all learners with high-quality opportunities to learn, at every level, and not just in compulsory schooling, but upper secondary and further into adult education. Moreover, all students in education have access to a health team made up of a school doctor, school nurse, school welfare officer, at no extra cost. Significantly, when the education act (Skollagen, 2010, p. 800) legislated that preschool become part of the education system in its own right, where the same legal framework applied to both public-aided and grant-aided independent schools. The act had implications on enforcing rights of the children in need of special support being integrated into comprehensive schools, and importantly, a higher qualification criteria for teaching. The consequence of this, according to Lundqvist, Allodi Westling, and Siljehag (2016), is that all schools should be providing the necessary support to learners with particular needs. Their recent study suggested children receiving inequitable experiences. Differing resources, materials, and quality of staff varied greatly between comprehensive and special schools. Preschool

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children with special needs were identified in comprehensive schools as receiving better quality teaching than those in the special schools. Furthermore, according to Lundqvist, Allodi Westling, and Siljehag (2016), there is now a need to justify segregated options for this group of children. Finally, as municipalities run independently from each other, one can find a variation in tests being used or promoted. The government provides downloadable assessments for mathematics between Grades 1–9, which provide research-based guidance and some suggested methods for development. Alternatively, there are also materials called Diamant diagnosing test, but they are used differently from teacher to teacher and school to school. The materials currently assess some areas of the mathematics curriculum, but not all, offering teachers some planning and structuring guidance in the teaching to whole classes or groups. The material offered also contains oral tests that are made in preschool class (Löwing & Fredriksson, 2009). However, there is no support or guidance to teachers in how they might approach different didactic strategies for children who are specifically suspected to have special needs. Since 2016 in special needs schools in Sweden, the compulsory use of assessment support in Year 1 has been introduced. This is once again to identify children who have particular difficulties in mathematics learning. Intervention Methods and Tools As reported by Denmark, Sweden has a holistic approach to cooperation and partnership between the child, family, and professionals, where the family is a priority. Schools work with the child and their families to suggest action and any intervention, offering guidance and counselling (Skolverket, 2015b). When reporting on what school intervention programs are used, apart from the official compulsory assessments made available by the government (Skolverket), it is difficult to be precise. Many schools, it seems, use the National Centre for Mathematics (NCM) book (see McIntosh, 2008). This book was developed with researchers by the National Centre in order to provide diagnostic tests and teaching guidance to support teachers in the development of children in their class. The focus is number only, but more guidance is presented for teachers of younger than older children. Teachers tend to use their own interventions and materials alongside this guidance book, either as small individual research studies or a collaborative “lesson study” or “learning studies” through the support of the municipalities. Several studies have reported on these municipality projects (Pilebro, Skogberg, & Sterner, 2010; Jeppson, Kiiskinen, & Käck, 2013; Lundqvist, Nilsson, Schentz, & Sterner, 2011; Sjöberg, 2006), all of which have indicated strongly that intervention, or intense programs, are successful. Sjöberg (2006), also tried to

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provide some strategies for teachers to use in their teaching, but this has not developed on a national scale, remaining at the municipality level. In sum, there appears to be no national approach or guidance on intervention, and so schools rely on identifying their own instruments from publishers. Moreover, in general, current practicing teachers are not aware of many intervention tools. The focus of most publishers, if not all, relates to the time and structure of appropriate instruction, and not necessarily providing teachers with didactical tools to use in their classrooms. Teacher Education Currently, although teacher education has introduced the idea of mathematical learning difficulties to general preservice teachers throughout their courses, (preschool, primary, F–3, 4–6, 7–9, out of school, subject teachers, upper secondary, and vocational) they have varied and limited access to tools and instruction on how to deal with those students. The trend currently is for all qualified special needs teachers to have some experience and qualification in teaching in order to study to be a specialist in special needs teaching of mathematics. The training is a postgraduate diploma in special needs; for example, Stockholm University offers a variety of professional competency qualifications (masters/half masters): deafness or hearing impairment, intellectual disabilities, vision or vision impairment and subject areas: language, reading and writing development, and mathematical development. The subject specialisms, language and mathematics are very popular and oversubscribed. DISCUSSION The aim of this chapter was to describe and compare the identification and support processes related to mathematical learning difficulties in three Nordic countries: Finland, Denmark, and Sweden. With this knowledge, we aimed to learn how equally Nordic education systems are treating children who have problems in mathematics learning; and we will, perhaps, identify important scientific and educational practice questions to be solved in the future. The Role of National Assessments The use of national assessment as a screener to identify children with learning difficulties has it challenges. On the one hand, the scales used

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mostly measure main topics of the mathematics curriculum, and are not necessarily specific enough to detect students with learning difficulties in mathematics. In addition to the topics measured, the national assessment scales very seldom have the psychometric properties needed in the identification of learning difficulties (Dowker, 2004). On the other hand, the strength of national assessment is the size of the sample and representativeness. This is why they are powerful in describing the performance level in children at the national level. The challenges in the use of national tests in Denmark has been its reliability and problems with the design of the digital test. As with Finland, the scales are measuring some of the main topics of the curriculum in Denmark (it does not measure mathematical competencies), but not all topics are the most relevant for detecting students with learning difficulties. In Finland, the national tests are held every 10th year and are randomized samples. The aim of this assessment is to investigate the learning outcomes at certain grade levels in Finland, and not to inform on student or school level, neither are they used for individual schools’ accountability on their performance against other schools. In Denmark, all students in Grade 3 and 6, and since 2018, Grade 8, are tested by means of the National Test. It is a digital adaptive testing and the aim is to measure school, class, and students’ performance in three topics of the curriculum. In Sweden, in 2017 a compulsory screening test in Year 1 was introduced to identify children in need of mathematical support, and this year 2018, the introduction of a compulsory preschool screening test was introduced. However, their national tests in Grades 3, 6, and 9 are not yet digitalized, as in Denmark, for they consist of different ways of testing; for instance, an oral scale is used to measure students’ mathematical language and conceptual understanding. Even though national tests are held in all of the three Nordic countries, their form, their use, and their aims are different in some ways—with few similarities between the countries’ national tests. Assessment Materials for Educators Although all three countries’ education authorities provide assessment materials for teachers to use, the level of funding, the type of assessment tool, the level of research behind these instruments, and whether they are compulsory or not, differs somewhat. In Finland, the Ministry of Culture and Education has provided funding to develop free to use evidence-based assessment tools for educators. The assessment tools are not compulsory for the teachers to use, and the teachers are given free access to print the test from the internet. As the instrument is only for the younger children, there is a need for teachers to access such tests for the older students in

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the future. There is a need to digitalize the screening, to make it easier to use and analyse. The Swedish Ministry of Education also funded the development of research-based assessments in Grade 1 to screen for children’s difficulties, but also their progress over the first year of schooling. The view is that children will be identified early and adjustments can be made in the teaching of children who were found to be in difficulty with particular mathematical competences. Neither the Finnish optional test, nor the Swedish compulsory test are tools provided in the Danish context. However, the Ministry of Education in Denmark in 2015, started funding the development of a dyscalculia test. When published it is expected to screen students with dyscalculia in all age groups. The stronger term “dyscalculia” is used in Denmark, which indicates that the possible neurological dysfunctions is a focus of the Danish Ministry of Education when it comes to detecting students with difficulties. This deviates somewhat from the other two countries’ government funded assessment tools, which are more closely related to “low performing” or “mathematical learning difficulties.” The consequence of focusing in this way, might result in a different student population being identified as having mathematical learning difficulties. Research literature internationally speaking, as Geary (2013) highlights, shows that children who score at or below the 10th percentile on standardized mathematics achievement tests for at least 2 consecutive academic years can be thought of as having mathematical learning disability (i.e., dyscalculia) and children scoring between the 11th and 25th percentiles, inclusive, across 2 consecutive years can be identified as low performers (i.e., low achievers, students with mathematical learning difficulties). As a result of the definition of dyscalculia, Denmark will report on a smaller number of children being identified as having problems in mathematics learning than in Sweden and Finland, which focus on children with low performance. Currently, there appears to be several private publishers in both Denmark and Sweden who distribute assessment tools (or additional assessment tools, respectively). Sweden draws on smaller research studies where the NCM, which publishes a book of number related tests (McIntosh, 2008)— or the municipality, or individuals—use their published work, their write up materials and sell them to teachers, schools, and other municipalities. Thus, a quite varied and ad hoc experience might be presented to the students in different areas of Sweden. However, there are some consistent resources available by the Department of Education. In slight contrast, Danish private publishers are the main distributors of assessment tools for educators. There appears to be a large private market for test material distribution in Denmark compared to Finland and Sweden. These private published Danish testing materials are not necessarily created according to criteria, such as AERA, APA, and NCME (AERA, APA, & NCME, 2014), which offer some benchmark of validity. Some of the private published assessment tools are

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made by researchers (e.g., Lindenskov & Weng, 2013), but these are not common. It is concerning that there is no validation or reliability study done with such tests. Hence, the main source of information about each test is based on the private publishers’ or the authors’ descriptions. Consequently, it can be discussed whether some Nordic education systems are treating children with learning difficulties in mathematics equally. Research-based assessment batteries offer teachers’ reliable background with their norms to make informed, good decisions on whose learning is not progressing well enough and might need special educational support (Fletcher, Lyon, Fuchs, & Barnes, 2006). Such quantitative foundations to such materials are essential if a teacher needs to know how an individual child is performing in relation to other children of the same age group. These commercial tools also create other problems regarding equity and children’s experiences, in that each school’s economic status could determine what assessment tool is available for its teachers to use. Intervention Programs and Teaching Materials In Finland, the Ministry of Culture and Education has funded evidencebased intervention tools free to use for educators. The focus has been in early years of school, but there is a desire for teachers to be provided with materials in older age groups, too. Providing free to use evidence-based knowledge via the internet is a clear attempt to treat teachers and students in an equal way, no matter where they are located. At the moment, in Denmark, intervention programs for low-performing students funded by the Danish Ministry of Education are primarily focused on the students’ final years of compulsory school (seventh, eighth, and ninth grade) which are different from the Finnish approach of highlighting the early grades. Intervention programs (e.g., TMTM) and teaching materials for early grades in Denmark are either funded by private organizations or private publishers. Somewhere in between these two perspectives, lie the intervention programs in Sweden, which are ad hoc and rely on individual municipalities, schools, or teachers investing time, energy, and resources into supporting children in difficulty with mathematics. The Swedish NCM provides some materials, information or training, although much of it is for purchase, and little is for free. The national knowledge base and intervention tools provided by researchers funded by the Finnish Ministry of Education and Culture is not present at the same level as it is in Denmark. Although the ministry of education does provide inspiration and guidance for teachers, there is no research-based information or tools provided for teachers to be used with students struggling with mathematics learning. Furthermore, the high level

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of local autonomy suggests that locally generated knowledge from developmental work and other initiatives is often not collected or shared outside the local community. This makes it difficult to construct a reliable overview of both the knowledge and the activities of schools and municipalities, and to create a common knowledge-base. In sum, the teachers’ access to assessment and intervention materials for students with learning difficulties in mathematics varies between the three countries. Denmark, has no national guidance or materials for educators to work with, therefore, schools must purchase their own assessment materials from publishers with an unknown evidence-based quality. Sweden has a compulsory screening test in preschool and Year 1 which is expected to indicate to teachers where their future planning and teaching adjustments lie. In particular, for those individuals who are struggling with different areas of mostly number, only few resources are provided centrally to support that aim. Finally, although Finland has produced evidence-based, free screeners and materials for all teachers in the early years of schooling, it has not provided anything for the older grades yet. Thus, children with difficulties in learning mathematics are reliant on the priorities of both their educational systems’ funding and their educational aims. CONCLUSIONS Relevance for Research Field As we reflect on the chapter written here, we believe we have highlighted the importance of collaboration between Nordic researchers in the field of mathematical learning difficulties, where we can provide our own context for study and information to guide further research, compare and contrast materials, and inform colleagues both in research and in the classroom. The cultural similarities between the three Nordic countries give researchers unique possibilities to look and learn from both the results of national projects and the international testing tools that seem to dominate politicians’ economic views of mathematics education. Relevance for Educational Policy and Practice What appears to be apparent is that at the national level it is important to secure an equal treatment of children who have learning difficulties in mathematics. How a child is identified and given educational support should be an equal matter for any educational system, for example, where the child lives and who is teaching him/her. National strategy might be an

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important element in this process. We, as researchers, should provide relevant evidence-based knowledge and design tools to assess children’s skills and competences, and support those who need extra support. It would be good if the national educational authorities would understand the importance of evidence-based materials. REFERENCES AERA, APA, & NCME. (2014). American Educational Research Association, American Psychological Association, & National Council on Measurement in Education. Standards for educational and psychological testing. Washington, DC: American Educational Research Association. Ansari, D. (2015, September 29). No more math wars – An evidence-based, developmental perspective on math education. Education Canada. Retrieved from www.cea-ace. ca Aunio, P., & Räsänen, P. (2016). Core numerical skills for learning mathematics in children aged five to eight years – A working model for educators. European Early Childhood Education Research Journal, 24(5), 684–704. BEK [Bekendtgørelse]. no. 1068. (2015, September 8). Bekendtgørelse om uddannelsen til professionsbachelor som lærer i folkeskolen [Executive Order on the Bachelor’s degree program as a teacher in primary and lower secondary school]. Retrieved from https://www.retsinformation.dk/forms/r0710.aspx?id=174218 BEK [Bekendtgørelse]. no. 742. (2017, June 14). Bekendtgørelse om obligatoriske test i folkeskolen [Executive Order on compulsory tests in the Folkeskole]. Retrieved from https://www.retsinformation.dk/Forms/R0710.aspx?id=191997 Beuchert, L. V., & Nandrup, A. B. (2014). The Danish national tests: A practical guide. Aarhus: Department of economics and business economics, Aarhus University. Economics Working Papers, 2014–25. The Danish National Tests: A Practical Guide. Retrieved from http://econ.au.dk/fileadmin/site_files/ filer_oekonomi/Working_Papers/Economics/2014/wp14_25.pdf Bundsgaard, J., & Puck, M. R. (2016). Nationale test: Danske lærere og skolelederes brug, holdninger og viden [National test: Danish teachers and school leaders’ use, attitudes and knowledge]. Copenhagen, Denmark: DPU. Danmarks Matematikvejleder Netværk. (2017). Organisation. Retrieved from http:// matnet.dk/ Dowker, A. (2004). What works for children with mathematical difficulties? (Vol. 554). London, England: DfES. Retrieved from https://webarchive.nationalarchives .gov.uk/20110202102730/http://nationalstrategies.standards.dcsf.gov.uk/ node/174504 Dreyøe, J., Michelsen, C., Hjelmborg, M. D., Larsen, D. M., Lindhardt, B. K., & Misfeldt, M. (2017). Hvad vi ved om undersøgelsesorienteret undervisning i matematik [What we know about study-oriented teaching in mathematics]. KIDM. Retrieved from https:// vbn.aau.dk/ws/portalfiles/portal/265213870/Forunders_gelsesrapport _delrapport_3_matematik_2.PDF

154 P. AUNIO et al. Dyssegaard, C. B., Bondebjerg, A., Sommersel, H. B., & Vestergaard, S. (2015). Litteraturstudie om intensive læringsforløb [Literature study on intensive learning]. Copenhagen, Denmark: Dansk Clearinghouse for Uddannelsesforskning. Fletcher, J. M., Lyon, G. R., Fuchs, L. S, & Barnes, M. A. (2006). Learning disabilities: From identification to intervention. New York, NY: Guildford. Geary, D. C. (2013). Learning disabilities in mathematics. Recent advances. In H. L. Swanson, K. R. Harris, & S. Graham (Eds.), Handbook of learning disabilities (2nd ed., pp. 239–255). New York, NY: Guilford. Gersten, R., Chard, D. J., Jayanthi. M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A metaanalysis of instructional components. Review of Educational Research, 79(3), 1202–1242. Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2008). Mathematics instruction for students with learning disabilities or difficulty learning mathematics. A synthesis of the intervention research. Portsmouth, NH: RMC. Hansen, H. C., Jess, K., Pedersen, B., & Rønn, E. (2006). Der er mere end ét svar: Matematik og specialundervisning [There is more than one answer: Mathematics and special education]. Copenhagen, Denmark: Alinea. Hellstrand, H., Korhonen, J., Räsänen, P., Linnanmäki, K., & Aunio, P. (2018). Validating the theoretical model of core numerical skills for learning mathematics in children aged five to eight years. Manuscript submitted for publication. Hirvonen, K. (Ed.). (2012). Onko laskutaito laskussa? Matematiikan oppimistulokset peruskoulun päättövaiheessa 2011 [Learning outcomes in mathematics in the final 9th grade of basic education in the spring of 2011]. Helsinki, Finland: The Finnish National Board of Education. Jeppson, E., Kiiskinen, J., & Käck, B. (2013). Intensivundervisning i matematik i Gävle kommun [Intensive teaching of mathematics in the municipality of Gävle]. Nämnaren, 2013(1), 1–7. Jimerson, S. R., Burns, M. K., & VanDerHeyden, A. M. (2007). Handbook of response to intervention: The science and practice of assessment and intervention. New York, NY: Springer. Kajamies, A., Vauras, M., Kinnunen, R., & Iskala, T. (2003). MATTE—Matematiikan sanallisten tehtävien ratkaisutaidon ja laskutaidon arviointi [MATTE—Assessment of mathematical problem solving and numeracy skills]. Turku, Finland: Oppimistutkimuksen keskus. Larsen, L. B., & Bengtsson, S. (2013). Talblindhed [Dyscalculia]. Copenhagen, Denmark: SFI—Det Nationale Forskningscenter for Velfærd. LBK [Lovbekendtgørelse]. no. 748 (2016, June 20). Bekendtgørelse af lov om dag-, fritids- og klubtilbud m.v. til børn og unge [Announcement of the Act on day, leisure and club offers etc. for children and young people]. (Cp. 2). Retrieved from https:// www.retsinformation.dk/forms/r0710.aspx?id=182051 LBK [Lovbekendtgørelse]. no. 1510 (2017, December 14). Folkeskoleloven [Primary education]. (Kap. 1). Retrieved from https://www.retsinformation.dk/ forms/R0710.aspx?id=196651 LiA Studieordningen. (2016, August 1). VIA Læreruddannelse Studieordning Læreruddannelsen Aarhus, Del 3: Fag og moduler [VIA Teacher Education Curriculum Teacher

Identification and Educational Support for Students With Learning Difficulties 155 Education Aarhus, Part 3: Subjects and modules]. Retrieved from https://www.via .dk/uddannelser/paedagogik-og-laering/laerer/tidligere-studieordninger Lindenskov, L., & Lindhardt, B. (2015, November 19–20). Developing test materials for developmental dyscalculia for Danish pupils in grade 4. Peer-reviewed paper for NORSMA 8 Conference 2015, Special Needs Education in Mathematics: Connecting research and practice, Kristianstad, Sweden: Kristianstad University. Lindenskov, L., Tonnesen, P. B., & Weng, P. (2016). Matematikvanskeligheder på de ældste klassetrin [Mathematical difficulties in the upper grades]. Viborg, Denmark: Dansk Psykologisk Forlag A/S. Lindenskov, L., & Weng, P. (2013). Matematikvanskeligheder—tidlig intervention [Mathematics—Early intervention]. Viborg, Denmark: Dansk Psykologisk Forlag A/S. Lundqvist, J., Allodi Westling M., & Siljehag, E. (2016). Characteristics of Swedish preschools that provide education and care to children with special educational needs. European Journal of Special Needs Education, 31(1), 124–139. Lundqvist, P., Nilsson, B., Schentz, E-G., & Sterner, G. (2011). Intensivundervisning med gott resultat [Intensive education with good results]. Nämnaren, 1, 44–50. Löwing, M., & Fredriksson, M. (2009). Diamant [Diamond]. Stockholm, Sweden: Skolverket. Mazzocco, M. M. M. (2009). Defining and differentiating mathematical learning disabilities and difficulties. In M. M. M. Mazzocco & D. B. Berch (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 29–48). Baltimore, MD: Paul H. Brookes. McIntosh, A. (2008). Förstå och använda tal [Understand and use speech]. Göteborg, Sweden: NCM. Metsämuuronen, J. (Ed.). (2013). Perusopetuksen matematiikan oppimistulosten pitkittäisarviointi vuosina 2005–2012 [Learning outcomes in mathematics of students who completed their second grade]. Helsinki, Finland: Finnish National Board of Education. Metsämuuronen, J., & Salonen, V. (2017). Matemaattisen osaamisen piirteitä ammatillisen koulutuksen lopussa 2015 ja pitkän ajan muutoksia [Mathematical performance in the end of vocational education and long-term changes in performance]. Helsinki, Finland: Finnish Education Evaluation Centre. Michelsen, C., Dreyøe, J., Hjelmborg, M. D., Larsen, D. M., Lindhardt, B. K., & Misfeldt, M. (2017). Forskningsbaseret viden om undersøgende matematikundervisning. KIDM [Research-based knowledge of investigative mathematics teaching]. Retrieved from https://vbn.aau.dk/ws/portalfiles/portal/265214078/Videns notat_matematik_2.PDF Ministeriet for Børn, Unge og Ligestilling. (2016, February 26). Børne- og Undervisningsudvalget 2016. Notat om undersøgelse af de nationale tests reliabilitet [Children’s and Education Committee 2016. Note on the study of the reliability of the national tests]. Retrieved from http://www.ft.dk/samling/20151/almdel/BUU/ bilag/119/1607357.pdf Mogensen, A., Rask, L., Lindhardt, B., Østergaard, K., & Rostgaard, P. (2014). Fagteamsamarbejde og matematikvejlederfunktion i grundskolen [Professional co-operation and mathematics supervisor function in primary school]. Copenhagen, Denmark: Undervisningsministeriet.

156 P. AUNIO et al. Mononen, R. (2014). Early mathematics interventions—Supporting young children with low performance in mathematics (Doctoral dissertation). University of Helsinki, Faculty of Behavioural Sciences, Helsinki, Finland. Mononen, R., & Aunio, P. (2012a). Improving mathematics skill in the first grade. An intervention program. Helsinki, Finland: University of Helsinki. Retrieved from https://thinkmathglobal.com/english/materials/ Mononen, R., & Aunio, P. (2012b). Improving mathematics skill in the second grade. An intervention program. Helsinki, Finland: University of Helsinki. Retrieved from https://thinkmathglobal.com/english/materials/ Mononen, R., & Aunio, P. (2014). A mathematics intervention for low-performing Finnish second graders: Findings from a pilot study. European Journal of Special Needs Education, 29(4), 457–473. Mononen, R., & Aunio, P. (2016). Math intervention for low-performing first graders. South African Journal of Childhood Education, 6(1), 407. https://doi. org/10.4102/sajce.v6i1.407 Mononen, R., Aunio, P., Koponen, T., & Aro, M. (2014). A review of early numeracy interventions: Effectiveness and pedagogical implementation. International Journal of Early Childhood Special Education, 6(1), 25–54. Mononen, R., Aunio, P., & Leijo, S. (2019). Matemaattisten taitojen harjoittelun vaikutukset matematiikan taidoiltaan heikkojen esiopetusikäisten lasten osaamiseen [The effects of mathematical intervention on low-performing children’s performance in kindergarten]. Manuscript submitted for publication. National Board of Education. (2011). Erityisopetuksen strategia [Strategy for special needs education]. Helsinki, Finland: Finnish National Board of Education. National Core Curriculum for Basic Education. (2014/2016). Basic education. Helsinki, Finland: Finnish National Board of Education. Retrieved from https:// www.oph.fi/english/curricula_and_qualifications/basic_education Niemi, E. K., & Metsämuuronen, J. (Eds). (2010). Miten matematiikan taidot kehittyvät? Matematiikan oppimistulokset peruskoulun viidennen vuosiluokan jälkeen vuonna 2008 [Learning outcomes in mathematics among students who had finished the 5th grade of basic education in 2008]. Helsinki, Finland: Finnish National Board of Education. Pilebro, A., Skogberg, K., & Sterner, G. (2010). Intensivundervisning [Intensive Training]. Nämnaren, 2010(4), 54–59. Professionshøjskolerne. (2017). Studieordningen for den pædagogiske diplomuddannelse [The curriculum of the program for diploma in education]. (7th ed.). Professionshøjskolerne. Retrieved from https://diplom.uc-dk.dk/wp-content/ uploads/2013/08/pd_studieordning_1-8-2018-min.pdf Räsänen, P. (2004). RMAT—Laskutaidon testi 9-12-vuotiaille [Skills test for ages 9–12]. Jyväskylä, Finland: Niilo Mäki Instituutti. Räsänen, P. (2005). BANUCA—Lukukäsitteen ja laskutaidon hallinnan testi [Test for reading and calculation skills]. Jyväskylä, Finland: Niilo Mäki Instituutti. Räsänen, P., Salminen, J., Wilson, A., Aunio, P., & Dehaene, S. (2009). Computerassisted intervention for early number skills in children with low numeracy: A short review and a randomized controlled trial. Cognitive Development, 24(4), 450–472.

Identification and Educational Support for Students With Learning Difficulties 157 Rautapuro, J. (Ed.). (2013). Hyödyllinen pakkolasku. Matematiikan oppimistulokset peruskoulun päättövaiheessa 2012 [Learning outcomes in mathematics in the final 9th grade of basic education]. Helsinki, Finland: Finnish National Board of Education. Richardson, U., & Lyytinen H. (2014). The GraphoGame method: The theoretical and methodological background of the technology-enhanced learning environment for learning to read. Human Technology: An Interdisciplinary Journal on Humans in ICT Environments, 10(1), 39–60. Riley-Tillman, T. C., & Burns, M. K. (2009). Single case design for measuring response to educational intervention. New York, NY: Guilford. Salminen, J., Koponen, T., Leskinen, M., Poikkeus, A-M., & Aro, M. (2015). Individual variance in responsiveness to early computerized mathematics intervention. Learning and Individual Differences, 43, 124–131. Salminen, J., Koponen, T., Räsänen, P., & Aro, M. (2015). Preventive support for kindergarteners most at-risk for mathematics difficulties: Computer-assisted intervention. Mathematical Thinking and Learning, 17(4), 273–295. Schmidt, M. C., Tonnesen, P., Overgaard, S. S., Nissen, S. K., Weng, P., Lindenskov, L., & Allerup, P. (2016). TMTM2014: Tidlig Matematikindsats Til Marginalgruppeelever—rapport til Egmontfonden [TMTM2014: Early Mathematics Efforts for Marginal Groups Pupils—Report to the Egmont Foundation]. Copenhagen, Denmark: Metropol. Sjöberg, G. (2006). Om det inte är dyskalkyli-vad är det då?: En multimetodstudie av eleven i matematikproblem ur ett longitudinellt perspektiv [If it’s not dyscalculia, then what is it? A multimethod study of students and problem solving from a longitudinal perspective]. (Doctoral dissertation). Umeå University, Department of Mathematics Technology and Science Education, Umeå, Sweden. Skollagen, Sverige. (2010). Lag (2010:801) om införande av skollagen (2010:800) [Law (2010: 801) on the introduction of the Education Act (2010: 800)]. Stockholm, Sweden: Norstedts juridik. Skolverket. (2011). Curriculum for the Preschool Lpfö 1998. Stockholm, Sweden: Skolverket. Skolverket. (2014a). Arbete med extra anpassningar, särskilt stöd och åtgärdsprogram [Work with extra adaptations, especially support and action programs]. Stockholm, Sweden: Skolverket. Skolverket. (2014b). Stödinsatser i utbildningen: Om ledning och stimulans, extra anpassningar och särskilt stöd [Support measures in the education: About management and stimulation, extra adaptations and special support]. Stockholm, Sweden: Skolverket. Skolverket. (2015a). Bedömning för lärande i matematik årskurs 1–9 [Assessment for learning in mathematics, grades 1-9]. Stockholm, Sweden: Skolverket. Skolverket. (2015b). An assessment of the situation in the Swedish school system 2015: [Summary of report 421]. Stockholm, Sweden: Skolverket. Sunde, P. B., & Pind, P. (2016). Comparison of two test approaches for detecting mathematical difficulties. In L. Lindenskov (Ed.), Special needs in mathematics education. CURSIV 18 (pp. 141–156). Aarhus, Denmark: Aarhus University. Retrieved from http://edu.au.dk/fileadmin/edu/Cursiv/CURSIV_18_-_Udgivet_version.pdf

158 P. AUNIO et al. Taguma, M., Litjens, I., Makowiecki, K., & Early, Q. M. (2013). Quality matters in early childhood education and care Sweden. Jerusalem, Israel: OECD. Trygfondens Børneforskningscenter. (2017, August 3). Forsøg med turboforløb for fagligt udfordrede elever [Experiments with turbo-course for students who have been challenged by the profession]. Retrieved from http://childresearch.au.dk/ forsoeg-med-turboforloeb-for-fagligt-udfordrede-elever/ Undervisningsministeriet [Ministry of Education, Denmark]. (2014). Fælles Mål for faget matematik [Common objectives of the subject mathematics]. Retrieved from https:// arkiv.emu.dk/modul/matematik-f%c3%a6lles-m%c3%a5l-l%c3%a6seplan -og-vejledning Undervisningsministeriet. (2017a, October 26). Regler om inklusion [Rules of inclusion]. Retrieved from https://uvm.dk/folkeskolen/laering-og-laeringsmiljoe/ inklusion/regler-om-inklusion Undervisningsministeriet. (2017b, December 11). Specialundervisning [Special education]. Retrieved from https://uvm.dk/statistik/grundskolen/elever/ specialundervisning Undervisningsministeriet. (2017c, October 25). UVM. Vejledning om de nationale test—til lærere i alle fag [UVM. Guidance on the national tests—for teachers of all subjects]. Retrieved from https://uvm.dk/folkeskolen/elevplaner-nationale-test -og-trivselsmaaling/nationale-test/vejledninger Undervisningsministeriet. (2017d, December 13). Pulje til løft af fagligt svage elever [Lifting academically challenged students]. Retrieved from https://www.uvm .dk/folkeskolen/viden-og-kompetencer/pulje-til-elevloeft/pulje-til-loeft -af-fagligt-svage-elever Undervisningsministeriet. (2017e, March 20). Tilmelding til satspuljeprojekt om turboforløb for fagligt udfordrede elever i udskolingen [Enrollment in an intensive cource for academically challenged students]. Retrieved from https://uvm.dk/ puljer-udbud-og-prisuddelinger/puljer/puljeoversigt/tidligere-udmeldte -puljer/grundskole/tilmelding-til-satspuljeprojekt-om-turboforloeb-for-fagligt -udfordrede-elever-i-udskolingen van den Heuvel-Panhuizen, M., & Becker, J. (2003). Towards a didactic model for assessment design in mathematics education. In M. C. A. J. Bishop (Ed.), Second international handbook of mathematics education (pp. 89–716). Dordrecht, The Netherlands: Kluwer Academic. Van Luit, J., Van de Rijt, B., & Aunio, P. (2006). Lukukäsitetesti [Early numeracy test]. Helsinki, Finland: Psykologien Kustannus Oy. Wilson, A. J., Dehaene S., Pinel, P., Revkin, S. K., & Cohen D. (2006). Principles underlying the design of “The Number Race,” an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Functions, 2(1), 1–14. Wilson, A. J., Revkin, S. K., Cohen, D., Cohen L., & Dehaene S. (2006). An open trial assessment of “The Number Race,” an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Functions, 2(2), 1–36.

CHAPTER 8

PARENTAL INVOLVEMENT AND EQUITY IN MATHEMATICS Susan Sonnenschein and Brittany Gay

There continue to be large and significant income-related differences in the percentage of children in the United States earning age-appropriate math scores as they go through elementary school (Reardon & Portilla, 2016). On the 2017 U.S. National Assessment of Educational Progress, only 22% of fourth graders (youngest grade the test is given) eligible for free or reduced lunch (a marker of low income in the United States) received proficient or higher scores in math compared to 52% of those not eligible for lunch subsidies (National Center for Educational Statistics, 2018). And, as Duncan and Magnuson (2011) discuss, low-income children in the United States entering kindergarten score, on average, one standard deviation lower on math tests than their more affluent peers. School children in the United States also routinely earn lower scores on math tests than do children from other industrialized countries. For example, on the 2015 Test of International Math and Science Studies (TIMSS; Provasnik et al., 2016), U.S. fourth graders ranked 14th in math among children from 49 different industrialized countries. Decreasing achievement gaps in math, as well as more generally improving children’s math skills, are important for

Equity in Mathematics Education, pages 159–178 Copyright © 2019 by Information Age Publishing All rights of reproduction in any form reserved.

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children’s academic and subsequent vocational well-being (Blevins-Knabe, 2016; National Mathematics Advisory Panel, 2008). Given that demographic group-related differences in children’s academic achievement are present at the start of school and generally continue or increase over time (Burchinal et al., 2011; Cheadle, 2008; Sonnenschein & Sun, 2016), we need to consider what math learning opportunities children have at home, even before the start of formal schooling. As Ginsburg, Lee, and Boyd (2008) have noted, even very young children should have many such opportunities. However, just because a child’s home environment affords the possibility of such opportunities, it does not mean that these potential opportunities are utilized well or at all. Differences in available opportunities can contribute to the inequitable development of math skills. Before turning to the main content in this chapter, we need to note that although there are large differences in academic achievement associated with both race/ethnicity and income (NCES, 2018), these two variables are strongly correlated (e.g., Hill, 2001; Reardon & Galindo, 2009) and the income achievement gap in the United States is now larger than the racial/ ethnic one (Reardon, 2011). In addition to the strong association between race/ethnicity and income, income and parents’ education are highly correlated (Krieger, Williams, & Moss, 1997). Thus, some of the incomerelated differences in socialization practices may reflect differences in parents’ race/ethnicity and education levels. This chapter considers the home learning opportunities available for children’s math development. We focus primarily on children from low-income backgrounds because of the critical need to narrow or close achievement gaps. Such a focus allows us to identify areas of relative need as well as potential strengths upon which to build, thus promoting equity (Cabrera, Beeghly, & Eisenberg, 2012). As we will show, both low- and middle-income parents view math as less important than literacy and are less knowledgeable about how to socialize their children’s math skills than their literacy skills (Blevins-Knabe, 2016). Nevertheless, despite such commonalities across low- and middle-income parents, there also are key differences in socialization practices between such lowand middle-income parents as well as societal differences (e.g., quality of school attended) to support children’s math acquisition. Such differences are potential sources of inequities in children’s math development. We consider parents’ beliefs and practices because these are key aspects of parents’ academic socialization and pertinent for children’s academic development (Serpell, Baker, & Sonnenschein, 2005; Sonnenschein, Metzger, & Thompson, 2016). Income-related differences in any of these components can result in differences in children’s math skills. We take the view that parents’ beliefs and practices foster their children’s interest in math which, in turn, is associated with the frequency of their engagement in activities and the type of activities engaged in. Such engagement

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then is associated with children’s math development. We focus primarily on children prior to the start of formal schooling and shortly thereafter because, as said, group-related differences emerge during preschool and the home influence may be greatest in the early years (Aikens & Barbarin, 2008; Jeynes, 2012). Unless otherwise noted, the research included in this chapter is based on children in the United States; however, we include research with children from other countries as appropriate. We begin with a short summary of the relevant theory that underpins our review and then present pertinent research. THEORETICAL BACKGROUND Parents’ socialization of their children’s math development includes parents’ attitudes, values, goals, expectations, and beliefs about education, as well as the opportunities and activities parents make available to their children (Green, Walker, Hoover-Dempsey, & Sandler, 2007; Puccioni, 2015; Taylor, Clayton, & Rowley, 2004). Such socialization by parents can be expressed through beliefs explicitly or implicitly conveyed to their children, through differential rewards for certain behaviors, parents’ reactions to children’s academic successes and failures, provision of artifacts and opportunities to engage in activities, and children’s observation of parents as role models of positive engagement in academic endeavors (Sonnenschen, Metzger, & Thompson, 2016). Socialization beliefs and practices not only provide children with learning opportunities, they also convey to children the importance parents attach to their children’s education and academic progress (Sonnenschein, 2002). Parents’ socialization is associated with children’s academic development (Puccioni, 2015; Sonnenschein & Galindo, 2015) through children’s interest and engagement in activities (Sonnenschein & Dowling, 2019). The nature of parents’ academic socialization is grounded in cultural models shared by members of a cultural group (Keels, 2009; Wong & Hughes, 2006), although some socialization beliefs and practices may also reflect family income and parents’ educational levels (Sonnenschein, 2002). Income-related differences in what Lareau (2003, 2011) called concerted cultivation also has been used to account for socialization differences and related achievement gaps. Lareau (2011) studied 12 low- and middle-income families. Middle-income parents in Lareau’s (2011) study engaged in concerted cultivation whereby they actively and purposely fostered their children’s growth through the provision of academic and leisure activities. In contrast, according to Lareau (2011), low-income parents engaged in a philosophy of child-rearing more consistent with the “accomplishment of natural growth.” Rather than parents seeking enrichment activities for their children, the children engaged in more spontaneously occurring activities or “hung out”

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with their families or other children. Although Lareau’s notion has found support (e.g., Bodovski & Farkas, 2008), some researchers question whether the income-related differences in approaches reflect differences in access to economic opportunities more than differences in child-rearing philosophy (Sonnenschein, Metzger, & Gay, 2018; Yeung, Linver, & Brooks-Gunn, 2002). Although the home has a major influence, children’s development also occurs in other overlapping contexts which can directly and indirectly influence what occurs at home and, therefore, impact children’s development (Bronfenbrenner & Morris, 2006; Garcia Coll et al., 1996). As will be discussed in a subsequent section, outreach by children’s schools and establishing shared understandings between schools and families is particularly important for promoting children’s academic development (Epstein, 2001; Green et al., 2007; Mapp & Kuttner, 2013). Parents’ Academic Socialization Given that income-related differences in children’s math skills are evident at the start of school, it is reasonable to assume they may be due, at least in part, to the amount, type, or nature of home-based experiences (Vandermaas-Peeler & Pittard, 2013). Galindo and Sonnenschein (2015), using a large nationally representative data set, found that home learning opportunities (a composite of parents’ expectations for their children’s development, involvement at child’s school, and frequency of child’s engagement at home in various academically related activities) significantly attenuated the relation between income and kindergarten children’s endof-year math scores, after controlling for children’s math skills at the start of kindergarten. However, Galindo and Sonnenschein (2015) did not directly compare the amount of home learning opportunities children from different income groups experienced. Research which has done such comparisons has not found a consistent pattern of income-related differences in the frequency with which children engage in math-related activities. For example, Tudge and Doucet (2004) did not find significant income-related differences in children’s home-based math experiences whereas Ramani and Siegler (2008) did. Given such inconsistencies across studies, we need to consider not only the frequency of children’s math-related experiences but other components of parents’ academic socialization. Parents’ Beliefs Parents have specific beliefs about children’s development and their role in the development that predict the experiences they make available to

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their children (Sonnenschein, Metzger, Dowling et al., 2016) which subsequently predict children’s development (Keels, 2009; Serpell et al., 2005). Much of the research on parents’ beliefs has been based on middle-income families or has not distinguished between middle- and low-income families. However, there do seem to be some systematic differences associated with income in parents’ expected timelines of their young children’s math acquisition and parents’ beliefs about how children acquire math skills. Research by Sonnenschein and colleagues showed that parents from economically diverse backgrounds strongly support the importance of having their children engage in math activities at home and assisting their children with such engagement (Sonnenschein et al., 2012; Sonnenschein, Metzger, Dowling, Gay, & Simons, 2016). However, these parents more strongly endorsed the importance of such engagement for reading than math (see Blevins-Knabe, 2016 for review). Sonnenschein et al. (2012) found a positive relation between a diverse group of parents’ endorsement of the importance of children engaging in math activities at home and children’s doing such activities. Relatedly, Skwarchuk (2009), with a small sample of middleincome families (N = 25), found that mothers’ views about how good they were at math and how enjoyable they found math was positively related to their preschool children’s math scores on a measure administered by the researchers (see also Skwarchuk, Sowinski, & LeFevre, 2014, N = 183). There are differences in the expectations that parents from low- and middle-income children have for when and how their children will acquire math skills. Summarizing across several studies, parents (typically mothers) from low-income families have less realistic timelines for when their children should be displaying specific math skills than parents from middle-income families (DeFlorio & Beliakoff, 2015; Starkey et al., 1999). Such unrealistic expectations may lead to practices not wholly consistent with children’s knowledge base and, therefore, lead to more stressful interactions. Sonnenschein and colleagues looked at three approaches parents report taking to fostering their children’s math skills. These were adapted from work on literacy socialization by Serpell and colleagues (2005). An engagement approach focused on making activities interesting for the child, a skills approach emphasized using flashcards, workbooks, and similar activities to foster skills acquisition, and a daily living approach focused on using everyday activities available in the environment (e.g., teaching math through setting the table). The majority of low-income families (60%) emphasized a skills approach. Such an approach was not positively related to children’s math acquisition. Far fewer low-income families emphasized the other two approaches (engagement 19%; daily living 20%) for fostering math skills (Sonnenschein, Metzger, Dowling et al., 2016). Parents who emphasized using daily living to foster their children’s math skills reported their children were more likely to engage in math activities at home (Sonnenschein et

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al., 2012). Stipek, Milburn, Clements, and Daniels (1992) found that lowincome parents favored a more traditional didactic means of instruction for their young children than middle-income parents. And, Starkey and colleagues (1999) reported that low-income parents, in contrast to middle-income parents, view teachers as more responsible than the home for young children’s math instruction (see also Serpell et al., 2005). Home-Based Opportunities We consider three forms of learning opportunities, parents’ as role models (Simpkins, Fredricks, & Eccles, 2015; Sonnenschein, Metzger, & Thompson, 2016), children’s engagement in math or math-related activities (Galindo & Sonnenschein, 2015; LeFevre et al., 2009), and the amount of math talk children hear (Gunderson & Levine, 2011; Levine, Suriyakham, Rowe, Huttenlocher, & Gunderson, 2010). Parents as Role Models Although the importance of social learning has long been acknowledged (Bandura, 1986), an understanding of its role in children’s math acquisition is limited. Research on math, with a primarily middle-income sample, shows a positive association between parents serving as role models of math engagement and the frequency with which preschool through first graders engaged in math activities at home (Sonnenschein et al., 2012). However, that study did not include measures of children’s math skills. Sonnenschein, Metzger, and Thompson (2016), using a sample of low-income Black and Latino parents of preschool through first-grade children, found that children’s engagement in math activities was significantly associated with parents serving as role models of engagement. Although parents reported that their children observed them engage in math activities, on average, several times a week, most parents did not view their children observing them engage in math activities as a source of learning math skills. Thus, they may be missing an opportunity to socialize their children’s math development. Children’s Engagement in Math Activities It is important to understand the nature of children’s math engagement at home because many researchers, but not all, have found positive relations between the frequency of math engagement and children’s subsequent math skills (see Blevins-Knabe, 2016 for a review). Most of the studies

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used primarily middle-income children (e.g., LeFevre, Polyzoi, Skwarchuk, Fast, & Sowinski, 2010; Skwarchuk, 2009) but Sonnenschein, Metzger, and Thompson (2016) also found positive associations between frequency of math engagement at home and children’s math skills in preschool with a low-income sample. However, it may not be just the frequency of math engagement that matters, but which specific activities the child engages in. Nguyen et al. (2016), using a large dataset with low-income and minority children, found that preschool children’s counting skills (e.g., recognizing that numbers represent quantities and have magnitudes, one-to-one correspondence, knowing number names and their order, and cardinality) was the best predictor of their fifth-grade math skills. More specifically, children’s advanced counting skills (e.g., being able to count from a number not one) was a better predictor than more basic counting skills. Relatedly, Skwarchuk (2009) found that parents’ reports of children engaging in tasks that involved more complex numeracy activities, as opposed to simpler ones, was related to children’s math scores. There are three issues to consider about children’s math engagement at home: the frequency of engagement, the types of activities, and the nature of interactions. Much of the research on frequency of engagement asks parents to indicate which math activities children engage in and then computes some summary score (e.g., DeFlorio & Beliakoff, 2015; Sonnenschein et al., 2012). Such scores show that children engage in math activities monthly to weekly, with low-income children engaging in these activities less frequently than middle-income children (DeFlorio & Beliakoff, 2015). Tudge and Doucet (2004) conducted one of the few observational studies of the frequency with which children engaged in math activities. They observed low- and middle-income preschool-age children for a week at home and at child care. Although the overall frequency of engagement in math activities was quite low, there were no income-related differences. The infrequent engagement in math activities found by these various researchers may not be sufficient to optimize children’s math skills. Based on research on children’s literacy development, Serpell et al. (2005) found that children whose reading skills improved significantly from the beginning of first grade to the end of third grade engaged in daily reading at home with a variety of different genres. Another way of considering children’s math engagement is to look at the type of math activities children engage in. LeFevre and colleagues (e.g., LeFevre et al., 2009; Skwarchuk et al., 2014) explored children’s engagement in what they called formal and informal math activities. Formal activities involved direct instruction in numbers or some form of numerical knowledge. Informal activities were board games or activities that could involve numbers but that were not the main purpose of the activity. Both

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forms of engagement predicted children’s math skills, albeit different aspects of their math skills. Research suggests that both low- and middle-income children may not frequently (or frequently enough) engage in the math activities most conducive to their math development. For example, Skwarchuk (2009) asked 25 parents to rate the frequency of various math activities their children engaged in. Such activities were not in keeping with the ones experts rated as most likely to foster children’s math skills. On the other hand, counting is a commonly reported activity. Sonnenschein, Metzger, and Thompson (2016) asked Black and Latino low-income parents to rate the frequency with which their children engaged in various activities thought to involve some form of math. Fifty-seven percent of each group reported that their children engaged in counting activities each day. And, as Nguyen et al. (2016) found, preschool children’s counting skills are predictive of their math skills later in elementary school. Other activities noted by these parents were asking about quantities, using tv remotes, and watching math television programs. DeFlorio and Beliakoff (2015) also found counting to be one of the most common activities among preschool low- and middle-income children. Despite the prevalence of counting, we do not know exactly what the children did when they were counting with their families at home or what skills they acquired. Relatively few studies have compared the nature of home math activities children from low- and middle-income families engage in. Ramani and Siegler (2008) found that low-income children reported playing linear board games significantly less frequently than did middle-income children. Playing such games foster children’s number sense. In fact, increasing low-income children’s exposure to linear board games in a school-based intervention closed the income gap in children’s number sense (Ramani & Siegler, 2011). Saxe, Guberman, and Gearhart (1987) found that low-income families of preschoolers were less likely to engage in more complex math activities than were middle-income families. And, DeFlorio and Beliakoff (2015) found that low-income children engaged in made-up games involving math, used math in the home routine, read math-related books, and used computers with math software less frequently than middle-income children. As research on literacy shows, the nature of the interaction matters. Sonnenschein and Munsterman (2002) found that children whose reading interactions were affectively positive were more likely to engage subsequently in reading activities which, in turn predicted their reading skills (see also Baker, Mackler, Sonnenschein, & Serpell, 2001). Not much work has looked at the quality of math interactions that children have at home. This is particularly important because knowing that an activity can involve math does not necessarily mean parents focus on that when interacting with their children. Metzger, Sonnenschein, Galindo, and Patel (2015) asked first

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through fourth graders to describe what they did when engaging in cooking and grocery shopping, two activities thought to have opportunities for fostering math development. Most of the children reported that they engaged in non-math activities (reading labels, picking out items) when they assisted their parents with cooking or grocery shopping. Relatedly, research by Vandermaas-Peeler, Boomgarden, Finn, and Pittard (2012) found that interactions between parent and child in activities that should involve math are more likely to happen if parents are explicitly told to focus on math and how to do so (will be discussed in more detail in a subsequent section; see also Vandermaas-Peeler, Ferretti, & Loving, 2012). Lukie, Skwarchuk, LeFevre, and Sowinski (2014) showed that preschool children were more likely to engage in math-related activities with their parents when both were interested in the activity and engaged in collaborative types of interactions. Math Talk Although children from low-income backgrounds generally score lower on math tasks than middle-income children (DeFlorio & Beliako, 2015), such differences are more likely to occur on verbally-based math tests than on ones with lower language demands (see Ramani, Rowe, Eason, & Leech, 2015 for review). And, as Purpura and Reid (2016) have shown with a preschool sample, the amount and type of talk specifically focusing on math that children hear is positively related to their early math skills (see also Elliot, Braham, & Libertus, 2017). In fact, such talk at home predicts preschool children’s math skills even one year later (Susperreguy & Davis-Kean, 2016). Therefore, it is important to consider the specific math language stimulation that children receive at home because low-income children generally receive significantly less exposure to oral language than their middle-income peers (see Hindman, Wasik, & Snell, 2016 for review). Three key findings emerge from research on parents’ use of math language at home (see Ramani et al., 2015 for review). One, there is significant variability across parents, regardless of income group, in how much math talk they engage in, although even at its most frequent, it is relatively low. The few studies that have compared the amount of math talk between low- and middle-income groups find that middle-income parents engage in more math talk (e.g., Blevins-Knabe, 2016; Vandermaas-Peeler, Nelson, Bumpass, & Sassine, 2009). In addition, parents’ views of their math skills are related to the type and amount of math talk they engage in with their young children (Elliott et al., 2017). Two, the type of math language used by parents (e.g., numeral identification, cardinal numbers) is positively related to children’s specific math skills. For example, Gunderson and Levine (2011), in a longitudinal study, documented the math talk of mothers when

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children were between 14 and 30 months old. Children then were tested on their knowledge of cardinal values when they were 40 months of age. Parents who engaged in more math talk involving counting or labeling sets of objects, particularly larger sets of objects (between 4 and 10 visible items) had children who subsequently displayed greater knowledge of cardinal values. Three, parents are most likely to use math language when given instruction or guidance on what to do, when reading math books with their children, and when the topic being discussed does not readily lend itself to other forms of discourse. For example, Vandermaas-Peeler, Boomgarden, and colleagues (2012) observed a group of 25 middle- to high-income mothers engage in a cooking task with their preschool children. Mothers who were told to include additional mathematics in the activity provided significantly more numeracy guidance and created more opportunities for their children to practice advanced mathematics than mothers in a control group. This is consistent with findings from Cannon and Ginsburg (2008) who found that middle-income mothers reported not knowing what to do to teach their young children math. PARENTING IN CONTEXT The larger context of children’s environments, such as the surrounding community, also needs to be considered. Many children from low-income backgrounds live in homes where there are more toxins, have poorer health care, experience food insecurity or poor nutrition, live in neighborhoods higher in crime and lower in supportive resources, and are under more chronic stress (e.g., Evans, 2004; Evans & Kim, 2013; Duncan & Murnane, 2015; Fiese, Gunderson, Koester, & Washington, 2011; Rothstein, 2013). These factors negatively impact a child’s availability for learning and for parents being able to offer their children optimal learning opportunities. Moreover, the stresses associated with financial issues may negatively impact the quantity and quality of parents’ interactions with their children (family economic stress model; Conger & Elder, 1994; Masarik & Conger, 2017). Relatedly, parents from low-income backgrounds may have fewer economic resources to devote to their children’s development (Yeung et al., 2002). In fact, as Duncan and Murnane (2015) suggest, more affluent parents, unlike low-income ones, now devote a larger absolute and relative amount of their incomes to facilitating their children’s development than they did 25 years ago (see also Reardon, 2011). Despite the importance of the home for children’s development, what occurs at school and relations between home and school should also be considered. As many have noted, there are opportunity gaps—conceptualized as limitations in the resources and experiences available at home and

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school—for many children from low income backgrounds (Flores, 2007; Welner & Carter, 2013). Schools serving primarily low-income children tend to have fewer qualified teachers, relatively more children with behavior problems and inattention (Duncan & Murnane, 2015), and lower quality instruction than schools serving more affluent families (Sonnenschein, Thompson, Metzger, & Baker, 2013). Such factors limit the learning opportunities in school, and more pertinent for this chapter, the nature and outreach from schools to home. This is of particular concern for low-income children whose parents may rely more heavily than other parents on their children’s teachers for information about what math activities to do with their children (Sonnenschein et al., 2018). As Epstein (2001) argued, the home and school constitute “overlapping spheres of influence” on children’s outcomes. Schools provide important sources of information for parents about how well their children are doing, pertinent content, and information about how to engage children in learning (Sonnenschein, Stapleton, & Metzger, 2014). However, low-income families may feel less welcome than other families in their children’s schools, may have less available time to attend school events, have less knowledge of school customs and mores, and more generally, less social capital (Green et al., 2007; Mapp & Kuttner, 2013). INTERVENTIONS Research on ways to improve young children’s math skills have focused on math curricula in the schools (e.g., Klein, Starkey, Clements, Sarama, & Iyer, 2008), and ways to improve the math experiences that middle-income children (e.g., Niklas, Cohrssen, & Taylor, 2016; Vandermaas-Peeler, Boomgarden et al., 2012) and low-income children have at home (e.g., Sonnenschein, Metzger, & Thompson, 2016). We focus here on interventions involving low-income children. However, it is important to realize that many of the difficulties low-income parents face in implementing recommendations go beyond the actual math intervention. An intervention can improve children’s math skills, but it may not be compatible with parents’ beliefs, knowledge, available time, or other aspects of their lives (Furstenberg, 2011; Green et al., 2007). If that is the case, it will not be implemented or implemented with fidelity. There have been fairly few interventions focusing on what low-income parents can do to improve their children’s math skills. Berkowitz et al. (2015) compared math skills between a randomly assigned intervention and control group of 587 low- and middle-income first graders. Children and their families received iPads in which they read math stories and received a set of math questions to ask. Such interactions were associated

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with growth in children’s math skills from fall to spring. The authors suggested that by telling parents what questions to ask, they increased parents’ knowledge, especially important for parents whose own math knowledge may be limited, and thereby decreased their anxiety about math. In a related vein, Starkey and Klein (2000) found parents of Head Start children were willing and able to support their children’s math skills at home, once they had received sufficient training. The training included eight sessions over a 4-month period for parents and their children. Starkey, Klein, and Wakeley (2004) found that a combination of a special math curriculum, teachers’ professional development, and a parent training component was instrumental in improving low- and middle-income preschool children’s math skills. An intervention by Sonnenschein, Metzger, Dowling, et al. (2016) shows the difficulties one can have in adapting interventions, even when they have been successfully used before. They attempted to “send home” a highly successful classroom-based intervention by Ramani and Siegler (2008) who had preschool children play a linear board game with a trained researcher. Ramani and Siegler’s intervention lasted about an hour spread over several days. After training, these children did significantly better on various number sense tasks than children who did not play the game. In fact, after training, low-income children did as well as middle-income children. Sonnenschein, Metzger, Dowling, et al. (2016) trained low-income parents of preschoolers to play the linear board game Chutes and Ladders with their children using the special counting procedure used by Ramani and Siegler (2008). Just providing brief training for parents did not improve children’s scores on number sense tasks despite parents believing in the importance of their role in children’s learning, liking the task, and thinking their children’s math skills improved. Additional research conducted with focus groups suggested parents preferred that their children be trained at school and needed guidance getting their children to sit still to play a board game. When training children at school was combined with training for parents and giving parents guidance in the use of stickers to promote ontask behavior, children showed some small growth from pretest to post-test (compared to a control group). Summarizing across the studies, there are at least three issues that should be considered in optimizing the effects of math interventions for parents to do at home with their children. One, training needs to be of sufficient length. Two, it is helpful to combine school components with home-based components. Three, as we have discussed here and in the prior section, there may well be obstacles that go beyond the actual targeted intervention that can interfere with compliance. That is, parents may not have sufficient available time to do the required tasks.

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CONCLUSION This chapter explored reasons why low-income children in the United States, on average, start school with significantly weaker math skills than their more affluent peers (Duncan & Magnuson, 2011; Reardon & Portilla, 2016). We focused on children’s home experiences and documented similarities and differences in such experiences between low- and middle-income children. Two trends emerge from the literature. On the one hand, there are many similarities between low- and middle-income children and their families in the socialization of children’s math development. Regardless of income, children do not engage in math activities or talk about math at home as often as might seem optimal, parents view math as less important than reading, and many parents lack knowledge of what to do to foster their children’s math skills (Blevins-Knabe, 2016; Cannon & Ginsburg, 2008; Sonnenschein, Metzger, & Thompson, 2016; Vandermaas-Peeler, et al., 2012). On the other hand, despite some similarities across groups, there are many differences in the nature and amount of math socialization that occur, which favor middle-income children. In other words, low-income children experience gaps in the opportunities for learning math that are available to them at home and at school (Rothstein, 2013). At home, low-income parents, in contrast to middle income parents, may be less knowledgeable about math and how to facilitate it, engage in less math talk, know less about what to expect of their children, and emphasize a means of instruction (e.g., skills approach) that is less effective (DeFlorio & Beliakoff, 2015; Ramani & Siegler, 2008; Sonnenschein, Metzger, & Thompson, 2016). Unfortunately, the schools that low-income children attend, on average, are limited in the resources they can provide and do not compensate for some of the limited opportunities found at home (Rothstein, 2013). In addition, many low-income parents do not have the time to attend school events or feel comfortable there when they do (Green et al., 2007; Weiss et al., 2003). Despite these opportunity gaps found at home and school, there are strengths within the families that can be used as foundations to build upon (Cabrera et al., 2012). Parents view it as their role to assist their children with math activities at home (Sonnenschein, Metzger, Dowling et al., 2016) and they turn to teachers for knowledge about what to do (Sonnenschein et al., 2018). In designing interventions for parents to use to facilitate their children’s math development, we need to make such interventions congruent with parents’ beliefs about their role in their children’s learning, involve schools in the training, and provide enough training for parents to feel confident in their mastery of the relevant skills (Mapp & Kuttner, 2013; Sheldon & Epstein, 2005). However, we need to keep in mind that low-income families experience many significant daily stressors, such as working several jobs to

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pay bills, living in high crime areas, and others (Duncan & Murnane, 2011; Rothstein, 2013) that may make math socialization a less compelling priority. In fact, some have suggested that more systemic forms of remediation are needed than just academic training, if one is to close the income-based achievement gap (Curto, Fryer, & Howard, 2011; Duncan, Magnuson, & Votruba-Drzal, 2014; Furstenberg, 2011). Limitations in Current Research and Future Directions Although home influences on children’s math development is a rapidly growing area, the research is still limited in scope. Research has generally focused on mothers without fully considering the role that fathers play or considering socialization within a family system. Much of the research is based on parents’ self-reports and does not include observation of actual interactions. At least some of the locus of income-related differences in children’s math skills may be due to differences in the quality of interactions children have with their parents. The research reported in this chapter focused primarily on children from low-income backgrounds in the United States. Although race/ethnicity is often conflated with income (Reardon & Galindo, 2009), it would be beneficial to take an intersectionality approach when feasible. That said, Sonnenschein, Metzger, and Thompson (2016) found few differences in beliefs or practices related to math socialization between a small sample of Black and Latino low-income parents of preschoolers. Furthermore, more research should compare socialization practices of children from different countries. Despite issues for future research to explore, the available research does suggest areas that are ripe for intervention. Low-income children in the United States arrive at school with more limited math skills, at least in part, due to limitations in the amount and quality of the math experiences they have had at home. As parents think math is important and are willing to play a role in fostering such skills, interventions for parents’ math involvement is one avenue to promote equity in math prior to school entry. REFERENCES Aikens, N. L., & Barbarin, O. (2008). Socioeconomic differences in reading trajectories: The contribution of family, neighborhood, and school contexts. Journal of Educational Psychology, 100(2), 235–251. Baker, L., Mackler, K., Sonnenschein, S., & Serpell, R. (2001). Parents’ interactions with their first grade children during storybook reading activity and home achievement. Journal of School Psychology, 39(5), 415–438.

Parental Involvement and Equity in Mathematics 173 Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. New York, NY: Prentice Hall. Berkowitz, T., Schaeffer, M. W., Maloney, E. A., Peterson, L., Gregor, C., Levine, S. C., & Beilock, S. L. (2015). Math at home adds up to achievement at school. Science, 350(6257), 196–198. Blevins-Knabe, B. (2016). Early mathematical development: How the home environment matters. In B. Blevins-Knabe & A. M. Berghout (Eds.), Early childhood mathematics skill development in the home environment (pp. 7–28). Cham, Switzerland: Springer International. Bodovski, K., & Farkas, G. (2008). “Concerted cultivation” and unequal achievement in elementary school. Social Science Research, 37(3), 903–919. Bronfenbrenner, U., & Morris, P. (2006). The bioecological model of human development (6th ed.). In R. M. Lerner and W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (Vol. 1; pp. 793–828). Hoboken, NJ: John Wiley. Burchinal, M., McCartney, K., Steinberg, L., Crosnoe, R., Friedman, S. L., McLoyd, V., . . . NICHD Early Child Care Research Network. (2011). Examining the Black-White achievement gap among low-income children using the NICHD Study of Early Child Care and Youth Development. Child Development, 82(5), 1404–1420. Cabrera, N. J., Beeghly, M., & Eisenberg, N. (2012). Positive development of minority children. Introduction to the special issue. Child Development Perspectives, 6(3), 207–209. Cannon, J., & Ginsburg, H. P. (2008). “Doing the math”: Maternal beliefs about early mathematics versus language learning. Early Education and Development, 19(2), 238–260. Cheadle, J. E. (2008). Educational investment, family context, and children’s math and reading growth from kindergarten through the third grade. Sociology of Education, 81(1), 1–31. Conger, R. D., & Elder, G. J. (1994). Families in troubled times: Adapting to change in rural America. Hillsdale, NJ: Aldine. Curto, V. E., Fryer, Jr., R. G., & Howard, M. L. (2011). It may not take a village: Increasing achievement among the poor. In G. J. Duncan & R. J. Murnane (Eds.), Whither opportunity? Rising inequality, schools, and children’s life chances (pp. 483–505). New York, NY: Russell Sage Foundation. DeFlorio, L., & Beliakoff, A. (2015). Socioeconomic status and preschoolers’ mathematical knowledge: The contribution of home activities and parent beliefs. Early Education and Development, 26(3), 319–341. Duncan, G. J., & Magnuson, K. (2011). The nature and impact of early achievement skills, attention skills, and behavior problems. In G. J. Duncan, & R. J. Murnane (Eds.), Whither opportunity? Rising inequality, schools, and children’s life chances (pp. 47–70). New York, NY: Russell Sage Foundation. Duncan, G. J., Magnuson, K., & Votruba-Drzal, E. (2014). Boosting family income to promote child development. The Future of Children, 24(1), 99–120. Duncan, G. J., & Murnane, R. J. (2015). Restoring opportunity: The crisis of inequality and the challenge for American education. Cambridge, MA: Harvard Educational Press.

174 S. SONNENSCHEIN and B. GAY Elliott, L., Braham, E. J., & Libertus, M. E. (2017). Understanding sources of individual variability in parents’ number talk with young children. Journal of Experimental Child Psychology, 159, 1–15. Epstein, J. L. (2001). School and family partnerships: Preparing educators and improving schools. Boulder, CO: Westview Press. Evans, G. W. (2004). The environment of childhood poverty. American Psychologist, 59(2), 77–92. Evans, G. W., & Kim, P. (2013). Childhood poverty, chronic stress, self-regulations, and coping. Child Development Perspectives, 7(1), 43–48. Fiese, B. H., Gunderson, C., Koester, B., & Washington, L. T. (2011). Household food insecurity: Serious concerns for child development. SRCD Social Policy Report, 25(3), 1–27. Flores, A. (2007). Examining disparities in mathematics education: Achievement gap or opportunity gap? The High School Journal, 91(1), 29–42. Furstenberg, F. F. (2011). The challenges of finding causal links between family educational practices and schooling outcomes. In G. J. Duncan & R. J. Murnane (Eds.), Whither opportunity? Rising inequality, schools, and children’s life chances (pp. 465–482). New York, NY: Russell Sage Foundation. Galindo, C., & Sonnenschein, S. (2015). Decreasing the SES math achievement gap: Initial math proficiency and home learning environments. Contemporary Educational Psychology, 43, 25–38. Garcia Coll, C., Lamberty, G., Jenkins, R., McAdoo, H. P., Crnic, K., Wasik, B. H., & Garcia, H. V. (1996). An integrative model for the study of developmental competencies in minority children. Child Development, 67(5), 1891–1914. Ginsburg, H. P., Lee, J. S., & Boyd, J. S. (2008). Mathematics education for young children: What it is and how to promote it. SRCD Social Policy Report, 22(1), 3–24. Green, C. L., Walker, J. M. T., Hoover-Dempsey, K. V., & Sandler, H. M. (2007). Parents’ motivations for involvement in children’s education: An empirical test of a theoretical model of parent involvement. Journal of Educational Psychology, 99(3), 532–544. Gunderson, E. A., & Levine, S. C. (2011). Some types of parent numbertalk count more than others: Relations between parents’ input andchildren’s cardinalnumber knowledge. Developmental Science, 14, 1021–1032. Hill, N. (2001). Parenting and academics socialization as they related to school readiness: The roles of ethnicity and family income. Journal of Educational Psychology, 93(4), 686–697. Hindman, A. H., Wasik, B. A., & Snell, E. K. (2016). Closing the 30 million word gap: Next steps in designing research to inform practice. Child Development Perspectives, 10(2), 134–139. Jeynes, W. (2012). A meta-analysis of the efficacy of different types of parental involvement programs for urban students. Urban Education, 47(4), 706–742. Keels, M. (2009). Ethnic group differences in early head start parents’ parenting beliefs and practices and links to children’s early cognitive development. Early Childhood Research Quarterly, 24(4), 381–397. Klein, A., Starkey, P., Clements, D., Sarama, J., & Iyer, R. (2008). Effects of a prekindergarten mathematics intervention: A randomized experiment. Journal of Research on Educational Effectiveness, 1(3), 155–178.

Parental Involvement and Equity in Mathematics 175 Krieger, N., Williams, D. R., & Moss, N. E. (1997). Measuring social class in US public health research: Concepts, methodologies, and guidelines. Annual Review of Public Health, 18, 341–378. Lareau, A. (2003). Unequal childhoods: Class, race, and family life. Berkeley, CA: University of California Press. Lareau, A. (2011). Unequal childhoods: Class, race, and family life (2nd ed.). Berkeley, CA: University of California Press. LeFevre, J.-A., Polyzoi, E., Skwarchuk, S.-L., Fast, L., & Sowinski, C. (2010). Do home numeracy and literacy practices of Greek and Canadian parents predict the numeracy skills of kindergarten children? International Journal of Early Years Education, 18(1), 55–70. LeFevre, J.-A., Skwarchuk, S.-L., Smith-Chant, B. L., Fast, L., Kamawar, D., & Bisanz, J. (2009). Home numeracy experiences and children’s math performance in the early school years. Canadian Journal of Behavioural Science, 41(2), 55–66. Levine, S. C., Suriyakham, L. W., Rowe, M. L., Huttenlocher, J., & Gunderson, E. A. (2010). What counts in the development of young children’s number knowledge? Developmental psychology, 46(5), 1309–1319. Lukie, I. K., Skwarchuk, S.-L., LeFevre, J.-A., & Sowinski, C. (2014). The role of child interests and collaborative parent-child interactions in fostering numeracy and literacy development in Canadian homes. Early Childhood Education Journal, 42(4), 251–259. Mapp, K. L., & Kuttner, P. J. (2013). Partners in education: A dual capacity-building framework for family-school partnerships. Austin, TX: Southwest Educational Development Laboratory. Masarik, A. S., & Conger, R. D. (2017). Stress and child development: A review of the Family Stress Model. Current Opinion in Psychology, 13, 85–90. Metzger, S., Sonnenschein, S., Galindo, C., & Patel, H. (2015, March). Children’s beliefs about the utility of math and how these beliefs relate to their home math engagement. Poster presented at SRCD, Philadelphia, PA. National Center for Education Statistics. (2018). National achievement-level results. Retrieved from https://www.nationsreportcard.gov/math_2017/#/nation/ achievement?grade=4 National Mathematics Advisory Panel. (2008). Foundations for success: The final report to the National Mathematics Advisory Panel. Washington, DC: U. S. Department of Education. Nguyen, T., Watts, T. W., Duncan, G. J., Clements, D. H., Sarama, J. S., Wolfe, C., & Spitler, M. E. (2016). Which preschool mathematics competencies are most predictive of fifth grade achievement? Early Childhood Research Quarterly, 36, 550–560. Niklas, F., Cohrssen, C., & Taylor, C. (2016). Parents supporting learning: A nonintensive intervention supporting literacy and numeracy in the home learning environment. International Journal of Early Years Education, 24(2), 121–142. Provasnik, S., Malley, L., Stephens, M., Landeros, K., Perkins, R., & Tang, J. H. (2016). Highlights from TIMSS and TIMSS Advanced 2015: Mathematics and science achievement of U. S. students in grades 4 and 8 and in advanced courses at the end of high school in an international context (NCES

176 S. SONNENSCHEIN and B. GAY 2017-002). Washington, DC: U.S. Department of Education, National Center for Education Statistics. Retrieved from http://nces.ed.gov/pubsearch Puccioni, J. (2015). Parents’ conceptions of school readiness, transition practices, and children’s academic achievement trajectories. The Journal of Educational Research, 108(2), 130–147. Purpura, D. J., & Reid, E. E. (2016). Mathematics and language: Individual and group differences in mathematical language skills in young children. Early Childhood Research Quarterly, 36, 259–268. Ramani, G. B., Rowe, M. L., Eason, S. H., & Leech, K. A. (2015). Math talk during informal learning activities in Head Start families. Cognitive Development, 35, 15–33. Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children’s numerical knowledge through playing number board games. Child Development, 79(2), 375–394. Ramani, G. B., & Siegler, R. S. (2011). Reducing the gap in numerical knowledge between low- and middle-income preschoolers. Journal of Applied Developmental Psychology, 32(3), 146–159. Reardon, S. F. (2011). The widening academic achievement gap between the rich and the poor: New evidence and possible explanations. In G. J. Duncan & R. J. Murnane (Eds.), Whither opportunity? Rising inequality, schools, and children’s life chances (pp. 91–116). New York, NY: Russell Sage Foundation. Reardon, S. F., & Galindo, C. (2009). The Hispanic-White achievement gap in math and reading in the elementary grades. American Educational Research Journal, 46(3), 853–891. Reardon, S. F., & Portilla, X. A. (2016). Recent trends in income, racial, and ethnic school readiness gaps at kindergarten entry. AERA Open, 2(3), 1–18. Rothstein, R. (2013). Why children from lower socioeconomic classes, on average, have lower academic achievement than middle-class children. In P. L. Carter & K. G. Welner (Eds.), Closing the opportunity gap: What America must do to give every child an even chance (pp. 61–76). New York, NY: Oxford. Saxe, G. B., Guberman, S. R., & Gearhart, M. (1987). Social processes in early number development. Monographs of the Society for Research in Child Development, 52(2), 1–161. Serpell, R., Baker, L., & Sonnenschein, S. (2005). Becoming literate in the city: The Baltimore Early Childhood Project. New York, NY: Cambridge University Press. Sheldon, S. B., & Epstein, J. L. (2005). Involvement counts: Family and community partnerships and mathematics achievement. The Journal of Educational Research, 98(4), 196–206. Simpkins, S. D., Fredricks, J. A., & Eccles, J. S. (2015). The role of parents in the ontogeny of achievement-related motivation and behavioral choices. In Monographs of the society for research in child development (serial no. 317), 80(2), 1–169. Boston, MA: Wiley. Skwarchuk, S.-L. (2009). How do parents support preschoolers’ numeracy learning experiences at home? Early Childhood Education Journal, 37(3), 189–197. Skwarchuk, S.-L., Sowinski, C., & LeFevre, J.-A. (2014). Formal and informal home learning activities in relation to children’s early numeracy and literacy skills: The development of a home numeracy model. Journal of Experimental Child Psychology, 121, 63–84.

Parental Involvement and Equity in Mathematics 177 Sonnenschein, S. (2002). Engaging children in the appropriation of literacy: The importance of parental beliefs and practices. In O. Saracho & B. Spodek (Eds.), Contemporary perspectives in early childhood education (pp. 127–149). Greenwich, CT: Information Age. Sonnenschein, S., & Dowling, R. (in press). Parents’ socialization of their young children’s interest in math. In O. Saracho (Ed.), Contemporary perspectives on research on motivation in early childhood education. Charlotte, NC: Information Age. Sonnenschein, S., & Galindo, C. (2015). Race/ethnicity and early mathematics skills: Relations between home, classroom, and mathematics achievement. Journal of Educational Research, 108(4), 261–277. Sonnenschein, S., Galindo, C., Metzger, S. R., Thompson, J. A., Huang, H. C., & Lewis, H. (2012). Parents’ beliefs about children’s math development and children’s participation in math activities. Child Development Research Journal Online, 2012, 1–13. Sonnenschein, S., Galindo, C., Simons, C. L., Metzger, S. R., Thompson, J. A., & Chung, M. (2018). How do children learn mathematics? Chinese and Latina immigrant perspectives. In S. S. Chuang, & C. L. Costigan (Eds.), Parental roles and relationships in immigrant families: An international approach (pp. 111–128). New York, NY: Springer. Sonnenschein, S., Metzger, S. R., Dowling, R., Gay, B., & Simons, C. L. (2016). Extending an effective classroom-based math board game intervention to preschoolers’ homes. Journal of Applied Research on Children, 7(2), 1–29. http:// digitalcommons.library.tmc.edu/childrenatrisk/vol7/iss2/1 Sonnenschein, S., Metzger, S. R., & Gay, B. (in press). Concerted cultivation among lowincome Black and Latino families. In S. Sonnenschein & B. E. Sawyer (Eds.), Academic socialization of young black and Latino children: Building on family strengths (pp. 39–60). New York, NY: Springer. Sonnenschein, S., Metzger, S. R., & Thompson, J. A. (2016). Low-income parents’ socialization of their preschoolers’ early reading and math skills. Research in Human Development, 13(3), 207–224. Sonnenschein, S., & Munsterman, K. (2002). The influence of home-based reading interactions on 5-year-olds’ reading motivations and early literacy development. Early Childhood Research Quarterly, 17(3), 317–338. Sonnenschein, S., & Sun, S. (2016). Racial/ethnic differences in kindergartners’ reading and math skills: Parents’ knowledge of children’s development and home-based activities as mediators. Infant and Child Development, 26(5), e2010. https://doi.org/10.1002/icd.2010 Sonnenschein, S., Stapleton, L. M., & Metzger, S. R. (2014). What parents know about how well their children are doing in school. The Journal of Educational Research, 107(2), 152–162. Sonnenschein, S., Thompson, J. A., Metzger, S. R., & Baker, L. (2013). Relations between preschool teachers’ language and gains in low income English language learners’ and English speakers’ vocabulary, early literacy and math skills. NHSA Dialog: A Research-to-Practice Journal for the Early Childhood Field, 16(4), 64–87.

178 S. SONNENSCHEIN and B. GAY Starkey, P., & Klein, A. (2000). Fostering parental support for children’s mathematical development: An intervention with Head Start families. Early Education and Development, 11(5), 659–680. Starkey, P., Klein, A., Chang, I., Dong, Q., Pang, L., & Zhou, Y. (1999, April). Environmental supports for young children’s mathematical development in China and the United States. Paper presented at the Society for Research in Child Development, Albuquerque, NM. Starkey, P., Klein, A., & Wakeley, A. (2004). Enhancing young children’s mathematical knowledge through a pre-kindergarten mathematics intervention. Early Childhood Research Quarterly, 19(1), 99–120. Stipek, D., Milburn, S., Clements, D., & Daniels, D. H. (1992). Parents’ beliefs about appropriate education for young children. Journal of Applied Developmental Psychology, 13(3), 293–310. Susperreguy, M. I., & Davis-Kean, P. E. (2016). Maternal math talk in the home and math skills in preschool children. Early Education and Development, 27(6), 841–857. Taylor, L. C., Clayton, J. D., & Rowley, S. J. (2004). Academic socialization: Understanding parental influences on children’s school-related development in the early years. Review of General Psychology, 8(3), 163–178. Tudge, J. R. H., & Doucet, F. (2004). Early mathematical experiences: Observing young black and white children’s everyday activities. Early Childhood Research Quarterly, 19(1), 21–39. Vandermaas-Peeler, M., Boomgarden, E., Finn, L., & Pittard, C. (2012). Parental support of numeracy during a cooking activity with four-year-olds. International Journal of Early Years Education, 20(1), 78–93. Vandermaas-Peeler, M., Ferretti, L., & Loving, S. (2012). Playing The Ladybug Game: Parent guidance of young children’s numeracy activities. Early Child Development and Care, 182(10), 1289–1397. Vandermaas-Peeler, M., Nelson, J., Bumpass, C., & Sassine, B. (2009). Numeracyrelated exchanges in joint storybook reading and play. International Journal of Early Years Education, 17(1), 67–84. Vandermaas-Peeler, M., & Pittard, C. (2013). Influences of social context on parent guidance and low-income preschoolers’ independent and guided math performance. Early Child Development and Care, 184(4), 500–521. Weiss, H. B., Mayer, E., Kreider, H., Vaughan, M., Dearing, E., Hencke, R., & Pinto, K. (2003). Making it work: Low-income working mothers’ involvement in their children’s education. American Educational Research Journal, 40(4), 879–901. Welner, K. G., & Carter, P. L. (2013). Achievement gaps arise from opportunity gaps. In P. L. Carter & K. G. Welner (Eds.), Closing the opportunity gap: What America must do to give every child an even chance (pp. 1–9). New York, NY: Oxford. Wong, W. S., & Hughes, J. N. (2006). Ethnicity and language contributions to dimensions of parent involvement. School Psychology Review, 35(4), 645–662. Yeung, W. J., Linver, M. R., & Brooks-Gunn, J. (2002). How money matters for young children’s development: Parental investment and family processes. Child Development, 73(6), 1861–1879.

CHAPTER 9

PROMOTING EQUITABLE TEACHING IN MATHEMATICS TEACHER EDUCATION Marta Civil and Roberta Hunter

Ensuring equity for all learners in mathematics education has been on the international agenda for many decades with little evidence that it is being achieved. Moving education way beyond its traditional function of sorting remains challenging. Reliable and trustworthy evidence of what is important and makes a difference is important. However, access to such evidence is not sufficient. Evidence about the negative effects related to streaming and ability grouping (Schleicher, 2014), segregation by race, and socioeconomic factors and its by-product of having access to only lower qualified teachers in these schools, as well as the labelling and low expectations of the students maintained through deficit theorizing by teachers of the students and their families within these communities (Kitchen & Berk, 2016) has accumulated over decades. There is evidence of good intentions on the part of policy makers, researchers, teacher educators, and educators to address the huge disparities in access to, and achievement in mathematics which exists between the different groups of students (OECD, 2016),

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but good intentions are insufficient, change through transformative action is urgently needed (Alton-Lee, 2015, 2017). Moreover, Alton-Lee (2017) argues the need for careful monitoring of implemented actions to ensure that unintended outcomes do not emerge which continue to support what Martin and Larnell (2013, p. 376) describe as “failure focused” story lines; narratives which place blame for lack of achievement back on the learner rather than on the essential lack of understanding and incompatibility between teacher and student backgrounds, along with the institutionalized practices in the school setting which position these learners to fail. Our goal in this chapter is to present research-based evidence of transformative actions that draw on strength-based approaches. We begin by presenting our view on what we mean by equity. We then draw on Rubel’s (2017) four equity focused instructional practices as these capture the current developments in the topic of equity in mathematics teacher education. We will then use vignettes from our own research with teachers to illustrate some of these practices and offer suggestions for how equity can be promoted in teacher education. ABOUT EQUITY IN MATHEMATICS EDUCATION Several authors and position papers have provided definitions of equity in mathematics education. Here we focus on the definitions that more closely have influenced our view on this term. For example, Gutiérrez (2009) talks about the role of power in her view of equity and discusses four dimensions in her definition; access, achievement, identity, and power. In discussing different components of a working definition of equity, Gutiérrez (2007) highlights the following component of, “being unable to predict students’ mathematics achievement and participation based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language” (p. 41). Civil (2014) views the concept of participation as central to the idea of equity: “Who gets to participate in the mathematics classroom? What does it mean to participate in the mathematics classroom?” (p. 4). Association of Mathmatics Teacher Educators equity position statement (2015) reads: “equity as access to high quality learning experiences; inclusion for all learners, mathematics educators, and mathematics teacher educators; and respectful and fair engagement with others” (p. 1). In the National Council of Teachers of Mathematics (NCTM, 2014) position statement on access and equity in mathematics education, we read: Creating, supporting, and sustaining a culture of access and equity require being responsive to students’ backgrounds, experiences, cultural perspectives, traditions, and knowledge when designing and implementing a mathematics

Promoting Equitable Teaching in Mathematics Teacher Education 181 program and assessing its effectiveness. Acknowledging and addressing factors that contribute to differential outcomes among groups of students are critical to ensuring that all students routinely have opportunities to experience high-quality mathematics instruction, learn challenging mathematics content, and receive the support necessary to be successful. Addressing equity and access includes both ensuring that all students attain mathematics proficiency and increasing the numbers of students from all racial, ethnic, linguistic, gender, and socioeconomic groups who attain the highest levels of mathematics achievement. (p. 1)

What are the implications of these different views on equity in mathematics education for mathematics teacher education? Civil (2014) uses her view of equity as centered on the concept of participation to propose a four-element framework that has clear implications for teacher education. These four elements are: 1. Concept of status: What does it mean to be good at math? 2. Nature of the task: Whose knowledge and experiences are represented? 3. Approaches to doing mathematics: Whose and what approaches are valued? 4. Language(s) in the classroom: Which language(s) and forms of communication are privileged? Louie (2018) calls for the need to add a sociopolitical lens to the concept of teacher noticing as a way to address the “cultural and ideological obstacles to noticing students’ mathematical thinking and strengths” (p. 56). Jorgensen and Perso (2012) suggest that for the indigenous and other diverse students in Australia, success in mathematics should happen “regardless of their language, cultural background, gender or geographical location” (p. 131). Atweh and Brady (2009) extend our thinking towards considering that the importance of mathematics for students needs to be seen at three possible levels “conforming, reforming, or transforming society” (p. 720). EQUITY-FOCUSED INSTRUCTIONAL PRACTICES Rubel (2017) identifies four approaches to teaching mathematics that are equity-focused; namely, standards-based mathematics instruction; complex instruction; culturally relevant pedagogy; and, teaching mathematics for social justice (p. 69). From an equity point of view, we interpret “standards-based mathematics instruction” as that which focuses on the two components called for by

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Hiebert and Grouws (2007) that promote conceptual understanding: an explicit focus on concepts both by teachers and students; and, opportunities for students to “struggle with important mathematics” (p. 387). At the heart of Complex Instruction (CI; Cohen, 1994; Cohen & Lotan, 1997; Featherstone et al., 2011) is the concept of status, “an agreed-upon social ranking where everyone feels it is better to have a high rank within the status order than a low rank. Cohen (1994) outlines how “group members who have a high rank are seen as more competent” (p. 27) while those with lower status most often participate less. However, Cohen (1998) explains how the failure to participate is situational, not dependent on the personality of the low status child and therefore the problem sits within classroom interactions and not with individuals. This makes teachers changing the social situation of classrooms a key step towards equity. For culturally relevant pedagogy (CRP) we draw directly on Ladson-Billings (1995): (a) Students must experience academic success; (b) students must develop and/or maintain cultural competence; and (c) students must develop a critical consciousness through which they challenge the status quo of the current social order. (p. 160)

But we bring in a more recent elaboration, that of culturally sustaining pedagogy (Paris, 2012): The term culturally sustaining requires that our pedagogies be more than responsive of or relevant to the cultural experiences and practices of young people—it requires that they support young people in sustaining the cultural and linguistic competence of their communities while simultaneously offering access to dominant cultural competence. (p. 95)

Finally, for teaching mathematics for social justice, we draw on Gutstein’s (2016) work on reading and writing the world with mathematics (RWWM): “RWWM means that students use mathematics to comprehend and change the world—and through the process, deepen their knowledge of both mathematics and their social reality” (p. 455). The work of Gutstein is a common referent in the literature of mathematics for social justice. We build on this but also bring in issues around home language versus language of instruction as aspects of teaching mathematics for social justice, particularly because often these language issues are tied to policy and often reflect certain political orientations (e.g., assimilation of immigrants rather than diversity as a resource; Alrø, Skovsmose, & Valero, 2005; see Civil, 2012 for a critical overview of educational policies that position diversity as a deficit that has to be overcome; Wright, 2005).

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We argue that the first two teaching practices (standards-based and CI) focus on pedagogical approaches that promote rich mathematical discussions and participation in challenging tasks that call for diverse students’ strengths. But the focus could be primarily on mathematics for the sake of mathematics. The other two practices take on a sociocultural-political orientation. These practices require teachers to build on students’ background and experiences; design learning experiences that engage students in using mathematics to solve problems that are relevant and meaningful to the students. The Association of Mathematics Teacher Educators (AMTE) Equity Committee conducted a survey to gain a better understanding of how teacher education programs are addressing the indicators focused on equity, diversity, and social justice that are embedded throughout the Standards for Preparing Teachers of Mathematics (AMTE, 2017; https://amte.net/sites/default/files/SPTM. pdf). Findings from this survey point to the fact that while respondents seem generally comfortable with building on students’ mathematical ideas (which would correspond more to the first two teaching practices that we just outlined), they found it much harder to address issues of power and privilege, and working with families and communities in their teacher preparation programs (these would correspond to the sociocultural-political orientation of the other two practices; AMTE Equity Committee, 2018). In this chapter we argue for the need to integrate the different equityfocused practices that Rubel (2017) discusses. While documents such as the Common Core State Standards for Mathematics, in the United States, outline a vision for standards-based mathematics, these documents often do not address sociocultural-political contexts (Bartell et al., 2017; Gutstein, 2010). In so doing, the implicit message seems to be that as long as we provide rich mathematical experiences to all students everything else will work out. But we join others (Bartell et al., 2017) in saying that this is not the case and that in fact, policy documents that develop strong vision for teaching rich mathematics but pay no attention to issues of power and privilege may actually exacerbate inequities. In what follows we use two well-established programs of research in mathematics teacher education in two different parts of the world (New Zealand and the United States) to illustrate how these different equity-focused instructional practices can be integrated for a mathematically rich experience for students who have traditionally been marginalized (e.g., Pāsifika students in the New Zealand context, Mexican American students in the U.S. context). EQUITY BASED ACCESS TO MATHEMATICAL DIALOGUE The importance of all students being able to participate in communicating mathematically has received considerable attention over the past 2 decades.

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Explicit links have been made between those students who have opportunities to participate in the dialogue and the positive outcomes this has on them developing rich and deep understandings in mathematics (e.g., Civil, 2014; Cobb, Wood, & Yackel, 1993; Manouchehri & St. John, 2006; Nathan & Kim, 2007; Wood, Williams, & McNeal, 2006). But, in recent times more direct focus has been placed not just on having the students participate in mathematical talk but on the more specific forms of mathematical talk students participate in—those used within mathematical practices (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). There are many mathematical practices, some examples of which include making mathematical explanations (the beginning of a mathematical argument), justification and argumentation, generalizations and representations (RAND Mathematics Study Panel, 2003). As a key equity issue, of particular interest in this section is how teachers can provide all students with opportunities to participate and engage in developing justifications, generalizations, and mathematical argumentation as a powerful way to reason. To do this we look at a case of a group of Pāsifika students in Aotearoa, New Zealand who have a history of underachieving in mathematics as a result of the institutionalized practices and structural inequities they encounter in the schooling system (Hunter & Hunter, 2017a). We explore ways to support teachers in a long-term professional learning and development program called Developing Mathematical Inquiry Communities (DMIC). Engaging in collaborative interaction and using the discourse of inquiry and argumentation is not something many students can accomplish easily without specific adult intervention. However, for some students the challenges extend beyond learning the social and mathematical norms (Sullivan, Zevenbergen, & Mousley, 2002) in the classroom of how to engage in the mathematical dialogue inherent in mathematical practices to include a set of values and beliefs which differ from those held by the more dominant groups of students of Western origin in their classrooms and those of their teachers. In the case of the Pāsifika students (and other diverse groups who hold similar values and beliefs) these values and beliefs, if not considered, have been shown to cause the students a high level of discomfort and cause them to withdraw from engaging in mathematics. Without direct teacher actions, both culturally responsive and culturally sustaining, these students are effectively precluded from questioning, challenging, and engaging in mathematical argumentation and thus developing rich connected understandings (see Hunter & Anthony, 2011; Hunter & Hunter, 2017b; Hunter, Hunter, & Bills, in press).

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COMING TO UNDERSTAND DIFFERENT WORLDVIEWS We use here the term Pāsifika peoples to refer to a multi-ethnic group of people from different Pacific Island nations who speak a wide range of languages and may include a group who are second or third generation born in Aotearoa, New Zealand. What binds this heterogeneous group are a set of cultural commonalities founded within some core values and beliefs which shape their interactions and ways of being as peoples of Pacific Island nations. These values, which include reciprocity, respect, service, inclusion, family, relationships, spirituality, leadership, love, and belonging (Anae, Coxon, Mara, Wendt-Samu, & Finau, 2001) are firmly set within the norms of collectivism and communalism. These values support a communal view, one in which the success of the fānau (extended family or group) is considered more important than the success of individuals. This adds another dimension to classroom interactions, which when seen as a strength by teachers supports stronger and more collaborative relationships. The challenge is to have teachers recognize that Pāsifika students and other diverse students may hold a different worldview than their own and that these values and beliefs shape their interactions and the way these students communicate and participate in mathematical practices. More importantly, teachers need to understand what a rich basis these offer as a foundation to build a collaborative classroom mathematics community in which a respectful but rich mathematics dialogue can occur. Byrd (2016) argues that teachers need to develop cultural competency so that they understand their diverse students’ communities and home lives. We argue that this needs to be extended to them gaining understanding of their different students’ core values and beliefs, and how these can be used to gradually induct all students into mathematical practices which provide them with rich connected mathematical understandings. When working with teachers with Pāsifika students our initial challenge is to have them interrogate their own values and beliefs which have been shaped by their upbringing and past experiences. Commonly, teachers in Aotearoa, New Zealand when asked to name some of the core beliefs and values they were raised with, identify factors which suggest a competitive and individualistic focus such as “be the best,” “the neatest,” and “hard work makes you a winner.” A professional development activity which engages teachers in mapping how these values shape classroom interactions against those held by their Pāsifika students offers them opportunities to begin to see how their individualistic values and beliefs may collide with the more communal outlook of some of their learners. It also offers a way to have them critically reflect on their perceptions of the students; students often considered as low status because they are positioned to participate and contribute less. Rather, the teachers are provided with a means to start

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building an understanding of why some students are reluctant to interact, and question, or challenge. SUPPORTING TEACHERS TO SCAFFOLD STUDENTS TO ACCESS MATHEMATICAL PRACTICES Teachers too, are often novices in such classrooms having learnt mathematics themselves in more traditional settings where transmission of mathematical knowledge was the norm. To address both support for teachers and them scaffolding students into the use of mathematical practices, as part of our work we use a communication and participation framework (for more information on the framework see Hunter & Anthony, 2011). This framework provides a set of communicative and performative actions which teachers can use flexibly to induct their diverse students into the use of a range of mathematical practices in culturally responsive ways. At the same time, importance is placed on scaffolding student use of questions to elicit further explanatory information or justification of reasoning. The use of the communication and participation framework supports teachers to build a mathematical inquiry community within the collective beliefs and values of the diverse students. Within the performative actions, the students are scaffolded to construct a joint explanation explained within a cultural metaphor understood by all members. For example: Teacher: You are all in the same waka (canoe), paddle together in unison.

Performative actions require that all group members are responsible for their own sense-making while also responsible for the reasoning of each other. At the same time, teachers are able to gradually reposition themselves from a central position of “mathematical authority”—a role which Pāsifika students commonly place teachers within their value system—to that of “participant in the dialogue.” In shifting from a central position of transmitter of knowledge to one of facilitator, the teachers are provided with opportunities to become more reflective and “notice” the interconnect between teaching and student learning. However, this extends past the common focus of noticing students’ mathematical reasoning to also include “noticing” student participation and access to the mathematics in the classroom, and provides them with the means to address the status of individual students. This adds the sociopolitical lens promoted by Louie (2018) and many other equity focused researchers (e.g., Hand, 2012; McDuffie et al., 2014; Wager, 2014) who argue that teachers need to not only respond to cognitive responses but also use “noticing” of participation in order to grapple with issues of power, privilege, and who has opportunities to access the mathematical dialogue.

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Time is an important component if teachers are to ensure equitable participation in the mathematics dialogue. For our Pāsifika students, they gradually come to realize that they have multiple voices, appropriate to different settings, and gain confidence to use them. In the next section, I illustrate the way in which a teacher enacted aspects of the communication and participation framework, drawing on it as a flexible support. She began by developing the students’ skills to work collectively to build mathematical explanations. She placed focus on the need for all group members to engage in construction of mathematical explanations and be able to explain them to a wider audience. Listeners were also required to examine, analyze, and make sense of the explanations offered by others: Teacher: They might say I think it is 59. That’s cool but they have to back it up, explain how they came up with it. They have to say why. I want you before you even begin to go around in your group and actually talk about it. Someone in your group may ask you a question. For example, that’s an interesting solution, why do you think that? Could you show us how you got it?

Through this statement she has provided opportunities for members of the small groups to practice constructing, explaining, and in turn questioning and clarifying explanations step-by-step. In doing so, for those students who previously had been reticent to speak, or question, being provided with opportunities to practice safely within a small group is important. At the same time, making explicit what is involved in active listening and sense-making is provided for those who need it. However, if students are to construct deep connected reasoning, then they need to engage in dialogue which extends beyond explaining. For many this requires specific teacher guidance and explicit discussions about how this might be done. In the following statement she asks: Teacher: Yes you could be agreeing with what the person says . . . but are you always agreeing, do you think?

Through her statement the students are pressed to consider the idea of disagreement as an appropriate part of mathematical dialogue. She also discusses mathematical argumentation directly: Teacher: Arguing is not a bad word . . . I am talking about arguing in a good way. If you do not agree with what someone else has said you can say that as long as you say it in an okay sort of way. If you don’t agree then a suggestion could be that you might say I don’t actually agree with that. Could you show that to me? Could you perhaps write it in numbers? Could you draw something to show that idea to me? That’s fine because sometimes when you go over and you do that again you think . . . oh maybe that wasn’t quite right and that’s fine.

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In this statement she has recognized that those students not used to engaging in mathematical argumentation as a mode of communication needed opportunities to practice so that they were strengthened in both challenging and responding to challenge. A second set of questions related to a student need to be convinced through mathematical argumentation was also introduced and the students were positioned to prepare responses: Teacher: Think about the questions that you might be asked. Practise using some of those questions like why does that work, or how can you know that is true?

Within this climate of intellectual autonomy and with a press toward the need for convincing through mathematical argumentation the students were provided with many opportunities to engage in higher levels of cognitive reasoning more equitably. Task design goes hand in hand with students having opportunities to participate in mathematical practices. There are three required aspects if tasks are to support equity. These include multiple points of entry and exit, access to key mathematical ideas, and their connectedness to the students’ lives. Henningsen and Stein (1997) explain that tasks that allow teachers to maintain high cognitive levels of challenge not only support multiple entry and exit points but can also be solved in a range of different ways using different representations, and support students drawing on a range of resources developed around both in-school and out-of-school. For our Pāsifika students providing them with such tasks broadens their opportunities to participate and allows teachers to “notice” their contributions. Exposing teachers to what students can do also increases their expectations of the students; an important aspect because many teachers hold low expectations towards their Pāsifika or Māori students (Rubie-Davies, 2016). One way to support their development of complex high level problematic tasks within the known Pāsifika contexts of the students in DMIC is for the teachers to meet with parents and ask them to tell them where they see mathematics within their home contexts. In both listening to families talk about mathematics in the community and considering the values and beliefs of the Pāsifika community, the teachers have become learners able to see the strengths these offer. FUNDS OF KNOWLEDGE IN MATHEMATICS TEACHER EDUCATION In this section we draw on over 25 years of research on the application of the concept of funds of knowledge to the teaching and learning of

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mathematics. The Funds of Knowledge for Teaching Project (González, Moll, & Amanti, 2005) is a strong example of CRP, with some unique features. The original work took place in bilingual classrooms and a major goal was the development of rich learning experiences that would support students’ biliteracy. As Moll (1992) writes, A sociocultural approach to instruction presents new possibilities in bilingual education, where the emphasis is not solely on remediating students’ English language limitations, but on utilizing available resources, including the children’s or the parents’ language and knowledge, in creating new, advanced instructional circumstances for the students’ academic development. (p. 23)

The Funds of Knowledge Project is not only an example of CRP but also of the more recent idea of culturally sustaining pedagogy (Paris, 2012) as it certainly aims to sustain “the cultural and linguistic competence of their communities while simultaneously offering access to dominant cultural competence” (Paris, 2002, p. 95). The idea behind funds of knowledge is that all communities and households have resources, knowledge, experiences on which they draw to get ahead and thrive. As Moll, Amanti, Neff, and González (1992) write, “we use the term ‘funds of knowledge’ to refer to these historically accumulated and culturally developed bodies of knowledge and skills essential for household or individual functioning and well-being” (p. 133). One of the main components in funds of knowledge work is that teachers become learners from the families they visit. That is, teachers conduct in-depth ethnographic home visits to learn about the funds of knowledge in the household. Then, in teacher study groups, teacher-researchers and university researchers discuss the findings from the household visits and develop learning modules based on this knowledge. As Amanti (2005), a teacher-researcher in the Funds of Knowledge for Teaching Project writes, “This process gives students and their families a sense that their experiences are academically valid. For far too long, the homes of working class and minoritized students have been constructed as deficient and lacking in sufficient stimulation for academic success” (p. 138). Amanti also brings up a key point about what a funds of knowledge approach is, “It is not about replicating what students have learned at home, but about using students’ knowledge and prior experiences as a scaffold for new learning” (p. 135). This idea of “using students’ knowledge and prior experiences as a scaffold for new learning” is at the center of the work done with teachers in the extension of the Funds of Knowledge for Teaching Project to focus on mathematics teaching and learning. Civil (2007, 2016, 2018) discusses the characteristics of this work. In particular, Civil (2018) argues that in a funds of knowledge teaching approach, the concepts of resources, valorization, and participation are central. The idea of resources is implicit in the

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definition of the concept “funds of knowledge.” Families have resources that teachers can draw on to develop rich mathematical experiences. For example, Civil (2007) describes a construction module (see also SandovalTaylor, 2005) where the teacher drew on the knowledge of her students’ parents and other family members to develop a module around the building of houses. Students (7 year olds) were actively engaged in mathematically rich discussions involving measurement and patterns (Civil, 2002; Sandoval-Taylor, 2005). Their knowledge about construction, as well as that of their parents was valued and participation happened in a natural way. Civil (2007, 2016) discusses how in everyday settings people often learn through participation in the practice. In mathematics classrooms that are grounded on students’ and their families’ funds of knowledge, participation looks more like what we may see in out-of-school settings, cooperative, apprenticeship-like, more like what Rogoff (2012) describes as the nature of learning in everyday life. In such an approach, teachers focus on the strengths that children bring to the classroom. For example, a child who is not yet proficient in the language of instruction can be positioned as a competent mathematics student, even in settings with language policies that restrict the use of home languages. This is the case of Tucson, Arizona. Yet many teachers with whom we have worked support their students’ use of their home language, which enhances their participation. The teacher plays a key role in encouraging and supporting that communication in Spanish, as this quote from a third-grade (8 year olds) teacher shows: Nadia: I have one little girl who arrived in September [from Mexico], and she’s really good at doing a lot of math. She’s really good at math and she knows that’s her strength so whenever they have something [to do in groups] the kids say, “Elsa, Elsa, come this way” because Elsa shares. In a very quiet way she shares. So when I ask her, “Ok, Elsa in your group you’re going to be the one sharing over here,” she’ll do it in Spanish and then she’ll turn around and she’ll look at me like, “Can you please translate for me so that they’ll understand what I’m talking about?” And then I’ll look at the group and I’ll say, “Did she cover everything you all talked about?” and they will say yes. Because they are talking about it in English not just Spanish.

This teacher is positioning Elsa as a capable student in mathematics (related to the concept of status in complex instruction). In so doing, the teacher is capitalizing on the student’s funds of knowledge (in this case her knowledge of Spanish) and on her knowledge of mathematics. We view the teacher’s actions in supporting and valuing Elsa’s use of her home language to contribute to the mathematical discussion as an example of teaching mathematics for social justice. Teaching within a funds of knowledge orientation can be challenging. It requires teachers to become real learners about their students’ and families’

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knowledge and experiences; to engage in the pedagogical transformation of the mathematics-based funds of knowledge for their implementation within the school curricula. As González, Andrade, Civil, and Moll (2001) write, “Although the households we interviewed certainly deployed mathematical concepts, the academic transformation of those concepts was elusive” (p. 120); and most importantly, it requires teachers to focus on the strengths that students bring to the classroom. Although the overall process can be challenging, teachers who have engaged in this work stress how rewarding and transformative it is. For example, here is what the teacher in the construction module we referred to earlier writes: The planning of this module was often frustrating. I knew my academic objectives, but I was not exactly sure how I would make them fit into this theme . . . I felt that I was in a state of uncertainty, yet I still had to cover certain areas of the curriculum. I had to make sure that my students learned the strategies and skills to be academically successful. In looking back I realize that . . . my students were catapulted to higher levels of literacy and numeracy during this module because I had provided them with multiple access to the content . . . My students flourished in ways that I did not expect. I believed my students had internalized what they were learning because what they brought from home surrounded and supported their learning. (Sandoval-Taylor, 2005, pp. 162–163)

And here are some excerpts from Leslie’s journal. Leslie was the teacher in the garden project (Civil, 2007; Kahn & Civil, 2001): Because I have taken the risk to ask parents to help with a real inquiry project they are rallying around me. This is not something that I pull out of my file cabinet every year in September. Having never done this before I have only some idea of where it is going. The parents know this and they are creating the curriculum within my frame. It is very exciting. And frankly I am hard to excite. [September] The kids weren’t kidding when they said we have been looking at perimeter and area for two months. That is not the only thing we have done in mathematics, but probably the one that they will remember because it is meshed with what they are doing naturally with their garden . . . The students and I are driving the curriculum together, or at least we are in the same car. [February] Are there things that my students haven’t gotten to mathematically? Yes, and that bothers me, but I have proven a point to myself. Much of the math that the kids did this year, was both authentic and valid for what we were doing. And that was the whole reason that I wanted to participate in this project. [April]

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IN CLOSING We end this chapter with some suggestions for how equitable teaching practices can be promoted in mathematics teacher education. We draw on the work we have just presented, as well as on other sources that suggest some practical equity focused activities in mathematics education for mathematics teacher education. Suggestion 1: Teachers Engaging in Action-Research or Design Research Both the DMIC in New Zealand and the funds of knowledge work in the United States have a model of professional development where teachers collaborate with university faculty. They become codevelopers and co-researchers. This is important to note, as this work requires time, trustbuilding, and respect for each other’s ideas (see Felton-Koestler, 2017 for a description of a teacher’s journey from traditional, teacher-centered instruction to equity-focused practices including venturing into exploring teaching mathematics for social justice). Both action-research and design research include elements of time through a process of iteration; relationships and trust are constructed, and mutual respect developed through the co-construction process inherent in the professional development model. Suggestion 2: Teachers Conducting Ethnographic Household Visits With a Focus on Learning From the Families This is the main component of the funds of knowledge work (González et al., 2005). A possible alternative to the household visits are community walks (e.g., Aguirre et al., 2013), where teachers (or preservice teachers) become familiar with the resources that exist in their students’ community. The goal of these practices (household visits, community walk) is for teachers to learn about and from their students’ everyday contexts to then draw on these as resources for school learning. Household visits allow for teachers to have an in-depth learning opportunity and the development of a strong relationship with the family. Community walks focus more on noticing the resources in the community that are likely to be familiar to their students. Teachers may want to interview community members to gain a better understanding of these resources. In both practices the main goal is for teachers to go in as learners.

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Suggestion 3: Teachers and Parents Engaging in Joint Explorations and Discussion of Mathematics This idea relates to the previous one of teachers learning from their students’ families but suggests the development of an authentic two-way dialogue between home/families and school/teachers (Civil, 2002; Civil & Andrade, 2003; Civil & Bernier, 2006). Through joint experiences, parents and teachers learn with and from each other not only about mathematics (e.g., immigrant parents are likely to bring different ways to do mathematics) but also about their expectations for the children’s education and for each other’s role. There are a number of ways this can be achieved. One way would be through teacher–parent/community meetings where the teachers and parents jointly engage in mathematical activity and then discuss the different ways of reasoning they bring to the activity. Another activity might be through the use of cameras. The parents and students are given cameras and asked to take images of when they are engaged in a mathematical activity outside of school. They then select and share the images with the teachers. Suggestion 4: Building Relationships Both in Classrooms and Within the Community To have all students participate in the high level cognitive dialogue of mathematical argumentation and other mathematical practices requires time, careful scaffolding (to ensure that the students are able to integrate their own belief systems with their new voice) and the building of trust (Civil & Hunter, 2015; Hunter & Anthony, 2011). Morrison, Robbins, and Rose (2008) outline the importance of all students being positioned for participation and pressed for academic success. Teachers can be supported through access to a flexible tool which outlines a set of possible actions teachers can use to establish a collaborative classroom environment (see Hunter & Anthony, 2011). Suggestion 5: Understanding Diverse World Views Seeing the strengths students bring from their family and community to the mathematics classrooms is one way to avoid the failure-focused stories of students who differ from the dominant groups of students in their classrooms and their teacher (Martin & Larnell, 2013). A key to understanding their own worldview and that of their students and how these are shaped by values and beliefs (Hunter et al., 2016) is having them engage in activities

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in which they actively discuss and explore their own values and beliefs. This provides them with opportunities to grow awareness of others’ differing views and participate in what Jilk (2016) describes as reculturing within “a culture of professional development in which the norm is to work from strengths rather than focus on deficits” (p. 197). Suggestion 6: An Important Strength That Students Bring Is the Home Language(s) Teachers can build on the diversity of languages in the classroom as a resource towards learning mathematics, as multiple languages and forms of communication can enhance mathematical discussions (Moschkovich, 2013). Within classrooms, students should have as a right to speak in the language which supports them to access deep and rich mathematics. There are many professional readings around how to empower teachers to support students to use their home language (e.g., Celedón-Pattichis & Turner, 2012; Planas & Civil, 2013). These could be used in teacher study groups and other forms of professional development. Suggestion 7: Equitable Noticing Extending Beyond the Cognitive Aspects of Mathematics Through shifting the focus beyond “noticing” students’ mathematical reasoning to include how individual students participate and communicate mathematical reasoning makes visible the power and status hierarchies which operate in most classrooms (Hand, 2012). A number of teacher professional development programs grounded in complex instruction (Cohen, 1998) have been shown to disrupt inequitable power relationships. These include intensive and ongoing professional development in classrooms with mentoring support, the development of communities of practice on equitable noticing, and engagement in video clubs (Jilk, 2016; Louie, 2018). Within this sociopolitical lens teachers are offered ways in which to consider culture and ideology as part of noticing (Louie, 2018). REFERENCES Aguirre, J. M., Turner, E. E., Bartell, T. G., Kalinec-Craig, C., Foote, M. Q., Roth McDuffie, A., & Drake, C. (2013). Making connections in practice: How prospective elementary teachers connect to children’s mathematical thinking and community funds of knowledge in mathematics instruction. Journal of Teacher Education, 64(2), 178–192.

Promoting Equitable Teaching in Mathematics Teacher Education 195 Alrø, H., Skovsmose, O., & Valero, P. (2005). Culture, diversity and conflict in landscapes of mathematics learning. In M. Bosch (Ed.), Proceedings of the fourth Congress of the European Society for Research in Mathematics Education (pp. 1141– 1152). Sant Feliu de Guíxols, Spain: FUNDEMI IQS, Universitat Ramon Llull. Alton-Lee, A. (2015, September). Disciplined innovation for equity and excellence in education: Learning from Māori and Pasifika change expertise. Invited paper for the World Educational Research Association focal session: Education of Diverse Students: A Multi Country Perspective. Budapest, Hungary. Alton- Lee, A. (2017). ‘Walking the talk matters’ in the use of transformative education CSE Seminar Series 268. CSE, Victoria, Australia. Amanti, C. (2005). Beyond a beads and feathers approach. In N. González, L. Moll, & C. Amanti (Eds.), Funds of knowledge: Theorizing practice in households, communities, and classrooms (pp. 131–141). Mahwah, NJ: Erlbaum. AMTE. (2017). Standards for preparing teachers of mathematics. Retrieved from https:// amte.net/sites/default/files/SPTM.pdf AMTE Equity Committee. (2018). Equity committee survey: Results and future directions. In B. M. Benken (Ed.), Connections (An official AMTE publication for the mathematics teacher education community). Avaiable at https://amte.net/connections/2018/03/amtes-equity-committee-survey -results-and-future-directions Anae, M., Coxon, E., Mara, D., Wendt-Samu, T., & Finau, C. (2001). Pasifika education research guidelines. Wellington, New Zealand: Ministry of Education. Association of Mathematics Teacher Educators. (2015). Position: Equity in mathematics teacher education. Retrieved from https://amte.net/sites/default/files/ amte_equitypositionstatement_sept2015.pdf Atweh, B., & Brady, K. (2009). Socially response-able mathematics education: Implications of an ethical approach. Eurasia Journal of Mathematics, Science and Technology Education, 5(3), 267–276. Bartell, T., Wager, A., Edwards, A., Battey, D., Foote, M., & Spencer, J. (2017). Toward a framework for research linking equitable teaching with the Standards for Mathematical Practice. Journal for Research in Mathematics Education, 48(1), 7–21. Byrd, C. M. (2016). Does culturally relevant teaching work? An examination from student perspectives. Sage Open, 6(3), 1–10. https://doi. org/10.1177/2158244016660744 Celedón-Pattichis, S., & Turner, E. E. (2012). “Explícame tu respuesta”: Supporting the development of mathematical discourse in emergent bilingual kindergarten student. Bilingual Research Journal, 35(2), 197–216. https://doi.org/10.10 80/15235882.2012.703635 Civil, M. (2002). Culture and mathematics: A community approach. Journal of Intercultural Studies, 23(2), 133–148. Civil, M. (2007). Building on community knowledge: An avenue to equity in mathematics education. In N. Nasir & P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 105–117). New York, NY: Teachers College Press. Civil, M. (2012). Mathematics teaching and learning of immigrant students: An overview of the research field across multiple settings. In O. Skovsmose &

196 M. CIVIL and R. HUNTER B. Greer (Eds.), Opening the cage: Critique and politics of mathematics education (pp. 127–142). Rotterdam, The Netherlands: Sense. Civil, M. (2014). Guest editorial: Musings around participation in the mathematics classroom. The Mathematics Educator, 23(2), 3–22. Civil, M. (2016). STEM learning research through a funds of knowledge lens. Cultural Studies of Science Education, 11(1), 41–59. https://doi.org/10.1007/ s11422-014-9648-2 Civil, M. (2018). Intersections of culture, language, and mathematics education: Looking back and looking ahead. In G. Kaiser, H. Forgasz, M. Graven, A. Kuzniak, E. Simmt, & B. Xu (Eds.), Invited lectures from the 13th International Congress on Mathematical Education (pp. 31–47). New York, NY: Springer. https:// doi.org/10.1007/978-3-319-72170-5_3 Civil, M., & Andrade, R. (2003). Collaborative practice with parents: The role of the researcher as mediator. In A. Peter-Koop, V. Santos-Wagner, C. Breen, & A. Begg (Eds.), Collaboration in teacher education: Examples from the context of mathematics education (pp. 153–168). Boston, MA: Kluwer. Civil, M., & Bernier, E. (2006). Exploring images of parental participation in mathematics education: Challenges and possibilities. Mathematical Thinking and Learning, 8(3), 309–330. Civil, M., & Hunter, R. (2015). Participation of non-dominant students in argumentation in the mathematics classroom. Intercultural Journal, 26(4), 296–312. Cobb, P., Wood, T., & Yackel, E. (1993). Discourse, mathematical thinking, and classroom practice. In N. Minick, A. Stone, & E. Forman (Eds.), Contexts for learning (pp. 91–119). New York, NY: Oxford University Press. Cohen, E. G. (1994). Designing groupwork: Strategies for the heterogeneous classroom (2nd ed.). New York, NY: Teachers College Press. Cohen, E. (1998) Making cooperative learning equitable. Educational Leadership, 56(1), 18–21. Cohen, E. G., & Lotan, R. A. (Eds.). (1997). Working for equity in heterogeneous classrooms: Sociological theory in practice. New York, NY: Teachers College Press. Featherstone, H., Crespo, S., Jilk, L., Oslund, J., Parks, A., & Wood, M. (2011). Smarter together! Collaboration and equity in the elementary math classroom. Reston, VA: National Council of Teachers of Mathematics. Felton-Koestler, M. (2017). “Children know more than I think they do”: The evolution of one teacher’s views about equitable mathematics teaching. Journal of Mathematics Teacher Education, 22(2), 153–177. González, N., Andrade, R., Civil, M., & Moll, L. C. (2001). Bridging funds of distributed knowledge: Creating zones of practices in mathematics. Journal of Education for Students Placed at Risk, 6(1–2), 115–132. González, N., Moll, L., & Amanti, C. (Eds.). (2005). Funds of knowledge: Theorizing practice in households, communities, and classrooms. New York, NY: Routledge. Gutiérrez, R. (2007). (Re)defining equity: The importance of a critical perspective. In N. Nasir & P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 37–50). New York, NY: Teachers College Press. Gutiérrez, R. (2009). Framing equity: Helping students “play the game” and “change the game.” Teaching for Excellence and Equity in Mathematics, 1(1), 5–8.

Promoting Equitable Teaching in Mathematics Teacher Education 197 Gutstein, E. (2010). The Common Core State Standards initiative: A critical response. Journal of Urban Mathematics Education, 3(1), 9–18. Gutstein, E. (2016). “Our issues, our people–Math as our weapon”: Critical mathematics in a Chicago neighborhood high school. Journal for Research in Mathematics Education, 47(5), 454–504. Hand, V. (2012). Seeing culture and power in mathematical learning: Toward a model of equitable instruction. Educational Studies in Mathematics, 80(1–2), 233–247. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549. Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Charlotte, NC: Information Age. Hunter, R. K., & Anthony, G. (2011). Forging mathematical relationships in inquirybased classrooms with Pasifika students. Journal of Urban Mathematics Education, 4(1), 98–119. Hunter, R., & Hunter, J. (2017a). Maintaining a cultural identity while constructing a mathematical disposition as a Pāsifika learner. In E. A. McKinley & L. Tuhiwai Smith (Eds.), Handbook of Indigenous education (pp. 1–19). New York, NY: Springer. Hunter, R., & Hunter, J. (2017b). Opening the space for all students to engage in mathematical talk within collaborative inquiry and argumentation. In R. Hunter, M. Civil, B. Herbel-Eisenmann, N. Planas, & D. Wagner (Eds.), Mathematical discourse that breaks barriers and creates space for marginalized learners (pp. 1–22). Rotterdam, The Netherlands: Sense. Hunter, R., Hunter, J., & Bills, T. (in press). Enacting culturally responsive or socially-response-able mathematics education. In C. Nicol, S. Dawson, J. Archibald, & F. Glanfield (Eds.), Living culturally responsive mathematics curriculum and pedagogy: Making a difference with/in indigenous communities. New York, NY: Springer. Hunter, J., Hunter, R., Bills, T., Cheung, I., Hannant, B., Kritesh, K., & Lachaiya, R. (2016). Developing equity for Pāsifika learners within a New Zealand context: Attending to the culture and values. NZ Journal of Educational Studies, 51(2), 197–209. Jilk, L. M. (2016). Supporting teacher noticing of students’ mathematical strengths. Mathematics Teacher Educator, 4(2), 188–199. Jorgensen, R., & Perso, T. (2012). Equity and the Australian curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen, & D. Siemon (Eds.), Engaging the Australian National Curriculum: Mathematics—Perspectives from the field (pp. 115–133). Brisbane, Australia: Mathematics Education Research Group of Australasia. Kahn, L., & Civil, M. (2001). Unearthing the mathematics of a classroom garden. In E. McIntyre, A. Rosebery, & N. González (Eds.), Classroom diversity: Connecting school to students’ lives (pp. 37–50). Portsmouth, NH: Heinemann.

198 M. CIVIL and R. HUNTER Kitchen, R., & Berk, S. (2016). Educational technology: An equity challenge to the Common Core. Journal for Research in Mathematics Education, 47(1), 3–16. Ladson-Billings, G. (1995). But that’s just good teaching! The case for culturally relevant pedagogy. Theory into Practice, 34(3), 159–165. Louie, N. L., (2018). Culture and ideology in mathematics teacher noticing. Educational Studies in Mathematics, 97(1), 55–69. https://doi.org/10.1007/ s10649-017-9775-2 Manouchehri, A., & St. John, D. (2006). From classroom discussions to group discourse. Mathematics Teacher, 99(8), 544–552. Martin, D. B., & Larnell, G. (2013). Urban mathematics education. In H. R. Milner & K. Lomotoy (Eds.), Handbook of urban education (pp. 373–393). New York, NY: Routledge. McDuffie, A. R., Foote, M. Q., Bolson, C., Turner, E. E., Aguirre, J. M., & Bartell, T. G. (2014). Using video analysis to support prospective K–8 teachers’ noticing of students’ multiple mathematical knowledge bases. Journal of Mathematics Teacher Education, 17(3), 245–270. Moll, L. C. (1992). Bilingual classroom studies and community analysis: Some recent trends. Educational Researcher, 21(2), 20–24. Moll, L. C., Amanti, C., Neff, D., & González, N. (1992). Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms. Theory Into Practice, 31(2), 132–14. http://dx.doi.org/10.1080/00405849209543534 Morrison, K. A., Robbins, H. H., & Rose, D. G. (2008). Operationalizing culturally relevant pedagogy: A synthesis of classroom-based research. Equity & Excellence in Education, 41(4), 433–452. https://doi.org/10.1080/10665680802400006 Moschkovich, J. (2013). Principles and guidelines for equitable mathematics teaching practices and materials for English language learners. Journal of Urban Mathematics Education, 6(1), 45–57. Nathan, M. J., & Kim, S. (2007). Regulation of teacher elicitations and the impact on student participation and cognition (WCER Working paper No. 2007-4). Madison, WI: University of Wisconsin-Madison. Retrieved from Available at https://amte.net/connections/2018/03/amtes-equity-committee-survey -results-and-future-directions National Council of Teachers of Mathematics. (2014). Access and equity in mathematics education: A position of the National Council of Teachers of Mathematics. Retrieved from https://www.nctm.org/uploadedFiles/Standards_and_Positions/Position_Statements/Access_and_Equity.pdf National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for mathematics. Washington, DC: Authors. OECD. (2016) PISA—Equations and inequalities: Making mathematics accessible to all. Paris, France: OECD. Paris, D. (2012). Culturally sustaining pedagogy: A needed change in stance, terminology, and practice. Educational Researcher, 41(3), 93–97. https://doi. org/10.3102/0013189X12441244 Planas, N., & Civil, M. (2013). Language-as-resource and language-as-political: Tensions in the bilingual mathematics classroom. Mathematics Education Research Journal, 25(3), 361–378. https://doi.org/10.1007/s13394-013-0075-6

Promoting Equitable Teaching in Mathematics Teacher Education 199 RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: Towards a strategic research and development program in mathematics education. Santa Monica, CA: RAND. Rogoff, B. (2012). Learning without lessons: Opportunities to expand knowledge. Infancia y Aprendizaje: Journal for the Study of Education and Development, 35(2), 233–252. http://dx.doi.org/10.1174/021037012800217970 Rubel, L. H. (2017). Equity-directed instructional practices: Beyond the dominant perspective. Journal of Urban Mathematics Education, 10(2), 66–105. Rubie-Davies, C. (2016). High and low expectation teachers: The importance of the teacher factor. In S. T. P. Babel (Ed.), Interpersonal and intrapersonal expectancies (pp. 145–157). Abington, Oxon: Routledge. Sandoval-Taylor, P. (2005). Home is where the heart is: A funds of knowledge-based curriculum module. In N. González, L. Moll, & C. Amanti (Eds.), Funds of knowledge: Theorizing practice in households, communities, and classrooms (pp. 153– 165). Mahwah, NJ: Erlbaum. Schleicher, A. (2014). Equity, excellence and inclusiveness in education: Policy lessons from around the world. Paris, France: OECD. Sullivan, P., Zevenbergen, R., & Mousley, J. (2002). Contexts in mathematics teaching: Snakes or ladders. In B. Barton, K. Irwin, M. Pfannkuch, & M. Thomas (Eds.), Mathematics education in the South Pacific: Proceedings of the 25th annual conference of the Mathematics Education Research Group (pp. 649–656). Sydney, Australia: MERGA. Wager, A. A. (2014). Noticing children’s participation: Insights into teacher positionality toward equitable mathematics pedagogy. Journal for Research in Mathematics Education, 45(3), 312–350. Wood, T., Williams, G., & McNeal, B. (2006). Children’s mathematical thinking in different classroom cultures. Journal for Research in Mathematics Education, 37(3), 222–255. Wright, W. E. (2005). The political spectacle of Arizona’s proposition 203. Educational Policy, 19(5), 662–700.

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CHAPTER 10

“PIE π IN THE SKY” Imaginative Possibilities to Foster Diverse Diversities in Primary School Mathematics James Biddulph and Luke Rolls

Diversity that somehow constitutes itself as a harmonious ensemble of benign cultural spheres is a conservative and liberal model of multiculturalism that . . . when we try to make culture an undisturbed space of harmony and agreement where social relations exist within cultural forms of uninterrupted accords, we subscribe to a form of social amnesia in which we forget that all knowledge is forged in histories that are played out in the field of social antagonisms. —McLaren (in bell hooks, 1994, p. 31)

Privilege works in a peculiarly seductive way in relation to whiteness, which is seen to be rooted in a whole range of things other than ethnic difference and skin colour. —Reay et al., (2007, p. 1041)

Equity in Mathematics Education, pages 201–228 Copyright © 2019 by Information Age Publishing All rights of reproduction in any form reserved.

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INTRODUCTION: THE STORY OF AN INDIAN GENIUS Srinivasa Ramanujan claimed that most of his mathematical ideas came in his dreams. In 1913, having dreamt a lot, the 23-year-old shipping clerk from Chennai (then Madras), India, wrote to the eminent (and perhaps “undreaming mathematician”) Cambridge professor of maths, Godfrey Harold Hardy, explaining that he had devised a formula that calculated the number of primes up to a hundred million with almost no error. When Ramanujan was eventually invited to Cambridge to study with Hardy, the process was one of “taming his creativity” (Béla Bollobás, 2016) and teaching Ramanujan the “current thinking of the field.” Whilst we accept that in any field of study it is necessary to learn the tools, discourse, and practices to be able to engage within it (Lave & Wenger, 1991; Csikszentmihalyi, 1990), we use Ramanujan’s story as an example of the tensions that exist across, between, and within ethnic and cultural diversities; between the mystical imagination of an untrained Indian genius and the solid foundations of a robust Cambridge academic’s thinking; between the vegetarian Brahman who could not cross water for religious reasons, to the meat-eating, sherry drinking culture of the Dons’ common room. Accepting that the social world is quintessentially “storied” (Atkinson, 2015, p. 100) and that children engage with maths in their social world, we see Ramanhujan’s story as a metaphor of the power-laden relationships that exist now in our modern classrooms, the hidden assumptions made about the “Other,” ethnic minorities, and marginalized groups. His story provokes questions about our own positioning as teachers, asking whether we too try to “tame” the thinking and mathematical creativity of the children whom we teach or rather if we allow space for their “pie in the sky imaginative-mathematical-thinking?” “Pie in the sky” is an English phrase used to describe or refer to something that is pleasant to contemplate but is very unlikely to be realized. In this chapter, we present one way of considering social justice in maths education to see whether it is merely a pleasant idea or can be realized in the primary maths classroom. We draw from Smyth (2011) who argues for necessary rethinking about teachers as intellectuals, who actively and critically develop a social justice pedagogy. We also refer to Bourdieu’s (Bourdieu & Passeron, 2000) notion of habitus as a thinking tool to consider the disjuncture between school and out-of-school mathematics. From these theoretical foundations, we share how, at the University of Cambridge Primary School,1 we have tried to evoke an inclusive mathematical culture as well as critiquing the very practice that we hold up to be inclusive. The example we use arose as a result of a lesson study activity in the school. The data comes from this observed lesson which was then critically reviewed with peers and school leaders; the purpose was to unravel the

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pedagogy beneath the intended plans and outcomes for the session. As well as evidencing what we consider to be a social justice discourse, we view the lesson through the lens provided by Smyth (2011). Though it is but one example, and there will be other lenses through which to consider the data, we see this very act of teacher-led criticality working within social justice principles. The chapter therefore is divided into two sections: the first, a view of social justice in multicultural primary education (with a particular U.K. focus). In this section, we problematize notions of culture and diversity, and consider Smyth’s provocations as a lens through which we can make sense of the differences that could lead to social inequality and injustice. In the second section, we provide an in-depth presentation of the lesson study example of an inclusive maths session with critique. We recognize that there will be other interpretations of this session and invite the reader to bring their own cultural awareness to problematize the assumptions we have made. Before we turn to the substantive matters, it is important that we position ourselves because as Bourdieu and Passeron (2000) implies, who we are is how we see. Who are we to write about diversity? Both authors are White, male, middle-class professionals who have authority, in respect of the leadership roles they have at the University of Cambridge Primary School. We both have postgraduate qualifications from highly regarded academic institutions. One could say, therefore, that we are privileged and have a certain degree of power. We have both worked in inner-city primary schools, in which there were over 90% of children who spoke English as an additional language and between 40–50 home languages. Our personal pathways also have some synergy in that we have lived and worked in various countries: Luke in Ghana, India, China, and Japan; James in Nepal, India, South Africa, and Namibia. Such experiences bring to the fore our interest in the multiplicity and rich diversities in human life. We write about diversity because we have seen inequalities and injustices arise in the various contexts of our personal and professional lives. But we recognize also the limitations because we see these from a privileged perspective. We acknowledge that this is problematic and hold ourselves up for critique. THE PROBLEM WITH DIVERSITY We live in rapidly changing communities (Cantle, 2012; Xenofontos, 2015). Many Western countries have become increasingly multicultural due to immigration. For example, between 1993 and 2013 the foreign-born population in the United Kingdom more than doubled from 3.8 million to around 7.8 million (Vargas-Silva, 2014). The number of foreign citizens increased from nearly 2 million to nearly 5 million (Phillimore, 2011). Schools are

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microcosms of such social complexity as a result of the diaspora. Having both worked in multi-diverse schools, we felt, as class teachers, that the increased cross-cultural exposure transpired in ways that showed that diversity “rubs against and challenges our prejudices” (Hofvander Trulsson & Burnard, 2016, p.123)—but that the term needed problematization. The problem with the term diversity is that it is a “white word” (Bhopal, 2017). It is a term often used to describe the Other by a dominant group. It is another way in which those in power navigate the complexities of society—reinforcing their own dominant view of the world, and in doing so homogenize and try and make an ordered and harmonious ensemble of other cultural spheres (Bourdieu & Passerson, 2000). It is also a term that is increasingly challenged because of the fluid and ever shifting lexicon related to gender, ethnicity, class, sexuality, identity and culture, and so on which indicates that whilst diversity can homogenize the Other and reify the dominant group, the notion of “diverse diversities” (Biddulph, 2017) indicates the multifariousness of social and cultural ways of life. In relation to maths education, the social amnesia of teachers, to which McLaren refers, arises in our lack of awareness of the diverse diversities that coexist, not only in our classrooms but also beyond the school gates and through children’s front doors; how do we (mis)understand the histories and cultural habitus in which mathematical knowledge is forged? When do we listen to the mathematical stories in children’s lives? How do diverse diversities exist in maths classrooms? And how do children make sense of number, shape, geometry, measurement, algorithms, and so on, with awareness of their own social and cultural diversities? Is it all “pie in the sky”? Our focus is predominantly on social justice in terms of ethnicity and culture. The first task is to attend to definitions of “culture” and what this means for maths learning (Xenofontos, 2015). Culture is a particularly contested, complex, and diversely understood concept: It means different things to different people. For many, it seems linked with “ways of life” at home or in other social spaces (e.g., religious spaces), and entangled in their religious beliefs, languages, and practices. Traditionally, the term has been used in the humanities to describe the intellectual and artistic activities and artefacts of particular groups. This is also a specifically Western and romanticized view of the term (Burnard, 2012). Such a view of culture has an inherent hierarchy, especially in the arts and music, and especially in Western capitalist societies, which reifies some forms of cultural artefacts over others (Bourdieu & Passeron, 2000). One could also suggest that there is a hierarchy of knowledge which is valued in schools, and which links to accountability systems and to the content of national tests. From an anthropological standpoint, culture is related to theories of human nature. In other words, where we are and how we are positioned, and the structures of the social context influence our understandings of “our”

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culture—and therefore influence how we make sense of the activities, like maths, which occur within the culture. From this perspective, culture refers to a sense of belonging to a group; understanding the rules of the game (Bourdieu & Passeron, 2000) within the group; knowing the structures and positionality of oneself in the group. How far do children see themselves as belonging in maths learning at school? Is this important to them? Mathematics is a universal language, is it not? We found Oberg’s (1960) conception about culture and the members of a society which “live” the culture as a helping starting point. In Oberg’s view, culture is to humans what water is to fish; that is, humans grow in certain cultural contexts which are necessary for all their activities, just like fish need water to survive. Water is invisible for fish, in the same way culture is invisible to humans. In feeling “something different” in school, by being a “fish-out-of-water,” how do ethnic minority and immigrant children see and experience a cultural difference? Defining culture as, the way of life of people, the constant and complex process by which meanings are made and shared was useful for our thinking because we were keen to expand understandings about how children engaged in maths. Meaning is normally transmitted through language, and so we accept that mathematics learning is also mediated through language (Xenofontos, 2015, 2016). Knowing that as a discipline and school subject, the relations between number, category, geometry, variable, and so on, are abstract, how do children who do not speak English make sense and meaning? How do teachers understand the complexities of this learning for their young primary students? The complexities of diverse diversities have resulted in tensions in schools. In the current U.K. educational context, tensions manifest in the relentless high stakes testing of children’s mathematical knowledge, and in the pedagogies that encourage imaginative responses to mathematical problems. The tensions arise between teaching mathematics as fact, in terms of certainty, or in mathematical teaching that teaches children how to go about exploring mathematics, allowing spaces of uncertainty (Biddulph, 2017) to exist; between getting the answers right or in documenting the mathematical thinking that arrived at a solution that needs rethinking. Foucault (1991) describes how systems of micropower comprise the foundations of society, through the “tiny, everyday, physical mechanisms” (pp. 222–223): The dominant power prefers order and cognitive disciplining, to limit unpredictability and inefficiency. It is this “cognitive ordering” that limits the possibilities in diverse cultural responses to mathematics. Teachers themselves prefer order and discipline. However, essentially, in thinking of mathematics education in terms of social justice, the notion of agency and children’s voice must be attended to. This means rejecting the instrumentalist transmission view of education in which experts determine content, which is then relayed to passive learners; it means, “accepting that

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knowledge does not exist independently of the meaning and significance which students attach to it by virtue of their previous experiences, their class and their culture” (Smyth, 2011, p. 26). It means allowing for spaces of uncertainty to arise. Therefore, the question is: How far do teachers make sense of their children’s class, language, and culture in considering how they plan mathematics learning and teaching? Where is the unpredictability? Where is the disorder? Most recently, as Kulz (2017) evidences, the roots of every educational system are deeply linked to its (dominant) culture. But the experiences that children bring with them from beyond the school gates are significant, although not always acknowledged, let alone understood (Biddulph, 2017). Children’s home and home culture are powerful factors in children’s lives (Cieraad, 2006). There is a significant body of research which shows that school cultures differ from that of the home (see Hall, Cremin, Comber, & Moll, 2013) resulting in a discontinuous process for both family and child. This seems socially unjust. Such tension is described by hooks (1994), who writes of her own Black-African-American-female experiences as she tried to fit in both at home and in school: To be changed by ideas was pure pleasure. But to learn ideas that ran counter to values and beliefs learned at home was to place oneself at risk, to enter the danger zone. Home was the place where I was forced to conform to someone else’s image of who and what I should be. School was the place where I could forget that self and, through ideas, reinvent myself. (hooks, 1994, p. 3)

The “danger zones” are the “uncertainties,” which arise when there is conflict, confusion, or difference. The notion of reinvention, of finding ways to fit in, also evidences the struggles children experience. The sense of “fitting in” and “sticking out” suggested the complex borders that surround individuals and groups and their spaces. Indeed, Bresler (2016) says that our human condition is one of boundaries; the imperceptible lines that draw across our varied and complex landscapes of experience. She suggests that borderlines are made perceptible through asking questions about difference that hope to build bridges, with the desire to overcome borders and learn about the Other without the neocolonial Othering. What does this mean for mathematics educators in the primary classroom? Asking questions reflects an awareness of the confusions and uncertainties that exist in diverse diversities. Despite advances in neuro and cognitive science, it is not sufficient to say that mathematics presents a universal language; instead, we argue as others have done in this book, that it is mediated through the experiences of children and teachers through their cultural habitus. Xenofontos (2015) reminds us that “the widespread view that mathematics is either acultural or pancultural has led to the common misconception that

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school mathematics is the subject least influenced by the diverse nature of contemporary classrooms” (p. 475) and that this must be challenged. A SOCIAL JUSTICE PEDAGOGY FOR MATHEMATICS EDUCATION Therefore, a social justice mathematical pedagogy must pay attention to not only children’s experiences but also the tensions and disjuncture that exist, often unseen and unheard. Without trying to make sense of these experiences, teachers are in danger of propagating their version of maths education; therefore, critique is essential. Moreover, if teachers are seen as passive recipients of government schemes, and robbed of their creativity and initiative, there is less opportunity for discussion about the inequitable nature of the society we live in, and therefore the possible inequitable nature of education in schools (Smyth, 2011). Mathematics education in the United Kingdom has been subject to much change since 2010 with conservative governments looking to other countries to solve the issues inherent in the system; for example, encouraging the use of a Chinese-inspired “mastery” approach to mathematics learning that is about in-depth knowledge, which has been critiqued for not always giving equal weighting to developing productive dispositions, adaptive reasoning, and strategic competence in young mathematicians (Kilpatrick et al., 2001). The outcomes of such initiatives are yet to be researched or reported. The consequence of pedagogies being decided by people outside education is that teachers become habituated into a situation of “epistemological consumerism” (Smyth, 2011, p. 26)—uncritical and unaware of the potential inequalities and injustices which exist, and which they may be propagating further through their pedagogy. Where there is some consensus amongst the mathematics community in the United Kingdom is that the 2014 National Curriculum has too much content in it and many other countries’ curricula tend to have much less content and so are able to go progressively much deeper (Seleznyov, 2018). The unintentional inability of maths educators to look beyond the school gate can limit the potential to improve opportunities for children because if we “block the process that lets the teacher into the secrets of the child’s life, how can they find ways that will inspire and acknowledge the child’s existence?” (Cannatella, 2008, p. 40). We argue, that to build an equitable society through education, including through mathematical education, there is need for people to see “a multitude of different perspectives that would make it rich in the arts of living experience” (Cannatella, 2008, p. 40). As Greene (1985) theorizes, once teachers engage with their own understandings, assumptions and lived experiences, they can begin to attend to the

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children and provide experiences for them to make sense of their own lived experiences. If we accept that mathematical education does not occur in a “pure” vacuum, but rather in the social and cultural situations in which it is experienced, then a social justice pedagogy must start with developing teachers as intellectuals. They must become critics of their own and wider educational practices. The aim is not to harmonize or still conflicts of value. Instead, we need common institutions in which many forms of life can coexist. Could this be intercultural mathematics space-making? Could this be about finding opportunities for differences to coexist? Is this “intercultural mathematics”? Giroux (1985) argues that teachers need to be active political intellectuals who question how they encounter teaching and learning, how they understand it, and how they feel about it. In relation to mathematics education, what types of questions should be generated by those engaging in the classroom to challenge the assumptions made about mathematics learning: How do children encounter mathematics, how do they understand it, and how do they feel about it? Building from questions asked by Smyth (2011): • What counts as school mathematics knowledge? • How is such knowledge selected and organized? • What are the underlying interests in this knowledge and the need for children to acquire it? What is the hidden curriculum enacting? • How is access to such knowledge determined and by whom? • What cultural formations of mathematics are disorganized and delegitimated by dominant forms of school knowledge? In considering teaching as an intellectual form of labor, teachers can regard their own classrooms and schools as sites of inquiry and research. They become intellectual professionals who are prepared to ask questions about the assumptions that so often perpetuate myths about what children can and cannot do, what they should and should not do, and consider what might cause limiting imaginative possibilities. In becoming such professionals, there are several dramatic and direct implications for the learners, “the most obvious is a bringing of students’ lives, perspectives, cultures, and experiences into the center of the curriculum in a way that involves students as co-constructors and co-creators (rather than passive consumers) of the curriculum, along with teachers” (Smyth, 2011, p. 34). However, to problematize teaching requires challenging habits and methods taken for granted and also questioning cherished assumptions. Smyth says that this occurs through a collective and collaborative process of teachers working with one another. The authors’ school and workplace is rooted in such principles. The University of Cambridge Primary School is the first primary University Training

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School in the United Kingdom. Inspired by the “laboratory schools” in the United States, the school has three functions: to be a brilliant primary school, to support initial teacher education, and to be a research-informed and research-generating school. The professional ethos of the school is based on trust and responsibility, which is consistent with the social justice theoretical position described in this chapter. Of central importance is the professional community we strive to nurture. In developing a social justice critical approach, what seems essential is to operate in dialogic ways. Drawing from an example of a school analyzed by Bigelow (1990), the essence of what is involved in critical teaching emerges from the everyday life of the classroom, summarized as: • Teachers engaging students with questions that have relevance beyond the classroom. • Working with students in ways that enable them to delve more deeply into content that is not normally presented to them. • Schools do not operate in competitive ways. • Changing mindsets to “how to do it” approaches. • Listening to voices that originate from within the classrooms. • Using personal experience. • Questioning the authority of the teacher. • Students themselves becoming important sources of knowledge and theorizing learning. How could these broad statements be applied to the context of a primary mathematics classroom? Where are the inequalities in mathematics education and how do, or how can, teachers see these and pose questions to challenge their assumptions? Drawing from Smyth’s (2011) synthesis of the layered moments that bring about a collaborative professional culture in schools, we approach our thinking and learning at the University of Cambridge Primary School. Smyth says that there are three key features of a social justice professional dialogue: 1. Describing—the virtues of their work, efficacy, and situational benefits; teachers look for similarities and differences, patterns, regularities, discontinuities: they become informed. 2. Informing—Teachers unravel the complexity of their classrooms; through discourse they articulate legitimate knowledge about teaching: they begin to theorize. 3. Confronting—In ascertaining how things came to be and what broader forces influence these, they pursue questions about personal assumptions and values, what causes them to be steadfast in

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their view, and what constrains the imaginative possibility thinking (Craft & Chappell, 2014). As Smyth suggests, “By the time teachers have begun to grapple with such questions and the forces that are shaping what they do, they are starting to think about how to act in different ways and are moving towards reconstructing parts of their teaching and the contexts and structures within which they teach” (p. 53). In the next section, we present one example of the ways we have enacted a social justice approach in a mathematics classroom. We use the principles described above to navigate the rich diversities that arose. ENACTING A SOCIAL JUSTICE PEDAGOGY FOR MATHEMATICS EDUCATION For the important cause of social justice within mathematics education, there has been focus in recent years on using so-called “real-world” contexts where children use their mathematical knowledge to calculate meaningful data; for example, to calculate or problem-solve in relation to “fairtrade,” or a country’s voting system (Wright, 2016). The purpose is to situate mathematics within a wider sociological context; to explore issues of injustice that exist in the world by understanding the mathematics related to it. While we see these as consistent to values-based knowledge of the world, relevant within the broader ethos of a values-led school curriculum and where appropriate, as contexts for maths learning, we argue here that it is ultimately only through a transformative and embedded social justice pedagogy within children’s daily experiences of maths teaching that more equitable outcomes for students can be achieved. In this section, we present and then critique one lesson in a Year 2 (ages 6–7) class. As such, it is not possible to present the many repertoires and experiences of learning maths that children need in order to learn maths and learn as mathematicians. However, through recordings of children’s responses, teacher observations, teacher post-lesson discussion and gathering children’s voice, some of their awareness of mathematics learning is identified and questions arise. These are considered and problematized in their potential as features of equitable practices for mathematics learning. Despite there being a high percentage of speakers with English as an additional language in the class, most were advanced fluent English speakers. From rich ethnic diverse diversities in the class, children with families from Palestine, Columbia, India, and the Caribbean were represented in the post-lesson pupil interview group.

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Mathematics Learning at UCPS The ethos of mathematics learning at the school is situated within a belief and expectation that all children have an entitlement to deep and sustainable mathematics learning, and that challenge and support between teachers, children, and peers is key to achieving this. Teachers aim to nurture in children the dispositions where they freely make conjectures, query what they do not understand and explain, and justify their thinking generously and reciprocally to their peers; empowering their agency. A successful mathematician at our school is considered not only the learner who answers correctly, but, importantly, one who displays the longer-term positive habits of mind, evident in someone with a love and curiosity for mathematics. A successful mathematician is someone who can also ask their own questions, explain and justify their thinking, investigate mathematical ideas, and change their mind. We posit that this ethos allows for rich mistakes and learning to occur. Table 10.1 presents the lesson plan which guided the session discussed herein; it is the first principle identified by Smyth: to describe the planning, purpose, knowledge content, and expectations of a lesson. The curriculum is defined by the U.K. Government (DfE, 2013) but the teaching sequence is designed by the teacher. The lesson was planned mindful of social justice principles, namely to (a) consider the entry points into the mathematical content, (b) to help children articulate their understanding of the mathematical content with the use of “stem sentences,” and (c) to provide space for children to enact their agency. As such, as well as teaching the knowledge content, the teacher’s role was to “set up” opportunities for collaboration that required children to be problem solving, querying, and to explain their understanding. In this way, it was hoped that children would bring their own unique cultural responses to the mathematics because they were given the freedom to present their own mathematical understanding (rather than completing a teacher-defined worksheet or “cloze-procedural” task). What follows are the three opportunities that we hoped would be commensurate with a social justice pedagogy. We raise questions and critique the assumptions beneath each opportunity. The First Opportunity: Using Stories as an Entry Point to Mathematical Understanding Based on a series of books by Jon Klassen, I want my hat back and We have found a hat, the lesson used both stories that explore the theme of wanting to have something that belonged to someone else. Willingham (2009) points to the “psychologically privileged” position of narrative as a

Addition and subtraction

Addition and subtraction using two-digit numbers without re-grouping; collaborative problem-solving. Prior to the lesson, children have experienced a sequence of lessons based on additive reasoning, including considering and contrasting the structures of subtraction as partitioning, “take away” and the concept of “difference” using one- and two-digit numbers, not yet with any regrouping. Children have become familiar and confident with using representations and manipulatives, such as double-sided counters, cuisenaire rods, dienes, straws, and place value counters.

To understand subtraction as difference (subtracting one- and two-digit numbers without re-grouping)

In this session, the children will explore two cooperative learning tasks: one role-play based task with content that reviews previous learning around subtraction as difference and the other a logic reasoning task where multiple representations of a subtraction equation is explored. Dialogic areas of focus: elaboration and querying.

• Hennessy, S., Rojas-Drummond, S., Higham, R., Torreblanca, O., Barrera, M. J., Marquez, A. M., García Carrión, R., Maine, F., Ríos, R. M. (2016). Developing an analytic coding scheme for classroom dialogue across educational contexts. Learning, Culture and Social Interaction, 9, 16–44. • Howe, C., & Abedin, M. (2013). Classroom dialogue: A systematic review across four decades of research. Cambridge Journal of Education, 43(3), 325¬–356. • National Research Council, & Mathematics Learning Study Committee. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press. • Care, E., Griffin, P., Scoular, C., Awwal, N., & Zoanetti, N. (2015). Collaborative problem-solving tasks. In Assessment and teaching of 21st century skills (pp. 85–104). Dordrecht, The Netherlands: Springer. • Fuson, K. (2017) Teaching progressions. Teaching and learning mathematics [Online]. Available at http:// karenfusonmath.com/teaching-progressions.html

Unit Title

Place of Lesson in the Sequence

Learning Intention

Story of the Lesson

Learning From the Study of Instructional Materials

(continued)

How can collaborative problem-solving encourage querying and elaboration in maths dialogue?

Research Question

TABLE 10.1 Maths Lesson Plan

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Episode 2

Episode 1

(b) Children practice the role-play in pairs. (continued)

3. Demonstration: (a) At demonstration table (children sit around a large table for the learning task to be modeled), demonstrate with a child as partner, the simple role-play and stem-sentence reasoning using character cards and cut-up manipulatives of hats (in tens and ones), one child (the Bear) counts their hats, using “quantity value” of the number, knowing 11 is 10 and 1 and then looks away. Their partner pretends to be the Hare stealing the hats and takes a certain amount. On looking back, the partner is challenged to calculate how many hats the Hare has stolen. The Hare confirms or denies whether they were correct and the Bear queries how they know. The partner must explain their thinking and the pairs then reverse roles. –– Introduce example of 0 where no hats are stolen.

(b) Hold on. I had 24 hats and now there are only 11. There is a difference of? [subtracting a two-digit number with regrouping] What connections can you make?

2. Reasoning: (a) Elicit reasoning and querying from class using the following examples and modeling/teaching the stem sentence— “There is a difference of…”: Hare: Oh…good morning. Bear: Hold on, I had 24 hats and now there are only 14. There is a difference of 10 here. You stole my hats! [Attending to the structure of subtracting ten] Hold on. I had 24 hats and now there are only 12. There is a difference of? [subtracting a two-digit number without regrouping] What connections can you make?

1. Consolidation: Re-read excerpt of “I Want My Hat Back” by Jon Klassen, re-imagining what might happen next. What’s your conjecture of what will happen as the Hare finds the Bear sleeping? Worried that his hats will get stolen again, the Bear becomes insecure about losing his hat and decides to buy lots. Bear: ‘I’ve got lots of hats now. I’ll show you’ [24]. How many has he collected? How did you know that without counting each? But yet again, after waking up in the morning, he finds some of them have been taken.

TABLE 10.1 Maths Lesson Plan (continued)

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3. Do you observe any cases of querying in group/whole-class context? Where are these more/less effective?

2. How do children respond to the query/explainer characters? Do they use either/both?

1. How do children interact with the stem sentences of difference within the role-play (Episode 1)?

Focus questions for observers in relation to the research question:

5. Whole-class: Compare multiple representations and solutions. (a) Orchestrate choice of anticipated solutions—children to present to class (Fuson, 1992). What’s the same? What’s different about these two solutions? (b) Class encouraged to challenge presenters where explanations are not understood.

(b) At demonstration table, model collaborative problem-solving norms of negotiating and agreeing, sharing out (unequal number of) cards using scrabble-type boards, taking turns to read out their clues, give reasons for their responses, building on ideas and looking for opportunities to query. When queried, partners give further elaboration/explanation. (c) Partners to attempt at tables. Once completed, compare methods and draw a representation and solution for the 24 – 12 key difference clue and rehearse presentation to class. (d) Extension: Order the clues in order of importance. Children to prepare a simple reasoned presentation of their solutions to the rest of the class.

He loves chocolate. Animal illustration heights: Deer: 52 cm; Armadillo: 14 cm; Snake: 8cm; Fox: 24 cm; Hare: 24 cm; Tortoise: 12 cm; Bear: 68 cm. Children need to determine which clues are irrelevant and can be eliminated (thief loves chocolate), compare which clues are interdependent to solving problem (it cannot be the Hare because he has stolen a hat before), and calculate to find the difference (height of thief in relation to tortoise).

4. Back to the story: (a) Another hat has gone missing, but this time it is the tortoise’s hat. Show the three scenes of sunset, night, and morning, where the hat has now disappeared. Let’s find out who the thief was. Clues: –– The height of the thief is less than 50 cm. –– He has fur. –– He has never stolen a hat before. –– He is not wearing a hat. –– He doesn’t have a shell. –– The height of the thief is equal to double the length of the tortoise.

TABLE 10.1 Maths Lesson Plan (continued)

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pedagogical tool for learning, and as a favorite story of the class, it served to build on their interests and humor. In capturing the children’s voice and views about the lesson, the class reported favorably on the utility of stories as a way-in to maths learning. Two children said:

L: I think stories are good for learning maths. If you don’t know what the story is, you might not understand and get stuck. E: I also think that it’s good to start a lesson off with a story because we get more detail and understanding of what we are doing and what we are not doing. The teacher does not have to repeat what we’re doing. [Post-lesson pupil interview]

Those who shared their ideas referred to stories helping them bridge their thinking to the abstract forms of mathematical knowledge in lessons.

J: I think it’s a good way to start a lesson because once someone tells you the story I understand the maths better. So, when you do the learning, you understand it. It helps you know what you need to calculate. M: I think stories help me quite a lot. The story could be related to the maths lesson, like the “I want my hat back” story—the story can tell you about the maths learning that you’re doing. It can help you learn. D: I agree with M and F because the story turns into a maths story. Then the story turns into something you can calculate with. [Postlesson pupil interview]

Children also spoke about “seeing” maths more broadly in stories, suggesting that they were beginning to experience maths not solely within classroom lessons but also in other aspects of their lived world: F: I find reading books helps me learn maths because I can see subtraction and things in the stories. If I get it wrong, I can think about what is happening in the story and understand what to do next time. [Post-lesson pupil interview]

However, in our critique, we recognize that (a) we made an assumption that children would make meaningful links between the text and the mathematics, and (b) despite anticipating common “methods” for calculating difference, we were by no means able to anticipate the diversity of connections made by children at different points in the lesson. Returning to Smyth’s questioning: The knowledge was defined by the National Curriculum and it was then organized by the teacher with an intention to give children access to the purpose of the mathematics learning. Children then responded both with taught skills, but also their own mathematical

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imaginations, making connections and considering different structures. We question how children’s natural “mathematical powers” (Johnston-Wilder & Mason, 2004) interact within these layers of power and how agency is fostered or marginalized within this context. During the post-lesson discussion, teachers noted high levels of participation, citing that the “story talk is very effective in framing a context for children to engage with the maths.” It seemed that the children engaged more meaningfully and curiously in mathematics learning because they were engaged in talking about the story. Whilst this is useful, we were mindful to problematize how the use of stories in maths could carry several potential limitations, such as: children becoming fixated on a context, children misunderstanding text or not being able to extrapolate the mathematics from the story, and not developing generalizations about mathematical structure. The use of variation (Gu, Hunag, & Marton, 2004) in the lesson attempted to reduce this fixed thinking so that learners were exposed to the invariance of the essence of a concept (Watson & Mason, 2006). In this way, stories might in fact strengthen opportunities for a flexible application of understanding as well as drawing children into the maths. The Maths in the City materials (Fosnot & Jacob, 2010) demonstrate well, for example, how contexts for learning can be designed coherently in a way that lead to key representations and mathematical structures. In this lesson, the contexts of quantity in Episode 1 which presented procedural variation through a series of discrete, cardinal amounts using the structure of 10, contrasted with that of Episode 2, which presented a comparison of using calculation connected to the strand of measurement. The Second Opportunity: Considering First Languages With the Use of Stem Sentences In our school there are 20 different languages spoken and 35% of the school community speaks English as an additional language (EAL). It can often be taken for granted that children have the necessary language to be able to translate their abstract mathematical thinking into articulated reasoning (Barwell, 2005). This is more problematic when considering the diverse first languages and how each language articulates abstract mathematical concepts (Monaghan, 2016). For this reason, and to establish a fair opportunity for all learners, the use of “stem sentences” was deployed in the lesson. These are sentence openers (or stems) that give the language structures that encourage the use of precise mathematical language. In our school, these are taught explicitly. For example: • 5 is the whole; 3 is a part, and 2 is a part • 6 subtract 1 is equal to 4 plus 1

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• If I know 3 ones plus 3 ones is equal to 6 ones, I also know 3 tens plus 3 tens is equal to 6 tens (which is equal to 60). • The multiplicand (the size of the group) is 3, the multiplier (the number of groups) is 2; the product is 6 • 2 + 2 + 2 + 4 = is equal to 2 x 5 because . . . Alongside these are non-domain specific sentence stems that children can draw on in the maths classroom. For example: • • • • •

Building on Gabrielle’s idea Could you explain that more? I agree/disagree because . . . I’ve made a connection here . . . I’ve changed my mind because . . .

After the research lesson, the children said the following about stem sentences: M: I think children benefit from using stem sentences. Some children wouldn’t know facts like 50 + 50, so by making it into easier numbers, they could add them. I: Stem sentences help you learn things because they give you a little start to start yourself off. Then you can try trickier things. E: I find them useful because it sometimes helps me remember things like, what is, say, for example, what is 80 + 20. It helps me remember 8 + 2 = 10 to use what I already know. [Post-lesson pupil interview]

While there is not yet a large amount of research literature on the use of stem sentences, certainly as envisaged within the new “mastery” approaches being adopted in the United Kingdom (NCETM, 2017), there is rationale for their use (Buffington et al., 2017). Widely documented is the finding that mathematical reasoning is a key predictor of numerical understanding (Nunes & Bryant, 1996) and that young children can struggle to articulate through their mathematical thinking (Nunes et al., 2015), a central aim of the national curriculum. Specifically, the technical language of mathematics is something that does not come intuitively to young learners as the words differ in conceptual meaning to the everyday uses of the words that they are familiar with (Schleppegrell, 2007). This is particularly problematic in the mathematical meaning of “difference” documented in this chapter, as in the United Kingdom, parents and often teachers refer to the concept of subtraction as “take-away,” highlighting the types of conceptual language barriers students, particularly those with EAL face (Neville-Barton & Barton, 2005).

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The Third Opportunity: Collaboration and Agency An important aspect of the lesson was to foster children’s agency and ability to collaborate. We considered this to be a “golden thread” to deepen the social justice culture we hoped to create (in this lesson and more generally in the culture of the school). Bearing in mind the diverse habitus that would be enacting within the classroom (involving all 30 children’s difference habitus as well as the teacher’s own power-laden habitus), we were keen to foreground how far children actually evoked their agency in the lesson and where they did not. Children in the class expressed positive attitudes towards collaborating with their peers to solve problems:

I: If you work by yourself, you might get stuck, but if you’re with someone they can help you. Your partner might also get stuck so then you can help them. I always try to explain it clearly and how I worked it out. E: I find it easier to work it out with a partner because the other person can explain it. [Post-lesson pupil interview]

And some recorded interactions suggested that their collaborations demonstrated “inter-thinking” (Littleton & Mercer, 2013) where their ideas built on one another’s:

F: If it doesn’t have a shell, it can’t be the tortoise because tortoises always have shells. R: And also, if he’s never stolen the hat before, it definitely can’t be the Hare. [Teacher observer notes]

Building up to the lesson, children had also become accustomed to regularly discussing and agreeing on ground rules for talk where they might, for example, discuss whether effective group dialogue is about taking turns or sharing thinking (Mercer & Sams, 2006; Wegerif, 2017). One such ground rule being established was the use of querying, where children agree to actively challenge others to clarify and explain their ideas where these are not mutually understood (Fuson, 2017). Rather than set group roles considered useful in collaborative learning (Boaler, 2008), children were made aware of two “querying” and “explaining” roles and were encouraged to employ them dynamically at any point. In the story, the children learnt that the Bear character embodied the power of “querying.” Central to the rationale of querying is that it prompts children to share their method and this has the potential to scaffold their partner’s thinking, particularly if they persist when still unsure. Examples of the children’s talk were gathered:

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R: Ok let me count my hats today. Oh some are missing. That’s weird; I thought I had 22. Right, let’s see . . . there’s a difference of . . . [pauses] Erm . . . 7! Hare!! F: How did you know that anyway? R: Because if you have 10 and 5 that equals 15. And if all I have is 22 then I would have 5 and 2 more, which would equal 7. [Teacher observer notes]

We saw this as creating a group culture in which not only was “getting the answer right” not the main purpose of the lesson, but that the agency of the children was central to creating successful mathematical thinking. And whilst developing a “querying protocol” within paired settings appeared effective, the interactions this provoked came with notable challenge. Querying required of the partner to elaborate on thinking which was sometimes in itself only emerging and intuitive. Positively, being questioned by a partner seemed to prompt children to articulate and make their ideas more explicit and known but where explanations were unclear to partners, they needed quite a persistent tenacity to keep seeking clarification. When responders found this difficult, there was variability in the amount of continued query prompts partners would give. One child in particular was praised for this in the post-lesson discussion for her numerous challenges of her partner, saying, “I’m very confused; could you explain more?” and “How did you get that?” Another child challenged their partner with, “Can you draw me something to show me?” Research into “productive failure” (Kapur & Rummel, 2012) and “desirable difficulties” (Bjork, 2017) suggests there is much opportunity learning within these times where learners find themselves in conflict and perplexity, and that children’s attempts to elaborate were valuable. The power of querying appears closely linked with elaboration; the latter being through a form of reasoning which is more cognitively and pedagogically demanding to create but also the key to unlocking the query. Perhaps, sometimes at least, both teachers and children can too readily accept responses with a sentiment of “good effort,” rather than allowing for further self-clarification: E: People do not always explain clearly enough. I might say, “How did you work it out?” Some people explain it better. In most learning, people help each other well to understand the answer. I: I think you should only learn with others sometimes. If it is something really hard, it’s good to work together; if it’s easier, you can do it alone. M: With a partner, you go slower, but you can also go deeper. [Postlesson pupil interview]

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In promoting inquiry in the classroom, there is a middle-class assumption, however, that (a) children understand the importance of challenging other people’s views, (b) that this agency was culturally appropriate for them (or was it the uncertainties that hooks [1994] described above), and (c) it encouraged children who were most like the teacher to achieve because they replicated the most dominant voice in the group. The awareness of power structures even within small group work is necessary to foreground, and requires of teachers to delve deeply and critically into the power structures and relationships that are embedded in the hidden curriculum and pedagogies. Again, we raise the question: Whose voice was most dominant? Who had most agency? Who did not? How were these activities exclusively fostering the attitudes of the dominant position (i.e., that of the teacher) within the mathematics lesson? Returning to McLaren, we believed the questioning and relentless querying vital to move towards the equity we wanted to foster in our classrooms. Diversity is a White word and in our problematizing we want to bring to the surface the potential underlying discords and disjunctures between what was intended and what was lived by the children. Towards a Social Justice for Mathematics Education Pedagogy Table 10.2 collates some of the salient features of the lesson to give examples of equitable practices which we suggest work within a social justice for mathematics education pedagogy. The social justice critique includes the questions that Smyth argues, makes for socially-just-aware teachers. We suggest how our decisions in the lesson are socially just, but in the last column we problematize these assumptions, which is the spirit of a socially just pedagogy. We describe, inform ourselves, and confront our assumptions. CONCLUSION In 1913, when the Indian genius came to Cambridge, the awareness of the diverse diversities that exist in human social life were not readily seen or problematized as much as they are now. In 2018, the term “diversity” and “immigration” are even more complex and yet conversely also homogenized, as the dominant group (White Western neoliberalism) attempts to see order in the chaos that McLaren describes. The language is often related to communities which do not “fit in” to the dominant community, like Ramanujan did not fit the model of a mathematician in the early 1900s.

Mathematical Teaching Knowledge

Lesson Features

• Who structures the coherence? • Who decides? • What are the underlying interests in this knowledge and the need for children to acquire it? • What is the hidden curriculum enacting? • What cultural formations of mathematics are disorganized and delegitimized by dominant forms of school knowledge?

Social Justice Critique Whilst a hierarchy of concepts is evident in maths learning, the “journey” of the small steps in a U.K. context is an under-researched and contested area.

There is a relationship between implicit and explicit variation (Watson & Mason, 2006) and it is likely that there is variability in the ways in which children attend to the maths structures. There is potential for dissonance between teacher intention and children’s sense making. This could lead to dominant viewpoints being reified, which could lead to self-limiting mindsets (e.g., I am not good at maths because I don’t understand what the teacher is saying). Time-poor teachers need access to high quality instructional materials or professional development that can support their understanding of mathematical development in children. Professional development opportunities for teachers are inconsistent and problematic (Cordingley et al., 2015).

Coherence: The lesson was part of a sequence of learning where the journey within and between lessons had been carefully considered. Small steps were designed to gradually build up children’s conceptual fluency. They were given ample opportunities to practise, consolidate, and make connections to other areas of maths. Variation: Procedural and conceptual variation in the lesson allowed children to maintain a focus on difference in various contexts, e.g., through different representations. This exposed them to the underlying structures of the maths.

Pedagogical content knowledge: Where teachers are aware of and can anticipate misconceptions, they are able to pre-empt or orchestrate effective discussion around typical student responses to mathematics content. Teachers are able to carefully select examples for students to present to the class and contrast with other solutions.

(continued)

Problematization

Examples of Equitable Practices

TABLE 10.2 Using Smyth’s Notion of Describing, Informing, and Confronting to Critique Our Practice

“Pie π in the Sky” 221

Social Justice Critique

• How do children make sense of the text and story? • What is the cultural literacy and how does this effect how children extrapolate mathematical thinking?

• What are the underlying interests in this knowledge and the need for children to acquire it? • What is the hidden curriculum enacting?

Lesson Features

Playful Enquiry: Using Stories and Role-Play as a Bridge to Mathematics

Reasoning: Precise Mathematical Language, Stem Sentences, Querying, Elaboration, Justification, Conjecture, Explanation, Generalizing A focus on querying provided increased opportunities for children to express when they needed clarification and to develop their reasoning skills and convince others of their thinking through their developing skills of elaboration.

Children used stem sentences and seemed to find these helpful as access points to the technical language as well as the concepts involved.

Broader opportunities for such play have the potential to allow for imaginative selfexpression, enjoyment, and productive dispositions towards maths learning (Craft & Chappell, 2014; Biddulph, 2017; Kilpatrick et al., 2001).

Using role-play appeared to increase student participation, engagement, and motivation in the task.

Playfulness is a natural state for children. A playful approach intended to engage and allow curiosity and meaning making to occur.

Examples of Equitable Practices

(continued)

We assume it culturally universal that querying and challenging the “authority” of the teacher and the subject knowledge being taught. There is a potential for a neo-liberal white-wash where other diverse representations are not understood by the dominant view and therefore delegitimated. These can be subtle and not always seen by the dominant view.

There needs to be a balance between given technical prescribed language structures and applying adaptive and strategic reasoning to problems (Kilpatrick et al., 2001). It is also important to consider other nonverbal ways to articulate meaning (e.g., through art/drawing/mind-mapping)

A potential danger is that children become too involved in the role-play context and it limits their attention to the mathematical content.

The stories used are in English and the subtle nuances of text will be interpreted differently. Cremin et al. (2015) identifies the complexities of reading in diverse contexts. There is a view often taken for granted that children will understand a story in the way that we intend them to.

Problematization

TABLE 10.2 Using Smyth’s Notion of Describing, Informing, and Confronting to Critique Our Practice (continued)

222 J. BIDDULPH and L. ROLLS

Social Justice Critique

• What are the assumptions about shared habitus in a class context? • Who has the most agency and therefore makes the decisions about the nature of the habitus?

Lesson Features

Collaborative Problem-Solving

There is evidence to suggest teachers find collaborative problem-solving challenging and difficult and professional development and knowledge can be lacking (Luckin et al., 2017). Because of this, learning tasks could be superficial and based on types of mathematical tasks preferred by the teacher.

Group roles were used dynamically where children could change their role according to the stage of dialogue. This allowed for children not to be limited to one role. This enforced the notion that everyone take leadership. Mathematical reasoning skills such as logical deductive thinking is not well-represented in the National Curriculum. Collaborative problem-solving is widely advocated as a key competency for the 21st century (Ananiadou & Claro, 2009).

There is a difference between learners articulating an understanding of appropriate talk agreements, and the more challenging feat of embedding these in practice and so a whole-school and long-term approach is likely to be needed. It is also important to consider who is not represented.

Problematization

Established “rules for talk” (Wegerif, 2017) provided the conditions for effective cooperation within groups where a shared understanding of expected behaviors is negotiated. This gives authority to the group and empowers children’s agency.

Examples of Equitable Practices

TABLE 10.2 Using Smyth’s Notion of Describing, Informing, and Confronting to Critique Our Practice (continued)

“Pie π in the Sky” 223

224 J. BIDDULPH and L. ROLLS

Extending beyond the mathematics classroom, the sociopolitical climate and responses toward immigrants and diversities in the United Kingdom, in recent times, has led to increasingly negative, stereotyping, and discriminatory discourse. The children and their families, in our schools, are, by association, homogenized and demonized. They were the perceived problem. Yet, each day in our classrooms, we see diversity as richness and opportunity, as a resource to envision a better world. This is not couched in terms of an Orientalism or neocolonial exoticism, but rather in a real interest in the children and their cultural well-being and inclusion in school. In relation to maths education, we see the need to explore such diverse diversities, as a “crucial step in helping to change discriminatory attitudes, in creating welcoming communities and in developing an inclusive society” (UNESCO, 1994, pp. 6–7). We see potential in understanding the untapped richness of diversities in schools to work towards a socially just response to them (Reay, 2012). In this chapter, we attended to the “call to arms” evoked by Smyth (2011) and other critical theorists who advocate for teachers to develop themselves as intellectuals who engage in critical reflection. We examined one lesson through which we intellectualized how we “brought” children to the mathematics, in this case through stories and role-play, sentence structuring through stem sentences and evoking their agency, and then how we asked questions to problematize the opportunities and assumptions that lay beneath the structures of the lesson. With awareness of the notion of habitus, we became aware of the power structures within the lesson plan, the outcomes of the lesson and in hearing how the children responded. We looked for ways that children would feel excluded. We also attempted to situate the intellectualization of our practice and experiences, aware that nothing exists in a vacuum. Our experience of critique builds on Biddulph’s (2017) work in the field of creative learning: How do we create spaces of uncertainty through which children can engage openly, culturally appropriately, and develop their own sense of agency? How often can teachers step back to intellectualize their work? Is the sense of agency in a mathematics classroom understood by the children and how do they embrace the opportunities afforded them? And how do teachers make themselves aware of the injustices in their classrooms when agency is not seen, enacted, or felt? When we make the mathematics classroom one with a culture of disturbances and disagreement, we begin to subscribe to a social imagination in which we remember that all knowledge is forged in the social histories and cultures of the children with whom we work; and our social justice response intellectualizes, problematizes, and reminds us to stay discontent as we work our way through the fields of social antagonisms (remembering McLaren’s [in bell hooks, 1994, p. 31] quote at the outset of this chapter). Teachers in their own contexts will have to consider how this could be done. When a teacher takes off the “mantle of the

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expert” and looks imaginatively, with an open-hearted desire to connect, and is prepared to critique assumptions, then subtle shifts in thinking can occur. By engaging the social imagination (Greene, 2000), we can recreate the world, both by becoming uncomfortably disturbed by the status quo, and by being stimulated and pushed to envision a better society. NOTE 1. www.universityprimaryschool.org.uk

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LIST OF CONTRIBUTORS

Peter Appelbaum is a professor of education at Arcadia University in the United States. He has been the vice president of the International Commission for the Study and Improvement of Mathematics Education, president of the American Association for the advancement of Curriculum Studies, and is the director of http://yomap.org, the Youth Mathematician Laureate Project. The author of Embracing Mathematics: On Becoming a Teacher and Changing With Mathematics, Appelbaum’s current scholarship uses post-colonial and psychoanalytic theories to co-create curriculum for global crises of the Anthropocene with all who would join him. He works with community arts groups, school innovation, and public pedagogy. Pirjo Aunio is a professor of special education at University of Helsinki. She is also a visiting professor (mathematical learning difficulties) at University of Oslo (Norway) and a visiting professor (early childhood education) at University of Johannesburg (South Africa). She is a member of the editorial board for two international scientific journals, and regular reviewer for international scientific journals and research grant foundations. Since early 2000, Professor Aunio has done research about the development and learning of mathematical skills, learning difficulties in mathematics, assessment of mathematical performance and development, and mathematical and related cognitive skills interventions. She has done unique work in developing evidence-based assessment and intervention tools to be used by teachers. She has published more than 60 peer-review articles and books related to her research area.

Equity in Mathematics Education, pages 229–235 Copyright © 2019 by Information Age Publishing All rights of reproduction in any form reserved.

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James Biddulph (PhD Cantab) lived and worked in Nepal and India in primary and secondary schools following a degree in English and music. He pursued his career following a PGCE at the Faculty of Education, University of Cambridge. In 2002, his creative approach to teaching gained him Advanced Skills Teachers (AST) status in creativity and in 2003 he was awarded Outstanding New Teacher of the Year for London. Having transformed two failing schools in East London, U.K., as deputy headteacher, he was the inaugural headteacher of a new Hindu-based primary school. He is now the first headteacher of the University of Cambridge Primary School, the first primary University Training School in the United Kingdom. He completed his PhD, which focused on creative learning in ethnic minority immigrant children’s homes. He is a founding fellow of the Chartered College of Teaching. Vasiliki Chrysikou has a PhD in mathematics and special education from the University of Thessaly (Greece). Her bachelor is in mathematics, while she holds a master’s degree in modern learning environments and design of teaching materials and another one in special education. She has specialized in mathematics education of students with intellectual disabilities in secondary education. Over the last 9 years she has been working in mathematics special education, and in the last 2 years, she has been working as a collaborating teaching staff in a postgraduate study program in special education. She has also been professor assistant for teaching mathematics to undergraduate students. Her research interests focus mainly on mathematics teaching and learning, teachers’ education, special education and inclusion of students with special needs in general education. Marta Civil is a professor of mathematics education and the Roy F. Graesser Chair in the Department of Mathematics at The University of Arizona in the United States. Her research looks at cultural, social, and language aspects in the teaching and learning of mathematics; participation in the mathematics classroom; connections between in-school and out-of-school mathematics; and parental engagement in mathematics. Her work is situated in working-class, Mexican-American communities and is based on a funds of knowledge orientation. She has led several funded projects working with children, parents, and teachers, with a focus on developing culturally responsive learning environments in mathematics education. Currently, she is exploring how to apply lessons learned from her work in equity in K–12 settings to undergraduate/entry level mathematics teaching and learning. Peter Gates worked at the University of Nottingham for 25 years until he retired in 2017, but he still works on the politics of mathematics education, teachers’ professional development and social justice and equity in

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education—as well as walking, reading all the books he never had time to read whilst working full time, learning to fly, photography, and just smelling the flowers. More recently, he has become interested in the social class implications of visuospatial pedagogy. He was on the International Committee of the International Group for the Psychology of Mathematics Education and was the joint founder of the Mathematics Education and Society group. He has written widely on social justice and equity, and has undertaken research on mathematics education and social exclusion, access, widening participation, and so on. He is a member of both the Cuba and Palestine Solidarity Campaigns, and the Association of Teachers of Mathematics. He is on Twitter (@petergates3) and Facebook (peter.gates237). Brittany Gay is a doctoral student in the Applied Developmental Psychology Program at University of Maryland, Baltimore County (UMBC). She has an MA in applied developmental psychology, which she received from UMBC in 2018. Her research interests include the impact of poverty on educational contexts and, more specifically, the promotion of educational equity. She is also interested in program evaluation and the translation of research to policy. Roberta Hunter is a professor of Pasifika educational studies in the Institute of Education at Massey University in New Zealand. Over the past 15 years Roberta has actively engaged in research and professional development projects which support teachers to develop culturally sustaining mathematics pedagogy, particularly with Pasifika and Maori students, and other diverse students in high poverty areas. The focus of her research and practice is in honoring the students’ culture and language to build students’ positive cultural and mathematical identity. She works with teachers to draw on the students’ values and cultural backgrounds to make mathematics real and meaningful but also engage their participation in mathematical reasoning and communication. Luis A. Leyva is an assistant professor of mathematics education in the Peabody College of Education & Human Development at Vanderbilt University. Leyva uses intersectionality from Black feminist thought to examine how members of historically marginalized groups construct their identities while navigating educational contexts as STEM majors. He is the principal investigator for a National Science Foundation grant on equity-oriented instruction in undergraduate mathematics as well as a senior research fellow in the Women of Color in Computing Research Collaborative sponsored by the Center for Gender Equity in Science & Technology. His work has been published in the Journal for Research in Mathematics Education and Journal of Urban Mathematics Education. Leyva was honored with a dissertation

232 List of Contributors

fellowship from the National Academy of Education and Spencer Foundation in 2015 and with the 2018 Early Career Publication Award from the Research in Mathematics Education special interest group of the American Educational Research Association. Mapula G. Ngoepe is an associate professor and chair of the Department of Mathematics Education (COD) at the University of South Africa, Pretoria. She holds a doctorate of mathematics education from Curtin University of Technology, Perth, Australia. Her doctoral research focused on investigating classroom practices in multilingual classrooms of disadvantaged schools. Her master’s degree in mathematics education is from the University of Birmingham in the United Kingdom. Her undergraduate and postgraduate studies are from the University of the North in Polokwane. Her career started as a high-school teacher, a lecturer at a college of education, a statistician at the department of education, and a postgraduate assistant. She leads a community engagement project “Mathematics Teaching and Learning Intervention Programme” (MTLIP) and is a member of the United Nations number sense project. Her research interests include: Teaching practice research in open distance learning (ODL), classroom practice in multilingual contexts, professional development of mathematics teachers, curriculum studies, mathematics education pedagogy and mathematics teacher education. Pernille Ladegaard Pedersen is a PhD student at Aalborg University, Denmark. Her research focuses on fractions, learning difficulties, and test development. Pernille teaches mathematics at the Department of Teacher Education at VIA University College, specializing in children with special needs and the guidance and implementation of professional development processes in schools. Pernille is a licensed teacher with years of experience of teaching mathematics to students with special needs in elementary school. Additionally, Pernille is the project manager of several large-scale research and development projects for the Danish Ministry of Education, focusing on intervention materials and assessment tools. Núria Planas is an ICREA-academia professor at the Division of Mathematics Education of Universitat Autònoma de Barcelona, and a research fellow at the Department of Mathematics Education of the University of South Africa. When she began teaching mathematics in secondary schools of Barcelona, she found that many of her learners did not speak the language of instruction and were unfamiliar with the local routines. Her choices as a researcher are very much influenced by this experience. Her research concerns various aspects of multilingual mathematics teaching and learning, including the use of language in classroom interaction and contributions

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to equity in educational practice. She has led many developmental projects in collaboration with school communities and has served as advisory expert in several international projects investigating mathematics education settings of poverty. Núria’s intellectual interests lie within social theories of learning and discourse analysis applied to the understanding of the production of learning in school mathematics. She has published widely on these questions. Inger Ridderlind is a university lecturer at Stockholm University. She has been supervisor and a specialist teacher in mathematics and teacher education for many years, and is responsible for the education of teachers’ mathematics special education programs. She has been working in the research group called PRIM since 2000 with various assignments, mainly as a test developer for national tests in mathematics in Grade 6 and Grade 9. She has been one of the authors of several writings published by the National Agency for Education, including Assessment for Teaching and Learning in Mathematics Grades 1–9. Ridderlind has a master’s degree in mathematics didactics and has written Student Perspective on Assessment for Learning. She is a licensed teacher with years of experience from elementary school and special needs education in mathematics. Luke Rolls lived and taught overseas for several years in a village school in Ghana, an international school in China and various public infant and junior high schools in Japan, having completed his bachelor’s degree in Psychology. Returning to London, Luke spent several years teaching in primary schools in East London, completing a master’s in teaching and a postgraduate degree in primary mathematics teaching. Luke trained as part of the first cohort of NCETM Mastery Specialist Teachers, is a specialist leader of education and a current member of the Early Years and Primary Group for the Advisory Committee for Mathematics at the Royal Society. Luke is passionate about the importance of high-quality professional development as an entitlement for all teachers. Judy Sayers is a lecturer in education at the University of Leeds. She has been involved in primary and early years’ mathematics education since being a primary teacher several years ago. As a researcher, she began in 2002 with the mathematics education traditions of Europe (METE; 2003) a study of 5 European countries: Finland, Flemish Belgium, Spain, England, and Hungary, looking at ways teachers teach. Judy has worked along with colleagues in both Finland and Sweden in comparing teachers’ perspectives on their subject and profession, and has given several talks and workshops to teachers and researchers of special education in mathematics. She publishes in professional journals offering classroom ideas to practicing

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teachers. More recently, she has been part of an ongoing project FONS (Foundational Number Sense), which is looking at both teachers’ and parents’ perspectives in the teaching of numbers in the first year of formal schooling. Having published several papers, presentations and conference proceedings from the work so far, she is currently researching what parents do to support their children’s learning at home. Susan Sonnenschein, PhD, is a professor and graduate program director in the Applied Developmental Psychology program at the University of Maryland, Baltimore County. Her research focuses on ways to promote the academic success of children from different demographic backgrounds (e.g., low income, minority, ELL). Much of her research has focused on children’s reading, language, and math development. Pernille Bødtker Sunde is currently engaged in research at VIA University College, while finishing a PhD study at Aarhus University on primary students’ development of strategies in arithmetic. Her research focus is learning difficulties in mathematics and assessment of mathematic achievement and development. She holds a master’s of science in biology and is a licensed teacher in mathematics with more than 10 years of experience in teaching students with special needs in mathematics at elementary school. Pernille has developed assessment and intervention tools for primary students in Denmark, awarded by the Danish ministry of education. Charoula Stathopoulou is a professor of didactics of mathematics in the Department of Special Education at the University of Thessaly. Professor Stathopoulou completed a bachelor’s degree in mathematics and pursued additional postgraduate study in the didactics and methodology of mathematics at the University of Athens. She studied social anthropology and completed her doctorate in ethnomathematics at the Aegean University. Her main research focuses include: ethnomathematics and sociocultural factors of mathematics teaching/learning in and out of school; language and mathematics teaching; and prison education. She participated in the LLP Comenius European Project M³EaL as local coordinator and is a participant in the Project for Roma Children Education, presently she is the coordinator of the Horizon project: CoSpiRom (Commons spaces for the integration of Roma). She is the author of the book, Ethnomathematics: Exploring the Cultural Dimension of Both Mathematics and Mathematics Education (Atrapos Publications, in Greek) and co-editor of the Greek journal Research in Mathematics Education. Panayiota Stavroussi is currently an assistant professor of “Intellectual Disability: Psychological Approach and Applications in Special Education” in the Department of Special Education at the University of Thessaly, Greece.

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She holds a BA in philosophy, education, and psychology—with a specialization in psychology (Faculty of Philosophy, Aristotle University of Thessaloniki, Greece); and a master’s degree and a PhD in psychology (School of Psychology, Faculty of Philosophy, Aristotle University of Thessaloniki, Greece). Her research interests include: cognitive and social skills of persons with intellectual disabilities, genetic syndromes associated with intellectual disability, education of (and support for) children/adolescents with intellectual disability, social inclusion and transition from school to adult life and work. Constantinos Xenofontos (PhD Cantab) is a lecturer in mathematics education, at the University of Stirling (Scotland, United Kingdom). Prior to his current appointment, he was a school teacher and academic in Cyprus. His research interests revolve around the areas of mathematical problemsolving, mathematics teacher education (both initial teacher education and in-service professional development), and sociocultural and sociopolitical issues in mathematics education.