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Transforming mathematics teacher education: an equity-based approach
 9783030210168, 3030210162

Table of contents :
Intro
Preface
Contents
Contributors
Part I: TEACH Math: The Story, Modules, & Our Reflections
Chapter 1: The Story of the Teachers Empowered to Advance Change in Mathematics Project: Theoretical and Conceptual Foundations
Introduction
Conceptualizing the Project
Design Principles
Theoretical Foundations
Sociocultural Theory of Teacher Learning
Connecting to Children's Mathematical Thinking
Connecting to Children's Home- and Community-Based Funds of Knowledge
Integration of Multiple Mathematical Knowledge Bases
Conclusion
References Chapter 2: Teachers Empowered to Advance Change in Mathematics: Modules for preK-8 Mathematics Methods CoursesOverview of the Modules
Mathematics Learning Case Study Module
Community Mathematics Exploration Module
Classroom Practices Module
Using the Modules in Your Practice
References
Chapter 3: Preparing to Use the Teachers Empowered to Advance Change in Mathematics Modules: Considerations for Mathematics Teacher Educators
Getting Started with the Modules in My Mathematics Methods Course
Familiarize Yourself with the Modules Consider Connections to Big Ideas in Mathematics EducationAdaptation to Context
Timeline Adaptations
Adaptations for Specific Local Contexts
What Tensions Might MTEs Expect?
Some Things Have to Go
Entry Points versus Depth
Deficit Perspectives and Resistance
Positionality, Power, and Privilege Dynamics
Concluding Remarks
Appendix A: Timelines for Teachers Empowered to Advance Change in Mathematics Module Adaptations in Methods Courses
References
Part II: Community Mathematics Exploration Module Chapter 4: Crafting Entry Points for Learning about Children's Funds of Knowledge: Scaffolding the Community Mathematics Exploration Module for Pre-Service TeachersAuthor Positionality
Crystal
Maria
Setting the Tone for Helping PSTs to Learn about Children's (Mathematical) Funds of Knowledge
Strategies and Scaffolds to Support Implementation of the Community Mathematics Exploration with PSTs
Establishing Routines to Learn about Children's Funds of Knowledge: Suggested Reading List and Video Clips
Starting with Mini-explorations of Children's Funds of Knowledge Scaffolding a More Intentional Exploration of the CommunityCME Module: Frequently Asked Questions
How Do These Ideas Fit into a Mathematics Content Course for PSTs?
What if Our PSTs Have Trouble Finding Time to Do the Community Walk or Have Trouble Walking in the Area?
What if PSTs Are Not Assigned to Work Directly with Children and thus Do Not Have a Community to Explore?
Challenges and Next Steps with the CME Module
Appendix A: Resources to Support Incorporating Students' Funds of Knowledge in Mathematics Instruction
References

Citation preview

Tonya Gau Bartell · Corey Drake  Amy Roth McDuffie · Julia M. Aguirre  Erin E. Turner · Mary Q. Foote Editors

Transforming Mathematics Teacher Education An Equity-Based Approach

Transforming Mathematics Teacher Education

Tonya Gau Bartell  •  Corey Drake Amy Roth McDuffie  •  Julia M. Aguirre Erin E. Turner  •  Mary Q. Foote Editors

Transforming Mathematics Teacher Education An Equity-Based Approach

Editors Tonya Gau Bartell Department of Teacher Education Michigan State University East Lansing, MI, USA

Corey Drake Department of Teacher Education Michigan State University East Lansing, MI, USA

Amy Roth McDuffie College of Education Washington State University Pullman, WA, USA

Julia M. Aguirre College of Education University of Washington Tacoma Tacoma, WA, USA

Erin E. Turner Department of Teaching, Learning, and Sociocultural Studies University of Arizona Tucson, AZ, USA

Mary Q. Foote Department of Elementary and Early Childhood Education Queens College, CUNY Flushing, NY, USA

ISBN 978-3-030-21016-8    ISBN 978-3-030-21017-5 (eBook) https://doi.org/10.1007/978-3-030-21017-5 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the six moms

Preface

This book is intended for all mathematics educators committed to transforming ­mathematics teacher education with an explicit focus on equity. We (the editors) provide guidance and examples to those working with prospective and practicing teachers of mathematics who want to create positive, culturally responsive, and equitable mathematics experiences for our nation’s youth. The ideas and approaches discussed in this book build on the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project,1 which had specific goals focused on developing a new generation of preK-8 mathematics teachers to connect mathematics, children’s mathematical thinking, and community and family knowledge – or what we have come to call children’s multiple mathematical knowledge bases – in mathematics instruction. Part of our work in the TEACH Math project included the development of three instructional modules (for mathematics methods courses2) to support the project’s goals (see Chap. 2 for details about the modules). The TEACH Math project leaders, who are the editors of this book, used and refined these modules over eight semesters, after which, in Fall 2014, a dissemination conference was held to share our material with other mathematics teacher educators from a variety of universities across the United States. The ideas for this book emerged at the TEACH Math writing conference in June 2016. This conference brought together the mathematics teacher educators who had attended the dissemination conference and who had used and adapted the modules in their own contexts (including, but not limited to, mathematics methods courses). The authors represent diverse programs and geographical contexts and teach prospective and practicing teachers from a variety of socioeconomic and ethnic backgrounds. This book shares the experiences of these mathematics teacher educators.  The TEACH Math project was funded by a grant from the National Science Foundation (Grant No. 1228034). Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 2  Mathematics methods courses in the United States tend to be one or two quarters or one semester in duration as part of one’s undergraduate teacher education program, with a specific focus on the teaching and learning of mathematics. 1

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The authors provide accounts of supports, challenges, and tensions that they have encountered in implementing equity-based mathematics teacher education. We view these chapters as providing rich evidence and illustrative examples for how other mathematics teacher educators and professional developers3 might make the modules work for their unique practices, courses, workshops, and teachers. Part I of the book includes three chapters written by TEACH Math project leaders, and we see these three chapters as providing a foundation for understanding the work described in the rest of the book. Specifically, Chap. 1 describes the story of the TEACH Math project and highlights the project’s conceptual and theoretical underpinnings. Chapter 2 describes each of the three modules in detail, including each module’s goals and distinct activities. Chapter 3, then, draws on our experiences as TEACH Math project leaders to highlight various decisions mathematics teacher educators may likely need to make as they prepare to use the modules in their own contexts. Part II focuses on work around the Community Mathematics Exploration (CME) module. This section begins with Crystal Kalinec-Craig and Maria del Rosario Zavala (Chap. 4) sharing the ways they used the CME in their own contexts and identifying important entry points and scaffolds for prospective teachers across various moments during their courses. In Chap. 5, Craig Willey and Weverton Ataide Pinheiro share their experiences in implementing the CME in their urban elementary teacher education program and describe its impact on prospective elementary teachers’ outlooks on culturally relevant mathematics teaching with urban youth. Considering other ways the CME might impact prospective teachers, Kathleen Stoehr (Chap. 6) examines prospective teachers’ reflections after engaging in a CME, demonstrating that the CME can be a valuable tool to support their understanding of how to make connections in their mathematics teaching to their student’s home and community experiences and practices. Concluding this section, in Chap. 7, Zavala and Stoehr analyze prospective teachers’ CME work to better understand how they bring forth ideas of social justice into mathematics tasks that have strong connections to students’ communities. The chapters in Part III center on work with activities in the Classroom Practices Module. In Chap. 8, Amy Parks and Anita Wager describe how the video lenses tool in the module can support teacher educators preparing PK-3 teachers, providing suggestions for extending the video lenses tool with questions particular to early childhood contexts. In Chap. 9, Julie Amador and Darrell Earnest consider the Curriculum Spaces Analysis tool as a support for prospective teachers’ curricular noticing, particularly with respect to analysis of curriculum materials in ways that enable the integration of students’ mathematical and community resources. Part IV considers ways that identity and positionality work shape our praxis as mathematics teacher educators. In Chap. 10, Crystal Kalinec-Craig, Theodore  Although the modules were developed for mathematics methods courses, we have used the materials with practicing teachers in graduate courses and in professional development contexts. We encourage the readers to consider and adapt the modules for other teacher learning and development courses and workshops. 3

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Chao, Luz Maldonado, and Sylvia Celedón-Pattichis analyze their own and their prospective teachers’ work with a Mathematics Autobiography assignment (an activity we used in TEACH Math as a precursor to the Mathematics Learning Case Study Module) as a means of supporting more mathematics teachers and teacher educators to adopt a humanizing stance toward mathematics education. In Chap. 11, these same authors describe various activities they use to engage prospective teachers in critical reflection on their identities and positionalities before they engage in specific TEACH Math module activities. These activities include Photovoice Interviews (Chao), Numbers about Me Posters, and Instagram Math Trails (Maldonado), exploring Rights of the Learner (Kalinec-Craig), and Mathematics Autobiographies (Celedón-Pattichis). Finally, Gladys Krause and Luz Maldanado (Chap. 12) analyze and use scenarios shared by bilingual prospective teachers to consider the implications for teacher preparation programs as they strive to provide prospective teachers with situational awareness and instruction attuned to bilingual students’ needs. This book is the product of the experiences and expertise of 19 individuals who are committed to the improvement of mathematics teacher education. We are indebted to the authors who have contributed to this book, and we appreciate their willingness to share their work. East Lansing, MI, USA  Tonya Gau Bartell East Lansing, MI, USA  Corey Drake Pullman, WA, USA  Amy Roth McDuffie Tacoma, WA, USA  Julia M. Aguirre Tucson, AZ, USA  Erin E. Turner Flushing, NY, USA  Mary Q. Foote

Contents

Part I TEACH Math: The Story, Modules, & Our Reflections 1 The Story of the Teachers Empowered to Advance Change in Mathematics Project: Theoretical and Conceptual Foundations������������������������������������������������������������������    3 Corey Drake and Erin E. Turner 2 Teachers Empowered to Advance Change in Mathematics: Modules for preK-8 Mathematics Methods Courses��������������������������������������������������������������������������������������   15 Amy Roth McDuffie and Mary Q. Foote 3 Preparing to Use the Teachers Empowered to Advance Change in Mathematics Modules: Considerations for Mathematics Teacher Educators ��������������������������   23 Tonya Gau Bartell and Julia M. Aguirre Part II Community Mathematics Exploration Module 4 Crafting Entry Points for Learning about Children’s Funds of Knowledge: Scaffolding the Community Mathematics Exploration Module for Pre-Service Teachers��������������������������������������������������������������������������   43 Crystal Kalinec-Craig and Maria del Rosario Zavala 5 Supporting Prospective Urban Teachers to Access Children’s Multiple Mathematical Knowledge Bases: Community Mathematics Explorations������������������������������������������������   57 Craig Willey and Weverton Ataide Pinheiro 6 Prospective Teachers’ Reflections Across the Community Mathematics Exploration Module������������������������������   77 Kathleen Jablon Stoehr

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7 From Community Exploration to Social Justice Mathematics: How Do Mathematics Teacher Educators Guide Prospective Teachers to Make the Move?����������������������������������   91 María del Rosario Zavala and Kathleen Jablon Stoehr Part III Classroom Practices 8 Focusing the Video Lenses Tool to Build Deeper Understandings of Early Childhood Contexts��������������������������������������  107 Amy Noelle Parks and Anita A. Wager 9 Integrating Curriculum and Community Spaces ��������������������������������  119 Julie M. Amador and Darrell Earnest Part IV Identity, Positionality, & Praxis 10 Reflecting Back to Move Forward: Using a Mathematics Autobiography to Open Humanizing Learning Spaces for Pre-Service Mathematics Teachers��������������������������������������������������  135 Crystal Kalinec-Craig, Theodore Chao, Luz A. Maldonado, and Sylvia Celedón-Pattichis 11 Preparing Pre-Service Elementary Mathematics Teachers to Critically Engage in Elementary Mathematics Methods ����������������������������������������������������������������������������  147 Theodore Chao, Luz A. Maldonado, Crystal Kalinec-Craig, and Sylvia Celedón-Pattichis 12 Our Linguistic and Cultural Resources: The Experiences of Bilingual Prospective Teachers with Mathematics Autobiographies����������������������������������������  161 Gladys H. Krause and Luz A. Maldonado Index������������������������������������������������������������������������������������������������������������������  177

Contributors

Julia  M.  Aguirre  is associate professor of Education at the University of Washington Tacoma. Her research interests include equity and social justice in mathematics education, teacher education, and culturally responsive mathematics pedagogy. A primary goal of her work is preparing the new generations of teachers to make mathematics education accessible, meaningful, and relevant to today’s youth. She is coauthor of the book The Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices. Julie M. Amador  is associate professor of Mathematics Education at the University of Idaho. She is interested in the integration of technology into mathematics education for prospective and practicing teacher learning. Her research centers on professional noticing, specifically around curriculum, students’ mathematical thinking, and the use of technology. She also researches methods to provide mathematics professional development to rural teachers via distance technologies. She is coeditor of a department for Mathematics Teaching in the Middle School. Tonya Gau Bartell  is associate professor of Mathematics Education at Michigan State University. Her research focuses on issues of culture, race, and power in mathematics teaching and learning with particular attention to teachers’ development of mathematics pedagogy for social justice and pedagogy integrating a focus on children’s multiple mathematical knowledge bases. She is a coeditor of the Journal of Teacher Education and editor of the monograph book Toward Equity and Social Justice in Mathematics Education. Sylvia Celedón-Pattichis  is senior associate dean for Research and Community Engagement and professor in the Department of Language, Literacy, and Sociocultural Studies at the University of New Mexico. Her research interests focus on studying linguistic and cultural influences on the teaching and learning of mathematics, especially with emergent bilinguals, and on preparing teachers to work with culturally and linguistically diverse students. She serves on several advisory boards for National Science Foundation-funded projects and on the Editorial Boards xiii

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for the Journal of Latinos and Education and the Bilingual Research Journal. Her most recent work includes Access and Equity: Promoting High Quality Mathematics in Grades PreK-2 and Grades 3-5. Theodore Chao  is assistant professor of Mathematics Education in the Department of Teaching and Learning at The Ohio State University. His research agenda involves empowering all students and teachers regardless of social identity (race, gender, socioeconomic status, etc.), to learn and teach mathematics, particularly through technology. He uses photovoice interviews to explore how mathematics teachers of color connect their mathematics teacher identities with racialized social identities. He also builds mobile app technology to help children share mathematical strategies with each other, opening up windows for teachers to listen to children’s mathematical thinking. He has published in journals such as Investigations in Mathematics Learning, Contemporary Issues in Technology and Teacher Education, Mathematics Teacher Education and Development, Digital Experiences in Mathematics Education, and Education Sciences. Corey Drake  is professor and director of Teacher Preparation at Michigan State University. Her research interests include teachers learning from and about curriculum materials, as well as the roles of policy, curriculum, and teacher preparation in supporting teachers’ capacity to teach diverse groups of students. Her work has been funded by the National Science Foundation and the Spencer Foundation and is published in venues including Educational Researcher, Journal of Teacher Education, and Journal of Mathematics Teacher Education. Darrell  Earnest  is assistant professor of Education at the University of Massachusetts, Amherst. His research focuses on cognitive development in mathematics and the relationship of learning with culture and power. In particular, his research explores the role of representations and tools in learning and instruction. In addition to his focus on elementary mathematics teacher education, he currently is involved in investigating the teaching and learning of time and the development of time literacy among students from elementary to undergraduate years. Mary Q. Foote  is professor emerita of Mathematics Education from the Department of Elementary and Early Childhood Education at Queens College, City University of New York. Her research attends to equity issues in mathematics education and broadly examines issues in mathematics teacher education. More specifically, her interests are in cultural and community knowledge and practices and how they might inform mathematics teaching practice. She is currently involved in researching and developing/facilitating two professional development projects: one supports teachers to teach mathematical modeling using cultural and community contexts in Grades 3–5, and the other supports teachers to develop more equitable instructional practices through action research projects that incorporate an examination of access, agency, and allyship in mathematics teaching and learning.

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Crystal Kalinec-Craig  is assistant professor of Curriculum and Instruction at the University of Texas at San Antonio. Her research examines issues of (in)equity in mathematics education and teacher education. As a graduate student who worked with the TEACH Math project, she uses children’s mathematical thinking and funds of knowledge as the cornerstones to her teaching and research in elementary and middle grades contexts and the notion of children’s Rights of the Learner to help teachers adopt democratic, strength-based practices that interrupt issues of unequal status and participation among students. As the co-director of the Support and Enrichment Experiences in Mathematics (SEE Math) project, her prospective teachers use the TEACH Math Case Study Module to elicit and support children’s thinking as it relates to their home and community knowledge. Gladys H. Krause  is assistant professor of Mathematics Education at William & Mary College. Her research centers on teacher knowledge and children’s mathematical thinking and how these two areas interact in classroom settings which involve multilingual and multicultural dynamics. She focuses on creating a consistent and robust framework for conceptualizing teacher knowledge of children’s mathematical thinking, situated in the practice of anticipating student strategies for fraction problems, and how teachers select numbers for problems to support fraction understanding. Her work also extends to work with bilingual parents and communities to support the development of a more equitable mathematics pedagogy. Luz A. Maldonado  is assistant professor of Bilingual Mathematics Education at Texas State University in San Marcos. She earned her PhD in Mathematics Education with a doctoral portfolio in Mexican American Studies from the University of Texas at Austin. She conducts professional development sessions on Cognitively Guided Instruction with elementary teachers from Texas, Arkansas, and Florida. Her primary research interests follow the mathematical learning experiences of the bilingual learner, from elementary student to prospective teacher, in particular, documenting empowering teaching and learning practices. She is currently interested in understanding translanguaging in the bilingual elementary classroom, in which students are encouraged to develop their bilingual identities and utilize their entire linguistic repertoires to engage in mathematics. Amy Roth McDuffie  is professor of Mathematics Education and associate dean for Research for the College of Education at Washington State University. Her research focuses on teachers’ professional development with attention to teachers’ practices related to equitable pedagogies and curriculum use in mathematics education. In addition to TEACH Math, she has served as co-PI on two other National Science Foundation-funded projects: Developing Principles for Mathematics Curriculum Design and Use in the Common Core Era and Mathematical Modeling with Cultural and Community Contexts. She was the series editor for the National Council of Teachers of Mathematics’ Annual Perspectives in Mathematics Education (2014–2016).

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Amy  Noelle  Parks  is associate professor of Early Childhood Education in the Department of Teacher Education at Michigan State University and author of Exploring Mathematics Through Play in the Early Childhood Classroom (Teachers College Press). She has recently worked on parent education videos about early childhood mathematics. These can be seen at http://video.wkar.org/show/wkar-specials/shorts. Weverton Ataide Pinheiro  is a second year PhD student and associate instructor of Mathematics Education at Indiana University Bloomington. His research is focused on Critical Mathematical Studies (MathCrit), with an emphasis on mathematical identity. More specifically, his interests are in understanding how students majoring in mathematics construct their mathematician identity, focusing on the phenomenon in the United States leading to the majority of mathematicians being straight men. He is also interested in understanding how the intersectionality between race and gender influences students’ mathematical identity development. Currently, he is involved in a research project investigating how college level students generalize in advanced mathematical domains and how teachers learn to design and implement specific curriculum that supports secondary students’ generalization. Maria del Rosario Zavala, PhD  is assistant professor of Elementary Education at San Francisco State University, specializing in mathematics and bilingual education. Her research focuses on mathematics identity development of students and teachers, the role of racial and linguistic identities in learning mathematics, and racial justice in mathematics. Kathleen Jablon Stoehr  is assistant professor of Mathematics Education at Santa Clara University. Her research interests include issues that relate to prospective and early career teachers’ processes and understandings of learning to teach mathematics. Using narrative inquiry, she has explored equity and social justice issues of language, race, culture, and gender that occur in the mathematics classroom. Her current research designs and studies a model of parental engagement in mathematics that is based on a two-way dialogue between home and school. Her work is published in journals such as the Journal of Teacher Education, Journal of Urban Mathematics Education, Journal of Mathematics Teacher Education, and School Science and Mathematics. Erin E. Turner  is professor of Mathematics Education in the Teaching, Learning, and Sociocultural Studies Department at the University of Arizona. Her research focuses on issues of equity and social justice in mathematics education, with particular attention to language, culture, and community in mathematics teaching and learning and to teachers’ understandings and practices related to children’s multiple mathematical knowledge bases. Her current research examines the teaching and learning of mathematical modeling with cultural and community contexts in Grades 3–5.

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Anita  A.  Wager  is professor of the Practice at Vanderbilt University’s Peabody College. She also serves as the associate chair of Teacher Education and director of Elementary Education in the Department of Teaching and Learning. Her research focuses on teacher education that supports culturally relevant and socially just mathematics teaching in early childhood and elementary school. She works with prospective and practicing teachers to develop mathematics pedagogy that draws on children’s multiple mathematical resources, including mathematical thinking, mathematics (and other) experiences in homes and communities, and mathematics children engage with in play. The recent publications are in journals such as the Journal for Research in Mathematics Education, Journal of Teacher Education, and Journal of Early Childhood Teacher Education. She is coauthor of the book Young children’s arithmetic: Cognitively guided instruction for preschool and kindergarten and coeditor of the book Teaching Mathematics for Social Justice: Conversations with Educators. Craig  Willey  is associate professor of Mathematics Education and Teacher Education at Indiana University-Purdue University Indianapolis, as well as coordinator of the Urban Elementary Education program. His research focuses on the following: (a) the teachers’ design and implementation of mathematics discourse communities with urban students, primarily students who are Latinx; (b) the ways teachers mine and leverage children’s community and cultural knowledge to make sense of math; (c) the development and incorporation of curricular features that provide bilingual learners better access to mathematical ideas and opportunities to engage meaningfully; and (d) the limitations and affordances of a school-university partnership model of urban teacher development. In addition, he is editor of the journal Teaching for Excellence and Equity in Mathematics (TEEM) and associate editor of the International Journal of Qualitative Studies in Education.

Part I

TEACH Math: The Story, Modules, & Our Reflections

Chapter 1

The Story of the Teachers Empowered to Advance Change in Mathematics Project: Theoretical and Conceptual Foundations Corey Drake and Erin E. Turner

Keywords  Teacher education · Mathematics education · Children’s mathematical thinking · Funds of knowledge

Introduction The Teachers Empowered to Advance Change in Mathematics (TEACH Math) project began as a funded project in 2010. We started developing the ideas for TEACH Math, however, in 2006, when we (Erin and Corey) attended a small conference funded by the Iowa State University and co-organized by Corey and two colleagues at Iowa State. The conference focused on the design of elementary mathematics methods courses, and one thing we learned from hearing about others’ courses was that, while many included a focus on the development of children’s mathematical thinking as well as attention to children’s funds of knowledge and issues of equity, very few courses addressed these topics in integrated ways. As a result, the courses (including ours) were potentially conveying the message to prospective teachers (PSTs) that children’s mathematical thinking and children’s funds of knowledge were separate knowledge bases that could (and perhaps should) be drawn on in isolation from each other in instruction. We began to wonder if we could imagine, design, teach, and study a more integrated approach to these topics. What would it look like to support PSTs in developing teacher practices that connected to both children’s mathematical thinking and children’s funds of knowledge? What kinds of

C. Drake (*) Department of Teacher Education, Michigan State University, East Lansing, MI, USA e-mail: [email protected] E. E. Turner College of Education, University of Arizona, Tucson, AZ, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_1

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experiences would PSTs need to develop these practices? What would we expect PSTs to know and be able to do as a result? This initial conference was followed by a National Science Foundation-funded conference in 2008.1 This latter conference brought together a larger group of elementary mathematics teacher educators with a specific focus on “Connecting Children’s Mathematical Thinking to Community and Family Funds of Knowledge in Elementary Mathematics Methods Courses.” At this conference, we began to further develop the conceptual foundations for integrating these knowledge bases and brainstormed ideas for activities and experiences in methods courses that could support PSTs in developing integrated connections. The TEACH Math group of six co-PIs emerged from this conference and the initial conceptualizations and activity brainstorming developed into the basis for the TEACH Math project, including the three TEACH Math instructional modules. In the remainder of this chapter, we discuss the conceptual and theoretical ideas that have framed the TEACH Math project and how those ideas have both informed and been informed by our design and implementation of the modules as well as our research on PST and novice teacher learning.

Conceptualizing the Project Our conceptualization of the TEACH Math project relies on three ideas, based on the literature and our own experiences. First, novice elementary teachers are often underprepared to teach mathematics to diverse groups of students (Hollins & Guzman, 2005; Howard, 1999; Nieto, 2010; Sleeter, 2001). Second, improved preparation of elementary teachers for teaching mathematics to diverse groups of students requires the development not only of teacher knowledge and beliefs but also of specific practices that support integrated attention and connections to children’s mathematical thinking and children’s funds of knowledge. This second idea was inspired in part by Grossman and colleagues’ call to action: It is not enough to prepare teachers with the dispositions to teach all students, or with knowledge of their students’ cultural and linguistic resources. Teachers need to know how to use such knowledge in order to help students develop intellectual skills and to succeed academically. (Grossman, McDonald, Hammerness, & Ronfeldt, 2008, p. 244)

Finally, because the six of us were teaching across six different teacher preparation contexts that varied along a number of dimensions—including teacher candidate and student demographics, geography, program goals and orientation, length of mathematics methods courses, and so forth—and because we wanted the materials we developed to be useful for mathematics teacher educators (MTEs) across a still wider range of contexts, we intentionally designed the modules from the beginning  National Science Foundation Award #0736964 (Corey Drake, Erin E.  Turner, and Alejandro Andreotti). 1

1  The Story of the Teachers Empowered to Advance Change in Mathematics Project…

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to be flexible and adaptable. From the beginning of the project, we also prepared both to research our implementation of the modules and to use those research findings to support MTEs in productively adapting the modules for their own contexts (See Chapters 2 and 10 for more information about the modules and our suggestions for preparing to implement the modules in different contexts). As we noted in our initial grant proposal: The diversity of sites will inform our understandings of how local instructor, course, program, university, and school contexts can impact the implementation of the instructional modules and professional development materials, thereby supporting the successful widespread dissemination of these products (Drake et al., 2010, p. 1).

Design Principles To enact our conceptualization of the project, as well as the eventual adaptation and dissemination of the modules, we focused on three key design principles (see Roth McDuffie et al., 2014, for more detail). The first principle was, whenever possible, to identify activities commonly implemented in elementary mathematics methods courses and then adapt those activities to focus specifically on the integration of children’s mathematical thinking and children’s home and community funds of knowledge. Common activities incorporated in this way in the modules include video analysis, problem-solving interviews with children, and the use of curriculum materials. Second, we designed the module activities to repeatedly focus PSTs’ attention on a small set of ideas related to mathematics teaching and learning, including the role of the task, the importance of power and participation, and most importantly, the integration of children’s mathematical thinking and children’s home- and community-­based funds of knowledge. Prompts related to these ideas are included in nearly every activity. Finally, we wanted the modules to be an integral component of MTEs’ methods courses, connected to other components of the courses. Thus, we designed the modules not only to attend to TEACH Math goals but also to incorporate key ideas about teaching and learning mathematics from other MTEs, including ideas related to cognitive demand and practices to support student discussion (Smith & Stein, 2011).

Theoretical Foundations Sociocultural Theory of Teacher Learning In our work with the TEACH Math project, we understood teacher learning to be a sociocultural activity wherein teachers develop knowledge, dispositions, and practices as they interact in multiple communities of practice over time (Lerman, 2001;

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Wenger, 1998). This sociocultural lens drew our attention to PSTs’ participation in various educational activities (e.g., planning lessons or discussing classroom videos with colleagues), including how they draw upon artifacts, tools, and other PSTs’ experiences and thinking to make sense of their experiences (Putnam & Borko, 2000). We also attended to PSTs’ engagement across multiple spaces (e.g., methods courses, elementary classrooms, communities) and considered how these spaces are each shaped by particular social, cultural, political, and historical forces. As we noted in an article in the Journal of Mathematics Teacher Education in 2012: While teacher education programs have traditionally focused on university courses and elementary school classrooms as spaces for PST learning (Cochran-Smith & Lytle, 1999), we aim to extend these spaces to include engagement with children’s families and communities. As PSTs interact with various participants across multiple spaces, we have considered how they develop knowledge, dispositions, and practices that position them as effective teachers of mathematics for diverse students. (Turner et al., 2012, p. 69)

In particular, in our design of the TEACH Math modules, we focused on supporting PSTs’ understandings and practices related to (a) children’s mathematical thinking and (b) children’s home- and community-based funds of knowledge in instruction, because these connections have been shown to support learning, participation, and identity development of diverse groups of students (Brenner, 1998; Gutstein, Lipman, Hernandez, & de los Reyes, 1997; Tate, 1995; Turner & Celdón-Pattichis, 2011).

Connecting to Children’s Mathematical Thinking A vast body of research documents the effectiveness of instruction that centers on children’s mathematical thinking, which includes attention to such things as children’s solution strategies, common understandings and misconceptions, and number choices and problem structures that would support an appropriate level of cognitive demand (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Stein, Smith, Henningsen, & Silver, 2000). This research, including decades of work on Cognitively Guided Instruction (e.g., Carpenter, Fennema, Peterson, Chiang, & Loef, 1989), has linked teachers’ knowledge of children’s mathematical thinking to productive changes in teachers’ beliefs, classroom practices, and student learning (Fennema et al., 1996). For example, when teachers have frameworks for understanding problem structures and progressions in children’s solution strategies, they can leverage these understandings to pose problems and adapt tasks in ways that support children’s learning. Research has shown that children in such classrooms perform better on problem-solving tasks and equally well on computation, when compared to students in classrooms where these frameworks do not guide instruction (Carpenter et al., 1989, 1998; Carpenter, Fennema, & Franke, 1996; Fennema et al., 1996). Specific to PSTs, research has identified experiences that can support learning about children’s mathematical thinking. For example, conducting and analyzing

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problem-solving interviews with children supports increased awareness of the diversity of students’ ideas and strategies, and an enhanced capacity to interpret children’s strategies, and to relate children’s reasoning to frameworks outlined in research (Jenkins, 2010; Philipp et al., 2007). Studies have also argued that PSTs need sustained and scaffolded opportunities to learn to elicit and respond to children’s mathematical thinking (Sleep & Boerst, 2012), as shifts in beliefs and understandings about children’s mathematical thinking are not always accompanied by parallel shifts in practice (Vacc & Bright, 1999). For example, supports such as specific protocols for interviewing children (McDonough, Clarke, & Clarke, 2002), examples of how to interpret children’s reasoning (Sleep & Boerst, 2012), opportunities to interact with the same children over time (Ambrose, 2004), and carefully selected video clips of the development of children’s mathematical reasoning (Maher, Palius, Maher, Hmelo-Silver, & Sigley, 2014) have all been shown to enhance PST understanding and practice. We designed our modules to include these scaffolds and to provide ongoing, yet varied opportunities to notice, elicit, interpret, and respond to children’s ideas. For example, we found that repeated enactments of a video analysis activity that included prompts designed to deepen PSTs’ noticing of children’s mathematical thinking (and other knowledge bases) supported PSTs in noticing student resources and understanding how these resources can support learning (Roth McDuffie et al., 2014). Similarly, we found that repeated opportunities to interact with a child (including through problem-solving interviews, observations, and informal conversations) supported PSTs in generating future instruction that connected to what they had learned about the child’s mathematical thinking (Bartell et  al., 2013; Turner et al., 2016). Together, these studies suggest that connecting to children’s mathematical thinking is accessible to PSTs and that it is a practice that is facilitated by sustained and scaffolded opportunities to interact with children and view representations of teaching.

 onnecting to Children’s Home- and Community-Based Funds C of Knowledge Other research has documented that historically underrepresented populations benefit from instruction that draws upon their cultural, linguistic, and community-based knowledge (Ladson-Billings, 1994; Lee, 2007; Silver & Stein, 1996; Turner, Celedón-Pattichis, & Marshall, 2008). This research has argued that teachers need to understand how students’ home- and community-based funds of knowledge—the knowledge, skills, and experiences found in students’ homes and communities— can support their mathematical learning (Civil, 2002; González, Andrade, Civil, & Moll, 2001; González, Moll, & Amanti, 2005; Moll, Amanti, Neff, & Gonzalez, 1992). For example, research in Alaskan Yup’ik communities has documented how teachers connected mathematics lessons to familiar cultural activities (i.e.,

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designing a fish drying rack) in ways that leveraged the knowledge and experiences of students, families, and community members, and deepened students’ understanding of important mathematical ideas (Lipka et  al., 2005). Yet in instances when teachers lack curriculum materials that include or prompt such connections, connecting to home- and community-based funds of knowledge in school mathematics lessons can be challenging (e.g., Civil, 2007; Gonzalez et al., 2001). Compared to the substantial body of work on PST learning related to children’s mathematical thinking, research on how PSTs learn to connect to home- and community-­based funds of knowledge in mathematics teaching has been limited (see Turner & Drake, 2016, for a thorough review of research in both areas). Some studies have described PSTs’ initial attempts to draw on children’s home and community experiences as they are planning mathematics lessons, and the connections they make are often superficial ones, changing names in problems, and adjusting contexts to reflect students’ interests (Nicol & Crespo, 2006). Other researchers have argued that PSTs need opportunities to interact with children, families, and community members outside of school, particularly for learning about children’s home- and community-based funds of knowledge (Evans, 2013; McDonald, Bowman, & Brayko, 2013). For example, Jurow, Tracy, Hotchkiss, and Kirshner (2012) noted that community-based spaces, which are less connected to PSTs’ preconceived ideas about children’s competencies, help PSTs “to ‘see’ learning and knowledge when it is expressed by children from nondominant backgrounds” (p.  158). Curriculum materials also seem to play an important role. Vomvoridi-­ Ivanović (2012) found that PSTs rarely made connections between textbook-like activities and students’ out-of-school experiences. Yet, during an extended project focused on recipes, PSTs regularly referenced family and cultural practices related to cooking and shopping for ingredients. This study suggests that mathematics tasks that relate to students’ experiences, which are unlikely to be found in a traditional curriculum, facilitate opportunities for PSTs to connect to children’s funds of knowledge. Finally, studies have suggested that while PSTs are often interested in connecting to children’s home and community knowledge in their mathematics lessons, they also express uncertainty about the sustainability of this kind of teaching (Leonard, Brooks, Barnes-Johnson, & Berry, 2010). We used these findings to guide the design of our modules, focusing in particular on providing opportunities for PSTs to interact with children, families, and community members across spaces (i.e., the mathematics classroom, the school, the community). In research related to these modules, we have found that while some PSTs begin with less demanding components of the practice of connecting to children’s home- and community-based funds of knowledge (e.g., adapting problem contexts to include community settings or children’s interests), other PSTs evidence more meaningful connections. For example, some PSTs are able to identify mathematical, cultural, and linguistic practices in children’s homes and communities and then consider ways to connect to those practices in their instruction (Aguirre et al., 2013; Turner et al., 2014). We contend that specific experiences in the modules supported these meaningful connections, including opportunities to (a) dialogue with families and community members about their everyday practices, (b) visit

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c­ ommunity locations and reflect with peers and instructors on the practices they observed, and (c) brainstorm possible activities and lessons that connect to what they have learned. Finally, we found that although most PSTs embraced the idea of considering how children’s out-of-school experiences might support mathematics learning, some PSTs described challenges related to their outsider status in the community (Aguirre et al., 2013).

Integration of Multiple Mathematical Knowledge Bases Given the benefits of instruction that builds on children’s multiple mathematical knowledge bases (MMKB) (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including children’s mathematical thinking and children’s home- and community-based knowledge), we see helping PSTs learn to draw upon these knowledge bases during instruction as critical to promoting equity. Research on how PST preparation might integrate these multiple knowledge bases, however, has been limited (e.g., Aguirre, 2009). To address this gap in extant research, our primary goal in the TEACH Math project was to support prospective and early career preK-8 mathematics teachers to develop understandings and practices that connect to children’s MMKB, specifically children’s mathematical thinking and children’s home- and communitybased funds of knowledge, through the development and implementation of instructional modules for elementary mathematics methods courses (Aguirre et al., 2012; Turner et al., 2012). While there was a dearth of prior research in this area, we drew on studies of how PSTs develop other kinds of integrated and complex teaching practices to inform and guide the design of the modules. For example, Garii and Rule (2007) examined how PSTs integrated both appropriately challenging content (mathematical goals) and meaningful connections to social justice issues (social justice goals) in their mathematics lessons. They found that PSTs struggled to balance these multiple dimensions in their lessons, with some PSTs abandoning mathematics content in favor of social studies–oriented investigations. Other studies also emphasized PSTs’ need for additional support to take up complex teaching practices. For instance, Parks (2008) examined how PSTs developed a mathematical lens (i.e., focus on content, tasks, and representations) and/or an equity lens (i.e., attention to equitable participation and children’s strengths, including cultural and linguistic resources) as they planned and taught a mathematics lesson. Despite prompting PSTs to consider these multiple dimensions in the lesson plan assignment, only one group of PSTs evidenced meaningful attention to both lenses in their lessons. The challenges that PSTs experience as they begin to take up complex, integrated practices raised questions about additional supports that might be needed. First, we conjectured that the persistent separation of practices (e.g., learning about eliciting children’s thinking, separate from learning about connecting to children’s

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o­ ut-of-­school experiences), possibly motivated by the goal of minimizing complexity for PSTs, may be counterproductive, making it harder for PSTs to develop this integrated practice. Additionally, features that have been shown to enhance practicing teachers’ efforts to connect to children’s mathematical thinking and children’s home- and community-based funds of knowledge in their teaching—time and supportive communities—are often lacking in PSTs’ preparation experiences. These insights guided the design of our modules, as we focused in particular on repeated opportunities to collaboratively engage with complex, integrated teaching practices throughout each of the modules. Early in our research into the implementation of the modules, we identified a conjectured learning trajectory to describe PSTs’ development of practices that integrate children’s MMKB (Turner et al., 2012). Building on the work of Mason (2008) and Jacobs et al. (2010), we developed multiple iterations of the trajectory, testing it against the evidence of PST learning that we were gathering over time. Ultimately, the trajectory suggests three key phases in PST learning: (a) Attention to, awareness of, and eliciting of children’s mathematical thinking and children’s home- and community-­based funds of knowledge; (b) Making emergent and then more meaningful connections to children’s MMKB; and (c) Incorporating children’s MMKB into instruction. We recognize that learning is not linear and that PSTs may enter this trajectory in many different places. An important implication of this trajectory work, however, is to consider, given a specific entry point, how to help PSTs progress and strengthen the practice of making connections to both children’s mathematical thinking and children’s home- and community-based funds of knowledge. Furthermore, the existence of different entry points suggests the importance of maintaining an integrated focus on children’s MMKB in methods courses. An integrated focus can support a range of PSTs—who themselves bring a variety of experiences and knowledge bases to mathematics methods courses—to extend their initial understanding of children’s MMKB to include other experiences and knowledge bases that support children’s mathematics learning.

Conclusion In conclusion, our commitment to the integration of MMKB in both teacher preparation and K-12 classrooms is grounded in our theoretical understandings of children’s and teachers’ learning. Over more than ten years, we have worked to develop, study, and disseminate a series of modules for elementary mathematics methods courses that reflect this commitment and support MTEs and PSTs in learning to enact this commitment. Many of the results of this work can be found in the remainder of this volume. Chapter 2 provides a more detailed description of the modules and how they have been used in mathematics methods courses across contexts. Sections II and III focus on the key components of two of the modules: (a) the community mathematics exploration and (b) the development of lenses for analyzing

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classroom video and curriculum as representations of classroom practice. This is followed by Section IV, which focuses on the work MTEs and PSTs can do to prepare themselves to engage productively with the modules. We are thrilled by the ways these chapters illustrate the use, adaptation, and improvement of our modules across a wide range of course, program, and community contexts. As with any successful research and development project, we are left not only with the findings and outcomes but also with new questions to explore in future work. First, we wonder about the kinds of supports and experiences that will enhance teachers’ understandings and practices for connecting to children’s MMKB as teachers move from teacher education programs and into the role of novice, and then more experienced, classroom teacher. We have begun to explore teachers’ early career practices related to children’s MMKB through a series of case studies (e.g., Kinser-Traut & Turner, 2018; Sugimoto, Turner, & Stoehr, 2017; Turner et al., in press), but more research is needed. We also wonder about how to further expand the construct of children’s MMKB to include neurodiverse students and students whose mathematical knowledge bases have been less documented in research. Further, we are interested in the ways in which the potential of mathematical modeling as an approach to mathematics teaching and learning, particularly when connected to meaningful contexts in children’s homes, schools, and communities, can support the integration of children’s MMKB in instruction. Finally, we are always seeking to identify ways to bring a more critical orientation to this work, so that the integration of children’s MMKB in elementary mathematics classrooms can become a means of disrupting privilege and oppression and creating more equitable, just, and humanizing schooling experiences for children.

References Aguirre, J. (2009). Privileging mathematics and equity in teacher education: Framework, counter-­ resistance strategies and reflection from a Latina mathematics educator. In B. Greer, S. Mukho-­ padhyay, S. Nelson-Barber, & A. Powell (Eds.), Culturally responsive mathematics education (pp. 295–319). New York, NY: Routledge. Aguirre, J., Turner, E., Bartell, T.  G., Drake, C., Foote, M.  Q., & Roth McDuffie, A. (2012). Analyzing effective mathematics lessons for English learners: A multiple mathematical lens approach. In S.  Celedón-Pattichis & N.  Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs (pp. 207–222). Reston, VA: National Council of Teachers of Mathematics. Aguirre, J. M., Turner, E. E., Bartell, T. G., Kalinec-Craig, C., Foote, M. Q., Roth McDuffie, A., et al. (2013). Making connections in practice: Developing prospective teachers’ capacities to connect children’s mathematical thinking and community funds of knowledge in mathematics instruction. Journal of Teacher Education, 64(2), 178–192. Ambrose, R. (2004). Initiating change in prospective elementary school teachers’ beliefs about mathematics and mathematics learning. Journal of Mathematics Teacher Education, 7(2), 91–119. Bartell, T. G., Foote, M. Q., Drake, C., Roth McDuffie, A., Turner, E. E., & Aguirre, J. M. (2013). Developing teachers of Black children: (Re)orienting thinking in an elementary mathematics methods course. In J. Leonard & D. B. Martin (Eds.), The brilliance of Black children in

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mathematics: Beyond the numbers and toward new discourse (pp. 343–367). Charlotte, NC: Information Age Publishing. Brenner, M. E. (1998). Adding cognition to the formula for culturally relevant instruction in mathematics. Anthropology & Education Quarterly, 29(2), 214–244. Carpenter, T., Fennema, E., & Franke, M. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97(1), 3–20. Carpenter, T., Fennema, E., Peterson, P., Chiang, C., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531. Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20. Civil, M. (2002). Everyday mathematics, mathematicians’ mathematics, and school mathematics: Can we bring them together? In M. Brenner & J. Moschkovich (Eds.), Everyday and academic mathematics in the classroom. Journal of Research in Mathematics Education Monograph #11 (pp. 40–62). Reston, VA: National Council of Teachers of Mathematics. 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. Cochran-Smith, M., & Lytle, S. (1999). Relationships of knowledge and practice: Teacher learning in communities. Review of Educational Research, 24(1), 249–305. Drake, C., Turner, E. E., Aguirre, J., Bartell, T. G., Foote, M. Q., & Roth McDuffie, A. (2010). Teachers empowered to advance change in mathematics (TEACH Math): Preparing PreK-8 teachers to connect children’s mathematical thinking and community-based funds of knowledge. Proposal submitted to the National Science Foundation. Evans, M. P. (2013). Educating preservice teachers for family, school, and community engagement. Teaching Education, 24(2), 123–133. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V., & Empson, S. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403–434. Garii, B., & Rule, A. C. (2007). Integrating social justice with mathematics and science: An analysis of student teacher lessons. Teaching and Teacher Education, 25(3), 490–499. 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.  C., & Amanti, C. (2005). Funds of knowledge: Theorizing practices in households and classrooms. Mahwah, NJ: Lawrence Erlbaum. Grossman, P., McDonald, M., Hammerness, K., & Ronfeldt, M. (2008). Dismantling dichotomies in teacher education. In M. Cochran-Smith, S. Feiman-Nemser, D. J. McIntyre, & K. E. Demers (Eds.), Hand book of research on teacher education: Enduring questions in changing contexts (pp. 243–248). New York, NY: Routledge. Gutstein, E., Lipman, P., Hernández, 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. Hollins, E., & Guzman, M. (2005). Research on preparing teachers for diverse populations. In M. Cochran-Smith & K. Zeichner (Eds.), Studying teacher education (pp. 477–538). Mahwah, NJ: Lawrence Erlbaum. Howard, G. R. (1999). We can’t teach what we don’t know: White teachers, multiracial schools. New York, NY: Teachers College Press. Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. Jenkins, O.  F. (2010). Developing teachers’ knowledge of students as learners of mathematics through structured interviews. Journal of Mathematics Teacher Education, 13(2), 141–154.

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Jurow, S., Tracy, R., Hotchkiss, J., & Kirshner, B. (2012). Designing for the future: How the learning sciences can inform the trajectories of preservice teachers. Journal of Teacher Education, 63(2), 147–160. Kinser-Traut, J., & Turner, E. (2018). Shared authority in the mathematics classroom: Successes and challenges throughout one teacher’s trajectory implementing ambitious practices. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-018-9410-x Ladson-Billings, G. (1994). The Dreamkeepers: Successful teachers of African American children. San Francisco, CA: Jossey-Bass. Lee, C.  D. (2007). Culture, literacy, & learning: Taking bloom in the midst of the whirlwind. New York, NY: Teachers College Press. Leonard, J., Brooks, W., Barnes-Johnson, J., & Berry III., R. Q. (2010). The nuances and complexities of teaching mathematics for cultural relevance and social justice. Journal of Teacher Education, 61(3), 261–270. Lerman, S. (2001). A review of research perspectives on mathematics teacher education. In F. L. Lin & T.  J. Cooney (Eds.), Making sense of mathematics teacher education (pp.  33–52). Dordrecht, Netherlands: Kluwer. Lipka, J., Hogan, M., Webster, J. P., Yanez, E., Adams, B., Clark, S., et al. (2005). Math in a cultural context: Two case studies of a successful culturally based math project. Anthropology & Education Quarterly, 36(4), 367–385. Maher, C. A., Palius, M. F., Maher, J. A., Hmelo-Silver, C. E., & Sigley, R. (2014). Teachers can learn to attend to students’ reasoning: Using videos as a tool. Issues in Teacher Education, 23(1), 31–47. Mason, J. (2008). Being mathematical with and in front of learners: Attention, awareness, and attitude as sources of difference between teacher educators, teachers and learners. In B. Jaworski (Vol. Ed.) & T.  Wood (Series Ed.), Handbook of mathematics teacher education (Vol. 4): The mathematics teacher educator as a developing professional (pp.  31–45). Rotterdam, Netherlands: Sense. McDonald, M., Bowman, M., & Brayko, K. (2013). Learning to see students: Opportunities to develop relational practices of teaching through community-based placements in teacher education. Teachers College Record, 115(4), 1–35. McDonough, A., Clarke, B. A., & Clarke, D. M. (2002). Understanding assessing and developing young children’s mathematical thinking: The power of the one-to-one interview for preservice teachers in providing insights into appropriate pedagogical practices. International Journal of Education Research, 37(2), 107–112. Moll, L., Amanti, C., Neff, D., & Gonzalez, N. (1992). Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms. Theory Into Practice, 31(2), 132–141. Nicol, C. C., & Crespo, S. M. (2006). Learning to teach with mathematics textbooks: How preservice teachers interpret and use curriculum materials. Educational Studies in Mathematics, 62(3), 331–355. Nieto, S. (2010). The light in their eyes: Creating multicultural learning communities. New York, NY: Teachers College Press. Parks, A. N. (2008). Messy learning: Preservice teachers’ lesson-study conversations about mathematics and students. Teaching and Teacher Education, 24(5), 1200–1216. Philipp, R. A., Ambrose, R., Lamb, L., Sowder, J., Schappelle, B., & Sowder, L. (2007). Effects of early field experiences on the mathematical content knowledge and beliefs of prospective elementary school teachers: An experimental study. Journal for Research in Mathematics Education, 38(5), 438–476. Putnam, R., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning? Educational Researcher, 29(4), 4–15. Roth McDuffie, A., Foote, M., Drake, C., Turner, E., Aguirre, J., Bartell, T.  G., et  al. (2014). Use of video analysis to support prospective K-8 teachers’ noticing of equitable practices. Mathematics Teacher Educator, 2(2), 108–140. Silver, E. A., & Stein, M. K. (1996). The QUASAR Project: The “revolution of the possible” in math ematics instructional reform in urban middle schools. Urban Education, 30(4), 476–521.

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Sleep, L., & Boerst, T. A. (2012). Preparing beginning teachers to elicit and interpret students’ mathematical thinking. Teaching and Teacher Education, 28(7), 1038–1048. Sleeter, C. E. (2001). Preparing teachers for culturally diverse schools: The overwhelming presence of whiteness. Journal of Teacher Education, 52(2), 94–106. Smith, M., & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. Thousand Oaks, CA: Corwin Press. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Reston, VA: National Council of Teachers of Mathematics. Sugimoto, A., Turner, E., & Stoehr, K. (2017). A case study of dilemmas when connecting middle school mathematics instruction to relevant real-world examples. Middle Grades Research Journal, 11(2), 61–82. Tate, W. F. (1995). Returning to the root: A culturally relevant approach to mathematics pedagogy. Theory Into Practice, 34(3), 166–173. Turner, E., & Celedon-Pattichis, S. (2011). Problem solving and mathematical discourse among Latino/a kindergarten Students: An analysis of opportunities to learn. Journal of Latinos in Education, 10(2), 146–168. Turner, E., Celedon-Pattichis, S., & Marshall, M. A. (2008). Cultural and linguistic resources to pro mote problem solving and mathematics discourse among Latino/a kindergarten Students. In R. Kitchen & E. Silver (Eds.), Promoting high participation and success in mathematics by Hispanic students: Examining opportunities and probing promising practices [Inaugural Research Monograph of TODOS: Mathematics for All] (Vol. 1, pp. 19–42). Washington, D. C.: National Education Association Press. Turner, E., Roth McDuffie, A., Sugimoto, A., Aguirre, J., Bartell, T., Drake, C., Foote, M., Stoehr, K., Witters, A. (in press, accepted in final form). Early career elementary mathematics teachers’ practices related to language and language learners. International Journal of Mathematical Thinking and Learning. Turner, E. E., Aguirre, J. M., Bartell, T. G., Drake, C., Foote, M. Q., & Roth McDuffie, A. (2014). Making meaningful connections with mathematics and the community: Lessons from pre-­ service teachers. In T. G. Bartell & A. Flores (Eds.), Embracing resources of children, families, communities, and cultures in mathematics learning [A Research Monograph of TODOS: Mathematics for ALL] (Vol. 3, pp. 30–49). San Bernardino, CA: TODOS. Turner, E. E., & Drake, C. (2016). A review of research on prospective teachers’ learning about children’s mathematical thinking and cultural funds of knowledge. Journal of Teacher Education, 67(1), 32–46. Turner, E. E., Drake, C., Roth McDuffie, A., Aguirre, J., Bartell, T. G., & Foote, M. Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematical knowledge bases. Journal of Mathematics Teacher Education, 15, 67–82. Turner, E. E., Foote, M. Q., Stoehr, K., Roth McDuffie, A., Aguirre, J., Bartell, T. G., et al. (2016). Learning to leverage children’s multiple mathematical knowledge bases in instruction. Journal of Urban Mathematics Education, 9(1), 48–78. Vacc, N.  N., & Bright, G.  W. (1999). Elementary preservice teachers’ changing beliefs and instructional use of children’s mathematical thinking. Journal for Research in Mathematics Education, 30(1), 89–110. Vomvoridi-Ivanović, E. (2012). Using culture as a resource in mathematics: The case of four Mexican-American prospective teachers in a bilingual after-school program. Journal of Mathematics Teacher Education, 15(1), 53–66. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge, UK: Cambridge University Press.

Chapter 2

Teachers Empowered to Advance Change in Mathematics: Modules for preK-8 Mathematics Methods Courses Amy Roth McDuffie and Mary Q. Foote

Keywords  Teacher education · Mathematics education · Mathematics methods courses · Community connections · Video case study · Curriculum analysis · Lesson planning · Student case studies The primary goals of the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project were to support and study ways to support prospective teachers (PSTs) in developing the knowledge, beliefs, dispositions, and practices needed to effectively plan, adapt, and implement mathematics instruction in culturally, linguistically, and socio-economically diverse schools. In an effort to meet this goal, the TEACH Math team of mathematics teacher educators (MTEs) and researchers (the editors of this book) designed and studied three instructional modules. We designed these modules with the specific aim of helping preK-8 PSTs learn to integrate mathematics, children’s mathematical thinking, and children’s cultural, linguistic, and community-based funds of knowledge in mathematics instruction. These modules are organized around key practices of teaching elementary mathematics, including developing mathematical tasks, lesson and curriculum analysis, and assessment of children’s mathematical thinking. Within each module are a collection of activities intended to provide multiple experiences related to the targeted learning goals. Throughout the TEACH Math project, we iteratively researched and refined these modules. Detailed descriptions of the modules, goals, lesson outlines, handouts, suggested timelines for sequencing activities throughout a term, and other materials for MTEs and their PSTs are available online at the TEACH Math project website (www.TEACHMath.info). The lesson materials are framed within the familiar Launch, Explore, and Summarize structure and include suggestions for A. Roth McDuffie (*) Washington State University, Pullman, WA, USA e-mail: [email protected] M. Q. Foote Queens College of the City University of New York, Flushing, NY, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_2

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MTEs for facilitating lessons with PSTs. We also include our reflections and advice to other MTEs based on our research using the modules in multiple preK-8 mathematics methods classes and with practicing teachers in professional development. Several published articles are also available and listed on the website, providing more information about individual modules and our research findings. From the start of the TEACH Math project, we designed the modules with the intent and hope that others would use whole modules and/or activities within modules not in a scripted way but by adapting, modifying, and improving them for their purposes and contexts. In doing so, the modules and activities provide a starting point for evolving and generative work in supporting PSTs’ and teachers’ efforts toward developing practices that support learners in culturally, linguistically, and socio-economically diverse schools.

Overview of the Modules Throughout this volume, chapters reference one or more modules and/or activities within modules. The next sections provide an overview of each of the modules to provide background for reading the chapters in the book. Each section below corresponds with a module and provides information about the goals and activities of the specified module. The three modules are Mathematics Learning Case Study, Community Mathematics Exploration, and Classroom Practices.

Mathematics Learning Case Study Module This module focuses on the learning, identity, and dispositions of a case study student. In working with the case study student over the course of the term, the PST considers how to use the knowledge gained from and with the student in mathematics instruction. The goal of the Mathematics Learning Case Study Module is to help PSTs learn to: • Observe and examine learning in more detail, to expand how PSTs think about children as mathematical learners, including which skills, knowledge, practices, and experiences they see as relevant to children’s learning. • Consider how the knowledge, skills, and competencies that children demonstrate in different contexts (e.g., school, after school, home, and community) might support their school mathematics learning. The activities within this module support PSTs to (a) conduct a mathematics “Getting to Know You” interview with a single student, (b) shadow a student for all or part of a day, (c) conduct one or more mathematics interview assessments with that same student, and (d) engage in written analysis and reflection on the first three activities in the module.

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Activity 1: Mathematics “Getting to Know You” interview  PSTs conduct an interview with one student in their practicum classroom in an effort to become more familiar with the student’s activities and interests, the student’s home and community knowledge, and the student’s home and community resources. We encourage PSTs to select a case study student that is of a different socio-cultural background than themselves as research documents that it is easier for teachers to develop positive relationships with children most like them (Nieto, 2004). In this way, this activity is intended to support PSTs in getting to know a student they may be least likely to otherwise know. In the “Getting to Know You” interview, PSTs ask students about their home and community with questions such as “If I were going to walk from school to your house, what are some things/places that I would see?” “Where do you like to go with family/friends? What do you do there?” and “What kinds of things do you do with family/friends at home – both regular routines (e.g., cooking) and things you enjoy (e.g., games)?” PSTs also ask students about their dispositions related to mathematics with questions such as “What are some things in math you really like?” “What about math do you not like?” “Do you think it is important to learn math? Why do you think so?” “Who do you know that is good at math? How do you know? What makes them good?” and “Why do you think it is hard for some people to learn math?” Knowledge gained from this interview can also be used to inform the Community Walk Activity in the Community Mathematics Exploration Module. Activity 2: Shadow a student  PSTs “shadow” a student for a period of time (ideally one full school day) in an effort to identify the child’s competencies across school contexts. A critical aspect of this shadowing experience is that not all shadowing occurs during math class and that shadowing does not occur while PSTs teach lessons or conduct activities with students. Rather, PSTs are encouraged to observe students in multiple classroom settings (e.g., art, gym, science) as well as in leisure settings (e.g., before or after school, lunch, recess). Similar to the interview, knowledge gained from this experience can also be used to inform the Community Walk Activity in the Community Mathematics Exploration Module. Activity 3: Problem-solving interviews  PSTs conduct one or more problem-­ solving interviews with one or more students in their practicum classroom. If PSTs are engaging in multiple activities from this module, then we recommend that at least one of the students should be the case study student from Activity 1. These interviews provide an opportunity to practice eliciting, interpreting, and assessing students’ thinking about mathematics, with a particular focus on children’s understanding of number concepts. Whole number interview protocols and guidelines were adapted from the work of Tom Carpenter and the Cognitively Guided Instruction (CGI) Group (Carpenter, Fennema, Franke, Levi, & Empson, 1999), as well as the work of Susan Empson and colleagues (Empson, Junk, Dominguez, & Turner, 2006). The fraction interview protocol was adapted from the work done by Edd Taylor for his own mathematics methods course.

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Activity 4: Synthesizing and connecting across activities  This assignment is designed to cut across the previous activities in this module. In this write-up, PSTs reflect across the multiple interviews and observations they completed with their case study student and consider how to use this knowledge in mathematics instruction (early thinking about leveraging students’ multiple mathematical knowledge bases [MMKB]). They synthesize and then apply this knowledge for further mathematics instruction in three ways: (a) design the next task they would pose to students, (b) integrate information gained into instructional practices that would benefit the child, and, in some iterations (c) provide points of discussion to be used in communicating with family members about the child’s mathematical learning.

Community Mathematics Exploration Module The goals of the Community Mathematics Exploration Module include supporting PSTs to: • Increase their knowledge of students’ out-of-school activities and practices, including the activities students engage in after school, and students’ perspectives on their own communities (what community locations are familiar to students, etc.). • Engage in students’ communities by visiting community locations and dialoguing with children, families, and community members about their home- and community-based activity. • Increase their knowledge and familiarity with students’ communities, particularly of activities and practices that might relate to mathematics instruction, and in doing so, challenge deficit-based or stereotypical assumptions about students’ communities. PSTs begin to see children as members of communities, and see communities as including home- and family-based activity, as well as broader community relationships, contexts, and activities. • Broaden their perspectives and understandings of students’ competencies (and the competencies of family members and community members) by recognizing ways that students see and use mathematics at home and outside of school. • Plan a problem-solving mathematics lesson or activity that draws upon knowledge and understanding of the practices, activities, and resources of students’ communities. This module consists of three activities: (a) a community walk, (b) the development of a mathematics lesson, and (c) a final write-up and reflection. PSTs typically work in pairs or small groups on the activities in this module. Activity 1: Community walk  PSTs conduct one or more visits (including virtual visits) to the community surrounding their field-placement school or to the community in which students may live. There are various options for these visits (e.g., tour guided by a child and/or family member in a small group with other PSTs),

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depending on the particular circumstances of each methods site. In general, during these visits PSTs look for and document evidence of mathematics, such as people using mathematics, mathematics concepts or principles “in action,” mathematical relationships, quantities, figures, and so forth. We have identified the conversations of PSTs with community members who work, play, visit, and/or make use of these community locations to be a key part of this activity. These visits may also involve posing and investigating a problem related to the community setting (e.g., local parks, businesses, bakeries, restaurants, gathering places, places of worship). Activity 2: Mathematics lesson development  PSTs draw on what they learned from the community visits to inform their instruction. Again, there are several options for how PSTs might think about the implications for instruction of what they learned on these visits and in these interactions, from planning (but not necessarily teaching) a problem-solving lesson related to the community visits, to planning a problem-solving task (or set of tasks) related to the community visits that might form the basis of a mathematics lesson. It is critical that the lesson, task, or set of tasks draw on community contexts, practices, issues, or activities to help students to deepen their understanding of a specific math concept. Activity 3: Final write-up and reflection  PSTs engage in a group discussion/ debriefing of the project (which may include presentations of their lesson plans and/ or problem-posing activities), followed by an individual written reflection (to be completed in class as a quick-write or out of class as a homework assignment). The purpose of this final reflection is to support PSTs in envisioning multiple ways to integrate community knowledge in mathematics instruction and to consider implications of this work for their future teaching.

Classroom Practices Module In this module, PSTs analyze myriad classroom practices through four lenses: 1 . Teaching (e.g., How does the teacher elicit students’ thinking and respond?); 2. Learning (e.g., What specific math understandings and/or confusions are indicated in students’ work, talk, and/or behavior?); 3. Task (e.g., What makes this a good and/or problematic task? How could it be improved?); 4. Power & Participation (e.g., Who participates? Does the classroom culture value and encourage most students to speak, only a few, or only the teacher?). The goals for the Classroom Practices Module include supporting PSTs in: • Identifying and analyzing aspects of mathematics teaching and learning that support eliciting and building on children’s MMKB.

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• Seeing mathematics classrooms in more detail, suspending everyday interpretation, and learning to look for patterns that otherwise go unnoticed at a conscious level. • Identifying and analyzing aspects of mathematics curriculum materials with respect to eliciting and building on children’s MMKB. This module consists of four activities, each of which can be enacted multiple times throughout a mathematics methods course. The four activities are: (a) analyzing video cases, (b) analyzing curriculum spaces, (c) analyzing lesson plans, and (d) analyzing an observed lesson or one’s own teaching practice. These activities can be completed in any order (though we typically start with Activity 1, analyzing video cases) and are designed to be repeated multiple times over the term to support PSTs’ learning. The module supports PSTs’ learning in part by introducing PSTs to two tools: Video Lenses and the Curriculum Spaces Analysis Table. The Video Lenses is used in Activities 1 and 4 to analyze enacted instruction (on video or in person), and the Curriculum Spaces Analysis Table is used in Activities 2 and 3 to analyze written representations of teaching (in textbooks or in lesson plans). Both tools focus PSTs’ attention on tasks, teaching, learning, and power and participation, as well as children’s MMKB. Activity 1: Analyzing video cases  PSTs analyze video cases using a quadruple “lens” approach (e.g., task, learning, teaching, power & participation) to analyze each case. This activity takes place over time such that PSTs analyze multiple videos using the various lenses in a rotating pattern. This activity supports PSTs to examine children’s mathematical ideas (e.g., concepts, skills, problem-solving strategies); knowledge resources (e.g., mathematical, family, community, cultural, linguistic, personal); participation and status issues; and instructional strategies that facilitate mathematical thinking and reasoning of students with varied cultural and linguistic backgrounds, math experiences, and confidences. Activity 2: Analyzing curriculum spaces  Teachers often use a particular mathematics curriculum series but still want to build on and connect to children’s MMKB. In this activity, PSTs use the Curriculum Spaces Analysis Table to analyze and adapt one or more lessons from commonly used curriculum materials to open “spaces” for eliciting, building on, and integrating children’s MMKB (Drake et al., 2015). Spaces PSTs consider are, for example, those which provide opportunities for students to make connections to lived experiences or prior knowledge, make sense of mathematics, develop their own solution strategies, or share and discuss their mathematical thinking. Activity 3: Analyzing mathematics lesson plans  PSTs use the Curriculum Spaces Analysis Table to evaluate a lesson plan they have written for its potential to open spaces for eliciting and building on children’s MMKB. The analysis is intended to inform instructional planning and lesson development.

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Activity 4: Analyzing mathematics lessons  PSTs use the Video Lenses to analyze enacted teaching from the aforementioned four perspectives: teaching, learning, task, and power & participation. This teaching could be another teacher’s practice observed by PSTs or video recordings of the PST’s own teaching.

Using the Modules in Your Practice These modules were designed to support preK-8 PSTs in learning to integrate children’s mathematical thinking and their cultural, linguistic, and community-based “funds of knowledge” (Gonzalez, Andrade, Civil, & Moll, 2005) in mathematics instruction. While each of the three modules is composed of multiple activities, the modules, or even the individual activities, can be used alone or in conjunction with activities in which you already engage with your PSTs. If, for example, you already have an assignment where your PSTs engage in problem-solving interviews with students, you could begin to use our materials by having them do the Getting to Know You Activity or the Community Walk Activity. These activities serve as nice entry points to learning more about a student, a student’s family, or a student’s community. Information gained during the interview or community walk can be useful in personalizing the problem-solving interview. In any case, whether you’re ready to dip your toe or dive right in, we hope you will find our materials can support you in your efforts.

References Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Drake, C., Land, T., Bartell, T. G., Aguirre, J. M., Foote, M. Q., Roth McDuffie, A., et al. (2015). Three strategies for opening curriculum spaces: Building on children’s multiple mathematical knowledge bases while using curriculum materials. Teaching Children Mathematics, 21(6), 346–352. Empson, S. B., Junk, D., Dominguez, H., & Turner, E. E. (2006). Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children’s thinking. Educational Studies in Mathematics, 63(1), 1–28. Gonzalez, N., Andrade, R., Civil, M., & Moll, L. (2005). Funds of distributed knowledge. In N. Gonzalez, L. Moll, & C. Amanti (Eds.), Funds of knowledge: Theorizing practices in households, communities and classrooms (pp. 257–274). New York: Routledge. Nieto, S. (2004). Affirming diversity (4th ed.). Boston: Pearson.

Chapter 3

Preparing to Use the Teachers Empowered to Advance Change in Mathematics Modules: Considerations for Mathematics Teacher Educators Tonya Gau Bartell and Julia M. Aguirre

Keywords  Multiple mathematical knowledge bases · Teacher educators · Mathematics education · Teacher education · Facilitation This chapter draws on our experience as co-PIs on the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project, highlighting various decisions for mathematics teacher educators (MTEs) as they prepare to use the TEACH Math modules in their courses (see Chap. 2 in this volume for module descriptions). We aim to support MTEs in making informed decisions to facilitate module implementation, adaptation, and integration in their own contexts. Specifically, we address the following topics: (a) getting started, (b) adaptations to my teaching context, and (c) common tensions.

 etting Started with the Modules in My Mathematics G Methods Course It is expected that most MTEs have already inherited or developed a functioning elementary mathematics methods course. Most likely these courses are informed by the predominant theories in the field (see Chap. 1). In particular, existing courses are likely to focus on developing PSTs’ mathematics content knowledge and pedagogical content knowledge with an intention of introducing PSTs to practices related to eliciting, drawing upon, and extending children’s mathematical thinking in a variety T. G. Bartell (*) Michigan State University, East Lansing, MI, USA e-mail: [email protected] J. M. Aguirre University of Washington-Tacoma, Tacoma, WA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_3

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of contexts (e.g., Cognitively Guided Instruction (CGI), Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Jacobs & Ambrose, 2008). If equity dimensions are included in mathematics methods, they usually appear in an isolated session in the middle or near the end of the course. This “math first” or “math only” approach limits attention given to mathematics instruction that is responsive to children’s cultural and communitybased knowledge and lived experiences (e.g., Civil & Khan, 2001; Gonzalez, Andrade, Civil, & Moll, 2005). The use of TEACH Math modules with existing course work aims to support the integration of a focus on meaningful mathematics, mathematical thinking, and students’ experiences and knowledge into PSTs’ development of mathematics teaching. We have learned that integration starts with the first class session and is intentional throughout the duration of a course. The modules are designed to honor this integrative focus and equity-based approach to mathematics teacher education.

Familiarize Yourself with the Modules It is helpful for MTEs to begin by familiarizing themselves with the modules, identifying the goals of each module and what the module might provide for PSTs’ learning and practice. We would encourage MTEs to also critically examine their current course goals to identify existing commonalities as well as gaps that may need further development. Reading the module descriptions in Chap. 2 of this volume and visiting the TEACH Math website (https://teachmath.info) are helpful places to begin learning about the modules. For example, in examining the Community Mathematics Exploration Module, MTEs may notice that one goal for PSTs is to increase knowledge and familiarity with students’ communities, as well as out-of-school activities and practices that might be incorporated in mathematics instruction, in part to challenge deficit-based or stereotypical assumptions about mathematics, students, or their communities. Perhaps this is already an explicit goal of a MTE’s course that they would like to further develop, or perhaps this is something they feel their course needs to incorporate in order to further support PSTs’ development. Or MTEs may notice that sometimes we ask PSTs to write a mathematics or cultural autobiography as a preliminary activity to heighten PSTs’ awareness of their own positionality with regard to mathematics and to the cultural experiences they bring to the classroom. As can be seen in later chapters of this book, many MTEs did, indeed, use the mathematics autobiography in their contexts in this way. We also recommend that MTEs read some of the readings they would assign their students (https://teachmath.info/additional-resources/readings/methods) as well as some of the readings aimed to support MTE knowledge development (https://teachmath.info/additional-resources/readings/teacher-education). For the latter, a starting point might be Erin Turner and Corey Drake’s (2015) Review of Research on Prospective Teachers’ Learning About Children’s Mathematical Thinking and Cultural Funds of Knowledge, which explicitly considers research related to how PSTs learn to connect to both children’s mathematical thinking and children’s funds of knowledge in mathematics instruction. Further, we encourage

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MTEs to engage in some of the activities associated with the modules. What might MTEs learn or notice using the four lenses with a video clip they traditionally show in their mathematics methods courses (Classroom Practices Module)? What might MTEs learn if they talked to children about their interests and activities prior to giving a problem-solving interview (Mathematics Learning Case Study Module)? What mathematical practices might MTEs identify on a community walk (Community Mathematics Exploration Module)? How might MTEs adapt a high cognitive demand task that engages students and connects to students’ prior experiences and out-of-school mathematics knowledge (Mathematics Learning Case Study Module and Community Mathematics Exploration Module)?

Consider Connections to Big Ideas in Mathematics Education Another helpful place to begin consideration of incorporating modules into an existing course is the identification of connections between the modules and big ideas in mathematics education. The goal in doing so is to help MTEs understand how the modules align with and potentially support achievement of other goals that guide their course. Table 3.1 provides a few examples of possible areas for connections, and we elaborate on two examples to illustrate what this process might entail. Table 3.1  Possible module connections to big ideas in mathematics education Big idea in mathematics education Cognitively Guided Instruction (e.g., Carpenter, Fennema, Franke, Levi, & Empson, 2015) Noticing/Video Analysis (e.g., Santagata & Guarino, 2011; Sherin, Jacobs, & Philipp, 2011) Teaching Mathematics for Social Justice (e.g., Aguirre & Civil, 2016; Wager & Stinson, 2012) Lesson Study (e.g., Hart, Alston, & Murata, 2011; Parks, 2008) Complex Instruction (e.g., Featherstone et al., 2011) Identity in Mathematics Education (e.g., Aguirre, Mayfield-Ingram, & Martin, 2013; Martin, 2000) Selection/Adaptation of Tasks (e.g., cognitive demand, Smith & Stein, 1998) Practice-Based Connections/Three Act Tasks (e.g., routines, Lampert et al., 2013; Lomax, Alfonzo, Dietz, Kleyman, & Kazemi, 2017; Van de Walle, Bay-Williams, Lovin, & Karp, 2014)

TEACH Math Module(s) Mathematics Learning Case Study Module Classroom Practices Module

Community Exploration Module, Mathematics Learning Case Study Module Classroom Practices Module Classroom Practices Module, Mathematics Learning Case Study Module Community Exploration Module, Classroom Practices Module Classroom Practices Module, Community Exploration Module Classroom Practices Module

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Perhaps one connection one might make is to ideas from Cognitively Guided Instruction (Carpenter et al., 2015). Many MTEs likely ask PSTs to conduct clinical interviews to support their eliciting and analysis of children’s mathematical thinking. The Mathematics Learning Case Study Module also includes such an interview, with explicit use of CGI problem types, and also requires PSTs to synthesize and make connections between the knowledge gained in the clinical interview with knowledge gained from a “Getting to Know You” interview with a child and/or from observation of the knowledge, skills, and competencies the child demonstrates in different contexts (e.g., other school settings, after school, home, community). A first step to using the TEACH Math modules within an existing course framework, then, may be to connect to existing course activities already connected to CGI, then expanding those to consider children’s multiple mathematical knowledge bases (MMKB; Turner, Drake, Roth McDuffie, Aguirre, Bartell, & Foote, 2012). In addition, many MTEs also use video of elementary mathematics teaching as a tool to support PSTs’ noticing of instructional strategies or student learning (e.g., Santagata & Guarino, 2011; Sherin, Jacobs, & Philipp, 2011). One way to integrate a focus on children’s MMKB might be to introduce our four-lens (learning, teaching, task, and power and participation) approach for viewing these videos (Roth McDuffie, Foote, Bolson, et al., 2014; Roth McDuffie, Foote, Drake, et al., 2014). To support PSTs’ learning, the Video Analysis Activity is designed as a decomposition and approximation of the practice of noticing (Grossman, Hammerness, & McDonald, 2009). We have found that the prompts and activity structure of the Video Analysis Activity supported PSTs in learning to notice across multiple foci and to notice at deeper levels (Roth McDuffie, Foote, Bolson et al., 2014). In this way, video analysis in elementary mathematics methods courses integrates a focus on equitable instructional practices in ways that do not overly complicate or overwhelm PSTs’ learning. Rather, PSTs are guided to notice a range of student resources and in analyzing how and why those resources support children’s learning of mathematics.

Adaptation to Context As with any teaching, it is likely that MTEs will need to adapt the modules to fit their context. For example, some MTEs have 10-week or 14-week semesters, while others have 6- or 8-week quarters. Some courses may meet for 90 minutes twice a week, while others have a weekly session of 2.5 hours. In addition, some MTEs teach primarily White, middle-class, English monolingual, female undergraduate students in their 20s, while others teach in a Master’s level program with second-­ career students generally older and with different life experiences, while still others may have PSTs teaching in and coming from primarily multilingual contexts or military-base contexts. Given that each of the six TEACH Math co-PIs taught in different contexts, here we share insights about adaptation learned from our

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own engagement in the process. Specifically, we consider how the modules might be adapted for different timelines as well as for specific PST populations and/or local contexts.

Timeline Adaptations One way to think about how to adapt the TEACH Math modules for a given MTE’s context is to consider which activities might be “entry points” into the work. We see this as similar to identifying connections to big ideas in mathematics education as outlined above, aligned with the MTE’s specific course goals. Another way to think about adaptation is with recognition that some things will “fit” and others will be omitted; what does one prioritize when thinking of adaptations? This, too, depends on an MTE’s particular course goals and needs. We urge MTEs to look for ways to enhance existing assignments and activities with activities from the modules. Here, we tackle the adaptation question by providing sample timelines in Appendix A for each individual module as well as for the integration of modules into a methods course. Appendix A includes sample timelines for each individual module’s incorporation into both a 10-week and 15-week course as well as three examples of incorporating all three modules into courses of different durations (2 quarters, 15 weeks, and 18 weeks). We hope that these sample timelines will help MTEs from a myriad of contexts consider ways to adapt the modules to fit diverse schedule and goals.

Adaptations for Specific Local Contexts As might be expected given national demographics in US teacher preparation programs (US Department of Education, 2017), most of us worked with PST populations of primarily White, female, undergraduate students in their 20s. There were other ways, though, that our contexts were unique and, as such, warranted adaptations to the modules. For example, one local context was a large city. In this case the majority of PSTs were either immigrants themselves or were the children of immigrants. Because of this, in preparation for the Community Exploration Module and the “Getting to Know You” interview in the Mathematics Learning Case Study Module, less time was devoted to developing understanding among PSTs of the ways in which schools often ignore the funds of knowledge brought to school by nondominant children; these PSTs had lived this experience. Another local context included a master’s level credential program at an institution serving multiple military bases. The Community Mathematics Exploration Module invited PSTs working in elementary schools that served the military communities to learn more about family activities on and off base.

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In these ways, we have begun to adapt the modules in consideration of our local contexts. Other chapters in this book highlight further ways local contexts informed module adaptation and implementation.

What Tensions Might MTEs Expect? As MTEs make decisions regarding how to integrate the modules with respect to their existing course goals and local contexts, various tensions may arise. We hope that sharing some of the tensions we have faced will help other MTEs both anticipate and navigate those tensions in their own work.

Some Things Have to Go A question we often asked ourselves, and which has been asked of us when we present our work, is how to decide what to cut if one cannot “fit” all aspects of the three modules into their mathematics methods course. What should be prioritized? What can be left out? There is no one, right answer with respect to this tension. Rather, we found that we made different decisions about what to foreground or what to omit based on our knowledge of our PST population, the local school contexts, and our goals for PSTs. For example, in one case the Community Exploration Module was moved to a second methods course—as part of a two-semester methods sequence in a year-long post-baccalaureate credential program—so that it would coincide with (a) the beginning of the school year efforts to get to know students, families, and communities and (b) an intensive internship experience where PSTs spent 4 days a week in the schools. In another case, in order to afford space for at least two rounds of whole class analysis of a shared lesson using the Curriculum Spaces Analysis Table, the instructor engaged in fewer Video Lens activities in class and instead assigned them as homework. To date, our research does not suggest a particular module order and has not examined the implications of different variations of what is included or removed from the course; we hope MTEs will begin to learn about this in their work in ways that can be shared with the field, thus continuing some of the work of this book.

Entry Points versus Depth As we promote simultaneous attention to children’s mathematical thinking and children’s’ cultural and community funds of knowledge, we encounter various degrees of openness to this dual focus from PSTs. It is important to acknowledge that PSTs in methods courses will be wrestling with deep feelings about mathematics learning and teaching from their own schooling experiences. Some will need

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more mathematical healing from previous negative experiences, while others will be wrestling with moving beyond their comfort zone of one correct way to solve math problems. MTEs should expect to be met with PST curiosity, fear, skepticism, and/or excitement. Anticipating this range, we try to meet PSTs where they are in attention to mathematics, children’s mathematical thinking, and cultural and community-based forms of knowledge. We recognize tensions in trying to meet the multiple, varied needs of PSTs. The TEACH Math modules afford multiple entry points for PSTs to begin their professional learning as teachers of mathematics. For example, the Case Study Module enables PSTs to get a more holistic portrait of one child’s mathematical, relational, and cultural strengths. This module explicitly asks PSTs to identify the child’s interests and positive family/peer relations, look for dispositional strengths inside and outside the classroom, and carefully analyze children’s mathematical strategies. This affords opportunities to create more holistic narratives of the child’s multiple mathematical strengths and growth areas as a foundation to build on, which in many cases may challenge cultural stereotypes held by PSTs. To support depth, this is also an opportunity for MTEs to provide PSTs with feedback on how to reframe one-dimensional portraits of students fueled by static labels. The Community Mathematics Exploration Module offers PSTs opportunities to get to know different communities served by a school. Tensions may surface as PSTs express uncertainty about visiting “certain neighborhoods” or question the relevance of the Community Mathematics Exploration because their students are “white and middle class.” Tensions may also arise from the math tasks and lessons designed by PSTs that may, to varying degrees, honor children’s mathematical thinking and community-based funds of knowledge in authentic and genuine ways. These tensions may manifest in somewhat superficial engagement with communities. We have found that requiring PSTs to talk with community members during engagement, asking them to re-visit sites, and linking what they learned and noticed about mathematical practices to a mathematics lesson they design are opportunities to support depth. Ultimately, our work with the modules has found that PSTs do need explicit prompts and multiple opportunities to notice and integrate children’s mathematical thinking with cultural and community knowledge into their developing practices. Our work also suggests that the modules may surface opportunities for MTEs to provide strategic supports to PSTs. We may not always fully meet the needs of PSTs with these modules, but as MTEs we acknowledge and fully commit to the hard and necessary work it takes to try.

Deficit Perspectives and Resistance One common experience we have all faced in implementing the modules is addressing the prevailing deficit language that inevitably emerged from some PSTs. The TEACH Math modules are specifically designed to highlight student

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mathematical thinking and family and community knowledge and experiences. This dual focus often challenges PSTs’ initial views of learning mathematics as well of some students, families, and communities. For instance, in the Case Study Module, PSTs are encouraged to find out more about a student, their likes/dislikes, favorite activities, and people in their lives. PSTs are also asked to conduct a clinical math interview with word problems with a student. If PSTs’ use deficit language in their written reflections on these experiences, it is helpful for MTEs to reframe that language by pointing out alternative evidence. For example, if a PST is focused on what a student does not seem to know or be able to do, the MTE can point out evidence of what the student does know and is able to do, perhaps noting a student-invented problem-solving strategy or the strengths of a specific representation used by the student. MTEs can also ask questions about deficit language, offering reframing language as a way to see what is problematic about a statement as well as provide a positive pathway for the PST to professionally grow. For example, if a PST notes that a child’s guardians did not come to student-teacher conferences and thus concludes that the guardians don’t seem to care much about their child’s education, the MTE might ask question such as, “What might be reasons for guardians to miss student conferences?” or “What evidence do you have that the guardians do or do not care about their child’s education?” Depending on one’s relationship with the PST using deficit language, an MTE might also cite relevant research. In this case, the MTE may note that research details how assumptions about appropriate time for student-teacher conferences (e.g., in the evening, presuming everyone works a 9:00 am to 5:00 pm job) reflect middle-class norms or that other research documents that guardians show care about their child’s education in a myriad of ways, such as helping with homework, making sure their child has good attendance, or believing that if something is not going well, a teacher will contact guardians so as to best support the child. It must be understood that prevailing deficit language is systemically supported by our current education system. Thus, MTEs who use these modules should have a strong commitment to eliminate the use of such language. MTEs must also prepare for a variety of resistance from PSTs when they are asked to engage with students, families, and communities as resources for mathematics learning and teaching. While many PSTs express a willingness to make mathematics more relevant and meaningful for/to students, they may also feel vulnerable because of their own lack of mathematical confidence or a prevailing curricular focus on computational fluency and application problems that are often culturally and linguistically biased. Notions that children lack background knowledge and lived experiences to problem solve must be vigorously challenged. The Community Mathematics Exploration Module, for example, is designed to facilitate a new opportunity for PSTs to identify math practices that occur outside of school and then provide instructional design opportunities to develop a math activity that would leverage those practices. Our research has shown that PSTs can design lessons that connect community funds of knowledge and children’s mathematical thinking in a variety of ways (Aguirre, Turner, et al., 2013; Turner et al., 2014).

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Positionality, Power, and Privilege Dynamics It is important to recognize that mathematics education is a racialized, gendered, and classed experience. In electing to use the TEACH Math modules, you are opening up opportunities to critically examine your own and your PSTs’ positionality in learning to teach mathematics—cultivating a positive mathematics learning space and experience for children, particularly children from historically marginalized communities. The gender, race, language, and class identities (among others) impacting the choices being made by MTE and PSTs’ responses to module activities need frequent reflection. Power dynamics are at play and are different given one’s positionality. For example, what might it be like for an MTE from a minoritized racial group (Latinx, Black, Indigenous, or Asian background) to work with primarily white, middle class, female PSTs to implement these modules? What criticisms from PSTs might she encounter that a white male or white female MTE may not face when working with a similar population? What about PSTs of color and their experiences with these modules? How are we as MTEs shaping conversations during these module activities that acknowledge our positionality and ask PSTs to do the same? It is important for MTEs to reflect on the power dynamics involved and what additional professional learnings related to power, privilege, and oppression are needed to implement these modules effectively (Aguirre et al., 2017). While our own research with these modules has not explicitly analyzed positionality, we have had various experiences that call into question our positionality as authority figures and racialized and gendered beings. We urge you to think critically about how your own positionality, including the roles of power, privilege, and oppression, may play out in the choices and enactments of the modules in your courses.

Concluding Remarks We are excited for you to get started with the TEACH Math modules in your courses. Given our own experiences and research we are confident that the modules build on familiar activities of methods courses and may bring new opportunities to deepen PSTs’ knowledge and practice for mathematics teaching. We recognize that an explicit equity focus that simultaneously attends to mathematics, children’s mathematical thinking, and cultural and community-based funds of knowledge may be ambitious for some and familiar to others; that is okay. We recognize multiple entry points are needed for this work, and we also know that with explicit frequent and structured support for this integrated focus, PSTs can and do create mathematical learning experiences that leverage children’s MMKB. By understanding the preparation, connections, and tensions related to enacting these modules, we are hopeful that MTEs will gain a deeper joy and understanding of equity-based mathematics teacher preparation.

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 ppendix A: Timelines for Teachers Empowered to Advance A Change in Mathematics Module Adaptations in Methods Courses Mathematics Learning Case Study Module Week 10-Week Course 15-Week Course 2 Introduce Case Study Module; prepare for Getting to Know You Interview and Shadowing a Student Introduce Case Study Module; prepare for Getting to 3 Prepare for Problem-Solving Know You Interview and Shadowing a Student Interview: learning about children’s strategies for addition and subtraction Share experiences in class, and submit report for 4 Share experiences in class, and submit report for Getting to Know Getting to Know You Interview and Shadowing a Student;prepare for Problem-Solving Interview: You Interview and Shadowing a learning about children’s strategies for addition and Student;continue learning about subtraction children’s strategies for addition and subtraction Learning about children’s strategies for 5 Learning about children’s multiplication and division strategies for multiplication and division 6 Share experiences in class, and submit report for Problem-Solving Interviews 8 Share and submit Quick Write on Synthesizing and Connecting Across Activities 9 Engage in Mock Parent/Family Conferences, and submit report 10 Share and submit report on Problem-Solving Interview 1 (Addition/Subtraction) 13 Share and submit report on Problem-Solving Interview 2 (Multiplication/Division) 15 Engage in Mock Parent/Family Conferences, and submit report on Synthesis

3  Preparing to Use the Teachers Empowered to Advance Change in Mathematics… Classroom Practices Module Week 10-Week Course 1 Introduce Classroom Practices Module: use Video Analysis Tool with Marshmallow Video 2 Use Video Analysis Tool with second video of your choice w/ focus on power and participation 3 Use Video Analysis Tool with third video of your choice w/ focus on task 4 Preparing for Analyzing Curriculum Spaces: What makes a good math task? Cognitive demand task sort 5 Analyzing Curriculum Spaces: analyze a common lesson together as a class 6 Use Video Analysis Tool with fourth video of your choice w/ focus on learning and teaching 7 Analyzing Curriculum Spaces: read Three Strategies for Opening Spaces TCM article 8 Analyzing Curriculum Spaces: analyze your lesson plan draft using Curriculum Spaces Analysis Table, and consider ways to “open spaces” 9 10

11 12 13

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15-Week Course Introduce Classroom Practices Module: use Video Analysis Tool with Marshmallow Video Use Video Analysis Tool with second video of your choice w/focus on power and participation

Preparing for Analyzing Curriculum Spaces: What makes a good math task? Cognitive demand task sort Analyzing Curriculum Spaces: analyze a common lesson together as a class Use Video Analysis Tool with third video of your choice w/focus on task Use Video Analysis Tool with fourth video of your choice w/focus on learning and teaching

Analyzing Curriculum Spaces: analyze another common lesson together as a class Analyzing Curriculum Spaces: analyze your lesson plan draft using Curriculum Spaces Analysis Table, and consider ways to “open spaces” Consider Video Analysis Tool for lesson study data collection and reflection Analyzing Curriculum Spaces: analyze another lesson draft using Curriculum Spaces Analysis Table

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Community Mathematics Exploration Module Week 10-Week Course 15-Week Course 2 Class readings related to Reading: Posing Problems that Matter; introduce Culturally Relevant Pedagogy Case Study of a Child project and Community Exploration Project 3 Readings related to Funds of Knowledge; Community Exploration Project: examination of lesson plan examples 4 PSTs engage in community exploration in local contexts 5 Class readings related to Teaching PSTs engage in community exploration in local contexts Mathematics for Social Justice/Posing Problems that Matter 6 Introduce Community Exploration PSTs engage in community exploration in local contexts; discussion of Community Exploration: Project: examine sample lesson What are you noticing? Brainstorm mathematical plans, brainstorm ideas with connections partner; PSTs engage in community exploration in local contexts Reading: Is Math Political? 7 Discussion of Community Exploration: What are you noticing? What questions do you have? PSTs engage in community exploration in local contexts 8 Community Exploration Project: revisiting sample lesson plans; drafting your own tasks; present tasks to peers for feedback Community Exploration Project: revisiting sample 9 Community Exploration lesson plans; drafting your own tasks; present tasks presentations and/or peer to peers for feedback feedback of draft lesson plans 10 11 Community Exploration presentations

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All Three Modules 19-Week Course Meeting Twice a Week for 1 Hour Each Meeting Week Course Topic Module Activities 1 Cohort Community Building Introduce Math Autobiography Assignment Share Math Autobiographies 2 Introduction to Teaching and Learning Mathematics for Understanding Watch Marshmallow Video with four square; 3 Teaching through Problem preparing for Getting to Know You Interview; select Solving in Diverse Classrooms: Case Study Student, and conduct Getting to Know Part I Focus on Language You Interview as homework Debrief Getting to Know You Interviews in class; 4 Teaching through Problem prepare for Problem-Solving Interview 1 in class; Solving in Diverse Classrooms: continue to shadow/observe Case Study Student Part II Focus on Status and Competencies Introduction to Assessing Children’s Mathematical Thinking Write up of Getting to Know You Interview Due; 5 Children’s Thinking about connect to Problem-Solving Interview 1 (addition/ Addition and Subtraction subtraction) in class; prepare for Problem-Solving Preparing for the Addition/ Interview 2 (multiplication/division); continue to Subtraction Problem-Solving shadow/observe Case Study Student Interview 6 Children’s Thinking about Summary of addition/subtraction interview Due Multiplication and Division (bring to class); debrief Problem-Solving Interview 1 in class; connect to Problem-Solving Interview 2; watch Amazing Equations with Task and Learning lenses only Debrief Problem-Solving Interview 2 in class; 7 Children’s Thinking about Base discuss requirements for final reflection; continue to Ten Concepts shadow/observe Case Study Student Planning Literature-Based Math Lessons 8 Children’s Thinking about Summary of Problem-Solving Interview 2 Due; in Multi-Digit Operations class work on literature-based lesson planning; watch Valentine’s Exchange video 9 No Class Meeting Teach literature-based math lesson PLC time: Community Math Project Field Trips; 10 Connecting Mathematics Field time: Informal conversations with students Teaching to Home and about community Community I (Cultural and Community Connections) 11 No Class Meeting (Spring Break) Continued Community Math Project Field Trips PLC time: Community Math Project Field Trips; 12 Connecting Mathematics Lesson planning based on Community Math Projects Teaching to Home and Community II (Connections to Equity and Social Justice) PLC time: Community Math Project Presentations; 13 Connecting Mathematics reflection on Community Math Project due end of Teaching to Home and week Community III: Presentations; Planning with Curriculum Materials

36 14 15

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T. G. Bartell and J. M. Aguirre Teaching and Learning Fractions Lesson Workshop and Equity Revisited (revise lesson plan for whole group math lesson with equity in mind) Teaching and Learning Geometry

17

Teaching and Learning Algebraic Reasoning

18 19

Whole Group Lesson Analysis Whole Group Lesson Analysis

Curriculum Materials Analysis: Fractions Critical analysis of own lesson plan with Curriculum Spaces Analysis Table

Curriculum Materials Analysis: Geometry Video own teaching; begin equity analysis Video lens activity with Equality video Review video of own teaching and work on equity analysis Student Presentations Student Presentations

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All Three Modules 15-Week Course Meeting Once a Week for 3 Hours Each Meeting Week Topic Module Activities Math Autobiography discussion 1 Course Introduction. Discussion Video Lens Marshmallow Video with 4-square of current issues in mathematics education Introduce Case Study Project; work with mentor 2 NCTM Standards; State teacher over the next week to select case study Standards; CCSSM; Early student; introduce Cognitive Demand Number Sense 3 Meaning for Operations Discuss Kersaint & Chappelle (2001) reading as preparation for Case Study Project Video Lens My Kids Can, Ch12, Get to 100 with 4 Teaching through Problem Solving: Selecting high cognitive Teaching and Task Lenses; CGI Video clips; conduct Getting to Know You Interview with case study demand tasks; anticipating student students’ thinking and methods Discuss Case Study Getting to Know You Interviews; 5 Lesson Planning: The three-part format; planning for diversity and Jigsaw readings on planning and discuss ways to plan for questioning (e.g., Herbel-Eisenmann & equity; questioning; facilitating Schleppegrell, 2008; Jacobs & Ambrose, 2008) discourse; including families specific students’ needs (e.g., Karp & Howell, 2004) as well as to include families in learning (e.g., Mistretta, 2013; Celedón-Pattichis & Ramirez, 2012) 6 Engaging students in learning so Classroom Practices Module: Cognitive demand task sort; jigsaw readings to connect to community that all students achieve; exploration module, critical mathematics, and culturally relevant pedagogy; mathematics for social justice (e.g., Drake et al., collaborative learning; adapting 2015; Featherstone et al., 2011; Gutierrez, 2009; curriculum materials for your Turner & Strawhun, 2007); use group time to begin students’ learning to discuss plans for community exploration 7 Mastering Basic Facts; Using Classroom Practices Module: discuss observation in Technology to Support Learning placement classroom using Teaching OR Task Lenses; Classroom Practices Module: analyze lesson from curriculum using Curriculum Spaces Analysis Table 8 Place Value; Assessing Thinking Preparing for Case Study Math Interview by viewing interview clips through Observing and Interviewing 9 Whole Number Computation and Community Math Exploration Part I—engaging the Estimation community; Video Lenses #3: Buying a Turkey with learning lens 10 More on Diversity and Equity in Case Study Module addition/subtraction interview report; discuss jigsaw articles to connect to topic Mathematics Teaching and Learning: language, culture, and (e.g., Aguirre & Bunch, 2012; Aguirre et al., 2013, Torres-Velasquez & Lobo, 2005; Witzel & Allsop, special needs 2007); prepare for Case Study multiplication/ division interview

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11

Algebraic Reasoning; Rational Number-Fraction Concepts

12

Rational Number—Operating with Fractions, Decimals, and Percent; Proportional Reasoning

13

Assessment; Data Analysis and Probability

14

Measurement Concepts; Exponents, Integers and Real Numbers Geometric Thinking and Concepts; Wrap-Up

15

Classroom Practices Module: discuss second observation of placement classroom using Power and Participation OR Learning Lens; debrief Community Math Exploration Part I; share ideas for Community Math Lessons; Video Lens #4: Equalities with Power & Participation Lens; work with your group on the Math in the Community Project Case Study Module: Multiplication/Division interview report; Curriculum Analysis with Curriculum Spaces Analysis Table on Fractions Lessons (other aspects of this module played out in course assignments that are not listed in this timeline); work with your group on the Math in the Community Project Math in the Community project presentations; Video Lens #5: Questioning Data (was assigned as homework) Math in the Community project presentations; debrief Community Exploration projects Mock Parent Conferences for Case Study Project

References Aguirre, J. M., & Bunch, G. C. (2012). What’s language got to do with it? Identifying language demands in mathematics instruction for English Language Learners. In S. Celedon-Pattichis & N.  Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs (pp. 183–194). Reston, VA: National Council of Teachers of Mathematics. Aguirre, J.  M., & Civil, M. (Editors) (2016). Mathematics through the lens of social justice: Special Issue. Teaching for Excellence and Equity in Mathematics, 7(1). Aguirre, J.  M., Herbel-Eisenmann, B., Celedón-Pattichis, S., Civil, M., Wilkerson, T., Stephan, M., et  al. (2017). Equity within mathematics education research as a political act: Moving from choice to intentional collective professional responsibility. Journal for Research in Mathematics Education, 48(2), 124–147. Aguirre, J. M., Mayfield-Ingram, K., & Martin, D. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. Reston, VA: National Council of Teachers of Mathematics. Aguirre, J. M., Turner, E. E., Bartell, T. G., Kalinec-Craig, C., Foote, M. Q., Roth McDuffie, A., et al. (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. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann. Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531.

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Celedón-Pattichis, S., & Ramirez, N. G. (2012). Beyond good teaching: Advancing mathematics education for ELLs. Reston, VA: National Council of Teachers of Mathematics. Civil, M., & Khan, L.  H. (2001). Mathematics instruction developed from a garden theme. Teaching Children Mathematics, 7(7), 400–405. Drake, C., Land, T. J., Bartell, T. G., Aguirre, J. M., Foote, M. Q., Roth McDuffie, A., et al. (2015). Three strategies for opening curriculum spaces. Teaching Children Mathematics, 21(6), 346–353. Featherstone, H., Crespo, S., Jilk, L., Oslund, J., Parks, A.  N., & Wood, M. (2011). Smarter together! Collaboration and equity in the elementary math classroom. Reston, VA: National Council of Teachers of Mathematics. Gonzalez, N., Adrade, R., Civil, M., & Moll, L. (2005). Funds of distributed knowledge. In N. Gonzalez, L. Moll, & C. Amanti (Eds.), Funds of knowledge: Theorizing practices in households, communities and classrooms (pp. 257–274). New York, NY: Routledge. Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education. Teachers and Teaching, 15(2), 273–289. Gutierrez, R. (2009). Embracing the inherent tensions in teaching mathematics from an equity stance. Science and Math: Equity, Access, and Democracy, 18(3), 9–16. Hart, L. C., Alston, A. S., & Murata, A. (2011). Lesson Study Research and Practice in Mathematics Education. Netherlands: Springer. Herbel-Eisenmann, B., & Schleppegrell, M. (2008). “What question would I be asking myself in my head?”: Helping all students reason mathematically. In M.  Ellis (Ed.), Mathematics for every student: Responding to diversity, grades 6–8 (pp. 23–37). Reston, VA: National Council of Teachers of Mathematics. Jacobs, V., & Ambrose, R. (2008). Making the most of story problems. Teaching Children Mathematics, 15(5), 260–266. Karp, K., & Howell, P. (2004). Building responsibility for learning in students with special needs. Teaching Children Mathematics, 11(3), 118–126. Kersaint, G., & Chappell, M. F. (2001). Capturing students’ interests: A quest to discover math ematics potential. Teaching Children Mathematics, 7(9), 512–517. Lampert, M., Franke, M., Kazemi, E., Ghousseini, H., Turrou, A. C., Beasley, H., et al. (2013). Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching in elementary mathematics. Journal of Teacher Education, 64(3), 226–243. Lomax, K., Alfonzo, K., Dietz, S., Kleyman, E., & Kazemi, E. (2017). Trying three-act tasks with primary students. Teaching Children Mathematics, 24(2), 113–119. Martin, D.  B. (2000). Mathematics success and failure among African-American youth: The roles of sociohistorical context, community forces, school influence, and individual agency. Mahwah, NJ: Lawrence Erlbaum Associates. Mistretta, R.  M. (2013). “We do care,” say parents. Teaching Children Mathematics, 19(9), 572–580. Parks, A. N. (2008). Messy learning: Preservice teachers’ lesson-study conversations about mathematics and students. Teaching and Teacher Education, 24, 1200–1216. Roth McDuffie, A., Foote, M. Q., Bolson, C., Turner, E. E., Aguirre, J., Bartell, T. G., et al. (2014). Using video analysis to support prospective K-8 teachers’ noticing of students’ multiple mathematical knowledge bases. Journal of Mathematics Teacher Education, 17(3), 245–258. Roth McDuffie, A., Foote, M. Q., Drake, C., Turner, E. E., Aguirre, J., Bartell, T. G., et al. (2014). Mathematics teacher educators’ use of video analysis to support prospective K-8 teachers’ noticing. Mathematics Teacher Educator, 2(2), 108–140. Santagata, R., & Guarino, J. (2011). Using video to teach future teachers to learn from teaching. ZDM, 43(1), 133–145. Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011). Mathematics Teacher Noticing. New York, NY: Routledge. Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3, 344–350.

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Torres-Velasquez, D., & Lobo, G. (2005). Culturally responsive mathematics teaching and English language learners. Teaching Children Mathematics, 11(5), 249–255. Turner, E. E., Drake, C., Roth McDuffie, A., Aguirre, J. M., Bartell, T. G., & Foote, M. Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematical knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67–82. Turner, E., Aguirre, J., Bartell, T., Drake, C., Foote, M., & Roth McDuffie, A. (2014). Making meaningful connections with mathematics and the community: Lessons from prospective teachers. TODOS Research Monograph, 3, 60–100. Turner, E. E., & Drake, C. (2015). A review of research on prospective teachers’ learning about children’s mathematical thinking and cultural funds of knowledge. Journal of Teacher Education, 67(1), 32–46. Turner, E. E., & Font Strawhun, B. T. (2007). Posing problems that matter: Investigating school overcrowding. Teaching Children Mathematics, 13(9), 457–463. U.S.  Department of Education. Institute of Education Sciences, National Center for Education Statistics. (2017). Characteristics of public elementary and secondary school teachers in the United States. Results from the 2015–2016 national teacher and principal survey. First look. Retrieved from https://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2017072rev Van de Walle, J. A., Bay-Williams, J. M., Lovin, L. H., & Karp, K. S. (2014). Teaching student-­ centered mathematics: Developmental instruction for grades 3–5 Boston, MA: Pearson. Wager, A. A., & Stinson, D. W. (2012). Teaching mathematics for social justice: Conversations with educators. Reston, VA: National Council of Teachers of Mathematics. Witzel, B.  S., & Allsop, D. (2007). Dynamic concrete instruction in an inclusive classroom. Mathematics Teaching in the Middle School, 13(4), 244–248.

Part II

Community Mathematics Exploration Module

Chapter 4

Crafting Entry Points for Learning about Children’s Funds of Knowledge: Scaffolding the Community Mathematics Exploration Module for Pre-Service Teachers Crystal Kalinec-Craig and Maria del Rosario Zavala

Keywords  Teacher education · Mathematics education · Funds of knowledge · Community exploration · Elementary teacher education Jim Cummins (1996), language development and bilingual education scholar, once noted: “Our prior experience provides the foundation for interpreting new information. No learner is a blank slate” (p.  75). Cummins’ words echo Freire’s (2000) belief that rejects the assumption that students are passive recipients of knowledge. Mathematics classrooms can also be spaces that reject this banking model of education: When mathematics teachers shift to seeing their students’ knowledge and experiences as assets to the practice of teaching mathematics, then their classrooms transform into places that emphasize equity and more participation by all students (Langer-Osuna & Nasir, 2016). When teachers foreground what their students already know, they create a familiar starting point for students’ mathematics learning in school (Turner et al., 2012). Gonzalez, Moll, and Amanti’s (2005) concept, “funds of knowledge,” helps frame the notion that students bring a wealth of knowledge and experiences from their homes and communities. Often, however, the connections between this knowledge and school mathematics is not immediately apparent to teachers. But to what extent are mathematics teachers, especially those learning to be mathematics teachers, prepared to recognize and foreground the experiences and knowledge of their students, students’ families, and students’ communities in their practice? What approaches can mathematics teacher educators C. Kalinec-Craig (*) University of Texas at San Antonio, San Antonio, TX, USA e-mail: [email protected] M. del Rosario Zavala San Francisco State University, San Francisco, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_4

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(MTEs) take within their mathematics methods courses to prepare pre-service teachers (PSTs) for effectively transforming their mathematics teaching in ways that connect to and build upon children’s knowledge and experiences? The purpose of this chapter is to share the ways that we, two MTEs, support PSTs as they learn to elicit and then incorporate children’s funds of knowledge in their mathematics instruction. We situate this work within the existing literature about children’s funds of knowledge and the broader notion of children’s multiple mathematical knowledge bases. Children’s multiple mathematics knowledge bases are defined as the “understandings and experiences that have the potential to shape and support children’s mathematics learning  – including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge” (Turner et al., 2012, p. 67). Specifically, we consider how we create scaffolds in our coursework for PSTs to think about children’s funds of knowledge in the context of the Community Mathematics Exploration (CME) Module. In addition, we discuss challenges that MTEs might face when leveraging the CME Module as an entry point for learning to teach mathematics for social justice and for supporting students’ positive mathematical identities (Aguirre, Mayfield-Ingram, & Martin, 2013). We chose to foreground our personal experiences with the CME to offer some collective case study knowledge for other MTEs. We begin our chapter sharing our positionality as it relates our work with the CME Module.

Author Positionality We believe it is important to start our work as MTEs by acknowledging our unique personal histories. Other colleagues in this volume have written about the role of mathematics autobiographies in helping PSTs be grounded in their personal stories as learners of mathematics (see Chaps. 10 and 12). Similarly, we begin with noticing how our particular histories and social identities influence our roles as MTEs and the ways that we have come to use the CME Module in our mathematics methods coursework.

Crystal I am a white MTE who prepares elementary mathematics PSTs at a HispanicServing Institution. I was raised along the coast of South Texas and have lived in numerous states in the south and in Europe for a short time. As a doctoral graduate research assistant for the TEACH Math research project from 2010 to 2012, my work as an MTE is guided by the intention of creating spaces for PSTs to learn and adopt equity-based mathematics practices. I openly acknowledge my privilege as a white MTE at a university who closely works with children, families, and teachers in the local community, many of whom come from Mexican and Central American

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backgrounds. I constantly push myself to seek new ways to interrupt and redistribute the power of teaching and learning so that it is shared in and outside of my classroom. My research explores children’s and teachers’ mathematical resources, strategies for implementing equitable group work, and integrating art and music with mathematics instruction. I have seen the power of connecting mathematics to the lives of children, particularly children who do not feel represented or acknowledged in many of our classrooms today.

Maria I am a Latina MTE and researcher and I teach elementary PSTs and practicing mathematics teachers at a large, urban, public university in California’s San Francisco Bay Area. I am the first American-born child of Peruvian immigrants. I research Latinx students and PSTs in mathematics contexts, often utilizing the lens of identity. Since studying mathematics in college, I have been fascinated with who feels like they have the right to be a mathematician, and the imperative around creating meaningful mathematical experiences for vulnerable student populations (e.g., emergent bilinguals, students living in poverty, students with learning differences). I believe that mathematics can and should have everyday meaning for children. School mathematics should be generated from students’ lives, and not the other way around. It is my goal to develop PSTs’ sense of advocacy and awareness for traditionally marginalized populations while also gaining competencies in teaching mathematics in culturally responsive ways, which includes teaching with attention to students’ multiple mathematics knowledge bases.

 etting the Tone for Helping PSTs to Learn about Children’s S (Mathematical) Funds of Knowledge Setting the tone is an important first step in helping PSTs learn about children’s funds of knowledge. Our courses set a tone through their framing, which in our case includes being informed by bodies of research examining teaching mathematics for social justice (Gutstein & Peterson, 2012), Cognitively Guided Instruction (Carpenter, Fennema, Franke, Levi, & Empson, 2015), Complex Instruction (Featherstone et al., 2011), and TEACH Math (Turner et al., 2012), to name a few. Each of these frameworks argues that teachers should view children with an asset-­ based approach that foregrounds the rich knowledge and experiences that children bring to the classroom. Children’s knowledge and experiences, which includes their community, cultural, and linguistic knowledge, can serve as an invaluable resource for PSTs as they learn to teach mathematics effectively and engagingly. We want PSTs to learn how to elicit and value knowledge not typically valued

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within the math classroom. Our intention is to help PSTs learn how to position students as competent within the mathematics classroom. As MTEs in an elementary education certification program, our first days of our methods classes highlight the issues children face when their mathematics classrooms do not necessarily value the resources and experiences rooted in their homes, communities, culture, and language. Because we acknowledge that not all PSTs are familiar with the children with whom they work, it is important to set the tone for an equity-minded approach to teaching mathematics. We also clarify for PSTs what we mean when we use the word “equity” in the mathematics classroom. We leverage Aguirre, Mayfield-Ingram, and Martin’s (2013) definition of equity, alongside principles of culturally responsive teaching: All students, in light of their humanity – their personal experiences, backgrounds, histories, languages, and physical and emotional well-being – must have the opportunity and support to learn rich mathematics that fosters meaning making, empowers decision making, and critiques, challenges, and transforms inequities and injustices. Equity does not mean that every student should receive identical instruction. Instead, equity demands that responsive accommodations be made as needed to promote equitable access, attainment, and advancement in mathematics education for each student. (p. 9)

We offer examples of the assumptions made about students in classrooms that are not equity-minded. For example, teachers of emergent bilinguals (García & Kleifgen, 2010) may assume that their students must first have fluency in English prior to learning any mathematics, which research has shown to be a deficit-based assumption (Aguirre, Mayfield-Ingram, et al., 2013). We also rely on the extensive research of Cognitively Guided Instruction (Carpenter et  al., 2015) to help PSTs learn to be curious about children’s existing mathematical knowledge and to not focus on what children do not know but rather what they do know. In addition, we frame PSTs’ experiences from the first day as that of culturally sustaining educators (Paris, 2012). We argue that this type of tone-setting with emphasis on “in light of their [students] humanity” (Aguirre, Mayfield-Ingram, et al., 2013, p. 9) establishes that teaching mathematics is not just about being good at explaining mathematical ideas  – rather, good mathematics teaching must be grounded in commitments to seeing your students as whole human beings.

 trategies and Scaffolds to Support Implementation S of the Community Mathematics Exploration with PSTs In the following sections, we discuss how we specifically use the CME Module in our elementary mathematics methods coursework and how this module aligns with our goal of emphasizing children’s funds of knowledge with our PSTs. We use a chronological approach to describing our work with the module, including the scaffolds we have created for PSTs who may be at varied points in their teacher preparation program. First, we define scaffolding as a process in which:

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teachers provide successive levels of temporary support that help students reach higher levels of comprehension and skill acquisition that they would not be able to achieve without assistance. Like physical scaffolding, the supportive strategies are incrementally removed when they are no longer needed, and the teacher gradually shifts more responsibility over the learning process to the student. (Hidden Curriculum, 2014)

Learning how to elicit and leverage children’s funds of knowledge is not easy for teachers, especially PSTs, and so the notion of scaffolding is important for them to sustain their learning after they have completed their teacher preparation program (Aguirre, Mayfield-Ingram, et  al., 2013). Figure  4.1 introduces the three incremental scaffolds that we use for the CME Module, which we will unpack further in this chapter.

 stablishing Routines to Learn about Children’s Funds E of Knowledge: Suggested Reading List and Video Clips In writing this chapter, we first reflected on how we situate PSTs’ learning about children’s funds of knowledge within a set of readings that are multi-faceted, research-based, accessible, and aligned with our notions of equity and social justice when teaching mathematics. We use Van de Walle, Karp, and Bay-Williams’ (2016) book for teaching methods and supplement this with other readings that serve multiple goals: (a) emphasize children’s funds of knowledge, (b) tackle issues of power and oppression, and (c) foreground the struggle of children from historically marginalized backgrounds to find success in the mathematics classroom. In Appendix A, we share readings and video clips that we have found useful when helping PSTs Fig. 4.1 Suggested scaffolds for PSTs to learn about children’s funds of knowledge

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learn about how mathematics teachers incorporate children’s funds of knowledge into their mathematics teaching. Appendix A in Chap. 3 is not intended to serve as an exhaustive or comprehensive list of resources but instead complements the TEACH Math website (https://teachmath.info) and reflects a living document that can change over time, context, and situation. As we thought about ways to create a space for PSTs to learn about children’s knowledge and experiences, we also thought about how these readings and video clips provide examples of routines that PSTs can take up in their own practice outside of the scope of our class and programs. We want to underscore routines that encourage a curiosity about children because we want to nurture teachers who foreground asset-based pedagogy as a foundation for lesson planning. We would argue that to delay talking with PSTs about children’s funds of knowledge until later in their program may privilege only one aspect of teaching mathematics, a unidirectional approach that only values the teacher’s mathematical knowledge base. We encourage PSTs to establish the routine that learning about children’s funds of knowledge is a fundamental core of any mathematics lesson. Furthermore, PSTs learn mathematical content while engaging with the readings and video clips because rich mathematical tasks are at their center.

 tarting with Mini-explorations of Children’s Funds S of Knowledge Once we have established an initial set of routines about eliciting and building upon children’s funds of knowledge, PSTs then engage in mini-explorations of that knowledge. One mini-exploration entails mathematizing the world through photographs. We acknowledge that the practice of mathematizing the world is not a quick or standardized process, but our goal is consistent: to help PSTs learn to see the mathematical world through the eyes of children. In one example of PSTs mathematizing their world through photographs, Maria brings a weekly photograph to share (via PowerPoint slide or document camera) from the surrounding city: the produce section of a local market, birds sitting on the telephone wires above a park, a woman making pupusas in front of her shop, to name just a few examples. PSTs are invited to describe what they see in the photograph, talk about where they have seen similar scenes in the city, and then craft word problems that could be generated from each photograph. A “What do you notice? What do you wonder?” protocol (Ray-Riek, 2013) helps PSTs bridge from something they see in the photograph to asking a question that could be answered using mathematics. Mathematizing a photograph is usually a short warm-up for class and models the routine of remaining curious about the lives and activities of their students in the community. Maria’s use of existing photographs coupled with a noticing and wondering protocol also helps her to scaffold the routine of mathematizing

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locations in the community before the PSTs engage in their own Community Walk (see Chap. 2 for a description of this activity). We also use the Mathematics Learning Case Study Module to support mini-­ explorations of children’s funds of knowledge. In the Mathematics Learning Case Study Module, PSTs are encouraged to focus on one student who may feel excluded from mathematics – the child may feel negatively towards mathematics, may come from a historically marginalized community, or may be labeled as a behavioral challenge by their current teacher. PSTs first conduct a Getting to Know You Interview about the child’s interests, hobbies, and practices that they do outside of school. Like the TEACH Math project MTEs did, we use the Getting to Know You Interview as a first step to the CME Module as it highlights the importance of getting to know a student as a learner and doer of mathematics both in and out of the classroom. The Mathematics Learning Case Study Module also serves as an early diagnostic tool to assess how PSTs learn to draw reasonable conclusions about students’ mathematical understanding from evidence collected in interviews. In coursework where PSTs may not be asked to complete a student case study, such as in Maria’s courses, they still conduct an interview and observe a focal student during mathematics class to tailor their CME work to a specific student’s funds of knowledge. Many MTEs who utilize the CME Module also utilize the Video Lens tool (see Chap. 2; Roth McDuffie et al., 2014) to engage in a mini-exploration of children’s cultural and linguistic resources while watching video clips of mathematics instruction. For example, in Annenberg Learner’s (Barzyk & Roche, 1997) Marshmallows video, PSTs identify the ways that the teacher, Ms. Torrejón, leverages her secondgrade students’ linguistic resources while engaging in solving a mathematics problem. Because teaching is a complex and dynamic practice, the Video Lens tool helps PSTs to focus on specific aspects of teaching and learning, which includes ways to elicit and honor children’s (mathematical) funds of knowledge.

Scaffolding a More Intentional Exploration of the Community Once we have established our classroom as a place where children’s fund of knowledge is a valued part of teaching mathematics, we move on to more intensive activities that explore the local community. We (Maria and Crystal) engage in this step somewhat differently, but with similar goals. Before Maria’s PSTs complete their own Community Walk Activity around their students’ school communities, they take a practice walk around the university campus. Maria’s class begins as a group on campus and then they are encouraged to walk around campus in pairs, mathematizing as they go, taking photographs and notes. One rule that has become useful through repeated iterations of this process is to require that PSTs speak to someone (a student, a staff member, etc.) on their walk. The PSTs practice before they begin their walks because Maria has found that once a PST sees how to approach someone in a friendly way and explain what they are up to, they are more likely to talk to someone on their own Community Walk off campus. After their walk, PSTs come

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together and discuss what they found and what mathematics they observed or learned about within those situations. This is also a first opportunity to look beyond “best-buy” scenarios and incorporate ideas of geometry, probability, fair wages, and other topics into potential contexts for mathematics. In total, this process takes about one hour of class time and gives PSTs a better vision of what a Community Mathematics Walk can be like. Using social media to engage PSTs in an exploration of the community is another way to approach this practice. For example, three doctoral students (Stephanie Garcia, Martina McGhee, and Traci Kelley), an MTE colleague (Emily Bonner), and Crystal have PSTs use a course-specific Twitter hashtag to document their mathematics and science community walks in real time. The PSTs post pictures that they see in the community and either include a claim about the mathematics and science in the photo or ask a new question they might pose to children (which is a practice that Luz Maldonado also does with her PSTs, as described in Chap. 11). Picture 4.1 is a PST’s tweet from Garcia’s Community Science Exploration walk and can be mathematized as well to discuss the proportions of the soil to sedimentary rock and how much of this mixed soil ends up in landfills.

Picture 4.1  Example of community exploration Twitter post

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The colleagues at Crystal’s institution are seeing how the various platforms of social media (i.e., Twitter, Facebook, and Instagram) help PSTs to immediately engage in a discussion about children’s funds of knowledge when they re-share or “like” the posts. Furthermore, when PSTs examine the same photograph(s), they can see how community contexts serve as multi-faceted opportunities to create interdisciplinary mathematical tasks that can help children learn mathematics. Finally, we include other scaffolds for more intensive community explorations that focus on discussing the features of lesson planning, analyzing features of mathematics tasks, reviewing assignment expectations, taking clarifying questions, and trouble-shooting specific challenges that vary by school. More specifically, Maria provides the following graphic as an additional scaffold for PSTs while they complete a CME assignment (see Fig. 4.2 below). Maria usually introduces the graphic after the students have taken their college campus math exploration practice walk.

Develop a Task

Explore Go on a math walk. Let what you know about your students guide this process. What do you notice? What do you wonder? Take photos. Talk to people.

Ask Questions Develop math questions based on what you wonder. It is okay if these initial questions lead to more questions. Try to answer the questions, notice what mathematics you engage in.

Describe the context and ask the questions. This should be something you could hand to kids, or the "script" for yourself if you are working with young children. Ask yourself: 1) What should student's spend time grappling with, or figuring out? 2) What information do students need to answer the question? 3) How are both mathematical knowledge and contextual or critical knowledge important for answering questions? 4) What images & diagrams will help students? 5) What is the product: a poster? a picture with an explanation? A position statement? Products should include justification.

Fig. 4.2  Steps to completing the CME assignment (Maria’s class)

Make the Lesson Plan(s) 1) What learning objectives align with this task? 2) How will you introduce the task? 3) What will you do as the students work: what kinds of scaffolding should you be ready to provide? What questions can you be ready to ask to monitor students' learning? 4) How will students share their work? How will you wrap up the lesson? What could this lesson lead to next? 5) How does your lesson plan explicitly connect ideas in the task back to issues in the community?

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Helping PSTs to recognize, value, and incorporate children’s funds of knowledge while resisting deficit thinking is a crucial component of the CME Module because of the time and sustained effort it takes for teachers to consistently implement these asset-based teaching practices. In the next section, we discuss some recurring concerns and challenges that we have come across or gathered from colleagues, and offer some advice based on our experiences.

CME Module: Frequently Asked Questions  ow Do These Ideas Fit into a Mathematics Content Course H for PSTs? As we think about how to create more opportunities for PSTs to learn about children’s funds of knowledge, we also are reminded about assumptions and critiques of this work outside of a methods classroom or a college of education. In response to an assumption that mathematics content classes should avoid talking about children’s funds of knowledge, there is a growing cadre of MTEs in mathematics departments who leverage issues of social justice and children’s cultural and linguistic backgrounds while teaching elementary mathematics content (e.g., Simic-­ Muller, Fernandes, & Felton-Koestler, 2015). When MTEs across multiple departments and colleges consider children’s funds of knowledge as intertwined with how children know and use mathematics in their lives, PSTs can learn more about children’s funds of knowledge even before they begin their methods coursework. We acknowledge a false divide between mathematics as a subject area and the mathematics that children and families use every day. We wonder: What would mathematics content courses consist of if they privileged both the knowledge of children and the mathematics children know and use? How might this shift affect the preparation, success, and number of culturally sustaining educators?

 hat if Our PSTs Have Trouble Finding Time to Do W the Community Walk or Have Trouble Walking in the Area? One goal of the CME is to approach the notion of space and children’s communities by helping PSTs to become more comfortable with exploring unfamiliar spaces. Crystal schedules the exploration of the community during lunch time so that PSTs can visit the community in small groups with a common goal of eating lunch. If PSTs are concerned they cannot physically travel to where their students live (as happens with some schools in which students are bussed long distances or students commute into the city with their parents to go to school), Maria has her PSTs walk around their student teaching school communities and discuss the local

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communities with children, faculty, and staff. As discussed earlier, MTEs can leverage the Getting to Know You Interview as a purposeful entry point to begin the Community Walk Activity. The weather has occasionally been an issue when conducting an actual walk of the community. Rescheduling the Community Walk is always an option, but there are other ways to “explore” the community. One option is to utilize online resources that evidence things such as businesses, parks, churches, and/or events that students and their families might access. Online explorations of the community can happen during or outside of mathematics methods class time, which is particularly useful for PSTs who may commute long distances to their field placement or student teaching campuses. Often PSTs in both of our classes find it useful to follow up a Community Walk with online explorations, to gather more details of features of the community, and to better visualize the neighborhood using a service such as Google Maps.

 hat if PSTs Are Not Assigned to Work Directly with Children W and thus Do Not Have a Community to Explore? Some mathematics methods classes do not have a field component and MTEs teaching such courses may find it challenging to support activities in the CME Module with their PSTs. MTEs may pivot by expanding the notion of “community” and having PSTs explore the broader communities in which they live and work. PSTs can conduct a Community Walk of their own neighborhoods and/or online. Ultimately, we have learned to also reframe for (and alongside) PSTs what community represents, what it means to do mathematics, and what strategies exist for incorporating children’s knowledge and experiences into the practice of teaching mathematics.

Challenges and Next Steps with the CME Module PSTs need time to test out and refine new ideas and strategies when learning about children’s funds of knowledge. As such, we acknowledge some potential challenges when supporting PSTs to incorporate children’s funds of knowledge into mathematics instruction. First, the CME Module activities could potentially reinforce PSTs’ harmful stereotypes and assumptions about children’s communities, particularly communities that have been historically marginalized in the past. Similarly, the CME Module activities may engage PSTs in creating math tasks based on assumed or inauthentic practices for children in their community (Aguirre, Turner, et al., 2013), thus not disrupting perspectives about school mathematics that foregrounds procedural fluency and efficiency. Notwithstanding these challenges, there

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is wide consensus that children need teachers who are responsive to their existing knowledge and experiences of mathematics and who honor children’s lives in their mathematics lessons. A part of learning to teach mathematics is learning to elicit and incorporate children’s (mathematical) funds of knowledge. When MTEs scaffold the CME Module activities through routines, mini-explorations, and rehearsals, PSTs can learn how to always be curious about the lives of their students when they have their own classroom. We hope that this chapter has highlighted some ways in which MTEs might help PSTs to learn how to see their students “in light of their humanity” (Aguirre, Mayfield-Ingram, et al., 2013, p.9).

 ppendix A: Resources to Support Incorporating Students’ A Funds of Knowledge in Mathematics Instruction

Readings Identity: Aguirre, Mayfield-Ingram, et al. (2013), Featherstone et al. (2011); Flores (2015) Teaching Mathematics for Social Justice: Berkman (2005), Peterson (2005), Varley-Gutierrez (2009) Community Connections: Cavazos (2014), Civil and Khan (2001), Leonard and Guha (2002), Richardson (2004), Simic-Muller, Turner, and Varley-Gutierrez (2009) Supporting English Language Learners: Celedon-Pattichis and Ramirez (2012), Oliveira (2012) Listening and Noticing: Gutierrez (2015), Oslund and Crespo (2014), Perkins and Flores (2002) Videos PayDay Loans, https://www.youtube.com/watch?v=yZ_WBLxat0k University of Arizona and Cholla High School Social Justice Education Project Rescuing Education, Part II https://www.youtube.com/watch?v=Uo_0ucXGZko University of Arizona and Cholla High School Social Justice Education Project. This film was created in response to a bill which threatened to eliminate Ethnic Studies classes from the Tucson Unified school district. Good Morning, Ms. Toliver! https://www.youtube.com/watch?v=dRm0aHNVCuk This video is an oldie but goodie and a great starting point for what mathematizing in your community around you could be like, with seventh-grade students. The Annenberg Learner Video Library on Teaching Mathematics, Grades K-4, https:// www.learner.org/resources/series32.html# This is another old but very good resource. In particular, we reference the Valentine’s Exchange video, included at this link, from a bilingual fourth-grade classroom (Note: this video collection was recently discontinued during the publication of this chapter)

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References Aguirre, J., Mayfield-Ingram, K., & Martin, D. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. Reston, VA: The National Council of Teachers of Mathematics. Aguirre, J. M., Turner, E. E., Bartell, T. G., Kalinec-Craig, C., Foote, M. Q., Roth McDuffie, A., et al. (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. Barzyk, F., & Roche, B. (1997). Marshmallows [video file]. Teaching math: A video library, K-4. Burlington, VT: Annenberg Learner. Berkman, R. (2005). Tracking PA announcements. In E. Gutstein & B. Peterson (Eds.), Rethinking mathematics: Teaching social justice by the numbers (pp.  130–131). Milwaukee, WI: Rethinking Schools. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann. Cavazos, R.  R. (2014). WOW! Mathematics convention: A community connection. Teaching Children Mathematics, 21(3), 154–160. Celedón-Pattichis, S., & Ramirez, N. (2012). Thoughts, stories and consejos (advice) from ELLs and their educators. In S.  Celedón-Pattichis & N.  Ramirez (Eds.), Beyond Good Teaching: Advancing Mathematics Education for ELLs (pp.  5–18). Reston, VA: National Council of Teachers of Mathematics. Civil, M., & Khan, L.  H. (2001). Mathematics instruction developed from a garden theme. Teaching Children Mathematics, 7(7), 400–405. Cummins, J.  (1996). Negotiating identities: Education for empowerment in a diverse society. Ontario, CA: California Association for Bilingual Education. Featherstone, H., Crespo, S., Jilk, L., Oslund, J. A., Parks, A. N., & Wood, M. B. (2011). Smarter together!: Collaboration and equity in the elementary math classroom. Reston, VA: The National Council of Teachers of Mathematics. Flores, N. (2015). What if we talked about monolingual White children the way we talk about low-­ income children of color? Retrieved July 14, 2016, from https://educationallinguist.wordpress. com/2015/07/06/what-if-we-talked-about-monolingual-white-children-the-way-we-talkabout-low-income-children-of-color/ Freire, P. (2000). Pedagogy of the oppressed (30th anniversary ed.). New York, NY: Continuum. García, O., & Kleifgen, J.  A. (2010). Educating emergent bilinguals: Policies, programs, and practices for English language learners. New York, NY: Teachers College Press. González, N., Moll, L.  C., & Amanti, C. (2005). Funds of knowledge: Theorizing practice in households, communities, and classrooms. Mahwah, NJ: L. Erlbaum Associates. Gutiérrez, R. (2015). HOLA: Hunt for opportunities–learn–act. The Mathematics Teacher, 109(4), 270–277. Gutstein, E., & Peterson, B. (2012). Rethinking Mathematics (2nd ed.). Milwaukee, WI: Re-thinking Schools. Hidden Curriculum. (2014, August 26). In S.  Abbott (Ed.), The glossary of education reform. Retrieved from http://edglossary.org/hidden-curriculum Langer-Osuna, J. M., & Nasir, N. S. (2016). Rehumanizing the “Other”: Race, culture, and identity in education research. Review of Research in Education, 40(1), 723–743. Leonard, J., & Guha, S. (2002). Creating cultural relevance in teaching and learning mathematics. Teaching Children Mathematics, 9(2), 114–118. Oliveira, L. (2012). The language demands of word problems for English Language Learners. In S.  Celedón-Pattichis & N.  Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs (pp. 195–206). Reston, VA: National Council of Teachers of Mathematics. Oslund, J. A., & Crespo, S. (2014). Classroom photographs: Reframing what and how we notice. Teaching Children Mathematics, 20(9), 564–572.

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Paris, D. (2012). Culturally sustaining pedagogy: A needed change in stance, terminology, and practice. Educational Researcher, 41(3), 93–97. Perkins, I., & Flores, A. (2002). Mathematical notations and procedures of recent immigrant students. Mathematics Teaching in the Middle School, 7(6), 346–351. Peterson, B. (2005). Teaching math across the curriculum. In E. Gutstein & B. Peterson (Eds.), Rethinking mathematics: Teaching social justice by the numbers (pp. 9–15). Milwaukee, WI: Rethinking Schools. Ray-Riek, M. (2013). Powerful problem solving: Activities for sense making with mathematical practices. Portsmouth, NH: Heinemann. Richardson, K.  M. (2004). Designing math trails for the elementary school. Teaching Children Mathematics, 11(1), 8–14. Roth McDuffie, A., Foote, M. Q., Bolson, C., Turner, E. E., Aguirre, J. M., Bartell, T. G., et al. (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. Simic-Muller, K., Fernandes, A., & Felton-Koestler, M. D. (2015). “I just wouldn’t want to get as deep into it”: Pre-service teachers’ beliefs about the role of controversial topics in mathematics education. Journal of Urban Mathematics Education, 8(2), 53–86. Simic-Muller, K., Turner, E.  E., & Varley-Gutierrez, M. (2009). Math club problem posing. Teaching Children Mathematics, 16(4), 206–212. Turner, E. E., Drake, C., Roth McDuffie, A., Aguirre, J., Bartell, T. G., & Foote, M. Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67–82. Van de Walle, J. A., Karp, K., & Bay-Williams, J. M. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Boston, MA: Pearson. Varley-Gutierrez, M. (2009). “I thought this US place was supposed to be about freedom”: Young Latinas engage in mathematics and social change to save their school. Rethinking Schools, 24(2), 36–39.

Chapter 5

Supporting Prospective Urban Teachers to Access Children’s Multiple Mathematical Knowledge Bases: Community Mathematics Explorations Craig Willey and Weverton Ataide Pinheiro

Keywords  Multiple mathematical knowledge bases · Community · Urban education · Teacher education · Mathematics education

Broadly speaking, mathematics teaching practices have not yielded equitable learning opportunities and outcomes for urban children, poor children, and children of color. In fact, some argue that the status quo mathematics teaching, influenced by increasingly strong pressure by accountability measures, perpetuates disparate levels of success in mathematics (Martin, 2015; Spencer, 2015). One way to address these inequities is to re-conceptualize mathematics instruction, and teacher education programs have a responsibility to engage the next generation of teachers in this re-conceptualization work. There is much unlearning to do, however, among mathematics educators in terms of interrogating mathematics teaching practices that currently serve to sort, alienate, and limit successful learning. Martin (2015) called on mathematics teachers and mathematics teacher educators to get serious about addressing persistent inequities among the Collective Black, and to carefully consider what it means to teach mathematics equitably. Mathematics teacher educators who engage in ongoing work to examine the sociopolitical landscape of mathematics teaching and learning are better positioned to skillfully support prospective teachers (PSTs) in unpacking and critiquing the dominant narrative about, and ideologies around, what it means to learn mathematics (Willey & Livers, 2018). It is critical that mathematics teacher educators design, implement, and scrutinize the results of learning activities in order to capitalize upon the knowledge bases of historically marginalized learners and evaluate progress toward an end goal: equity. This chapter explores the implementation of the Community C. Willey (*) · W. A. Pinheiro Indiana University-Purdue University Indianapolis, Indianapolis, IN, USA e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_5

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Mathematics Exploration (CME) Module, looking specifically at how prospective urban teachers (urban PSTs) re-think the richness of community and cultural resources that help children make sense of mathematics concepts. Specifically, we (a) outline the conceptual goals of the CME, (b) situate this activity in our context of an urban elementary teacher education program, (c) share examples of CME projects completed by our PSTs and draw conclusions about what these reveal about PSTs’ experiences, and (d) provide evidence of the impact of the CME Module on PSTs’ outlook on culturally relevant mathematics teaching (CRMT) with urban youth. This study is guided by the following question: In what ways did urban PSTs make sense of the CME Module? Specifically, our analysis aims to address: (a) How did urban PSTs describe their experiences with the CME Module activities, and (b) How did urban PSTs’ describe CRMT at the end of the semester? We conclude with a discussion of the relationship between PSTs’ described experiences and their conceptualizations of CRMT, which serves as the foundation from which equitable mathematics teaching practices can emerge.

 ultiple Mathematical Knowledge Bases and the Conceptual M Goals of Community Mathematics Explorations One approach to supporting urban PSTs—most of whom are White, monolingual, and not from urban contexts (U.S.  Department of Education, 2016)—to teach mathematics with equity in mind is to help them critically reflect upon their histories, understand the subtle and not-so-subtle discourse and messaging that normalizes and elevates White perspectives and rationales, and “see” the epistemological value and brilliance of children and families of color. Children undoubtedly maintain rich repositories of knowledge for learning mathematics, what have been referred to as multiple mathematical knowledge bases (MMKB) (Turner et al., 2012). Although these MMKB exist, they largely go unacknowledged and untapped, especially among children of color and children from other marginalized communities. Furthermore, discourse and teaching practices, and the ideologies that are represented through them, have significant implications for how we define mathematical success. For example, it is a marginalizing practice to limit mathematical success to students’ arrival at correct answers or competency performing traditional algorithms. On the contrary, success could be defined as growth in affinity toward mathematics, the development of positive mathematics identities, or an enhanced ability to explain mathematical thinking and critique the reasoning of others. Achieving these kinds of successes, then, depends on a more sophisticated understanding of the role of language and culture—the primary foundation of

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children’s MMKB—in mathematics teaching and learning processes. Collectively, as mathematics teachers, we have been slow to envision the role of, and draw on, children’s MMKB as we facilitate the mathematics learning process. Of course, steps in this direction are inhibited by external pressures (i.e., scope and sequence guides, benchmark assessments, teacher evaluations connected to student performance on standardized assessments) and historical approaches to mathematics teaching that have given us the (false) impression they have worked for many but, in reality, have failed many more. Well-intentioned efforts to incorporate children’s MMKB in mathematics instruction have been limited and might be characterized as contextualizing mathematical applications in scenarios that are familiar to children. To be clear, this is positive movement; however, it does not represent the dramatic transformation in teaching children of color that is needed in order to confront and redress decades of pedagogy designed for White, mainstream classrooms and the dominant cultural norms represented in them (Martin, 2015). Ladson-Billings (1995), invoking the work of Native American scholar Cornel Pewewardy (1993), argues that one of the reasons children of color “experience difficulty in schools is because educators have attempted to insert culture into education, instead of inserting education into culture” (p. 159). Although an important first step in making connections with children, it is not enough to merely glean a few insights about children’s likes and dislikes and use them as a hook to capture their interest. These connections likely amount to only a fraction of the cultural backdrop of mathematics lessons, which leaves the vast majority of the remaining backdrop to be defined by the teacher or curriculum developers. In all likelihood, this means that the context for mathematics learning is largely defined by White norms and frames of reference, whereas what Ladson-Billings is calling for is quite the opposite in the design and engagement of mathematics learning. But, how do we support novice teachers, who themselves are grappling with their own situatedness in dominant structures and practices, to believe in the power of, and develop a commitment to, such activities that represent a “dramatic” or “significant” departure from what’s “normal”? From our perspective, the CME Module leads us into conversations that combine what we can see about the treatment of children of color in schools and communities, raising alternative explanations for disparate learning outcomes, and designing mathematics instruction with the children in mind. In particular, this means thinking through how to access, and then leverage, children’s and families’ funds of knowledge (Gonzalez, Moll, & Amanti, 2005); recognizing value in community life; and proving to ourselves that mathematical sense making happens both inside and outside the classroom, and optimally sense making happens over time, when children make connections between the two worlds (school and out-of-school) and develop scientific concepts (Vygotsky, 1978). These are the concepts that underpin MMKB, and the CME Module is an exercise in practicing to “see” the mathematical practices of community members by mathematizing community spaces and visiting and interacting with community members.

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 ontext and Programmatic Design to Facilitate Racially C Conscious Mathematics Instruction We teach in an urban teacher education program that focuses heavily on confronting issues of race and disrupting patterns of White dominance. Mathematics instructional practices are not exempt from this focus; in fact, we operate on the assumption that mathematics teaching and learning are racialized experiences (Battey, 2013; Martin, 2006). This means that mathematics teaching and learning is not a space that is somehow free from the racial biases, stereotypes, and micro-­aggressions that exist in all other aspects of social discourses and interpersonal interactions in the U.S.; rather, they permeate mathematics classrooms in predictable ways, and also combine with prominent mathematical ideologies to influence the mathematics identities of learners of color in unique ways (Aguirre, Mayfield-Ingram, & Martin, 2013; Jackson, 2009; McGee & Martin, 2011). The TEACH Math modules are particularly helpful to lead us into necessary and rich discussions about epistemology, funds of knowledge, Whiteness and White dominance, and marginalization, among other oppressive forces and unearned privileges that are inevitably associated with our current racial order (Bonilla Silva, 2010). Still, there are structures that can help mediate the kind of growth toward consciousness and culturally relevant (mathematics) pedagogy (CRP) that we, teacher education faculty, collectively aim to develop among our PSTs.

Intentional Design and Use of Clinical Experiences The curriculum of our urban teacher education program is guided by Ladson-­ Billings’ culturally relevant pedagogy (CRP) (1995, 2009, 2014). The tenets of CRP (academic success for all students, cultural competence, and critical consciousness) may seem straightforward, but developing a stance toward teaching that embraces and utilizes these tenets requires deep reflection, study, and extended practice. For example, while teachers often agree with the idea of CRP, they are less able to incorporate it meaningfully into their actual work with students (Ladson-Billings, 2014; Young, 2010). The central ideas of CRP are often misunderstood and oversimplified without continued opportunities to consider how the theoretical underpinnings can be enacted in practice (Ladson-Billings, 2006). As such, we have designed a clinically centered program to support the development of CRP, where thoughtful (clinical) learning activities and skillful mediation of these experiences are central. We understand the significant gap between how CRP resonates with PSTs and their ability to design instruction in accordance with its tenets. Therefore, we see as our primary aim to help urban PSTs process, say, classroom observations, schooling and teaching practices, and school-sanctioned treatment of children, in relation to constructs of equity. Although clinical experience and mediation have long been identified as key features of teacher education programs aimed at teaching for social justice, there are other structural

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arrangements we have instituted in order to maximize the developmental benefits for PSTs as they work toward enacting culturally relevant mathematics pedagogy. For example, we have focused on designing a school–university partnership model that affords urban PSTs the opportunity to see, feel, and hear school life for children of color from a variety of angles. In order to provide an in-depth, immersive experience, we anchor a cohort of PSTs and a consistent instructional team at one urban school for the entire professional program (four semesters). We recognize that no partnership school will be perfect, and we acknowledge the complexity of schools and teachers’ professional lives. Still, we suggest a minimum set of criteria when preparing to engage in the mutually beneficial arrangement of the partnership: an urban location, abundant diversity in the student population (and preferably among teachers and administrators), and a shared understanding of how the partnership could and would serve children, families, teachers, and PSTs. In addition, it has become clear that the administrative team (at a minimum, the principal) needs to evidence sophisticated understanding of how racism functions in schools and share with their university partners a vision to create schooling spaces that are not harmful to children of color; this is critical and involves a willingness to be honest, transparent, and critically reflective in aspects of administrative work and relationship-building with faculty and PSTs. We also designed a clinically centered program that provides abundant opportunities to interact with children and teachers, presented in Appendix A.

 ommunity Mathematics Exploration Products and Learning C Outcomes This analysis primarily draws on two data sources: PSTs’ lesson plans based on their CME Module experiences and their Field Journals, where they were asked to document and analyze students’ mathematics learning, reflect on their field experiences, and describe CRMT. For this chapter, we analyzed Field Journals from two cohorts (n = 20), separated by 1 year, in order to understand PSTs’ approaches to designing and implementing a math lesson based on students’ community resources and, subsequently, PSTs’ learning from the field experiences and conceptions of CRMT. There are three primary findings from this analysis. First, instructor-initiated scaffolds seemed to help PSTs overcome struggles to access authentic, community-­ based mathematical practices (or opportunities for mathematical practices) beyond simply identifying contexts and spaces with which the children were familiar. Second, as a result of the extended experience designing and leading community-­ based mathematics lessons with children, PSTs expressed a marked shift in their outlook on teaching mathematics through a culturally relevant lens. Finally, the CME Module activities supported PSTs in developing a more robust understanding of CRMT, as evidenced by their insights about what constitutes CRMT and the conclusions they drew about the significance of the community-based mathematics lessons on children’s identities.

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Accessing Community-Based Mathematics Practices Analysis of the Field Journals from Cohort 1 revealed contention around what community-­based math lessons should look like. PSTs familiarized themselves with the vicinity of the school in order to engage their students in conversation around what they do when they are not at school. They reported that they learned specific things about the children’s families (e.g., whether they had siblings or pets, favorite restaurants), but, anecdotally, they also reported a tension around trying to find a common denominator that tied all children together. In many cases, there was uneasiness when small groups of PSTs landed on a topic for the design and implementation of the community-based math lesson, as there was the perception that the topic needed to have equal relevance to each child. Table  5.1 shows the CME Module lesson plan topics from six representative Field Journals. The choice of topics explored reflects the lines of inquiry with which the PSTs approached the students. Some topics are universal enough that they might be applied to many groups of fifth graders (e.g., planning a pizza party), while others reflect a specificity and level of detail that is unique to that specific group (e.g., music production). Some appear driven by the mathematics the PSTs wanted to cover (e.g., how children spend their day). While there is diversity among these six topics, overall the topics are one indicator that there was an element of uncertainty as the PSTs were tasked to engage children in dialogue about topics, and subsequently, there did not appear to be a clear path forward by which PSTs were to design community-based mathematics lessons that moved beyond children’s likes, dislikes, interests, or familiar contexts. Despite PSTs’ initial struggles to both engage children in exploring community-­ based practices and incorporate these into mathematics lessons, working with children in the CME Module provided PSTs opportunities to experiment with Table 5.1  Representative CME topics from Cohort 1 Teacher Topic Gretchen Community places (e.g., YMCA, McDonald’s) Jada Planning a pizza party Ashley

Basketball

Matt

How children spend the hours in their day Music Production (What’s Fair?: Record Labels, Artists, & Profit) Basketball, restaurants

Alexa

Sara

Mathematical concepts explored Division (whole number); decimals Multiplication; division; decimals Comparing fractions; converting fraction to decimals Fractions; part-whole relationship Operations with decimals

Social/community phenomenon explored Community math practices (emergent) Children’s interests

Fractions; operations with decimals

Children’s interests and familiar contexts

Children’s interests

Children’s familiar contexts Community math practices

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ways of engaging children in mathematics beyond what is laid out in commercial mathematics curricula. For example, Matt reported, “As the students worked, I overheard Edward tell Zahir, ‘Wait a minute. They have us doing math.’ The session was designed to get the students talking about their pets and slowly begin the lesson.” This eventually led Matt to draw conclusions about how children are socialized into mathematics and what counts as “doing” mathematics: “By the time many students reach intermediate grades, I observed that very few enjoy math, and their proficiency is defined by an ability to regurgitate teacher instruction.” Influenced by readings and classroom observations that illuminated the apparently detrimental effects of routine and direct instruction, Matt and his team enlisted new pedagogical moves. For example, they tried promoting student-tostudent discourse, which led to a “shared meaning” (Chapin, O’Conner, & Anderson, 2003) of the elements of a bar graph, and the creation of a studentcreated anchor chart depicting the features of bar graphs. In order to better support PSTs in Cohort 2 to consider and access community-­ based mathematics practices, Craig (the first author) made some adjustments. First, he recognized the need for practice mathematizing community spaces. PSTs, like most adults, do not see spaces and places and automatically consider their potential as sites of mathematical practices. As such, Craig took pictures of local places (e.g., paneria, library, bus stop) and facilitated brain-storming sessions about what mathematics might take place at each site. Second, he designed centers, or stations, at which PSTs would develop lines of questioning for children that moved beyond likes and interests, ones that provided insights into how the child engages with their family and how the family engages with the community. Table 5.2 shows some of the topics and connections from Cohort 2. While the connections to practice were not always apparent in the community-­ based math lessons of Cohort 1, they were more apparent in the lessons of Cohort 2. This is likely attributable to the strategic supports provided to the PSTs in Cohort 2 that were not afforded to the PSTs in Cohort 1. It seems that the ongoing Table 5.2  Representative CME topics for Cohort 2 CME topic Fire Station and Firefighters Being a Business Owner: Pizza Restaurant Designing and Creating New School Uniform Shirts Safe Community Spaces: Designing a Local Park Planning a Field Trip to Chicago

Math content Multiplication with whole numbers, decimals Adding, subtracting with decimals; making reasonable estimations; multiplication (partial products) Units of measure, measuring with precision, adding many quantities, multiplication Units of measure, measurement, estimation, area, perimeter, conversions Estimation, measurement (distance), rates, multiplication, decimals (money)

Connections to practice Purchasing, budgeting, promoting diversity in workforce Accounting for costs, establishing affordable prices while making a profit Tailoring for varying body types, fundraising Social considerations for design of public spaces; mapping spaces; materials and construction Budgeting time and money, accommodating large groups, fundraising

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conversations about the potential for mathematical practices in community spaces had a direct impact on how PSTs viewed community contexts as the source for mathematical explorations. Furthermore, it seems reasonable to conclude that the explicit support for how PSTs might engage children in conversations about how they interact with family and community translated into a more acute focus on community practices in the design and evolution of the community-based math lessons for Cohort 2. Still, there were PSTs who attempted to incorporate children’s lived experiences into math instruction, but the focus on specific mathematics concepts tended to override the maximization of the community context. This was evident in a pair of PSTs who struggled to fully develop the context of jobs the children’s parents did (i.e., food service, auto mechanic); in this case, the focus was largely committed to working with money, with little dialogue around how the context might be leveraged to help children make sense of mathematical ideas.

 STs’ Historical Orientations to, and New Conceptualizations P of, Mathematics Teaching Data from the PSTs’ Field Journals showed that the CME Module, to varying degrees, facilitated a shift in the PSTs’ conceptions of how mathematics ought to be taught in order to better advance the learning of urban children. In general, PSTs expressed some apprehension about teaching mathematics. They frequently referred to it as a “hard” discipline and, specifically, were not familiar with how to teach it to third-grade students. At the same time, however, there appeared to be a belief among PSTs that there was a need to diverge from the “traditional” forms of mathematics teaching that they had experienced as students and that has successfully served too few students. This shift, or tension, was related to the evolution of their identities as mathematics learners and teachers. For example, one of the PSTs, Mercedes, said, “Traditional math is what I am used to, and it did not work for me. I hated math and that has carried into my adult life, the last thing I want is for my students to feel like math is not for them.” Having been unsuccessful with mathematics might result in difficulty conceptualizing how to teach the subject because it is impossible to draw on experiences of mathematics pedagogy that one did not have. Such barriers with mathematics faced by the PSTs are socially and historically induced and need to be broken to ensure that teachers are able to teach in culturally relevant ways. Like Turner et al. (2012) findings, Mercedes’ statement provides further evidence that many teachers are not prepared to teach mathematics in ways that draw on children’s MMKB.  Still, Mercedes reports that she is “currently thinking about intermediate math in a different light.” She continues: Math doesn’t have to be so separated from what actually occurs outside the classroom … This is problem solving in its most raw form. You give students the tools to problem solve and throw a word problem at them and see how they come out on the other end. This is how we solve problems in the real world; we aren’t always equipped with the best or the right tools.

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Mercedes is describing a shift in the way she conceptualizes mathematics teaching with children of color. Another PST, Jackson, also expressed difficulty reconciling his experiences as a mathematics learner and the importance of drawing on children’s MMKB via CRMT: I struggled in math throughout my years because of the lack of relevancy. The most common question math teachers will hear is “When are we ever going to use this?” To avoid this confusion, I think it is extremely important to be transparent with our students about why we are learning math the way we do.

CRMT is not a simple undertaking; it involves a shared understanding among teachers and students about the utility and applications of mathematics in the community, as well as leveraging children’s MMKB to make sense of abstract mathematical ideas. As stated by PST Jennifer, “Using culturally relevant teaching in the classroom is beneficial for students and teachers, because it builds a strong sense of classroom community.” The community she is referring to is one that is collaboratively formed based on a respect for the resources and ways of understanding that children bring to the classroom. It is the teacher’s responsibility to draw upon these resources in order to help children feel a part of the community. The insights shared by the PSTs are promising signs that they are overcoming the developmental barrier created, in part, by dominant perspectives of what constitutes mathematical knowledge. They are on their way to imagining—and implementing, at least in ways reflective of their novice status—CRMT. Both Mercedes and Jackson show that they have no doubt that CRMT represents an opportunity to help children make meaning of, and develop affinity for, mathematics in ways that have not historically been afforded to them, especially children of color in urban contexts. Given mathematics teachers’ persistent struggle to provide abundant and appropriate opportunities for children of color to succeed in mathematics (Martin, 2015; Spencer, 2015), coupled with an ever-increasing diversity among K-12 children (Kirk, 2016), the significance of the shift in Mercedes’ and Jackson’s perspectives (among other PSTs) cannot be understated. Still, we recognize that implementing CRMT with students from non-dominant backgrounds with depth and fidelity will be neither simple nor automatic. There is also evidence in our data that PSTs are thoughtfully considering how to make CRMT work inside their future classrooms. Dorinda, for example, shares how it is important to get to know students in order to allow PSTs to teach things in ways that are appealing to students: I wanted him to see that even though we were learning new information that we had already been using all of our life that I wanted him to also have fun and that I cared about getting to know him … It is important to get to know the student and interact with them as much as possible because they will give you insights of what their life is and maybe, just maybe, somewhere along the road you will find a student with similar experiences and this way you will be able to connect and understand the new students in a new perspective.

PSTs have identified that being a culturally relevant mathematics teacher means being able to connect mathematics with children’s lived experiences, ultimately making clear the relevance of mathematics in their community. Another PST, Gayle,

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builds on this idea: “I have seen the importance of valuing students’ cultures. I now realize that if students don’t have a personal connection with their education, they will not learn.” PSTs also readily pointed to other factors that make it possible to make connections to mathematical practices to build cultural relevance for students. For example, teachers need to be cognizant of the language they are using in order to ensure effective communication, especially with emergent bilingual students, and maximize students’ access to mathematical ideas. As PST Lidia points out, it is helpful to situate the learning of new mathematics in rich contexts and problem-solving scenarios: We decided to push through and continue attempting the money problems. We realized that a couple of our students were non-native English speakers, and they were not familiar with our currency. So, we adjusted our lesson for the next time to review the coins and values. As soon as we did that, the students did really well with the money problems.

It is encouraging to see insights like these as a result of the CME Module, where PSTs can recognize their responsibility to use multi-modalities to support emergent bilingual students to process new disciplinary language and draw on cultural resources to make sense of mathematics concepts. As Gutiérrez (2015) argued, Latin@ students are often stereotyped as students, or framed through deficit-­ oriented language, when in reality they might need explicit supports to process a new linguistic register and other (dominant) cultural norms of the social context of the mathematics classroom. With recognition of students’ needs—but more importantly, with intentional instructional approaches like CRMT—teachers can contribute to a re-framing of learners from different racial/language backgrounds and help disrupt the perpetuation of negative stereotypes about their mathematical competency. As Lidia is beginning to understand, subtle shifts in how PSTs perceive and interact with children of color can make a significant difference in the affinity or alienation children experience with mathematics learning: [In] the first lesson, one student was quite quiet, but we thought she was just confused about problem-solving like the rest of the group was. When we realized that she just needed to hear the problem in Spanish to be successful, she took off with the lesson and started to show her classmates how to solve some of the problems that were presented to the group.

These CME Module experiences have helped PSTs identify specific elements of pedagogy that constitute CRMT.  In addition, they have pointed to particular readings, namely, Math in the Milpa (Barta, Sánchez, & Barta, 2009), Math Club Problem Posing (Simic-Muller, Turner, & Varley, 2009), and The Case of Isabelle Olson (Smith, Silver, & Stein, 2005), that have helped shape their perceptions of what’s possible, relevant, and critical to include in mathematics instructional spaces with historically marginalized learners. Jennifer expressed that “the readings that we did in class prepared me more than anything for field.” These articles were selected, among others, because they illuminate connections to community mathematics practices and promote a different way of engaging children and their cultural backgrounds. Keisha candidly reflected, “Initially I could not see how you

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Fig. 5.1  Personal and CME experiences

could make math culturally relevant. I was one of those people who assumed you could only be culturally relevant in literacy. That’s what I have grown to know and love about CRP – you can use it in all subjects.” Keisha debunks a long-standing myth about mathematics—that it is somehow culturally and politically neutral—and, in doing so, highlights the relationship between mathematics and CRMT. Figure 5.1 summarizes the personal experiences described by PSTs while engaging children in community-based math lessons. Most PSTs encountered some disequilibrium, given their strong socialization to construct mathematics in narrow ways, and, similarly, had limited ability to envision mathematical teaching in ways other than how they’ve experienced it themselves. Their words, however, suggest that many PSTs were inspired by different aspects of CRMT this semester.

PSTs’ Emergent Conceptions of CRMT As suggested above, PSTs frequently described that they did not have confidence to teach mathematics and did not identify as mathematically strong teachers in the beginning of the semester. The confidence developed by some PSTs comes from the new skills and insights gained through the process of designing, leading, and evaluating learning experiences through the CME Module. Moreover, presumably because of the comprehensive challenge presented to them in the module activities, PSTs were often motivated to innovate new organizational techniques. For example, “teacher outlines” helped Jennifer to improve her lessons by creating and utilizing a sketch of her lesson plan that highlighted the most important activities and key questions to be developed during the lesson:

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C. Willey and W. A. Pinheiro This semester I felt almost too prepared in a very good way. Before each lesson, I would make a teacher outline for the day. In the outline was every question that I would ask, what materials were needed and how much time I planned for each. It was a simpler lesson plan that I was able to look of during lessons to keep on track. There was always a ton planned, even if I knew we wouldn’t get to all of it, because I wanted to be prepared.

Jennifer is describing a robust, non-routine lesson, one that focuses not only on community-based content but also ways of engaging children. It appears as though the scaffolded supports for lesson planning and iterative opportunities to plan and implement community-based math lessons impacted PSTs’ comfort level and confidence with CRMT. Jackson stated, “This semester in field (CME) really saved my confidence when it came to my ability to teach. I came into this semester shaken and very uncomfortable because of my lack of confidence in my ability to interact with students.” During the work with CRMT, PSTs realized how important it is to allow students to work in groups and discuss the problems proposed to them in order to build mathematical knowledge. A key take-away from PSTs’ engagement with the CME Module is that mathematics always provides various ways to solve a problem, and thus, children usually present different solutions for the same problem. As Katrina puts it, “You always have to be even more prepared than you think ‘extra prepared’ is and to have ideas of all the possibilities that could happen during the lesson.” This comment about anticipating and managing students’ different contributions and different ways of reasoning about problems represents movement toward PSTs’ acknowledgment and valuing of children’s everyday experiences and positions PSTs to better leverage these experiences in the development of mathematical knowledge. Sabrina described this thought with the statement and graphic (Fig. 5.2): I chose this picture to show how different languages make dog sounds. None of these ways are wrong, but they are all different. However, they are all still the sound a dog makes. This reminds me of students’ prior knowledge with how their prior knowledge of the sound a dog makes might be different from mine, but they can still use their prior knowledge to know the dog is making a sound.

Also, PSTs suggested it was helpful to have an assortment of experiences working with one child prior to implementing community-based math lessons with small groups. Dorinda said that although it might be hard and take a long time to engage a student who was taken out from his classroom to be working alone with a teacher, this experience gave her the opportunity to practice getting to know a student in meaningful ways, which paved the way for her to devise activities that were interesting for students. Her commentary reveals her understanding of how to engage children in mathematics learning through making connections to community practices and contexts: During my focus student field experience, I really learned that it takes different activities to engage a student, especially when they are pulled from their classroom to work with a person who is a stranger to them. With my focus student I really focused on getting to know him, thinking that I was going to be working with this student for the remainder of the semester. However, by getting to know him a little, I was able to interact with him and do math problems that incorporated some of the things he liked.

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Fig. 5.2  Worldwide woofs and other animal sounds illustrated in different languages (Streams, 2013)

Dorinda’s commentary underscores the importance of scaffolding clinical experiences, which ultimately positions PSTs to maximize the CME Module experiences and extract meaningful insights. Working with one student allowed a depth of interaction that is needed before getting into the nebulous terrain of mining and leveraging children’s MMKB. Since the interactions are likely to get more complex when adding additional children, this one-on-one experience provides PSTs with the opportunity to feel what it is like to talk and know a child deeply, and ultimately understand the benefits of this relationship. As mentioned by Jennifer, it is only through experiencing this relationship with the student that the PST starts to see that the students’ needs, and knowledge, ought to come first. This is a departure from dominant forms of mathematics instruction, where lessons are created independent of any knowledge of or consideration for children’s MMKB. Moreover, the PSTs learned that, from these relationships with students, they can glean significant insights into children’s communities, ones that (eventually) move beyond simple likes, interests, and familiar contexts (see Table 5.2). Some PSTs described how important it was to have worked with third-grade students (compared to Kindergartners). They described it as a challenge, where more preparation was needed because things did not always happen as planned inside the mathematics classroom. Furthermore, if things did not go as expected, they noted that the teacher needed to have the ability to take control of the class and guide it in the right direction. Throughout the lessons during the semester, PSTs

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described how they have learned to manage themselves when students were facing a hard time in figuring out how to solve a mathematics problem. It was fundamental for PSTs to let students work, sometimes on their own, and believe in the capacity and competence of the students. PSTs identified that granting their students some autonomy was a key pedagogical move that allowed students to build knowledge and learn with their funds of knowledge. Sabrina notes, “Before this semester, I never realized how important it is to connect students’ prior knowledge to new knowledge to create a deeper understanding.” Similarly, Lidia mentioned: During the first group problem-solving lesson, the students really struggled with the problems we had created. We took an approach that we had read about, and I decided that ‘I didn’t show them what to do. I just used a context they were familiar with and told them that I wanted them to solve the problem in any way that made sense to them’ (Empson & Levi, 2011, p. 95). This proved to be quite difficult, because they kept looking at us to help them figure it out. This helped me grow as an educator because I had to challenge myself to not step in and try to help the students solve the problem in the way that I would solve it.

Not only did PSTs grow personally and professionally but they also gained an element of consciousness, where they are able to situate new, CRMT practices vis-á-­ vis pedagogical tendencies that are engrained in their conceptualizations about what mathematics instruction ought to look like. PSTs described the importance of working with children in a variety of formats (i.e., small groups or larger classes), which helped them to build new skills related to teaching. Larger classes, for example, required PSTs to design dynamic and sequential activities guided by a set of principles. PSTs have described that analyzing students’ work helped them to know the students better and possibly illuminate emergent needs students might have. This analytical work also helped PSTs to develop subsequent lesson plans that build on new insights about students and specifically address said needs (e.g., building context around problem-solving activities). Through this design–implement–analyze cycle, PSTs described a personal evolution from university students learning about education to teachers meaningfully processing the importance of re-thinking mathematics teaching and learning through a lens of CRP. In collecting new insights and developing new pedagogical tools for CRMT, the CME Module presented a critical opportunity for PSTs to strengthen the conceptual base from which they will work as novice teachers and, arguably, helped provide more clarity around their identities as urban teachers.

Conclusion Throughout this semester the PSTs have gone through a process of development that is introduced in Fig. 5.3. While no stage is neatly bound, we found it helpful to posit three general categories depicting the developmental process described by PSTs: Initial, Formation, and Enduring. First, PSTs started their teaching practice stating that they lacked confidence, strategies, skills, teaching tools, experiences, and mathematical knowledge. As the PSTs’ work with children became more

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Fig. 5.3  Development process

involved, supported by regular, scaffolded reflection and analytics, the PSTs started to develop confidence in their knowledge of CRMT and pedagogical skills and gain trust through relationships with children. These experiences and feelings contributed to the evolution of their teacher identities. We call this stage Formation. Finally, in the Final stage, after PSTs have had the opportunity to step back and reflect on the entirety of the experiences, they are not only able to articulate a brand of mathematics instruction that is appropriate for children of color but they are also able to consciously make choices about which principles of their mathematics instruction they plan to live with. This stage might also be called the Enduring stage, where certainly not everything from CRMT will be embodied in the PSTs’ pedagogy, but much will. This stage is marked by more confidence in skills and knowledge, a more solidified professional outlook, and a more strongly developed teacher identity. After engaging with the CME Module, PSTs realized that CRMT is fundamental for disrupting discouraging patterns of mathematics teaching and learning, where urban children are afforded fewer opportunities to engage in rich, meaningful mathematical activity; CRMT also represents a vision for equitable mathematics ­teaching and learning. CRMT facilitates the process of learning mathematics by making mathematics not only relevant but also worthwhile, which in turn helps students make sense of mathematics and might even serve as a source of motivation to invest more intellectual energy into mathematics. The adjectives relevant, important, meaningful etc. are some, among many other, positive adjectives that show how important PSTs think CRMT is in the learning of mathematics. However, if CRMT is so important and incites surprise among PSTs that mathematics instruction has deviated little from the traditional approaches to teaching from decades past, one may ask why isn’t it happening? As mentioned above, we know that there is a

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difference between teachers agreeing with the idea of CRMT and their ability to enact CRMT in classroom contexts where they manage competing demands. The six-­lesson format of our CME Module activities, coupled with preceding mathematics field experiences with children, seems to support PSTs’ conceptualizations of, and enduring commitment to, CRMT. PSTs have shown evidence of mathematics literacy, where they “indicate the ability to put mathematical knowledge and skill to functional use rather than just to master it within a school curriculum (OECD, 2000, p. 50)” (Stacey & Turner, 2015). Still, as the PSTs noted, CRMT is not very well explored in mathematical curriculum, textbooks, and inside the mathematics classroom. These experiences with the CME Module, however, could represent a turning point in students’ affiliation with mathematics and motivation to engage and persist, as mentioned by Jennifer: Culturally Relevant Teaching (CRT) should become a lifestyle when teaching and become a natural part of the curriculum. It should not be forced and rather than putting the CRT in the lesson, the lesson should be put into CRT. Using Culturally Relevant teaching in the classroom is beneficial for students and teachers, because it builds a strong sense of classroom community.

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 ppendix A: Clinical Experiences and Corresponding A Mediating Activities The table below lays out some of the clinical experiences of PSTS and the ways in which instructors perceive their responsibilities to mediate these experiences with an eye toward supporting critical consciousness. Clinical experience Literacy and Math One-on-­ One Focus Student Sessions (semesters 1 and 2) Curriculum Centers/ Invitations (Semester 1)

Asset-Based Community Survey (Semester 1)

Observations (Semesters 1–4)

What are they? Five to six sequential sessions (for each discipline) in which PSTs focus on problem-solving and student thinking (math) and emergent literacy and funds of knowledge (literacy) Three iterative, small group facilitations for literacy and three for math; pairs of PSTs design, implement and refine a Center based on dimensions of children’s early development of number sense and literacy PSTs spend several hours exploring the local community of their home school. They are organized in groups that focus on community history, resources, and experiences. PSTs are specifically asked to focus on assets and to develop curricular ideas that connect to what they find. Additionally, they look for evidence of inequities in the communities Periodic observations of classroom teaching that range from helping PSTs become familiar with instructional approaches (e.g., workshop model) to practices related to student engagement and behavioral support

Mediating activity Support for session planning; promotion of the focus on children’s thinking; development of analytical skills through debriefing exercises; working toward more expansive views and explanations of children’s behaviors Peer and instructor critique of conceptual plan; review of center lesson plan; reflection and accountability for improvements in between each iterative cycle; guidance in the production of documentation panel/notebook Instructor facilitates post exploration discussions. During these discussions PSTs identify their own expectations and assumptions about the community. PSTs work in groups to develop a multimedia presentation and paper that is shared with the class. School administrators are invited and discuss with PSTs how the teachers are supported to also explore the community and challenge deficit notions they may have about families and resources Arranging a variety of contexts to observe; preparation via developing a lens to focus on certain issues; debriefing around key topics; extending invitations to teachers/staff and facilitating Question & Answer sessions

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Clinical experience Community-­ School Events (CSE) (Semesters 1–4)

What are they? CSEs are opportunities for PSTs to interact with students and families in non-classroom spaces. Typically, these include: Family Fun Night, STEM Night, ISTEP testing and proctoring, District Curriculum Fairs, After School Clubs, After Care programs, front office, lunchroom and dismissal assistance, Book Fair, Hispanic Heritage Night, and others. PSTs are required to attend at least three of these events each semester and many attend more than that Science Inquiry Eight science inquiry-based Unit (Semester 2) lessons are developed by PST teams and taught to groups of 8–10 intermediate age students. The PSTs are required to reflect on how the lessons demonstrate culturally relevant teaching using the three tenets of Ladson-­ Billings (1995) Six, iterative small group Community facilitations; curriculum Mathematics development work stems from a Exploration community member- or (Semester 2) self-guided tour of the community in which PSTs become more acquainted with community spaces and explore the mathematical practices of the community Pairs of PSTs complete six days Paired Student of classroom observations, and Teaching then quickly assume full teacher (Semester 3) responsibilities for eight weeks Individual Student Teaching (Semester 4)

PSTs individually complete six days of classroom observations, and then quickly assume full teacher responsibilities for eight weeks

Mediating activity Connecting PSTs to the community/ parent listserv and phone distribution list that promotes events; providing time and space to make meaning of these events. These events make for productive class discussions regarding biases and stereotypes of urban families. PSTs are required to draw on these experiences for capstone projects (storytelling) each semester

Inquiry lessons are reviewed and supervised by the instructor. Classroom teachers are invited to provide feedback as well. In-class discussions focusing on culturally relevant teaching and critical pedagogy are facilitated by the instructor

Arrangement of a community walk led by a community member (e.g., parent); Support in interviewing children around their cultural resources; Peer and instructor critique of conceptual plan of lessons; Sustained dialogue from week-to-week; Strategic debriefing

Multiple informal and a minimum of two formal observations by University (Faculty) Coaches; Sustained dialogue; Weekly seminars; Support for mentor (classroom) teachers Multiple informal and a minimum of two formal observations by University (Faculty) Coaches; Sustained dialogue; Weekly seminars; Support for mentor (classroom) teachers

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References Aguirre, J., Mayfield-Ingram, K., & Martin, D. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. Reston, VA: The National Council of Teachers of Mathematics. Barta, J., Sánchez, L., & Barta, J.  (2009). Math in the Milpa. Teaching Children Mathematics, 16(2), 90–97. Battey, D. (2013). “Good” mathematics teaching for students of color and those in poverty: The importance of relational interactions within instruction. Educational Studies in Mathematics, 82(1), 125–144. Bonilla Silva, E. (2010). Racism without racists: Color-blind racism and racial inequality in contemporary America. Lanham, MD: Rowman & Littlefield. Chapin, S., O’Conner, C., & Anderson, N. (2003). Classroom discussions: Using math talk to help students learn: Grades (pp. 1–6). Sausalito, CA: Math Solutions Publications. Empson, S.  B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann. Gonzalez, N., Moll, L. C., & Amanti, C. (Eds.). (2005). Funds of knowledge. New York: Routledge. Gutiérrez, R. (2015). HOLA: Hunt for opportunities-learn-act. The Mathematics Teacher, 109(4), 270–277. Jackson, K. (2009). The social construction of youth and mathematics: The case of a fifth-grade classroom. In D. B. Martin (Ed.), Mathematics teaching, learning, and liberation in the lives of Black children (pp. 175–199). New York: Routledge. Kirk, A. (2016, January 21). Mapped: Which country has the most immigrants? The Telegraph. Retrieved from http://telegraph.co.uk Ladson-Billings, G. (1995). But that’s just good teaching! The case for culturally relevant pedagogy. Theory into Practice, 34(3), 159–165. Ladson-Billings, G. (2006). Yes, but how do we do it? Practicing culturally relevant pedagogy. In J. Landsman & C. W. Lewis (Eds.), White teachers/Diverse classrooms: A guide to building inclusive schools, promoting high expectations, and eliminating racism. Sterling, VA: Stylus Publishing. Ladson-Billings, G. (2009). The dreamkeepers: Successful teachers of African American children. San Francisco: Jossey Bass. Ladson-Billings, G. (2014). Culturally relevant pedagogy 2.0: aka the remix. Harvard Educational Review, 84(1), 74–84. Martin, D. B. (2006). Mathematics learning and participation as racialized forms of experience: African American parents speak on the struggle for mathematics literacy. Mathematical Thinking and Learning, 8(3), 197–229. Martin, D.  B. (2015). The collective black and “principles to actions”. Journal of Urban Mathematics Education, 8(1), 17–23. McGee, E. O., & Martin, D. B. (2011). “You would not believe what I have to go through to prove my intellectual value!” Stereotype management among academically successful Black mathematics and engineering students. American Educational Research Journal, 48(6), 1347–1389. Pewewardy, C. (1993). Culturally responsible pedagogy in action: An American Indian magnet school. In E. Hollins, J. King, & W. Hayman (Eds.), Teaching diverse populations: Formulating a knowledge base (pp. 77–92). Albany, NY: State University of New York Press. Simic-Muller, K., Turner, E.  E., & Varley, M.  C. (2009). Math club problem posing. Teaching Children Mathematics, 16(4), 206–212. Smith, M. S., Silver, E. A., & Stein, M. K. (2005). Improving instruction in geometry and measurement (Vol. 3). New York: Teachers College Press. Spencer, J. A. (2015). African American, minoritized student in school mathematics: New American or designated serf? In T. G. Bartell, K. N. Bieda, R. T. Putnam, K. Bradfield, & H. Dominguez (Eds)., Proceedings of the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. East Lansing, MI.

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Stacey, K., & Turner, R. (Eds.). (2015). Assessing mathematical literacy: The PISA experience. Cham: Springer International Publishing. Streams, K. (2013, October 24). Worldwide woofs and other animal sounds illustrated in different languages. [Web log comment]. Retrieved from https://laughingsquid.com/ worldwide-woofs-and-other-animal-sounds-illustrated-in-different-languages/ Turner, E.  E., Drake, C., McDuffie, A.  R., Aguirre, J., Bartell, T.  G., & Foote, M.  Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67–82. Vygotsky, L.  S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Willey, C., & Livers, S. (2018). Forging new terrain in critical mathematics teacher education: The role of collaborative reflective practice. Teaching for Equity and Excellence in Mathematics, 9(1), 6–16. Young, E. (2010). Challenges to conceptualizing and actualizing culturally relevant pedagogy: How viable is the theory in classroom practice? Journal of Teacher Education, 61(3), 248–260.

Chapter 6

Prospective Teachers’ Reflections Across the Community Mathematics Exploration Module Kathleen Jablon Stoehr

Keywords  Teacher education · Reflection · Community connections · Mathematics education · Engagement · School mathematics · Diversity · Equity · Teaching practices A growing body of research indicates that connecting mathematics instruction to familiar experiences in children’s homes and communities is important in supporting student learning (Civil, 2002; González, Andrade, Civil, & Moll, 2001; Wager, 2012). Many elementary prospective teachers (PSTs), and particularly those from minoritized communities, however, did not experience this type of learning of mathematics when they were students. For many PSTs, elementary mathematics teaching instruction was  strictly tied to a given textbook that incorporated generic real-world contexts that did not connect to any particular children’s multiple mathematical knowledge bases (MMKB) but was generally most familiar to white, middle-­class students. This creates a challenge for mathematics teacher educators (MTEs), as schools today are culturally, linguistically, racially, and socioeconomically diverse, requiring a myriad of different home and community connections (Foote et al., 2013; Kena et al., 2015). Therefore, MTEs need to offer opportunities that help PSTs learn how to connect their mathematics teaching to the everyday experiences of their students to foster successful mathematics learning (Leonard, 2008; Turner et al., 2016). The TEACH Math Community Mathematics Exploration (CME) Module was created to address this need. The CME Module offers PSTs the opportunity to learn about their students’ and their families’ home and community knowledge, investigate the mathematics in their students’ communities, and create a mathematics lesson that ties this together for the purpose of learning mathematics (see Chap. 2 of this volume). This chapter examines the reflections of 33 PSTs, who as part of their mathematics methods course, engaged in the Community Walk Activity of the CME Module in local communities during their teacher preparation program. As an MTE, K. J. Stoehr (*) Santa Clara University, Santa Clara, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_6

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I believe it is important to examine the reflections of PSTs, as their reflections can help to inform my teaching. Moreover, for PSTs, reflections can be a valuable learning tool that offers the opportunity to think back on one’s experiences as a means to understand what had occurred (Schon, 1983).

Instructional Context These 33 PSTs were enrolled in their second of two quarters of elementary mathematics methods at a university located in the western region of the United States. All of the PSTs were simultaneously student teaching in an elementary classroom 5 days per week and were seeking their Masters of Education degree and teaching credential. There were 30 women and 3 men. Twenty-four of the PSTs were White, five were Asian, two were Latina, and two were from India. Most of the PSTs were in their 20s; however, there were several candidates who were second career seekers and therefore older. The PSTs worked in small groups of three to five people to create a lesson that centered on the mathematics they observed in their students’ communities. The PSTs first talked to their students about the places they go with their families and then investigated two of the sites their students mentioned. The PSTs visited and talked with people at both sites. After these visits, the PSTs chose one of the locations for continued focus. PSTs were required to write a description of these community visits describing how their community visits informed their mathematical problem-solving lesson plan. PSTs then created a lesson plan that was tied to a specific grade level and was linked to the California Common Core State Standards in Mathematics (National Governors Association for Best Practices, Council of Chief State School Officers [NGA CCSO], 2010). The PSTs invited their university instructors, university staff members, and peers from the college to participate in a community fair experience to showcase their CME Module experiences. Guests walked from project to project where PSTs shared their work through visual representations including tri-board posters, power points, and artifacts they collected from their sites. The PSTs’ final task was to write an individual reflection on their CME Module experiences. Table 6.1 outlines the questions that PSTs were asked to reflect upon after completing their CME Module activities. Bolded are the questions that are a focus of this chapter.

Analysis My first analytic pass through the data focused on thoroughly reading each PST’s reflection. I reread each PST’s reflection a second time with specific attention on the two main categories of questions: (a) questions that pertained to the community visits and (b) questions that centered on lesson planning and mathematics teaching.

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Table 6.1  Community mathematics exploration module reflection questions Related to community visits Write about your experiences visiting the two places in the community What did you learn about the community, including the mathematical activity occurring in the community? What did you learn from your conversations with and observations of community members about their work and practices? How do community members frame the role of mathematics in their work and daily activity? What did you learn about your students and their families? What did you learn about yourself? As you participated in the different parts of this project, what surprised you? What did you expect before engaging in this activity? Did your expectations/preconceptions change because of this experience? In what ways?

Related to lesson planning/math teaching Write about your experiences drawing on what you learned during your community visits to design a mathematics problem-solving lesson. What were your goals in designing the lesson? What connections did you try to make? What do you think might be the benefits of this kind of math teaching (that draws on/connects to the mathematical funds of knowledge in students’ communities) for your students? What do you see as challenges related to this kind of mathematics teaching (that draws on/connects to the mathematical funds of knowledge in students’ communities)? How might you respond to those challenges? What are some other ways you might use the knowledge and understandings that you gained from the community visits to inform your mathematics instruction?

I then began an iterative analysis (Bogdan & Biklen, 2006) to sort the PSTs’ key ideas regarding their experience across all of CME Module reflection questions. Examples of these key ideas include the many ways that mathematics exists in communities, thinking about the role that students’ home and community plays in mathematics, and considering a future CME project in their mathematics classroom. I then separated the PSTs’ key ideas into five categories that captured the main ideas across all the questions. Within each category are themes that I titled using a composite of the PSTs’ words to encapsulate the essence of the reflections. The findings of this study suggest that PSTs found the CME Module to be valuable in helping them to think about how to make connections in their mathematics teaching to their students’ home and community experiences. In the next section, the findings of this study are organized by category and theme.

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Findings Opportunities to Learn About Students and Their Communities Getting to know students  By engaging in the CME Module activities, PSTs shared that they learned how important it was to get to know their students and their communities. Elsa discovered that the types of activities she thought her students would be interested in were quite different from her assumptions. For example, she learned from her students that they spent time with their families going to the library or visiting local museums. Elsa acknowledged that this was quite different from her experiences as a child as her weekends were spent playing sports. Without that knowledge, she learned that it would be difficult to make her mathematics lessons relevant to her students. As a result, she reported that the CME Module was a great way to “create a lesson that really sparked students’ interests.” After completing the module activities with her group, Clara shared that she learned how important it was for her to learn about the culture of the community in which she was teaching so that she could understand her students better. She reported that the CME Module activities offered her “a great way to increase my understanding and awareness of my students and my teaching practice.” This indicates that engaging in the CME Module activities helped PSTs to see that this could be a valuable way to get to know their students. Investment in the local community  The CME Module provided PSTs the opportunity to learn about the ways that students and their families spend time in their communities. Randi shared that upon learning her students and their families spent a significant amount of their free time at the local community center, she was able to “take their real life experiences and create enriching and meaningful mathematics tasks.” Marla learned that her students and their families enjoyed outings in their community, especially those that created memories of good times. Nina discovered that her students and their families appeared to spend their leisure time close to home, choosing to support the local businesses in their community. Ashley also found that her students and their families stayed close to home, spending time together at parks, hiking, and/or biking. The CME Module activities helped PSTs to see how and where their students and their students’ families spent time in their communities as well as how communities impacted students’ day-to-day lives. Embracing the community mathematics connection  Some PSTs stated they were hesitant to go out into the community and talk to people working/shopping/ participating in the local businesses and organizations in their students’ communities. They worried that people in the community would not have the time or the interest to talk with them. What PSTs found, however, was that many of the community members were excited about the CME project and were eager and willing to share the mathematics they used on a daily basis. Candy, whose group investigated a local farmer’s market and cupcake store, reported, “Everyone we spoke to was

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incredibly friendly and willing to share their thoughts on the role of math in their work … They expressed how large a role math plays in their daily business activities.” After Susan and her group visited a community library and a local zoo, she stated: Everyone that we spoke with was extremely supportive of our project and wanted to help in any possible way they could. I feel that this is an extension of their customer service perspective and the idea that they are providing services to better their community.

These types of positive interactions with community members may lead to PSTs continuing to learn more about the mathematical practices in the communities in which they will ultimately teach. Additionally, these interactions might also help PSTs to reframe how they see community members. For example, the PSTs learned that community members have great expertise, they support children’s learning, they want to collaborate with schools, and they engage in mathematics regularly. Limitless links  PSTs shared that the CME Module provided them with the opportunity to learn about the importance of linking their mathematics teaching to their students’ lives. Clara shared how she thought she understood what it meant to make connections to student understanding by relating the content she was teaching to her students’ funds of knowledge. Having previously created a non-fiction book about students’ favorite cultural foods, she assumed she had accomplished this goal. However, after the CME Module activities she stated: This project proved to be different, as it pushed me to think more deeply about authentic mathematics in the community. Instead of merely creating a community “theme” for a purely academic lesson, our group had to consider why students were learning particular mathematical concepts and how these concepts might be used in daily community life.

Susan shared that the CME Module supported her to consider all the different ways that mathematics can be represented in community businesses and organizations. She reported, “This project helped me to look at all of the opportunities that there are to create meaningful math lessons out of what could be perceived as an everyday experience.” Nancy found that by engaging in the CME Module she made connections for herself as well. She stated, “I had not thought about how many ways the use of math surrounds our lives, and it made me excited about moving forward with ideas for future lessons.” These reflections suggest that PSTs were viewing the CME Module as a means to make multiple types of connections in their mathematics teaching; connections they viewed as valuable to student learning. Moving beyond the borders of the CME Module  After completing the CME Module, many PSTs reflected upon ways in which they could extend or add other elements to the module to integrate students’ communities into their learning experiences in mathematics classrooms. Several PSTs expressed an interest in using a survey to find out more about their students’ home and community knowledge. As Ashley thought about her future classroom of students, she stated:

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Completing the CME Module activities supported PSTs to think of other ways (like a survey) they might gain information in order to make connections to their students and their students’ home and community knowledge.

Promotes Student Engagement Greater interest and much more  The PSTs had many thoughts about how teaching mathematics building on community mathematics practices and students’ community knowledge could benefit their students. All 33 PSTs believed that mathematics tasks that revolved around a local business or organization in their students’ communities had the potential to lead to greater student engagement. For example, Casey and her group created a mathematics problem-solving lesson around a local zoo that all of her students had visited. She said: I think the benefits for this kind of math lesson is how engaging and relatable it is. The students have been to this community place and can relate to the [mathematics] activity more. The students will be able to have a better understanding of what the facility is about and really contribute more to the tasks because they enjoy the place … It [the lesson] can hit so many different standards and subjects that it could create a great project.

After implementing her lesson that revolved around a skateboard park, Rhonda envisioned how engaging the mathematics tasks she and her group created could be for students. She believed this engagement could lead to reshaping students’ beliefs around mathematics competency. She stated: By connecting math to their community, students are more interested in the content. Math is commonly a subject where students think they are or are not “math people.” This can be debunked by enhancing their learning and interests with the content … This can be powerful in advancing students’ math knowledge.

Other PSTs thought that engaging in a mathematics lesson that was connected to their students’ communities could be viewed as more exciting and fun, thus perhaps resulting in greater mathematics learning. The PSTs experienced how many different ways a mathematics problem-solving lesson integrating knowledge of students and their mathematical practices at home and in the community could have an impact on their mathematics teaching. PSTs did not, however, make the connection to how a CME could provide an opportunity for students to access the mathematics being taught. Building classroom community  Many PSTs found that getting to know about the activities and places where their students and families spent time in the community

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helped to create a sense of community in their own classrooms. Clara stated that her group of PSTs informally polled their students to learn about the community spaces they enjoyed visiting with their families. She reported: I found that students were excited to share this aspect of their lives with me, often recalling happy family memories in the process. Other students would then gather around and chime in with their own experiences and suggestions. The data gathering process helped create enjoyable dialogue and a greater sense of classroom community.

Marla stated that as she learned more about what her students and their families liked to do in their community, she could use that information to illustrate what the students had in common with each other. Moreover, Marla learned that she enjoyed many of the same activities as her students, which helped to create a sense of shared community in her classroom. Janey shared that engaging in the activities of the CME Module supported her in building relationships with her students. She stated, “Asking students what activities they like to participate in on the weekends, welcomed me into their personal lives.” As a result, Janey believed that this could help to build a strong sense of community in her classroom. These three examples demonstrate the potential that the CME Module may have in supporting PSTs in establishing an engaging classroom community in their mathematics classrooms. Beaming with pride  Through the CME Module, PSTs experienced how powerful learning about what their students and families do and where they spend their time could be. Lannie stated: I really learned that I should take the time to ask my students what they like to do in their free time in order to find those connections to write lesson plans around. Upon asking my students what they enjoy doing, they opened up to me like never before and were genuinely thrilled that I was taking interest in their extracurricular activities.

Susan shared that her students “beamed with pride” when she mentioned a tradition or custom that they participate in or when she mentioned a place that had significance and meaning to them. Ashley reported, “Gathering information from my students helped show me how willing they are to share information about their lives with me.” These three examples illustrate how CME Module activities can help PSTs recognize the importance of learning about their students and their students’ families to make their mathematics teaching more engaging and relevant for their students.

Real Mathematics Versus School Mathematics Making mathematics matter  Some PSTs noticed that the CME Module activities created a bridge to “real math;” that is, math found in the “real world,” versus “school math,” or math found in textbooks. Upon reflection of his experience in creating a problem-solving lesson plan around a community center, Jack stated,

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“This connects math to things that really matter to students.” Angela experienced how her group’s mathematical tasks offered students the opportunity to learn about proportional reasoning by adjusting a recipe from ingredients purchased from a local farmer’s market. She reported: We could have designed a very similar lesson where students were practicing multiplying fractions on worksheets or with basic word problems … After a task like this, it is beneficial that students see they can do math even when they go to the grocery store or farmer’s market with their parents.

Some PSTs also found great value in the CME Module in that connecting mathematics to the real world was valuable when their students asked why they needed to learn a particular mathematics concept. Joan talked about her own experiences learning mathematics and stated that she did not see how it [mathematics] applied to her until she saw it as “real.” She said, “Math for me did not become interesting until I knew I needed it in my daily life. Once I saw that math could make my life easier, I was onboard.” Alan shared that after creating his lesson plan based on knowledge of his students and their communities, he could see the importance of teaching his students just how “real” math is in their everyday settings as well as the usefulness and practicality of mathematics. Taken together, these examples highlight how PSTs saw value in creating mathematics lessons that reflected ways that math is used in “real life.” So much math!  While visiting the different businesses and organizations in their students’ communities, the PSTs expressed great surprise in how prevalent mathematics appeared to be. They talked about the opportunities for finding mathematics “in the most unlikely places” as well as all the different ways in which mathematics was represented in community spaces. After Kate and her group visited a local taqueria and miniature golf site, she was amazed at how much mathematics was integrated in running these two businesses. She stated, “I have been to both of these locations before but never with an eye for math.” Randi and her group visited a local park and a community center. Upon reflection of the visits, she reported: As I think back on my experiences at [Local] Park, and the [Local] Community Center, I realize that there is a lot of math around us! I never really stopped to think about math in this way. I realize how powerful this idea is! Creating “an eye for mathematics” through engaging with the CME Module may support PSTs to incorporate relevant mathematics tasks in their future classrooms.

More ways to make mathematics matter  Sandy suggested that a career day could also be incorporated into the lesson planning based on CME Module activities. She reflected on her ideas as follows: In the future, it would be interesting to build a community-based math lesson off of a career day at a school. We could ask students to interview parents or relatives about how they use math at their job and develop a lesson around the information that students collect.

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Some PSTs also considered integrating the lesson plan based on CME Module activities with another curriculum area. For example, Jack thought about how he might incorporate a CME into a literature-based mathematics lesson. For some PSTs, the CME Module gave them additional ideas of how they could connect their students’ home and community knowledge to their mathematics teaching as well as with other content areas such as social studies and science. This suggests that the CME Module activities have the potential to help PSTs consider how important it is to make mathematics more meaningful to their students.

Springboard to Learning About Diversity and Equity Discovering diversity  PSTs talked about how the CME activities offered them the opportunity to learn about the diversity that made up their students’ communities. They were pleased to see not only different races and cultures of people but also how many diverse activities, restaurants, and business establishments there were in a community. After completing her CME Module activities Ann said, “It was a great experience to see cultural diversity of people and produce [cultural diversity of people and different varieties of produce].” Pete learned about the diverse services at a local community center and about the impact the center had on people of all ages. He reported, “I learned there is a big gym that people use for swimming, badminton, and basketball. There is a large theatre that hosts a local theatre organization.” Randi added that there was a vast amount of space for a variety of outdoor activities for families to utilize as well as a senior center. She said, “I realize that the community center is helping adults and children of all ages and backgrounds to build unique relationships in their neighborhood.” Although what the PSTs learned about their students’ communities in these activities were not deeply explored, it was a starting point for PSTs to build on communities’ strengths as well as connect to the mathematical richness in communities. Equity for all  Some PSTs expressed concern for choosing community sites or businesses that all students in their classrooms were familiar with and had access to, having spent a year in classrooms comprised of students from different language, race, cultural, and socioeconomic backgrounds. For example, Katrina reported: A main challenge for me for this lesson was the diversity of the students in my community, particularly the different levels of economic resources available to my students. It was challenging to find community locations that would be accessible, relevant, and familiar to all my students. My school serves a student population that is highly stratified in terms of economic status. For some of my students a new book or a special candy might be an easy indulgence, the cost of which their parents might not give a second thought. For others they [candy or book] might be a rarely indulged luxury. In many ways, this assignment highlighted for me the extent to which some of my students live in parallel worlds.

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Kate shared the following thoughts, “Bringing in the community is important but we want to make sure it is not making parts of the class feel left out or like they cannot access the information.” Jeri and her group also considered that all students may not be familiar with the community site addressed in their mathematics lesson. Referring to the gelato lesson her group created she said, “While this isn’t a huge roadblock to their [the students] achieving the learning goals of the lesson, these students [the students not familiar with the community site] may feel left out, or less able to share in a common experience of the class.” Taken together, these three examples demonstrate PSTs’ concern to create an accessible mathematics lesson for all their students. This was an important consideration for the PSTs, as the time spent in their student teaching classrooms revealed to them that not all students had access to the same out-of-school experiences.

A Guide to Teaching Practices Looking through a mathematical lens  After completing the CME Module, PSTs spoke of the importance of thinking carefully about how they wanted to teach mathematics in their future classrooms. Their reflections ranged from “this is a great way to immerse myself in the community” to “this project pushed me to think more deeply about authentic math in the community.” Emma reported, “My perceptions certainly changed because I learned how to view an ordinary, everyday setting in a mathematical way. My way of looking at my surroundings has shifted a bit for the better.” Franny and her group created a mathematics problem-solving lesson plan that revolved around a candy store. Feeling optimistic about her future mathematics teaching, Franny stated: I felt like a kid exploring a candy store with an overwhelming scent of sugar and the colorful wrappers all around me. If I would have had math assignments that were related to my community as a kid, I would have loved math.

Expanding the depths of mathematical learning  Some PSTs spoke of how a mathematics problem-solving lesson drawing on CME Module activities might instill in students a deeper appreciation of mathematics. As Alan reflected on this assignment he said: Math education ought to begin with the appreciation of math in our world and out of that wonder and appreciation begin to teach the building blocks of strong mathematical thinking … The community-based math lesson can be a means for the students to experience some of that appreciation … that math has to do with them.

Kristi thought about how mathematics learning that was connected to students’, families’, and students’ community experiences might “help foster an enthusiasm for exploration and lifelong learning.” Angela reflected on the importance of creating mathematical tasks that were linked to students’ communities so she could help

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prepare students for the mathematics they would experience throughout their lives. By completing the CME Module, Carol learned how everyday applications of mathematics might result in students becoming more aware of the importance of mathematics. She hoped that creating tasks around everyday situations students’ experience “might create new connections and develop new knowledge by exploring their world and problem solving.” Through completing the CME Module, PSTs envisioned how deep mathematics learning could be for their students and how their students might experience mathematics that “has to do with them.” Wrestling with the curriculum  Some PSTs gave careful consideration as to how community-based mathematics tasks could align with their required mathematics curriculum. When Emma thought about integrating this kind of teaching into her future mathematics class, she reported, “When teachers are stressed to get through their curriculums and keep up with the pacing guides, it can be hard to find the time to add these sorts of activities and problems into the schedule.” Kristi also pondered this issue and said, “I could see how this might be a challenge to implement based on our restrictions with being tied to a curriculum.” However, she added that integrating students’ home and community experiences was “worth pushing for.” When Cindy thought about integrating students’ community experiences into her mathematics teaching, she noted that it would require a lot of effort on her part to integrate it into the curriculum. She believed that if she could become familiar with the curriculum, she would be able to make this integration a reality in her mathematics classroom. By completing the CME Module during their teacher preparation program, PSTs had the firsthand experience of trying to imagine how they would incorporate what they learned through the CME Module into their own future practice. Although they were concerned with how and if they would be able to do so when they had the sole responsibility of planning and teaching mathematics, they experienced how valuable this sort of teaching could be. Ticking time clock  Although all 33 PSTs saw the benefits of creating mathematics lessons that were tied to their students’ communities, many of them wondered how they would be able to do so when they would be responsible for their own classrooms. Lannie shared that although she enjoyed the CME Module and learned a lot about her students, their families, and their communities, she found the project to be quite time consuming. Candi echoed Lannie’s thoughts and said, “Community-­ based math teaching can require a considerable amount of preparation time and work on the teacher’s part. At the same time, exploring and investigating the community in which students live is a valuable endeavor.” Sandy added that she saw the greatest challenge in creating a community-based math lesson as the initial time needed to learn about her students’ community but believed that “the time investment is worth it in the long run because it helps develop a real-world understanding of math.” Ashley’s concern was finding the time to create the community-based tasks as opposed to following the prescribed curriculum. She hoped, however, that she would be able to design a community-based activity bi-monthly. These examples illustrate that although PSTs expressed concern over the time needed to create

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mathematics tasks that were linked to their students’ home and community knowledge and experiences, they remained hopeful that they would be able to do so. Shaping my mathematics teaching philosophy  All 33 PSTs reflected on how the CME Module impacted their ideas regarding what it meant to teach mathematics. Kate talked about how lessons that revolved around community-based mathematics have “possibilities that are truly unlimited and really can have an amazing impact on math instruction in a classroom.” Rhonda reflected that she found this module “extremely enlightening and will be powerful in reforming my math teaching philosophy.” After completing the CME Module, Casey stated that one of her goals for her students was for them to “see that math will always be with them.” Jeri noted that one of the tenets of her mathematics teaching philosophy was to “help students develop a healthy and open attitude towards math.” Nola spoke of how her mathematics instruction would draw upon students’ community experiences so that she could “provide students with a measure of comfort in their math learning.” Nola also talked about drawing upon something that was familiar to students when introducing new topics so “they aren’t overwhelmed.” These examples demonstrate how engaging in the CME Module may positively influence PSTs’ commitment to incorporate students’ home and community knowledge into their mathematics teaching.

Conclusion The reflections of these 33 elementary PSTs indicate that the CME Module is one way in which MTEs may help PSTs to recognize the importance of making connections in their mathematics teaching that promote student learning. The reflections written by PSTs upon the completion of the CME Module suggest that this TEACH Math module has the potential to guide new teachers in their planning and implementation of mathematics lessons. Moreover, the CME Module offers opportunities for PSTs to learn about their students and their families, their students’ communities, and the benefits and challenges of mathematics teaching that is relevant to their students’ lives, as well as opportunities to learn about themselves. The CME Module activities can also help PSTs begin to think about other ways that they can make mathematics more accessible to their students for the purpose of supporting successful student learning.

References Bogdan, R.  C., & Biklen, S.  K. (2006). Qualitative research for education: An introduction to theories and methods (5th ed.). New York: Pearson. Civil, M. (2002). Culture and mathematics: A community approach. Journal of Intercultural Studies, 23(2), 133–148.

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Foote, M. Q., Roth McDuffie, A., Turner, E. E., Aguirre, J. M., Bartell, T. G., & Drake, C. (2013). Orientations of prospective teachers towards students’ families and communities. Teaching and Teacher Education, 35, 126–136. González, N., Andrade, R., Civil, M., & Moll, L. (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. Kena, G., Musu-Gillette, L., Robinson, J., Wang, X., Rathbun, A., Zhang, J., et al. (2015). The condition of education 2015 (NCES 2015–144). Washington, D.C.: U.S. Department of Education, National Center for Education Statistics. Retrieved April 19, 2016, from http://nces.ed.gov/ pubsearch Leonard, J.  (2008). Culturally specific pedagogy in the mathematics classroom: Strategies for teachers and students. New York: Routledge. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, D.C.: Author. Schon, D. (1983). The reflective practitioner: How professional think in action. New York: Basic Books, Inc. Turner, E., Foote, M., Stoehr, K., McDuffie, A., Aquirre, J., Bartell, T., et al. (2016). Learning to leverage students’ multiple mathematical knowledge bases in mathematics instruction. Journal of Urban Mathematics, 9(1), 48–78. Wager, A. A. (2012). Incorporating out-of-school mathematics: From cultural context to embedded practice. Journal of Mathematics Teacher Education, 15(1), 9–23.

Chapter 7

From Community Exploration to Social Justice Mathematics: How Do Mathematics Teacher Educators Guide Prospective Teachers to Make the Move? María del Rosario Zavala and Kathleen Jablon Stoehr

Keywords  Elementary education · Mathematics education · Social justice · Teacher learning · Teacher education · Mathematizing · Bilingual

The idea for this math task [on planning a march to support immigrant rights] came from conversations with my students. With them we have talked about where we come from. Many students mentioned that their parents came from other countries … some mentioned that they know people who are here illegally. This inspired me to write a task that responds to their concerns, showing how one can take action around one’s questions, using the neighborhood that they all know. – Carlos, prospective teacher [Translated from Spanish] We wanted to bring in a social justice/awareness piece to the [children’s zoo] project so that once students found data to support saving the rainforest, they could then use the data to potentially organize a fundraising drive or a letter writing campaign, something to get them involved in the cause if it was important to them. – Sara, prospective teacher In order to democratize math, students need to be explicitly shown that the everyday math that occurs around them counts - math does not belong only to highly trained specialists making complicated calculations. – Katrina, prospective teacher

In this chapter, we analyze mathematics tasks prospective teachers (PSTs) developed as part of the Community Mathematics Exploration (CME) Module and PSTs’ reflections about those tasks across two contexts located in urban areas of the San Francisco Bay. The PSTs represent three cohorts of prospective elementary school M. R. Zavala (*) San Francisco State University, San Francisco, CA, USA e-mail: [email protected] K. J. Stoehr Santa Clara University, Santa Clara, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_7

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teachers at two universities, each taught by one of the authors of this chapter. As mathematics teacher educators (MTEs), we were curious to understand in what ways the CME Module could support PSTs to design mathematics tasks that connect to students’ multiple mathematical knowledge bases (MMKB; Turner et al., 2012) and included issues of social justice relevant to students’ communities (e.g., rising cost of housing, immigrant rights, urban community’s access to healthy food and safe transit). MMKB are defined by Turner et al. (2012) as “the understandings and experiences that have the potential to shape and support children’s mathematics learning    –  including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge in mathematics instruction” (p.  68). We were particularly interested in how PSTs might include opportunities to connect to social justice issues in their mathematics tasks because these issues are a part of students’ MMKB yet are not often built upon in mathematics instruction. In each of our teacher education programs, social justice figures centrally as a key part of teaching mathematics for equity. Social justice in mathematics can be conceptualized in a variety of ways, and in our courses, it takes two primary routes: (a) explicitly utilizing mathematics to understand and act on an issue of social injustice (Gutstein, 2006; Gutstein & Peterson, 2013) and (b) transforming discourses of mathematics understanding and engagement for traditionally marginalized youth (Gutiérrez, 2013; Leonard & Martin, 2013). Typically, our PSTs draw on these two themes in different ways: first, with respect to how they explain the significance of tasks they create for their students, and second, in what ways these activities are thought to be meaningful to their students. We feel strongly that activities such as those in the CME Module can help PSTs to both develop their own understanding of what teaching mathematics for social justice can mean, as well as learn how to develop curriculum that supports bringing issues of social justice into the mathematics classroom. To that end, in our courses PSTs are encouraged, but not specifically required, to construct mathematics tasks for the CME Module with attention to social justice issues impacting their students. We use the CME Module as an entry point to embedding social justice mathematics in the curriculum. By this, we mean constructing mathematics tasks so that students think critically and make sense of mathematics in the service of understanding their communities in mathematical ways. The practice of “mathematizing,” which is “a process of inquiring about, organizing, and constructing meaning with a mathematical lens (Fosnot & Dolk, 2001)” (Hintz & Smith, 2013, p. 104), factors heavily in our definition. Beyond that, we see great potential in the CME Module activities to support PSTs to take what we might call a “next step” in their journey as social justice mathematics educators and engage in issues of fairness, power, and inequality through mathematics lessons that connect meaningfully to students’ lives and support students to use mathematics to understand and act on the world around them. We see this next step as important for all teachers, but especially for mathematics teachers and PSTs given that critical mathematics (Gutiérrez, 2013; Gutstein, 2006) should play a key role in a transformative experience around mathematics for all students.

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A variety of factors, however, influence whether PSTs are ready to take this next step within the context of their limited time in a mathematics methods class (see, e.g., Zavala, 2016). In one study, Aguirre, Zavala, and Katanyoutanant (2012) found that when PSTs rated their own mathematics lessons as containing strong ties to community knowledge, they were more likely to find ties to social justice as well. This suggests an important area for further research, part of which we take up here, which is to better understand how PSTs build on children’s community knowledge as an entry point in developing rich and rigorous mathematical tasks that may help children better understand a social (in)justice issue impacting their community. Identifying community contexts in which to situate mathematics can be challenging (e.g., Civil, 2007; Trexler, 2013). In addition, going beyond the superficial features of the context and connecting to a significant theme in students’ lives is challenging (e.g., Aguirre et al., 2013; Turner et al., 2012). The CME Module activities created opportunities for PSTs to develop their own understanding of their students’ communities as well as demonstrate and deepen their ability to connect to issues that can significantly impact their students. In our work, we think of PSTs as developing through a number of processes in which they learn to (a) mathematize the world around them, (b) demonstrate competencies in mathematics, and finally (c) raise their own awareness of how social justice issues connect to mathematics. In this chapter, we analyze PSTs mathematics tasks (n = 23) and reflections (n = 33), produced from the CME Module activities, to better understand how a sample of PSTs developed around the final component, specifically what PST work from CME Module activities tells us about their developing sense of how to integrate social justice into mathematics curriculum. To complete this analysis, we focus on mathematics tasks that have strong connections to social justice, as well as those that hold the potential to connect to social justice. We include those tasks with potential in order to deepen the discussion of how we as mathematics teacher educators (MTEs) might support PSTs to take the work they have done and revise it for future use, toward a mathematics task with stronger social justice connections. First, we describe the way CME Module activities are utilized in our courses, and then we describe the analysis we undertook as well as the results stemming from that analysis.

 ow We Use the CME Module in Our Teacher Education H Programs Both of us, Maria (first author) and Kathleen (second author), utilize the CME Module in our mathematics methods courses for multiple-subject (K-8) PSTs. There are many similarities in the ways we frame the activities around “funds of knowledge” (González, Andrade, Civil, & Moll, 2001), equity, and connecting to our PSTs’ own experiences learning mathematics as well as their own

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dissatisfaction with how mathematics is traditionally taught as disconnected from the real world. For the courses from which these data are drawn, both of us required our PSTs to do community walks around students’ neighborhoods, as opposed to driving around or riding the bus walks (see Chap. 2 for a description of the Community Walk Activity). Part of this is because of our physical location—the San Francisco Bay Area’s mild weather makes it practical to require they walk even in the winter months. There was slight variation in where PSTs walked due to differences in how local school districts set up attendance areas. Maria’s PSTs walked around the school community and locations near the schools that were described as significant to children, whereas Kathleen’s PSTs walked around spaces in their students’ communities that the students described as significant. PSTs from both universities were then required to create a math task that connected to the California Common Core State Standards in Mathematics (National Governors Association for Best Practices, Council of Chief State School Officers [NGA CCSO], 2010). PSTs then wrote reflections both on the experience of the community walk and of the design of the mathematics task, including their thoughts on how they might utilize a mathematics task based on knowledge gained in a community walk as part of their future teaching, if at all. The context in which Maria introduced the CME Module was a cohort of 12 bilingual PSTs within an intensive 1-year post-baccalaureate teacher credential program. The math methods course was an accelerated 40-hour course held over 9 weeks and occurred during the first half of the spring semester, after the PSTs had completed one fall semester of education coursework and just prior to full-time student teaching. This course was the only mathematics methods course PSTs were required to complete as part of their credential. PSTs had been in classrooms teaching children since the beginning of the school year on a part-time basis. Each PST completed a Community Math Walk after talking with students about significant features of their communities and finally selected a site in the community to further mathematize. Each PST was asked to complete their own write-up of the mathematics task including which California Common Core State Standards in Mathematics are addressed through the task as well as an individual reflection on the process. A lesson plan was not required due to time considerations. However, PSTs were asked to describe in their reflections how they would implement the mathematics task they designed, providing some aspects of lesson planning in their reflection. PSTs had 45-minute small-group discussions the week before the assignment was due to consult with each other around their ideas for math tasks. Maria, the instructor in the class, provided some guidance to the PSTs at that time, though not all PSTs were given feedback from the instructor before turning in their final assignment. For Kathleen, the CME task was undertaken by two sections of an Elementary Mathematics Methods course that 33 PSTs were enrolled in during their second 10-week mathematics methods course in a post-baccalaureate credential program. All PSTs were simultaneously student teaching 4  days per week for most of the school day. This project occurred during their winter quarter of a 1 or 2-year Masters of Teaching/Credential Program. Students worked in groups of 3  to  5 people to

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investigate two community locations that their students go to with their families. They then chose one of the locations, after talking with people at both locations, to inform their creation of mathematics tasks. They were required to write a description of their community visits and how their community visits informed their ­mathematics task development. PSTs then created a lesson plan that was tied to a specific grade level and was linked to the California Common Core State Standards of Mathematics. They presented their community site and lesson plans to their peers and other teacher educators in a community fair format and then individually wrote a reflection.

Where Is Social (In)Justice and What Does It Mean? Of the 23 tasks completed by PSTs across both of our programs, we found that they fell into three categories: (a) mathematics tasks with strong community and strong social justice ties (n = 6), (b) mathematics tasks with strong community ties with potential to be connected to social justice issues (n = 11), and (c) mathematics tasks with weak community ties (n = 6). Our analysis focused on the first two categories in order to better understand patterns in how strong social justice tasks were anchored in community issues. We also examined the possibility for feedback from MTEs for tasks that have the potential to connect to social justice issues. We provide a list in Appendix A containing paraphrased examples of the task cards/lesson plans (i.e., Carlos’ task, Small Zoo Group, Cupcake Store Group) to which we refer in this chapter. Because one cohort of PSTs created task cards to give to students and the other cohort of PSTs created lesson plans, there is some inconsistency in the information reported; however, we provide what we think gives the reader a sense of what PSTs intended children to engage in mathematically and contextually. Finally, we suggest conclusions and recommendations for MTEs.

Mathematics Tasks with Strong Social Justice Ties We found that of the 23 tasks, 6 had strong connections to social justice. The themes of these tasks included: • Involvement of a local cupcake store in community issues as a way to explore the kinds of issues a small business can help address in the community • A small local zoo as a context to learn about and contribute to saving the rainforest • The local farmer’s market and the impact of buying local, organic produce instead of food from the store • Travel to and from the Mission District neighborhood in San Francisco as a way to explore rising housing costs and gentrification

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• The growth of high-cost, dense urban housing in the school community as a way to explore gentrification and the rising cost of rent and home ownership • Mapping the route for a march for immigrants’ rights in San Francisco Each of these tasks held significant connections to a social justice issue and demonstrated a clear conceptualization of how students would explore the issue and make a plan for what they would do to address it. In Appendix A, Figures 1 and 2 are examples of tasks that we found to be particularly strong in their use of bridging local issues with social justice. Each of the six tasks in this category had a significant community issue to address through mathematics: deforestation, health, community activism, gentrification, and immigrant rights. In each case, PSTs started with a local space, and in the process of mathematizing, the space connected to a significant community issue. The six tasks also shared two features that we think are significant. First, though many of the contexts lent themselves to “best buy” scenarios, the PSTs pushed beyond this idea and instead, or in addition, incorporated a more significant aspect of the situation. For example, the group that used the cupcake store as a context (Appendix A, Fig. 3) crafted not only a cost-efficient scenario that asked students to figure out how they would spend money and how many cupcakes need to be baked per day but also engaged in the bigger question of how the small-business owner reached out to the local community. The PSTs embedded in their lesson plan the ways in which the business contributed to the community. This included giving the unsold cupcakes at the end of the day to church groups, nonprofit groups, and food banks/soup kitchens. Their intention was to discuss these issues with students, using their cost analysis as a starting point. These kinds of situations help us notice that PSTs can be supported to take multiple directions in what mathematics they focus on in a given context, but that they might need help in pivoting from cost-efficiency mathematizing to other mathematics such as, in the cupcake example, a case study of a local business owner’s social activism. Second, though each scenario in these six math tasks starts focused on a local issue, they ultimately connect to larger themes of equity and social justice. For example, Carlos (quoted in the introduction) developed a task focused on the mathematics of organizing a march to support immigrant rights (refer to Appendix A, Fig.  1). His task was developed directly from conversations with his Latino students, who all knew or had family members directly impacted by federal immigration policies and inspired by leaflets and posters in the community focused on this galvanizing issue. He used his conversations with students about their concerns to develop a mapping task, which utilized geometry and elements of graph theory, as a way to tie mathematics and activism together. While Carlos’ tasks could use some improvement in how the questions are asked to the students, including making the task itself more open-ended, the connections to the social justice issue are clear, and the mathematics he has the students work on could support them in designing and carrying out a march to support immigrants’ rights.

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Strong Community Ties, Not Explicit About Social Justice The majority of the math tasks PSTs created based on the Community Walk Activity (n = 11) demonstrated strong community ties but did not evidence explicit connections to social justice issues. Rather, these tasks had the potential to “pivot” toward social justice. By pivot, we mean that the context could be utilized as a platform to explore a social justice issue with some reworking of the focus of the task, such as the Gelato group’s task (e.g., shifting from a “best buy” scenario to a question about supporting local businesses versus large chain stores), or a task extension such as Alicia’s task (e.g., adding a second lesson in which students will go deeper into the issue after the first lesson). To be clear, we were very happy to see the way that PSTs successfully mathematized aspects of their students’ community experiences and do not consider these tasks inferior to those in the prior category. We are encouraged by the number of tasks in this category because of the potential to help PSTs move from their current task development toward embedding social justice in these tasks. The themes of these tasks include: • A local gelato store as a context for calculating how much gelato needed to be sold to pay for employee wages • A local candy store and the relative amounts of sugar in different products • A community theater box office as a context for balancing a budget • A local skate park and the geometry of traveling up and down surfaces • Transportation options for commuting around the San Francisco Bay Area • Services offered at a well-known tire shop, including tire repair • Window replacement for uniquely shaped panes, featured architecture of houses around the school • A well-known mom and pop honey and oranges stand as a context for buying snacks for the classroom • A local favorite ice cream shop as a context to explore flavor combinations and total gallons of ice cream produced every week • Family gatherings at a pizzeria comparing sizes of pizzas and slices per pizza • A local park with monkey bars for kids to play on during recess time There was a lot of variety in the extent to which the contexts selected for the tasks were significant to students versus being seen by PSTs as significant to the community. For the sake of our analysis, we grouped them together because the PSTs articulated why these sites were significant, and why the mathematics they would be exploring could be useful to students. These tasks also went beyond simple best-buy scenarios, asking something more complex of students such as having to account for multiple parameters. For example, in Sofia’s task (Appendix A, Fig. 5) on cost-­ efficient ways to get in and out of San Francisco, she provided information about bridge toll, BART (commuter rail) costs, and costs of gas per gallon (and in her bonus question, scooter rental costs), and students were asked to use the information to make an argument for the best way to travel between an address in the East Bay

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and San Francisco. Even though cost-efficiency was part of the goal, students could also use their funds of knowledge of getting around the city (such as city street congestion or riding BART and buses) to argue for a combination of methods that made the most sense. Some of the tasks in this category were on their way to having explicit social justice considerations, but were not there yet. In the mathematics task that revolved around a local gelato store (Appendix A, Fig.  4), the PSTs created mathematics tasks that examined not only how much gelato needed to be sold to make a profit but also what a reasonable wage might be for the employees working at the gelato store. The PSTs wanted their students to begin to consider this question from a variety of perspectives. One perspective is that of the employer—weighing why he or she might want to hire more people and the impact this decision has on costs. This group of PSTs commented in their reflection that in later lessons, students could examine the trade-offs (financially and otherwise) of paying higher versus lower wages to the employees. For example, the PSTs thought about having their students consider if paying the minimum wage required by law might make it more difficult to find steady, high-quality help if the employer is not paying a high enough wage. In addition, the PSTs created a lesson where students would take into account the personal ethics of employers paying a minimum wage versus a higher wage. Their plan to start to have students consider minimum wage as part of the task is a key way that this task could pivot toward the explicit examination of a social justice issue, namely, that of a living wage. One could imagine that further developing the ideas of living wage could make this task into a mini-unit of curriculum that could have a higher impact than the examination from the employer’s perspective alone.

Conclusions and Implications for MTEs Our goal is to push PSTs to transform “status quo” mathematics teaching, in part by supporting them to develop lenses through which to mathematize students’ lived experiences. In our review of 23 CME Module projects, it appeared that the contexts PSTs chose to focus on and the complexity of the mathematics involved in the tasks they designed indicate how able the PSTs were to conceptualize a mathematics task that was tied to their students’ MMKB and also included a social justice component. Key developmental steps we note from our analysis include seeing the mathematics of the real world as more than cost-benefit analyses and seeing complex problems as different from simply requiring multiple operations. These decisions, however, are influenced by what PSTs believe is best for their students, and what issues they see as impacting their students’ communities. If PSTs choose a context that they believe “all students” are familiar with, then we as MTEs can push them to see the mathematics and social justice beyond the presumably obvious. In our experience, PSTs often struggle with thinking that a good mathematics task based on student’s

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MMKB must draw on contexts familiar to all students. We encourage our students to both privilege the experiences of a marginalized group of students in their classroom setting and approach the CME Module as an opportunity to introduce all students to a context that only some of them are familiar with. When teachers intentionally utilize contexts that traditionally marginalized students have knowledge of, we also shift the power balance in the classroom to privilege the knowledge of these students. The six tasks with the strongest connections to social justice came from PSTs who mathematized an issue directly impacting their students. Local issues are social justice issues. Perhaps if, as MTEs, we help our PSTs privilege understanding the community issues over teaching to specific grade level standards, we can support PSTs to successfully mathematize these situations and still connect to required core mathematical practices and some content standards. If PSTs can embrace mathematics in a manner that’s truly meaningful for students, we as MTEs can help them prepare mathematics lessons that “matter” (Turner & Strawhun, 2007). To that end, MTEs need to have funds of knowledge of social justice issues impacting local communities and methods to scaffold PSTs to understand those issues more deeply before or after taking community mathematics walks. This could include encouraging PSTs to get involved in local issues that are pertinent to their students’ communities. At the time of this writing, living wages in the bay area, community displacement, and the ethics of the technology industry are significant topics, among many others. Social media and other resources could help PSTs to better understand local issues and provide much-needed information to fill in gaps. This is especially important, as making mathematics relevant to the students’ communities with which PSTs work may result in those students viewing the mathematics as valuable and worth their time. In terms of PST mathematical knowledge, there seems to be a connection between PSTs’ knowledge of how mathematical ideas are connected and therefore how such ideas may be part of mathematizing settings. This view supports understanding what Fosnot and Dolk (2001) call “the landscape of learning”—a dynamic movement of undulating terrain, not linear progression through specific concepts. This way of knowing mathematics involves understanding connections within and across mathematical domains and understanding landmark concepts and representations that lead to new understanding. This connected way of knowing mathematics supports PSTs to design tasks that have multiple entry and exit points. In our experience, it is PSTs who are less confident in their abilities who design overly constrained tasks that have the potential to squash the creative mathematical thinking from the task in the name of scaffolding students toward success. Alicia’s task (Appendix A, Fig. 6) is an example of this—a situation that students are very familiar with is given too many constraints, until the students are solving specific problems instead of crafting a whole solution to the big issue of how to get kids across the monkey bars in an equitable manner.

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Also, we were not surprised that the strongest mathematics tasks were created for fourth- and fifth-grade contexts, though we believe this may not always be the case. The Common Core State Standards for Mathematics for fourth and fifth grade contain multistep problem-solving and the use of all four operations (NGA CCSSO, 2010). Hence, PSTs may feel that such complex tasks were made more complex by social justice considerations and therefore belong in grades with more advanced mathematics standards. We believe this should be addressed more clearly in methods courses. One way to accomplish this could be to support PSTs in focusing on a diversity of standards across domains (measurement and statistics, operations and algebraic thinking, etc.), versus all four operations, as a way to make a multistep mathematics task. We need to support PSTs to understand solving multistep problems as a process that may involve multiple representations or trials versus multiple operations. Conceptualizing multistep or “complex” problems in this manner may help PSTs in the K-2 grades have more tools to design rich community-based tasks for these grade levels. This is a social justice issue for MTEs, because it may be that the narrow definition of what counts as doing the mathematics we often encounter with PSTs will limit their ability to design rich mathematics tasks for their students. We also know that sometimes our PSTs have explicit commitments to social justice for their students but struggle realizing these commitments through mathematics tasks. Mathematics is often seen as the last frontier for social justice infiltration, perhaps strongly tied to traditional views of both what counts as school mathematics and perceptions of what urban youth need (e.g., remedial skills development) but could also be connected to what PSTs believe are their teaching goals. We acknowledge a need to address beliefs about the nature of meaningful mathematics with PSTs, so that those with explicit commitments to social justice can see how those commitments play out in their own lesson design. For example, Carlos’ task in which students designed a march for immigrant rights is a good illustration of how a PST with clear commitments to social justice could use more support to develop his idea into a stronger mathematics task. In Maria’s feedback to this student, she encouraged him to do two things to modify this task: (a) add open-ended questions that would help students do more of the mathematical reasoning of how to plan the march and (b) consider where this would lead—was it feasible to have the class come to consensus and plan the march to actually happen? In this way, the PST’s ideas were met where he was at, and he was encouraged to continue to develop his abilities to design meaningful mathematics tasks without sacrificing the social justice component. As we get a sense of the landscape of PST learning, we gain a clearer sense of what different PSTs need to take steps in curriculum design in ways that draw on students’ MMKB. We then need to leverage those ideas in the service of PSTs’ own commitments to social justice. We need to set PSTs up for success mathematically with these tasks and support their development as they work toward mathematizing the social issues embodied in those contexts.

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Appendix A: Examples of Social Justice Mathematics Tasks The students are planning to march through the Mission District in order to advocate for compassionate immigration reform. They are marching from our school to the Bart Station. Every time they come to an intersection, they stop and wait 30 seconds for traffic to pass before crossing. ● If they want to march through as many different streets as possible, without ever going away from the direction of the Bart station, how many streets can they march through? ● If they want to cross as many intersections as possible, without ever going away from the direction of the Bart station, how many intersections can they cross? ● If you had to plan a route to get the most exposure, what would it be? Why? [not pictured, but provided to students: map of the local area from google maps] Fig. 1  Carlos’ task planning march for immigrant rights: strong ties to social justice

Background: The focus was on tasks that would immerse students not only in mathematics, but also connect to habitats and environments since this is one of the zoo’s main focuses. Tasks were created that related to saving the rainforest. This lesson challenges fourth graders to solve multi-step problems while connecting to the science of habitats and to the social justice principal of saving endangered species and their environments. ● How much money would it take to save 1 spider monkey? ● If Jack wanted to donate $5.00 to the Nature Conservancy, how much of each rainforest animal would he save? ● If all of the fourth grade classes at Jack’s school fundraises $1000 and donates it to the Nature Conservancy to protect rainforests, what could the class save with that $1000? Fig. 2  Small zoo group: strong ties to social justice

Background: The Employer of the Cupcake Store was committed to donating the daily unsold cupcakes to organizations in the community such as churches and soup kitchens. The PSTs in the group created a lesson plan that led to discussion of community involvement. In addition, they asked questions such as: ● You have a budget of $50 to buy cupcakes for your birthday party. You are planning to have 20 guests (including yourselves). You can buy the cupcakes from Safeway and/or Frost bakery. Individual cupcakes are $3.50 from Frost, or $39 for a dozen signature size. Frost also has a “kids” dozen option for $28. Safeway sells a dozen of chocolate or vanilla frosted cupcakes for $13.30, or 2 dozen for $21 .99. How would you spend the $50? Would you spend it all to buy cupcakes from Frost or from Safeway, or would you split it up and buy from both? How would you get the most for your $50? Fig. 3  Cupcake store group: potential to pivot explicitly to social justice

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Background: “Cost of living and what makes a living wage have been hot issues not only in the Bay Area, but nationally. We wanted our students to begin to consider this question from a variety of perspectives. One perspective is that of the employer - why might they want to hire more people and how much does that cost? After establishing this understanding, our students, in later lessons, can begin to consider how an employer might consider what to pay an employee and to examine the trade-offs, financially and otherwise, to paying a lower wage.” It’s a busy Saturday night and you are in charge of planning the number of employees that will be working at Gelato Classico. You know: Each cup costs $3.60, and it costs $15 an hour for each employee to serve customers ● It takes 60 minutes for 1 employee to serve 15 people in line. How many minutes does it take 2 employees to serve 28 people in line? ● If two employees serve twice as many scoops of gelato together but it costs twice as much to employ the extra server, how much more money would you make during six hours? If each gelato sold costs $3.60, how many do you need to sell in 1 hour to make a profit with 1 employee? With 2 employees? Fig. 4  Gelato group: potential to pivot explicitly to social justice

Like many of your family members, I live in the East Bay and work in San Francisco. Apart from the traffic there are parking meters, toll roads, and gas. It’s a good thing we live near good public transportation, but which is more cost efficient? Here are the facts: ● I commute into the city and back home 4 times a week. ● My car holds 12 gallons of gas and I would need to refill it 2 times per week. Gas costs $2.45/gal. In addition to gas, for each day I commute I need to pay a $6.00 toll one way. ● If I take public transit, it would cost me $16.00 per day to get to and from San Francisco from my house. What do you think is the most cost efficient way to commute into the city? Why? Fig. 5  Sofia’s task: community ties

Every week 22 students from the after school program walk over to the park. Although the park has many different play structures to choose from, all of the students only want to play on the monkey bars. The students only have 20 minutes to play at the park, and so far this hasn't been enough time for all students to have a chance to play on the monkey bars. The teachers want to design a system so all 22 students will have a turn to use the monkey bars within the 20 minutes. Only two students can cross the monkey bars at a time. Help these teachers understand and find a solution to this problem. ● First predict do you think it is possible or isn't possible for all the students in the class to play on the monkey bars, and why? ● How many pairs of students are there for everyone to cross the bars? ● How long should every pair take in order for everyone to have a chance at the monkey bars? [not pictured provided to children: the circular monkey bars from the park] Fig. 6  Alicia’s task: community ties

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References Aguirre, J. M., Turner, E. E., Bartell, T. G., Kalinec-Craig, C., Foote, M. Q., Roth McDuffie, A., et al. (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, 62(2), 178–192. Aguirre, J. M., Zavala, M., & Katanyoutanant, T. (2012). Developing robust forms of pre-­service teachers’ pedagogical content knowledge through culturally responsive lesson analysis. Mathematics Teacher Education and Development, 14(2), 113–136. 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: Teachers College Press. Fosnot, C., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, VA: Heinemann. González, N., Andrade, R., Civil, M., & Moll, L. (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. Gutiérrez, R. (2013). Why urban mathematics teachers need political knowledge. Journal of Urban Mathematics Education, 6(2), 7–19. Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York: Routledge. Gutstein, E., & Peterson, B. (2013). Rethinking mathematics: Teaching social justice by the numbers (2nd ed.). Milwaukee, WI: Rethinking Schools. Hintz, A., & Smith, T. (2013). Mathematizing read alouds in three easy steps. The Reading Teacher, 67(2), 103–108. Leonard, J., & Martin, D. B. (2013). The brilliance of black children in mathematics. Charlotte, NC: Information Age Publishing. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Trexler, L. (2013). Adventures of a beginning teacher with social justice mathematics. In E. Gutstein & B. Peterson (Eds.), Rethinking mathematics: Teaching social justice by the numbers (2nd ed., pp. 54–60). Milwaukee: Rethinking Schools. Turner, E., Drake, C., Roth McDuffie, A., Aguirre, J., Bartell, T.  G., & Foote, M.  Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children's multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67–82. Turner, E., & Strawhun, B.  F. (2007). Posing problems that matter: Investigating school over crowding. Teaching Children Mathematics, 13(9), 457–463. Zavala, M. (2016). Methods, maps, and meaningful mathematics. Teaching for Equity and Excellence in Mathematics, 7(1), 36–44.

Part III

Classroom Practices

Chapter 8

Focusing the Video Lenses Tool to Build Deeper Understandings of Early Childhood Contexts Amy Noelle Parks and Anita A. Wager

Keywords  Mathematics education · Centers · Early childhood · Elementary education · Place value · Play As mathematics teacher educators who focus on early childhood contexts and as former elementary school teachers who do research in early childhood classrooms, we have frequently found ourselves on the receiving end of requests from mathematics education colleagues to provide insight into common instructional practices in Prekindergarten—Grade 2 (PK-2) classrooms (like stations), common tools in PK-2 classrooms (like rekenreks), and common commitments of early childhood teachers (like developmentally appropriate practices). We have also occasionally found ourselves struggling to explain to colleagues why mathematics methods course assignments designed for middle-grades classrooms (like holding 20-minute discussions) might not be as productive in early childhood contexts. These experiences are, of course, not surprising. Many of the people teaching elementary mathematics methods courses have not, for a variety of reasons, ever taught in elementary schools (much less in the primary grades). They may not have ever taken a course on child development or elementary instructional methods and may not have ever conducted research in elementary schools. The number of mathematics teacher educators who have had any of these experiences in early childhood contexts is even smaller. So when we think about what teachers and teacher educators need to know in order to connect children with mathematics in meaningful ways, one concern that is always salient for us is the need to promote developmental, as well as cultural, responsiveness. The goal of this chapter is to support mathematics teacher educators in understanding early childhood contexts by providing examples of how the Video Lenses A. N. Parks (*) Michigan State University, East Lansing, MI, USA e-mail: [email protected] A. A. Wager Vanderbilt University, Nashville, TN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_8

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tool in the Classroom Practices Module (see Chap. 2) can be used to make sense of key ideas in early childhood mathematics education. We would also note that while our focus in this chapter is on highlighting issues related to early childhood, we believe the Video Lenses tool could be used in similar ways to dig into other contexts and issues, such as bilingual education, approaches to teaching fractions, or argumentation. Closely focusing on tasks, teaching, learning, and power and participation—the four lenses of the tool—slows down the experience of watching a classroom video and allows viewers to attend to multiple layers of the interactions. The analysis of each of the videos described in this chapter could be done as an activity in an early childhood or elementary mathematics methods course or in professional development to support prospective or practicing teachers’ abilities to engage young children in mathematics. Analyzing videos from early childhood contexts could also help teacher educators, including graduate students, who are not familiar with the primary grades to develop a deeper understanding of these instructional contexts. In the same ways that the four lenses support prospective teachers in identifying children’s mathematical and cultural resources, the four lenses can also be used to support teacher educators in developing a child-centered approach to teaching mathematics. For us, this means attending not only to the kinds of teaching practices recommended by mathematics education organizations like the National Council of Teachers of Mathematics but also attending to recommendations for developmentally appropriate practices by organizations such as the National Association for the Education of Young Children. These include such practices as concern for social, emotional, and physical development along with cognitive development; consideration of the typical interests, attention spans, and abilities of young children; and awareness of the relationship between children’s social, cultural, and developmental contexts (Copple & Bredekamp, 2009). In other words, we believe that recognizing and understanding childhood as a cultural context (Corsaro, 1997; Montgomery, 2008) is as central to enacting equity pedagogies as understanding linguistic, mathematical, racial, and ethnic contexts. When educators enact pedagogies that allow children to engage in mathematics in ways that are respectful of their developmental stage, they create classrooms that are more humane for children and that are more likely to provide opportunities for educators to learn about their students in holistic ways. To highlight the issues we have raised, we offer an extension of the Video Lenses tool by providing additional questions that can be posed in each of the lenses to focus attention on young children (Table 8.1). Our goal was to use the lenses to help our teacher candidates see the particular context of early childhood and to think deeply about what it would mean to provide a developmentally as well as mathematically productive instructional environment. The following two sections of this chapter are devoted to demonstrating how using this extended version of the four lenses of the Video Lenses tool can help educators build an understanding of child-centered developmentally appropriate practice for mathematics teaching. In particular, we focus on two key areas: (a) mathematics content (particularly early number) and (b) early childhood

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Table 8.1  Extending the Video Lenses tool for early childhood Video Lenses tool TASK: What makes this a good and/or problematic task? How could it be improved?

Early childhood-specific questions How does the task support children’s social, emotional, and physical development? How long are children expected to engage in a particular task? How does the task engage children’s interests? How does the task reflect children’s cultural/community funds of knowledge? How does the task support learning of early number? LEARNING: What specific math Are required representations developmentally understandings and/or confusions are indicated in students’ work, talk, and/or appropriate (e.g., are children expected to write numerals, number sentences, etc.)? behavior? TEACHING: How does the teacher elicit What are the child-teacher interactions around early number? students’ thinking and respond (e.g., moves, questions, responses to students’ What teacher moves encourage children to demonstrate one-to-one correspondence or correct answers/mistakes/partial cardinality? solutions, decisions)? How does the teacher attend to children’s social, emotional, and physical needs, in addition to their cognitive needs? Do all children have access to materials for various POWER & PARTICIPATION: Who participates? Does the classroom culture activities? Are materials developmentally appropriate so that all value and encourage most students to children have access to learning? speak, only a few, or only the teacher? Does the task design allow children to act like children?

i­nstructional practices. All of the videos discussed are publically available online. An appendix at the end of the chapter provides information about where each of the videos can be found and lists additional videos that educators interested in focusing on early childhood classrooms may find productive to analyze using the four lenses (see Appendix A).

 uilding Conceptions of Developmentally Appropriate B Mathematics Content: Early Number Mathematics education research and elementary methods courses have tended to underrepresent mathematical concepts central to early childhood, such as counting, number concepts, and descriptions of shapes, in favor of perennially popular topics such as division of fractions and story problems related to the four operations (Ginsburg, Jang, Preston, Van Esselstyn, & Appel, 2004; Parks & Wager, 2015). Analyzing videos of early childhood classrooms not only provides prospective teachers with opportunities to make sense of the mathematical understandings that must be in place before students learn to use equations to represent addition and subtraction but also allows them to build a repertoire of teaching strategies for addressing this mathematical content in their classrooms. Similarly, videos may

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give mathematics educators who are unfamiliar with early childhood classrooms a repertoire of stories that can be used to support prospective teachers who are working with the youngest learners. Part of creating a developmentally responsive classroom is ensuring that children have multiple opportunities to engage with the early counting skills that will support much of their later work in number. Young children need to develop an understanding of counting that extends beyond the rote recitation of the number sequence (although this is also important) to include one-to-one correspondence, cardinality (or recognizing that the last number in the counting sequence names the total set), and related concepts like subitizing (recognizing small quantities without counting), and relationships of more, less, and the same (National Research Council, 2009). It is equally important that prospective teachers develop a language for discussing these counting skills and the ability to recognize them and to design activities that will support their development. Research has shown that adults’ intentional engagement with children around these concepts, such as by counting and labeling sets simultaneously, significantly improves children’s understanding of early number (Gunderson & Levine, 2011; Mix, Sandhofer, Moore, & Russell, 2012). Below, we analyze a short video of children playing the game Hi Ho Cherry-O using the four lenses to highlight ideas around counting. In this video (available on the TeachMath website, https://teachmath.info), a teacher is playing the game with three prekindergarten children. The game includes a spinner with “trouble” pictures (dump out the bucket, a bird, a dog) and pictures of one, two, three, or four cherries, with each cherry labeled with a numeral. In this example, the children play by spinning and then putting that number of cherries back on the tree (which is the reverse of the way the game is typically played, with children picking cherries off the tree and placing them in the bucket). The video begins with the second player spinning. His spinner points to two cherries. He immediately announces “two” and places the correct number of cherries on the tree. The second child spins a “three.” He counts each of the cherries pictured on the spinner by pointing and saying “one, two, three” with each touch. He then places the three cherries on the tree one at a time, saying a number word each time he puts a cherry on the board. The last child also spins “three” but is unsure how to read the spinner. The teacher moves the arrow out of the way and circles the appropriate group of cherries with her finger, asking how many the child spun. The child initially says “one.” The other two children say “Three!” “Three!” The teacher maintains her focus on the third child, who eventually says “three.” In Table 8.2, we provide an example of how the video might be analyzed using each of the lenses from the Video Lenses tool. As is recommended in the Classroom Practices Module, we suggest having different groups of participants use a different lens to watch the video. Applying the four lenses of the Video Lenses tool productively highlights the complexity of the mathematical and social work occurring as the teacher and children play a popular game for young children. Particularly in relation to mathematical learning, the lenses help viewers to break apart the construct of counting to see it as a skill that is not simply present or not present, but as an umbrella that contains multiple skills (e.g., number words, cardinality, one-to-one correspondence, subitizing) that teachers must help children to develop.

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Table 8.2  Video Lenses tool: Hi Ho Cherry-O video Lenses TASK: What makes this a good and/or problematic task? How could it be improved?

Possible responses to the lenses The game allows for the children to practice counting in a variety of ways, using subitizing or one-to-one correspondence depending on the child. The rules of the game also require that children count out the appropriate number of cherries from the bucket after the spin, which is a different and more challenging counting task than counting a set already in front of them In addition to supporting mathematical development, the playing of a game also provides an opportunity for young children to learn social skills, such as managing turn taking and the feelings involved in winning and losing One challenging aspect of this task is that young children need a lot of support to play games; thus, an adult must be fully engaged with them the whole time. Perhaps, a simplified version of the game could be created that was more easily able to be played by young children without adult supervision The way the game is played in this video is somewhat problematic: the cherries are all different colors, which does not match children’s experience of cherries in the real world or in books. And the teacher’s decision to play by putting cherries onto the tree instead of taking them off also undermines any real-world connections to picking fruit The first child subitizes the quantity he spun without having to count, LEARNING: What while the second child counts using one-to-one correspondence, specific math touching each cherry as he counts and then again naming each understandings and/or confusions are indicated in cherry with a number word as he places each one on the tree. Both students’ work, talk, and/or children are able to hold the total they are counting toward in their heads and place the correct number of cherries on the tree behavior? The third child initially says that she spun “one,” although she spun three. She seems to be looking at the numeral 1 on the cherry. It is unclear whether her confusion is a difficulty in counting or in understanding the representations used by the game The teacher repeatedly asks questions designed to reinforce TEACHING: How does the teacher elicit students’ cardinality: “So how many do you have?” “So how many cherries thinking and respond (e.g., are you going to put in?” By asking children to rename the set after they have counted it out, she supports them in attending to this big moves, questions, idea of counting (that the amount of the set does not change after it responses to students’ correct answers/mistakes/ has been counted) Throughout the activity, the teacher manages the children’s partial solutions, behavior, such as by reminding children of whose turn it is, while decisions)? keeping the major focus on mathematics. She also helps the children without the fine motor skills needed to work the spinner, allowing them to have as much control as possible of the materials The video focuses on a small group of children; however, the noise POWER & of the video makes it clear that other children are moving around PARTICIPATION: Who the room engaged in other activities. One child approaches the participates? Does the game and asks what it is and one of the player’s answers classroom culture value The teacher encourages each child to take responsibility for the and encourage most mathematics and the materials during his or her turn, reminding the students to speak, only a children to allow the player whose turn it is to manipulate the few, or only the teacher? materials. She is fully engaged with each child during their turn, although she does not encourage the children to engage with each other’s thinking

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In addition, the power and participation lens calls attention to the ways that small group work supports engaged participation by all members of the group and allows for the teacher to enter into conversation with each child. Understanding early childhood contexts, and productive strategies for teaching mathematics in those contexts, requires educators to move beyond a focus on whole-class instruction or even cooperative problem-solving groups, such as might be used in upper elementary or middle grades settings, toward the small group and independent activities. Our second counting video, Counting Bugs, was taken during an opportunity for free play in a prekindergarten classroom. Unlike the small-group game playing described above, in which the teacher intended to focus on counting concepts, this video (also available on the TEACH Math website: https://teachmath.info) shows a teacher capitalizing on an opportunity to highlight counting by calling out to a child during play. The child had been lining up a collection of 11 plastic bugs on the floor. The video begins just after the teacher asks the child to count how many bugs he has. The child counts from one to five, using one-to-one correspondence correctly. At six, he double counts and then continues to count to 12 without matching his touch to his oral count. The teacher asks him to count again. He starts counting again in a different place, this time counting from one to four accurately using one-­ to-­one correspondence and then continuing to count to ten without matching his touch to his oral count. He finishes by saying “ten.” The teacher says “Good job. So how many do you have?” The child responds, “ten.” The whole video is only 20 seconds long, which would allow prospective teachers to watch it repeatedly, even during a methods class. Using the four lenses to analyze the video would allow prospective teachers to think carefully about both the kinds of mathematics the child is able to encounter in the play as well as the interventions made by the teacher (Table 8.3). Analyzing this video in contrast with videos taken in other instructional settings (such as those listed in Appendix A) would help prospective teachers to see the ways that counting objects that can be touched is fundamentally different from reciting the number sequence or even from counting objects that a teacher or another child is touching (both frequent practices in whole-group settings). In addition, the learning lens in particular demonstrates what a teacher might be able to learn quickly through informal assessment. The teaching lens opens space to discuss how teachers respond to play and what decisions are made in the moment about the kinds of questions to ask and whether to push for accurate answers. The power and participation lens raises questions about the ways that different kinds of instructional settings shape the ways that children can participate in mathematics in classrooms and about how children might be differentially supported in these mathematical engagements. These questions could be explored through the analysis of other videos or through observations in classrooms. Applying the four lenses to these and similar videos can help prospective teachers develop a more complex picture of early counting than is possible through readings or watching videos of clinical interviews, in which the counting tasks are much more carefully controlled

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Table 8.3  Video Lenses tool: counting bugs video Lenses TASK: What makes this a good and/ or problematic task? How could it be improved?

LEARNING: What specific math understandings and/or confusions are indicated in students’ work, talk, and/or behavior?

TEACHING: How does the teacher elicit students’ thinking and respond (e.g., moves, questions, responses to students’ correct answers/mistakes/ partial solutions, decisions)?

POWER & PARTICIPATION: Who participates? Does the classroom culture value and encourage most students to speak, only a few, or only the teacher?

Possible responses to the lenses The child’s task was not initially mathematical. He was interested in lining up the toy animals. By asking the child to count the animals, the teacher invited the child to engage in mathematical thinking. Her intervention created an opportunity for the child to practice critical early mathematics content without significantly interrupting his play. The child’s performance on the task—able to engage but not yet proficient—suggests that this task provided an appropriate level of challenge for the child The child in the video correctly rote counts from one to ten twice. He demonstrates one-to-one correspondence when counting up to four, but this breaks down as he counts to ten when he double counts some insects and skips others. The child demonstrates cardinality when he quickly repeats “ten” as the answer to the teacher’s final question, showing that he knows the quantity of the set remains the same after counting (whether he counted accurately or not) The teacher responds to the child’s incorrect count by asking him to count again, providing a second opportunity for the child to practice a still-developing skill. When he announces his total the second time, she responds by saying “Good job. So how many do you have?” This question provides the teacher with an opportunity to assess and promote cardinality in addition to rote counting and one-to-one correspondence. In addition, these kinds of questions have been shown to help children establish cardinality sooner. The teacher may or may not have known that the child’s final count was inaccurate. This interaction provides an interesting opportunity for discussion among teacher candidates around stopping with an inaccurate answer The child in the video actively participates in the counting task. It is impossible to tell from the video how the other children in the class are engaged, although one might conjecture that other children are similarly involved in their own play. More information would be needed to know if the teacher routinely prompts all children to engage with mathematics in similar ways

and scaffolded (and lack any of the pressures and challenges of working with 20 or more young children at the same time in a small space). Beyond using the four lenses to support prospective teachers to build conceptions of early number, they can also be used to understand child-centered approaches to teaching mathematics. In the next section, we describe ways we have used the lenses to examine instructional practices to identify equitable and developmentally responsive teaching.

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 uilding Conceptions of Developmentally Appropriate B Mathematics: Formal Instructional Practices Scholars who research early childhood mathematics are embroiled in an ongoing pedagogical debate about the role (or extent) of play in teaching and learning mathematics. We have argued in prior work that play is a critical component of early mathematics learning (Wager & Parks, 2016) but we recognize that children also need access to developmentally appropriate teacher-organized instruction (National Research Council, 2009). Young children should be provided with opportunities both to engage with mathematics during free play and to experience explicitly planned mathematics with their teachers. Our point here is that providing time for free play does not give teachers license to either ignore math time or ignore developmentally appropriate practices during teacher-organized activities. As described above, developmentally appropriate practices attend to mathematical learning opportunities and support for children’s social, emotional, and physical development. But recommendations for practices that are responsive to children’s developmental stage also incorporate the ideas that TEACH Math raises in the description of children’s multiple mathematical knowledge bases (Turner et  al., 2012). Instruction that considers the whole child by attending to children’s cultural and social resources is at the core of equitable and developmentally responsive practice. As a way to explore and understand how developmentally appropriate practices might be enacted in early childhood classrooms, we turn to a common practice in early and elementary classrooms—math centers (or stations). Well-organized and orchestrated centers provide an opportunity for children to collaborate in their learning while providing time for the teacher to work directly with small groups of children. A common question we get from both prospective teachers and colleagues who have not taught in elementary classrooms is, “How do you organize centers so that children are actively engaged in learning relevant mathematics without having a teacher at every center?” We have found that the Video Lenses tool provides an effective way for prospective teachers to assess how various center activities and structures support young children’s mathematical and other learning. In particular, the tool can shed light on what practices in the video provide evidence of developmentally appropriate practice, including the scaffolding needed to keep children engaged in mathematical tasks. In this section, we use the Video Lenses tool to unearth how classrooms that use these structures are (or are not) productive in supporting developmentally and culturally responsive teaching and learning. In our discussion, we focus in particular on those aspects of the videos that seem most salient for early childhood contexts. In this section, we analyze video #4 Place Value, from the Annenberg Library (https://www.learner.org/resources/series32.html#), in which a first-grade teacher is working with children to develop their understanding of place value (Table 8.4). The video follows a “launch, explore, close” lesson plan by beginning with a brief introduction to place value during which the teacher asks students how many days they have been in school. The teacher uses this to review the idea of bundling 10 ones to make ten. The children then move to one of four centers to work with partners on various place value tasks (6:50 in the video). In this explore portion of the video, the

Table 8.4  Video Lenses tool: Place Value video Lenses TASK: What makes this a good and/ or problematic task? How could it be improved?

Possible responses to the lenses Each center provides children with a different way to engage with place value concepts and reinforces counting (and in some cases measurement) skills. The primary goal in each center is to get children thinking about place value by bundling 10 ones to make ten and placing bundled tens and leftover ones on place value mats, but in order to do this, children were constantly counting. In each center, there is an appropriate number of tools to engage the children there and either an adult is present to scaffold the activity or the activity is simple enough that children can immediately engage independently. These task structures support the use of centers as an instructional routine In addition to mathematics, the task provided opportunities to learn social skills and work on fine and gross motor development. Social interactions in the centers varied from collaborative work, to independent work, to competition. The different sizes and shapes of materials for counting (unifix cubes, buttons, stones, and base ten blocks) provided a range of fine motor requirements, whereas stretching out on the floor and measuring each other supported gross motor. Some tasks were on the floor while others were at tables or desks We see evidence of various approaches to counting in LEARNING: What specific math each. In centers 1, 2, and 4, children use 1–1 understandings and/or confusions are indicated in students’ work, talk, correspondence to count to ten. Cardinality is reinforced as they group the 10 ones to make ten – counting to ten and/or behavior? and then saying ten. In the inventory center, the children count to fill cups with ten items and then count the cups to determine the number of tens. In center 3, we see one child skip count by two while counting the ones and one child skip count by tens while counting the ten-unit stick. In center 4, we see one child count to make a train of ten and then use that as a unit of measure to make other trains of ten. Throughout, the children are counting out loud and sharing their answers with each other or the teacher Although base ten mats were used in each center, children were provided with a variety of ways to represent their understanding TEACHING: How does the teacher The teacher set up stations that engage different ways of elicit students’ thinking and respond exploring place value. As she moves to different stations, (e.g., moves, questions, responses to the teacher asks children about their thinking. For students’ correct answers/mistakes/ example, in the inventory center, she asks questions such as, “How did you set up your buttons? What did you do?” partial solutions, decisions)? and then reinforces the idea of place value by asking how many tens and ones were made The questions posed by the teacher came from what the children were doing rather than from a script There are several ways that the teacher gets at student thinking, but there are times when she could have reinforced various counting activities and place value understanding. For example, in the inventory station, the teacher asks what 8 tens stands for but does not support the understanding that 8 tens equals 80 (continued)

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Table 8.4 (continued) Lenses POWER & PARTICIPATION: Who participates? Does the classroom culture value and encourage most students to speak, only a few, or only the teacher?

Possible responses to the lenses All the children have opportunities to participate throughout – sometimes independently (center 3), sometimes cooperatively (centers 1 and 2), and sometimes competitively (center 4). The type of questions the teacher asks during the closure focuses on the interests of the children and honors their ideas, such as when children notice who is taller and who is shorter Not surprisingly, not every partnership is as collaborative as others. In some situations, the child with the materials holds the power as they count independently and write down answers. For example in the measuring center, one child counts the cubes and records the findings while the child’s partner asks to be measured. In this situation, the counter/recorder has access to the math concept

centers are set up as follows: (a) children measure various items using unifix cubes and then break the cubes into tens and ones to record on place value mat, (b) children count collections by inventorying buttons and other small materials using cups to collect the tens, (c) the student teacher uses a “mystery box” to reveal a number on 100s chart and children illustrate the number using base ten blocks, and (d) children race each other to link together unifix cubes and then separate the cubes into tens and ones to determine who has the most. The teacher closes the activity by reviewing each of the centers asking children what they found interesting in order for them to “make connections” that they might not otherwise make with the free exploration alone. Using the Video Lenses tool enables us to attend to the math content and early childhood practices evident in the video. In examining the video through the four lenses, we are able to see how a teacher-organized activity provided space for complex mathematical thinking and social interactions. The task design is provided for movement throughout the math period, both moving between centers and at each individual center. The centers also provided options for children to decide how they wanted to be physically positioned. In terms of mathematics content, we see that the task designed to support place value understanding also reinforces counting and measurement. For example, the first center is a measurement activity that becomes a counting activity, whereas the last center is a counting activity that, for one child, became a measurement activity. The lenses also shed light on more than the mathematics by focusing our noticing on appropriate practices for young children. We see how the centers provide support for children’s socio-emotional growth by setting up various participation structures that require children to share materials and tools, work independently, or compete. After using the Video Lenses tool to identify how math centers can be organized to provide developmentally appropriate learning opportunities in the Place Value video, we suggest extending the activity by analyzing Annenberg Teaching Library video #40, Story-Based Centers. In that video, a second-grade teacher develops math centers based on the picture book, Caps for Sale (Slobodkina, 1940). This

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video provides an interesting contrast to the first video with respect to the extent to which developmentally appropriate practices and engagement with meaningful mathematics are present. Analyzing the two together might encourage viewers to reflect on the benefits of various structures and designs for math centers.

Conclusion In this chapter, we suggest that by adding an explicit focus on those aspects of lessons or tasks that are child-centered, the Video Lenses tool can be used to identify teaching practices that are developmentally responsive. In attending to interactions around early number, prospective teachers can develop a more nuanced understanding of counting and place value. By watching video to identify practices that support socio-emotional development, prospective teachers will have the opportunity to think deeply about practices that are developmentally appropriate and ways to create classrooms that do not push young children to behave like their older peers. We have analyzed three videos here to provide models for how one might engage prospective and practicing teachers in using the Video Lenses tool to identify early number and developmentally appropriate learning opportunities (Appendix A includes suggestions for additional videos).

Appendix A: Videos for Continuing Use of Video Lenses Tool While not all of the links listed to videos below may continue to be available, the resources listed in the chart provide for continuing use of the Video Lenses tool. Video name Website Counting Bugs TeachMath https://teachmath.info Hi Ho TeachMath Cherry-O https://teachmath.info #40 Story-­ Teaching Math: A video library K-4 based centers www.learner.org

Instructional setting Free play Prekindergarten Small group Prekindergarten Small group Second grade

#4 Place value centers #9 Domino math

Teaching Math: A video library K-4 www.learner.org Teaching Math: A video library K-4 www.learner.org

Leprechaun traps

The Teaching Channel www.teachingchannel.org

Small group First grade Whole group/small group First/Second grade Whole group 0.48–5:51 Small group 5:51 First grade

Key mathematics Counting Counting Patterns, problem solving, graphing, adding Counting, place value, measurement Counting, adding

Counting, adding, numeracy

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References Copple, C., & Bredekamp, S. (2009). Developmentally appropriate practice in early childhood programs serving children from birth through age 8. National Association for the Education of Young Children. 1313 L Street NW Suite 500, Washington, D.C. 22205-4101. Corsaro, W. A. (1997). The sociology of childhood. Thousand Oaks, CA: Pine Forge Press. Ginsburg, H. P., Jang, S., Preston, M., Van Esselstyn, D., & Appel, A. (2004). Learning to think about early childhood mathematics education: A course. In C. Greenes & J. Tsankova (Eds.), Challenging young children mathematically (pp. 40–56). Boston: Houghton Mifflin. Gunderson, E. A., & Levine, S. C. (2011). Some types of parent number talk count more than others: Relations between parents’ input and children’s cardinal-number knowledge. Developmental Science, 14(5), 1021–1032. Mix, K. S., Sandhofer, C. M., Moore, J. A., & Russell, C. (2012). Acquisition of the cardinal word principle: The role of input. Early Childhood Research Quarterly, 27(2), 274–283. Montgomery, H. (2008). An introduction to childhood: Anthropological perspectives on children’s lives. Chichester, UK: Wiley. National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, D.C.: National Academies Press. Parks, A. N., & Wager, A. A. (2015). What knowledge is shaping teacher preparation in early childhood mathematics? Journal of Early Childhood Teacher Education, 36(2), 124–141. Turner, E.  E., Drake, C., Roth McDuffie, A., Aguirre, J., Bartell, T., & Foote, M.  Q. (2012). Promoting equity in mathematics teacher education: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67–82. Slobodkina, E. (1940). Caps for sale: A tale of a peddler, some monkeys, and their monkey business. New York: Harper Collins Publishers. Wager, A.  A., & Parks, A.  N. (2016). Assessing early number learning in play. ZDM The International Journal on Mathematics Education, 48(6), 991–1002.

Chapter 9

Integrating Curriculum and Community Spaces Julie M. Amador and Darrell Earnest

Keywords  Curriculum analysis · Resources · Community connections · Everyday life · Curriculum adaptation · Teacher education · Case study · Mathematics education Early in their careers, teachers often face a tension between using the published curriculum materials assigned by their districts and being responsive to children’s lives or their community funds of knowledge—“the diverse cultural and linguistic knowledge, skills, and experiences found in children’s homes and communities” (Turner et al., 2016, p. 67). Incorporating community connections into mathematics instruction integrates children’s funds of knowledge together with mathematical ideas and symbol systems, thereby leading to a powerful vision of the meaning of mathematics (Aguirre et  al., 2013). In this chapter, we explore the use of the Curriculum Spaces Analysis Tool (CSAT) from the Classroom Practices Module (see Chap. 2 in this volume for a description of the module) designed by the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project. The CSAT aims to support analysis and reflection of opportunities—either present or missing—for connecting mathematics to students’ community funds of knowledge. Curriculum is a key resource that the vast majority of teachers utilize. One strategy for teachers to integrate students’ out-of-school experiences with mathematics lessons is to make authentic connections that are meaningful to students’ everyday experiences in their communities and in their lives (Drake et  al., 2015). Yet, the integration of mathematical and children’s everyday lived experiences in curriculum materials is not necessarily straightforward. Activities such as those featured in the Community Mathematics Exploration Module (see Chap. 2 of this volume) have the potential to reveal vital information for teachers about how mathematics is J. M. Amador (*) University of Idaho, Coeur d’Alene, ID, USA e-mail: [email protected] D. Earnest University of Massachusetts Amherst, Amherst, MA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_9

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embedded in the local practices of students’ communities and, complementary, that any and every community is rich with mathematical resources. At the same time, curriculum materials have a direct influence on teachers’ instructional design and enactment (Brown & Edelson, 2003). For this reason, we must prepare teachers to analyze curriculum in ways that enable the integration of students’ community resources and knowledge with mathematics instruction. The CSAT serves as a mechanism to connect the mathematics instruction delineated in textbooks with mathematical resources in students’ communities. Drake et al. (2015) identified three strategies for “opening” curriculum spaces to support this kind of integration, each was captured in our case study below: (a) rearranging lesson components, (b) adapting tasks, and (c) making connections authentic to children’s everyday experiences. We first consider aspects of the CSAT that provoke consideration of curriculum-community connections and frame this tool as a useful mechanism to support curricular noticing—meaning the extent to which teachers attend to, interpret, and make decisions to respond using curriculum materials (Amador, Males, Earnest, & Dietiker, 2017; Dietiker, Males, Amador, & Earnest, 2018; Jacobs, Lamb, & Philipp, 2010; Males, Earnest, Dietiker, & Amador, 2015). Drawing upon our use of the CSAT in our mathematics methods courses, we have analyzed how prospective teachers (PSTs) interact with the tool, with a careful focus on the adaptations they make to curricular materials (Earnest & Amador, 2017). We extrapolate the case study focus of this chapter from a larger project in which PSTs in methods courses at one of the two different universities over the last 3 years engaged with the CSAT (see Amador & Earnest, 2016; Earnest & Amador, 2017). To provide a more in-depth understanding (Yin, 2014) of how PSTs adapt curricular materials and consider the community context, we consider a case study of a trio of methods students who used the CSAT to analyze curriculum materials and, in using this resource, engaged with how materials might be adapted or interpreted to elicit, draw upon, and connect to children’s home- and community-based mathematical experiences.

The Curriculum Spaces Analysis Tool One goal of a mathematics methods course is to support PSTs’ capacity to identify and build on children’s multiple mathematical knowledge bases (MMKB) in their mathematics instruction. Turner et  al. (2016) consider MMKB to encompass an understanding of children’s interests and abilities given their “cultural, home and community-based knowledges and experiences” (p. 49). For the PST context, and for the purposes of this chapter, this goal includes supporting PSTs’ use and adaptation of curriculum materials to reflect children’s MMKB (Aguirre et  al., 2013; Turner et al., 2012, 2016). Because curriculum is written for any classroom in any context (Goldsmith, Mark, & Kantrov, 2000; Hirsch, 2007), teachers must engage in creative design work to adapt curriculum so that instruction integrates children’s MMKB (Drake et al., 2015).

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To support this analytic work, the CSAT provides specific prompts to provoke PSTs’ consideration of potential adaptations for leveraging and expanding curriculum spaces (Drake et al., 2015). The CSAT features a series of eight questions that ask about, for example, the central goals of the lesson, the cognitive demand of tasks featured, and the possible adaptations one may make (see Appendix A). Aligned with the goals of integrating community resources with inquiry and problem-­ solving, the CSAT’s questions are arranged according to the launch, investigate, and summarize components of a lesson, as well as “lesson peripheries,” or the elements of the textbook lesson that are not a part of the three portions just mentioned (e.g., a Teacher’s Note or Home-School Connection). Prompts on the CSAT include questions such as “Where are opportunities for activating or connecting to family/cultural/community knowledge in each phase of the lesson?” “How does each phase of the lesson open spaces for making real-world connections?” “Do students have opportunities to make their own connections?” and “Given your responses above, what kinds of adaptations might you make to each of the phases of the lesson?” More broadly, we see this tool as supporting the noticing of curriculum materials. Curricular noticing (Amador et  al., 2017; Dietiker et  al., 2018; Males et  al., 2015) is a critical aspect of mathematics instruction that is integral to ambitious teaching (Kazemi, Franke, & Lampert, 2009; Philipp, 2014). In the curricular noticing framework, teachers attend to particular curricular features, interpret these features, and then make decisions to draw upon such features to design instruction. Given that research has indicated curriculum as being consequential to the mathematics instruction children experience (Remillard, 2012), the components of the noticing framework—attend, interpret, and make decisions to respond—are illuminated through PSTs’ analyses of curriculum with the CSAT. In particular, unless teachers attend to opportunities for opening curriculum spaces for children’s MMKB (Aguirre et al., 2013; Drake et al., 2015), they cannot use this information to inform the design of lessons. We consider here how the CSAT supports curricular noticing in order to connect to students’ community resources.

Assignment Context We assigned the CSAT to support and expand PSTs’ sense of empowerment to adapt curriculum materials to connect to their students. During the curricular analysis process, the PSTs used the CSAT and answered the specific questions within the tool to identify opportunities for connections in the materials as well as to recognize opportunities for considering children’s MMKB. They then adapted the materials in accordance with their decisions based on their analyses, which resulted in a lesson plan modified from the original materials. PSTs’ adaptations were consistent with the three strategies for opening community spaces identified in prior research (Drake et al., 2015), as the PSTs (a) drew upon an activity located in the peripheries (front matter) of the unit, (b) adapted that task to bridge the gap between curriculum

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and community, and (c) made authentic connections based on the everyday experiences of local students. Conducting our study required selecting curricula for analysis and lesson design. Although coursework included discussion about various curricula, we focus here on EnVision, Math Realize Edition (referred to as EnVision from this point forward; Charles et al., 2015), which is a K-6 mathematics curriculum produced by a private entity. We chose a curriculum considered commonly used in U.S. elementary schools (Banilower et al., 2013). In particular, we focus on the introduction of fractions in Grade 3, a content area that is challenging from both teaching and learning perspectives (Lamon, 2007). We provided PSTs with all materials related to the given lesson, including the introductory front matter for the topic, connections to content and practice standards, unit scope and sequence, suggestions for differentiating instruction, key vocabulary, and the main parts of the intended lesson related to guided and independent practice.

Case Study Methodology Data collected for this project occurred at two university settings in regular coursework for PSTs. University A is a large public university in the northeastern United States. The elementary mathematics methods course is a part of a master’s and licensure program for Grades 1–6 and lasted for a total of 13 weeks. University B is a medium-size public university in the northwest region of the United States. The elementary mathematics methods course at University B is part of an undergraduate licensure program for Grades 1–6 and lasted for a total of 16 weeks. We, the authors of this chapter, were the instructors of record for the elementary mathematics methods courses from which we gathered data for this chapter. In prior analyses, we have found that PSTs translate curriculum materials in ways that variably include, adapt, or omit information provided. Our analyses revealed PSTs’ focus on fun ways to motivate engagement in the lesson, including adapting a fractions task such that the teachers used a pan of real brownies to appeal to students’ excitement about treats (Earnest & Amador, 2017). Such considerations did not necessarily reflect a real consideration of community resources. In the present chapter, we build on those findings to consider in particular how PSTs think about children’s MMKB, with an eye toward adaptations PSTs propose in order to tailor their lesson to the local community. In order to document how PSTs adapted curriculum materials as they considered children’s MMKB, we identified a group that demonstrated attention to this in their CSAT and lesson plan as they engaged with the curriculum. Using this criterion, we selected a group for case study analysis (Yin, 2014), which included three PSTs. The three PSTs, two female and one male, had recently graduated from college and were in their first semester of a 1-year master’s and teacher licensure program. Following a thorough review of all CSAT data completed by these PSTs across the 3 years they were each enrolled at the two institutions, we identified these PSTs as noticing the curriculum materials at a more

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in-depth level than their peers. Thus, they were designated as the focal case. Further, although they did not plan a lesson for the purposes of implementing it in an actual classroom (PSTs were all assigned a Grade 3 lesson on fractions, but individually had field placements across grades), they drew upon their field experience placements.

 ase Study: Integrating Community and Curricular C Resources To illustrate how the CSAT supported curricular noticing that considered children’s MMKB, the following section first describes the recognized opportunities for the inclusion of authentic connections in the materials. Next, we consider PSTs’ identified instances of opportunities for better supporting connections to authentic contexts that were made explicit in the CSAT but were not explicit in the original materials. We conclude with a discussion of adaptations made to the curricular materials, as evidenced in both the CSAT and in the written lesson plan. More specifically, we focus exclusively on one element of that lesson that appeared in a section prior to the main lesson pages in the unit front matter and was titled Math Project: Social Studies. This section gave the following “factoid,” which connects to the main emphasis of the lesson to understand fractions by dividing regions into equal parts: The flag of Nigeria is made up of three equal parts, or thirds. There is one row of three vertical bands. The left band is green, the middle band is white, and the right band is green. Adapted by Nigeria in 1960, the flag was designed for a competition in 1959 by student Michael Taiwo Akinkunmi. The green bands stand for Nigeria’s agriculture, and the white band stands for peace and unity.

This factoid was immediately followed by directions stating: Have students design a flag for your community that is made up of equal parts. They should choose the number of equal parts, the color for each part, and a meaning of the colors. Students should draw their flag and write a statement about the number of equal parts and the meaning of the colors they chose. (p. 217).

We focus our discussion in this chapter around this flag activity because of the central role the curriculum element played as a main focus for this specific trio of PSTs in both their written analysis in the CSAT and in their written lesson plan. Essentially, even with such a potential connection being located in materials describing the entire unit rather than in this particular lesson, the PSTs identified the flag activity as a key element of importance in the materials and centered their analysis on this activity within the broader lesson. We argue that the scaffolds provided in the questions on the CSAT likely supported the emphasis on authentic community connections, which may have influenced the curricular elements that became the most salient points of discussion for the PSTs.

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Curriculum Analysis Recognizing Opportunities for Connections in Materials As the trio read the curriculum materials and worked through the CSAT, they recognized opportunities in the materials for bridging the connection between mathematics and students’ communities and experiences. In this process, they attended to curricular elements that reflected possible knowledge and experiences students may have. For example, when asked about how the launch of the lesson may open spaces for making real-world connections, they considered the symbolism of a flag and recognized that students may understand that flags can represent communities: “Responses to the flag problem could be community related as the flag is a representation of the community.” Building from this, they also recognized that the “Flag activity could provide connection (provided as an option (sic) activity) to possibly serve as an introduction to fractions,” meaning they attended to the flag activity in the materials and highlighted it as an optional way to introduce fractions to students, even though the materials were not explicit that it would be an introduction. Further, in the process of using the CSAT to analyze the materials, the PSTs made comments referencing social studies and community connections to such a great extent that the connections often seemed to foreground that focus with the mathematics in the background. We recognize that the materials presented the flag activity as a social studies connection—the PSTs recognized this and even added to the connections in the curriculum materials. Specifically, when asked to identify opportunities for activating or connecting to family, cultural, and community knowledge in each phase of the lesson, they noted, “Students use the example of the Nigerian flag and design a flag for their own community.” They went on to recognize, “Symbolism of flag colors could make connections to other color symbolism.” They then considered students’ possible knowledge of countries and how that would relate to their knowledge of specific countries, “Flags could connect to knowledge of Nigeria” and “Possible connections to flags, if a student has a connection to one of these countries.” (Note the materials included graphics of the flags for Mauritius, Nigeria, Poland, and Seychelles.) In these examples, through specific questions, the CSAT scaffolded these connections for PSTs to such an extent that the comments of analysis about the materials minimally referenced the mathematics in their use. Although we emphasize the importance of maintaining the mathematical focus on equal parts in this lesson, we treat the PSTs’ attention and interpretation of the curriculum materials through the lens of community resources as an opportunity growing out of the CSAT’s structure.

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 ecognizing Unrealized Opportunities for Connections R in Materials To complement recognized opportunities presented in the materials, the PSTs also used the CSAT to identify unrealized opportunities for connections in the materials, meaning instances when more could have been included in the materials to form connections between the mathematics and students’ communities and experiences. Specifically, they recognized these areas within the presented content on the flags and also thought about how connections could extend beyond the flag activity. The wording of the questions on the CSAT supported consideration for noticing potential community connections in multiple ways (i.e., Where are opportunities for activating or connecting to family/cultural/community knowledge in each phase of the lesson?), which afforded PSTs opportunities to consider the lesson from multiple perspectives. When asked if students have opportunities to make their own connections, the trio considered students’ knowledge about countries and wrote, “These connections could be meaningless if symbolism of color choice is not fully understood. The teacher will need to be intentional in the explanation of this activity.” The PSTs recognized that the materials did not include a clear focus on emphasizing a color reflecting some aspect of the local context, yet another community connection. At the same time, the PSTs maintained cognizance of the role of mathematics in relation to community connections and more specifically in relation to the experiences of individual students. The trio wrote: Unless the teacher allowed for students to share their flag activity or come together with thoughts on the activity, there are not many avenues for students with varied mathematical and linguistic backgrounds/confidences to communicate their mastery on lesson objectives in the summarize phase.

In this example, the trio recognized the unrealized opportunities in the materials, meaning the materials did not overtly call for opportunities for students to share their mathematical understanding in relation to the flag activity. Even more specifically, the PSTs recognized that students will likely have varied mathematical backgrounds and would benefit from an opportunity to share their mathematical understandings. Much of the focus in the previous section on opportunity, and in this section on unrealized opportunity, is centered on the community connection. Finally, the flag example is not a connection to mathematical resources or practices in the community or to ways that children might use or engage in mathematical practices at home. Instead, the connection focuses on how the math concept (area model of fractions) relates to a (potential) community artifact (a flag). The absence of connections to activities and practices that involve mathematics seemed to be an additional missed opportunity. Despite recognizing multiple opportunities for connections to community, the PSTs also recognized that flags, and specifically those of countries geographically distant, may not provide authentic opportunities that meaningfully connect to students’ lives. The trio wrote, “Problem Solving section could provide a connection

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for students from the countries represented by these flags though it seems unlikely that a large number of students will have connections to these countries.” Further recognizing the lack of connections students may have to these flags, the PSTs voiced their reaction as they read the materials: Students might not connect to the country flags provided. Seems like there could have been a much better real-world connection problem, possibly to more commonly known objects or tasks (food, items from a classroom, etc.) that more students would recognize.

The CSAT provided opportunities for PSTs to critique what was present in the lesson; they considered ways the curriculum materials could have been adapted to improve connections for students. However, it is important to note that, despite this comment, they retained the connection to flags when they planned their lesson and considered adaptations. We find such considerations consistent with other groups in our courses: the focal trio considered how the community connection may not actually connect with all students’ experiences yet made decisions based on available information in the original curriculum materials. In doing so, they seem to acknowledge the possibility that children may not have a connection between flags and their communities at all.

Recognizing Adaptations to Support Connections As the PSTs considered adaptations in response to the CSAT, they wrote down possible changes they considered in order to improve the lesson. When asked about the adaptations they would make to each phase of the lesson, the PSTs noted they would, “shift the flag lesson to the summarize phase.” They also noted that they would, “ask students to create a classroom flag and take turns sharing each student’s response.” These adaptations—including both the aforementioned realized and unrealized opportunities—were further codified as they translated the work from the CSAT into a written lesson plan and emphasized connections to community. In the written lesson plan, the trio included connections to community and experiences that children of that particular age may have, many of which were noted on the CSAT. In the launch of the lesson plan, the trio planned to “introduce the new topic of fractions as an important mathematics knowledge for students’ futures (scientists, engineers, teachers, etc.).” They then had students work in pairs to “divide up something whole into equal parts” again, providing opportunity for students to consider their experiences and connect those to the classroom through encouragement to select items with which they were familiar. We note that the launch of the lesson plan did not focus on the flag activity. Rather, that activity was present in the summarize portion of the lesson and the PSTs maintained close fidelity to the curriculum materials in the investigate section of their lesson plan. We note that the investigate portion of the lesson included little, if any, connection to students’ funds of knowledge. Instead, students solved computation problems about fractional parts of geometric shapes. In the summarize portion of the lesson plan, the PSTs included

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the greatest emphasis on connections to students’ experiences as compared with any other parts of the lesson plan. It should be noted that these connections were to classroom experiences (i.e., flag for our classroom) and not to home- or community-­ based knowledge necessarily. The PSTs planned to include adaptations to the flag activity and wrote: Students will be given an 8 × 8 piece of grid paper and instructed to create a flag for our classroom. They will be told that it must be divided up into equal pieces, and that these pieces can be colored any color they choose- they should try to have a meaning behind the color choices.

The PSTs then noted that students: Will then write a short paragraph telling how many equal pieces are in their flag, what these parts are called, and why they chose the colors they did. They will then have the opportunity to share in front of the class.

And finally, the trio added an addition to recognize and value the created flags, an adaptation that was not present in the original materials or what they had written when completing the CSAT. One PST wrote, “I will later create a ‘Classroom Flags’ bulletin board to show off student work.” Together, the final written lesson plan, along with the CSAT, provides insights about what the PSTs noticed in the curricular materials (Dietiker et al., 2018; Males et al., 2015) and exemplify the adaptations this trio made to curricular materials as they translated them into a lesson. At the same time, the written narrative in the lesson plan provides evidence of the ways the PSTs, as a case, translated the curriculum materials into decisions about how to respond, which included their intended actions for drawing connections to students’ communities and experiences. These actions and decisions exemplify the trio’s curricular noticing in that they were able to communicate their thinking about that to which they attend and interpret in the materials (Dietiker et al., 2018; Males et al., 2015). The CSAT afforded an opportunity for the PSTs to consider, or attend to, the introductory fraction lesson in conjunction with possible opportunities for connections. The following section describes the role of the CSAT in supporting community connections.

Supporting Community Connections Although we cannot know the specific curricula our PSTs will use once in their first teaching positions, we can safely say that teachers are likely to have some form of curriculum. Given the diversity of materials and opportunities presented to teachers to engage with instructional resources, we argue for the importance of recognizing opportunities for authentic meaningful connections in the intersection between materials and students’ experiences (Drake et  al., 2015). Community funds of knowledge are critical in mathematics instruction and, therefore, in mathematics methods coursework. Yet, in order for teachers to draw upon community resources, they must first treat them as integral to the practice of teaching mathematics and

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must notice opportunities in the curriculum for said connections as they interact with materials. Ideally, such a treatment is already being embedded in instruction across mathematics methods courses; however, additional attention to the curricular noticing of PSTs may be instrumental for teacher educators to gain an even more in-depth understanding of PST thinking. At the same time, we wish to highlight the simple fact that the CSAT, coupled with a lesson plan, illuminates how teachers attend to particular information in curriculum materials and then interpret this information. In the case of our focal trio, the CSAT guided their analysis of the materials in ways consistent with tapping into children’s MMKB. Further, we contend that the CSAT supported PSTs in attending to curricular elements, and specifically elements about authentic connections. The specific responses the PSTs included in the CSAT about the flag activity reflect a consideration for curricular elements that may have otherwise not been read or analyzed. Although we cannot contribute the consideration of community connections exclusively to the CSAT, we are able to recognize the role of the CSAT in supporting considerations of these elements, such as the social studies connection found in the Envision materials. Further, the CSAT prompted the PSTs to recognize opportunities for connections and also recognize unrealized opportunities in the materials, to which they were able to consider adaptations in the lesson design process. As educators, whether prospective as in this chapter or practicing, interact with the CSAT, opportunities for identifying and building on children’s MMKB in the lesson design process surface (Drake et al., 2015). In the present example, the PSTs were able to consider how curricular elements could be coordinated with students’ lived experiences. In fact, the PSTs acknowledged not only that the specific countries may not allow for authentic mathematical connections but also that adaptations would further enable connections to children’s community. Their adaptation included a plan for children to create their own flags with color symbolism that connected to their lives—all the while focusing on the mathematical focus of equal parts. Essentially, the specific questions asked on the CSAT prompted the PSTs to consider children’s MMKB and provoked adaptations related to integrating curriculum and community spaces.

Conclusion The case presented in this chapter highlights one trio’s experience in reading, analyzing, and considering one introductory lesson on fractions. We emphasize that this is a case of PSTs who were mindful toward the consideration of MKKB as compared with other PST groups. Given the outcomes of this case, we argue for opportunities for PSTs to analyze curriculum materials with an eye towards mathematical community resources, and we view the CSAT as a particularly valuable tool to conduct such an analysis. For these PSTs, the CSAT became a support structure through which they were able to direct their analysis of materials in the process of considering connections to students’ experience. We contend that the CSAT can serve as a

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similar support structure for practicing teachers integrating curriculum and ­community spaces. Although curriculum is often written for any student in any classroom, the CSAT supports the inclusion of community connections in teaching that is vital for supporting students’ MMKB and providing authentic meaningful connections for mathematical understanding (Drake et  al., 2015; Goldsmith et al., 2000).

Appendix A: Curriculum Spaces Analysis Tool

Overall lesson What are the central mathematical goals or ideas of this lesson? Question for lesson phases Launch Explore Summary 1. What makes the task(s) in each phase of the lesson good and/or problematic? Consider:  Multiple entry points  Representations used  Level of cognitive demand  Language supports  Alignment with lesson goal(s) 2. Where are the opportunities for activating or connecting to family/cultural/ community knowledge in each phase of the lesson? 3. How does each phase of the lesson open spaces for making real-world connections? Do students have opportunities to make their own connections? 4. Where are the opportunities for students to make sense of the mathematics and develop/use their own solution strategies and approaches? 5. What kinds of spaces exist for children to share and discuss their mathematical thinking with the teacher and the class? 6. Where does the math authority reside in the lesson (e.g., only with teacher, only with textbook, only a few students, shared among teacher and students)?

Lesson peripheriesa

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Possible lesson adaptations Given your responses above, what kinds of adaptations might you make to each of the phases of the lesson? What kinds of adaptations might you make to the overall lesson structure or order? Lesson peripheries are anything in the textbook lesson that is not part of the “main” launch/ explore/summarize lesson. The peripheries are often ideas for differentiation or extension, typically found in the margins of the teacher’s guide and/or at the top or bottom of the page and/or set apart from the rest of the text by a box, shading, or other formatting

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References Aguirre, J. M., Turner, E. E., Bartell, T., Kalinec-Craig, C., Foote, M. Q., Roth McDuffie, A., et al. (2013). Making connections in practice: How prospective elementary teachers connect children’s mathematics thinking and community funds of knowledge in mathematics instruction. Journal of Teacher Education, 64(2), 178–192. Amador, J., & Earnest, D. (2016). Lesson plan-imation: Transforming preservice mathematics teachers’ lesson design experiences with animation. In M. Niess, S. Driskell, & K. Hollerbrands (Eds.), Handbook of research on transforming mathematics teacher education in the digital age (pp. 241–271). Hershey, PA: Information Science Reference. Amador, J., Males, L., Earnest, D., & Dietiker, L. (2017). Curricular noticing: Theory on and practice of teachers’ curricular use. In E. Schack, M. Fisher, & J. Wilhelm (Eds.), Building perspectives of teacher noticing (pp. 427–444). New York: Springer. Banilower, E.  R., Smith, P.  S., Weiss, I.  R., Malzahn, K.  A., Campbell, K.  M., & Weis, A.  M. (2013). Report of the 2012 national survey of science and mathematics education. Chapel Hill, NC: Horizon Research, Inc. Brown, M., & Edelson, D. (2003). Teaching as design: Can we better understand the ways in which teachers use materials so we can better design materials to support their changes in practice? (Design Brief). Evanston, IL: Center for Learning Technologies in Urban Schools. Charles, R.  I., Fennell, F., Caldwell, J.  H., Schielack, J.  F., Copley, J.  V., Crown, W.  D., et  al. (2015). enVisionMATH Common Core, Realize Edition. Glenview, IL: Pearson Education Inc. Dietiker, L., Males, L., Amador, J., & Earnest, D. (2018). Curricular noticing: A comprehensive framework to describe teachers’ interactions with curriculum materials. Journal for Research in Mathematics Education, 49(5), 521–532. Drake, C., Land, T. J., Bartell, T. G., Aguirre, J. M., Foote, M. Q., Roth McDuffie, A., et al. (2015). Three strategies for opening curriculum spaces. Teaching Children Mathematics, 21(6), 346–353. Earnest, D., & Amador, J. (2017, online first). Lesson planimation: Preservice elementary teachers’ interactions with mathematics curricula. Journal of Mathematics Teacher Education. Goldsmith, L., Mark, J., & Kantrov, I. (2000). Choosing a standards-based mathematics curriculum. Portsmouth, NH: Heinemann. Hirsch, C.  R. (2007). Curriculum materials matter. In C.  R. Hirsch (Ed.), Perspectives on the design and development of school mathematics curricula (pp.  1–5). Reston, VA: National Council of Teachers of Mathematics. Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.

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Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. In R. Hunter, B. Bicknell, & T.  Burgess (Eds.), Crossing divides: Proceedings of the 32nd Annual Conference of the Mathematics Education Research Group of Australasia. Wellington, New Zealand. (Vol. 1, pp. 12–30). Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 629–668). Charlotte, NC: Information Age. Males, L., Earnest, D., Dietiker, L., & Amador, J. (2015). Examining K-12 prospective teachers’ curricular noticing. In T. G. Bartell, K. N. Bieda, R. T. Putnam, K. Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 88–95). East Lansing, MI: Michigan State University. Philipp, R. (2014, April). Using representations of practice in survey research with mathematics teachers. In Symposium conducted at the National Council of Teachers of Mathematics Research Conference. New Orleans, LA. Remillard, J. T. (2012). Modes of engagement: Understanding teachers’ transactions with mathematics curriculum resources. In G.  Gueudet, B.  Pepin, & L.  Trouche (Eds.), From text to ‘lived’ resources: Mathematics curriculum materials and teacher development (pp. 105–122). New York, NY: Springer. Turner, E., Foote, M., Stoehr, K., Roth McDuffie, A., Aguirre, J., Bartell, T., et al. (2016). Learning to leverage children’s multiple mathematical knowledge bases in mathematics instruction. Journal of Urban Mathematics Education, 9(1), 48–78. Turner, E. E., Drake, C., Roth McDuffie, A., Aguirre, J. M., Bartell, T. G., & Foote, M. Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67–82. Yin, R. K. (2014). Case study research: Design and methods (5th ed.). Los Angeles: Sage.

Part IV

Identity, Positionality, & Praxis

Chapter 10

Reflecting Back to Move Forward: Using a Mathematics Autobiography to Open Humanizing Learning Spaces for Pre-Service Mathematics Teachers Crystal Kalinec-Craig, Theodore Chao, Luz A. Maldonado, and Sylvia Celedón-Pattichis

Keywords  Mathematics teacher education · Funds of knowledge · Culture · Community · Elementary teacher preparation

The pervasive ideologies that dominate traditional mathematics classrooms suggest to students that mathematics is a culture-free, apolitical field that values hard work and natural talent as markers of success. Similar ideologies also perpetuate the myth that only some students can be successful in mathematics. Recent research in mathematics education pushes back on these traditional ideologies by reframing mathematics classrooms as humanizing spaces (e.g., Aguirre, MayfieldIngram, & Martin, 2013; Gutiérrez, 2013) that value diverse ideas of what it means to do mathematics and who can do mathematics. Teachers who learn to view mathematics as a socially constructed field (which also undergoes constant reconstruction over time) (Ernest, 1998) can also learn to see how their culture, native language, race, geographical location, socioeconomic status, gender identification, and religion play a role in how they teach (hooks, 1994). Teachers who adopt the C. Kalinec-Craig (*) University of Texas at San Antonio, San Antonio, TX, USA e-mail: [email protected] T. Chao The Ohio State University, Columbus, OH, USA e-mail: [email protected] L. A. Maldonado Texas State University, San Marcos, TX, USA e-mail: [email protected] S. Celedón-Pattichis University of New Mexico, Albuquerque, NM, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_10

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stance that mathematics is neutral (Gutiérrez, 2013; Martin, 2015), unbiased, or unaffected by prior experiences can also learn to identify and interrupt potential inequities that arise in their classrooms. Mathematics teachers seeking to adopt a humanizing stance about their practice might first examine their own prior experiences and positionalities in the context of their current stance. Teachers might engage in self-reflection to consider how they replicate (or resist) those practices they experienced as students (Lortie, 1975). For example, Drake, Spillane, and Hufferd-Ackles (2001) found that teachers’ prior experiences as students informed their stance about teaching mathematics and literacy. Some teachers leveraged their prior negative experiences as to not replicate similar negative experiences for their students as they learned mathematics: “the combination of early negative and early positive experiences learning mathematics enabled these four teachers to envision, desire, and create more positive mathematics learning environments for their students” (Drake et al., 2001, p. 10). Teachers’ beliefs about what it means to do mathematics can also inform the actions they take, words they speak, and humanizing spaces they create (or do not create) for students learning mathematics. For example, if teachers believe that students should learn mathematics by only practicing procedures through a series of worksheets, then their students might come to believe that only these skills are valued in the classroom. Moreover, those same students may come to believe that their mathematical knowledge and experiences brought from home are not valued. In contrast, if teachers believe that institutional racism plays a role in students’ success in mathematics, then they may create opportunities that help students to question, push back, and resist those forces in their classrooms. Mathematics teacher educators’ (MTEs’) prior experiences and positionality inform how they prepare new teachers (e.g., Celedón-Pattichis & Ramirez, 2012; Kalinec-Craig & Bonner, 2016; White, Crespo, & Civil, 2016). Aguirre (2009), for example, shared her positionality and identity as a bilingual Latina mathematics teacher educator with her mathematics methods pre-service teachers (PSTs). Aguirre described how she would push back against her PSTs who used deficit language about any student, especially Latinx students or emerging bilinguals. When MTEs model for PSTs an introspective stance about how their prior experiences and positionalities inform their practice, they can help PSTs adopt a similar curiosity about their own emerging practice; reflecting back to move forward in the practice of adopting a humanizing stance in mathematics education. We, the authors of this chapter, situate our work within the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project and Freire’s (1970) framework for praxis which he defines to be the “reflection and action directed at the structures to be transformed” (p. 126). With respect to the TEACH Math project, we use the mathematics autobiography, an activity sometimes used by TEACH Math teacher educators prior to engaging their PSTs in the work of the TEACH Math modules. We assign the mathematics autobiography to help our PSTs reflect on who they are as mathematics students and as future teachers as they learn to create humanizing spaces to do and learn mathematics. We also use it as an entry point for our PSTs to adopt a stance of curiosity about their students and their

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lives, especially when they conduct a walk of the community surrounding the school (see Chap. 2 for more information about the Community Mathematics Exploration Module). While writing this chapter, we first shared our own mathematics autobiographies with each other and engaged in a dialogue about what we learned from each other’s stories. Our discussion considered how we use the mathematics autobiography as a tool for helping PSTs to engage in praxis as it relates to their own prior experiences. We also considered the challenges we might encounter in asking others to share their mathematics stories. This following chapter is the result of this conversation.

Author Mathematics Autobiographies We are four MTEs who represent a wide range of backgrounds and experiences, and who all engaged in the TEACH Math work in some way. Crystal has been a part of TEACH Math since she was a doctoral student at the University of Arizona. Luz and Theodore also helped develop initial drafts of some of the Problem-Solving Interviews that are part of the Mathematics Learning Case Study Module as doctoral students at the University of Texas at Austin. Sylvia was an Advisory Board member for TEACH Math since the project began and while at the University of New Mexico had been part of the work before the project was conceptualized. We have each used, augmented, and reflected upon our use of the TEACH Math modules in our own elementary mathematics methods courses. In this section, we will briefly share our own autobiographies, then engage in a discussion about what we noticed from each other’s autobiographies and how the mathematics autobiography accomplishes multiple goals for us as MTEs using the TEACH Math modules.

Crystal’s Mathematics Autobiography For much of my life as a mathematics student and during my first few years as a teacher, I felt smart in mathematics. I spoke the language of my teachers and could relate to the mathematical contexts my teachers used in mathematics class. It was not until years later (after I gained more experience as a teacher) that I realized I based my self-perception on an assumption that all of mathematics could be simplified to a procedure that could be replicated. I assumed anyone could be successful in mathematics if they tried hard and followed the prescribed steps. Yet over years, I questioned my perspective on (and approach to) mathematics teaching and learned to incorporate more problem-solving tasks that connected to my students’ lives. As a doctoral student, I learned to see mathematics and mathematics education as inherently gendered, racialized, class-based, and political. I honestly share my journey as an MTE with my PSTs and admit to them that my professional growth as a teacher is ongoing. I hope that by sharing my prior experiences and positionality

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with my PSTs I can create a safe space for them to share their own autobiographies. I state that my class is one where we will challenge assumptions, resist deficit language, and take a stance of curiosity about mathematics education.

Theodore’s Mathematics Autobiography My own identity as a mathematics teacher colors the way I listen to and work with mathematics teachers. As a child, I excelled in mathematics, yet felt embarrassed to fit into the model minority myth that purports Chinese American students, like me, are “gifted” at mathematics (Hartlep, 2013; Lee, 1996). So, I developed a disinterested disposition to mathematics, while secretly solving mathematics problems at home. It didn’t work. My teachers kept promoting me into the “honors” mathematics classes and engineering clubs. After a short career in software engineering, I taught middle school mathematics, bringing with me all the traditional recall and procedural techniques from childhood into my classroom. I taught mathematics to seventh- and eighth-grade students at a middle school serving a primarily Black and Latinx population in Brooklyn, New York. It did not take me long to realize how poorly these techniques worked in actually helping children learn, love, and be successful in mathematics. This revelation, coupled with my passion for social justice and equity, helped me forge an identity as a mathematics teacher focused on helping students see the beauty of mathematics in the world around them and to use their mathematics knowledge to fight oppression.

Luz’s Mathematics Autobiography There are pieces of my mathematics autobiography that come out during my semester teaching an elementary mathematics methods course. In sharing relevant numbers about myself during the first day of class, PSTs often learn I was born in the border city of El Paso, Texas and that I did not enter teaching through traditional coursework and degree programs, but through alternative certification. They learn about how I was a first-grade bilingual teacher in Houston, Texas, and about how my students called themselves ingenieros y científicos (engineers and scientists) by the end of the year. My PSTs learned about how my drive to understand how children better learn mathematics led me to graduate school in New York City and a job in Harlem working with students who were the same, yet not, as those that I taught in Houston. I shared how two years in a master’s program was not enough time to fully understand why children would often change from excited, risk-taking, participatory mathematical kindergarteners, to silent, error-averse, and withdrawn math-­disliking fifth graders. I further shared how I continued my ­journey back to Texas where I would spend close to a decade completing my

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doctoral studies, being involved in research around children’s mathematical thinking, particularly bilingual children’s thinking, and in working with prospective and practicing teachers around our common questions. And finally, they learned how my experiences led to my big aha moment: teachers are the key! Teachers are a key to what happens in the mathematics classroom, from the tasks that children are asked to solve, to the ways in which children are allowed or expected to participate, and finally, to the decision of what “counts” towards being a “good” mathematician. I often share with PSTs how I too shared in the traditional type of mathematics teaching that many of them experienced (and carry with them into adulthood), and then use this shared experience as a common goal in changing mathematics teaching and learning for the better.

Sylvia’s Mathematics Autobiography I was born along the border in Texas and raised in México. Having been schooled in México up to the third grade, I developed strong Spanish literacy and mathematics knowledge and skills. In the mid-1970s, my parents obtained a visa, and our family moved to California. Going to school for the first time in the United States and listening to teachers speak English all day were experiences I will never forget. I was immediately labeled an English Language Learner (ELL) and fortunately I was not placed in a lower grade level. This ELL label followed me until ninth grade, when a white English teacher, Ms. Barbara Osuna, asked the counselor that I be placed in College Prep English during my second semester of ninth grade, instead of the ELL English class. Her husband, Mr. Sabas Osuna, was a mathematics teacher who influenced my love of mathematics through his Pre-Algebra and Calculus courses taken during my freshman and senior year, respectively. Because Ms. Barbara Osuna challenged my placement in ELL courses, I was placed in college prep courses for my remaining high school years. Mr. Osuna advised me on which mathematics courses I needed to enroll in to progress from Pre-Algebra to Calculus by the time I was a senior. These major life-changing events opened up a path for me to attend the University of Texas at Austin and to major in secondary education with an emphasis in mathematics and Spanish. The work that I have conducted for the past 20 years as a mathematics/bilingual teacher educator and the 4 years of teaching high school mathematics in the same school I graduated from is rooted in the advocacy and strong support that Mr. and Mrs. Osuna, as well as other teachers, provided for me as a second language learner of English. In my elementary mathematics methods course, I use the mathematics autobiography to have bilingual/teaching English to speakers of other languages (TESOL) PSTs share stories related to their experiences learning mathematics, including people who may have had a major influence in their lives. After reflecting on these mathematical experiences, PSTs are then asked to discuss how these experiences, feelings, and beliefs might impact the kind of mathematics teacher

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they will be or the kind of teacher that they want to be. I share my own experiences with PSTs about my learning of mathematics. hooks (1994) refers to this process as engaged pedagogy, which empowers both students and teacher and involves taking risks and engaging in tense conversations when addressing equity and social justice issues.

Discussion and Dialogue About the Mathematics Autobiographies As a part of the process of writing this chapter, we first shared our mathematics autobiographies with one another and then engaged in a discussion about what we noticed. The purpose of the discussion was to unpack our own experiences and journeys to become MTEs, which in essence were four mini-Getting to Know You Interviews (see activity description in Chap. 2 of this volume) about our prior experiences. We also discussed how we see these mathematics autobiographies unfold in our teacher preparation coursework, namely with elementary PSTs. As a result of our discussion about our autobiographies, we noticed three overarching themes that emerged with respect to the purpose of the mathematics autobiography in our methods classes and will briefly discuss each of them below: • Leveraging the power of testimonials/testimonios to elevate historically silenced voices; • Using autobiographies as an entry point for PSTs to develop a critical lens for teaching mathematics; and • Framing the mathematics autobiography as a tool to support PSTs’ curiosity about getting to know their students in more humanizing and holistic ways.

 everaging the Power of Testimonials/Testimonios to Elevate L Historically Silenced Voices As we discussed our own mathematics autobiographies, we realized particular stories are typically elevated over others in research and in the classroom. The narratives of white, middle class, native English-speaking PSTs (who make up the majority of our PST population) have been shared in many spaces before and these experiences are ones that are typically valued in traditional American classrooms. While reading the mathematics autobiographies of each other and unpacking them through discussion, we, in turn, recognized the implications from this process: there is power in sharing of the testimonial/testimonies, which can elevate stories that might have been silenced in the past. As Crystal said during our small group discussion:

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These stories and testimonials help our PSTs see the power they have as future teachers. Look at the first chapter from Sylvia and Nora’s book (Celedón-Pattichis & Ramirez, 2012). These stories cannot come from my personal experience. This is real life, so it’s good to have these book chapters that talk about the stories our PSTs might not necessarily be familiar with.

Coupled with the Mathematics Learning Case Study Module, the mathematics autobiography can create potential spaces to help elevate stories of emerging bilinguals in our classrooms. By elevating Sylvia’s story, MTEs can also show examples of how teachers can reimagine mathematics classrooms as humanizing spaces that value children’s native language and experiences and resist deficit thinking about students.

 sing Autobiographies as an Entry Point for PSTs to Develop U a Critical Lens for Teaching Mathematics We noticed how the mathematics autobiographies also serve as an entry point for PSTs to develop a critical lens for teaching mathematics. When PSTs share stories of being stereotyped as being less (or more) capable of doing mathematics because of their native language, race, class, gender, and/or sexual orientation, MTEs can support PSTs to ask the broader question of what purpose these stereotypes served (and continue to serve) and for whom. As Sylvia stated: I don’t want my PSTs to see that English Language Learner (ELL) is a bad label, because then our children won’t get the support they need. We want our PSTs and students to be supported. For me, having bilingual support, it would have been more helpful. In California, I didn’t get native language support and my parents did not speak English, so I had no help with homework. I want my PSTs to question the labels like “bilingual, special education, and ELL.” The labels impact people directly like the placements they get. It gets into tracking and can set up the context of different lenses and a critical framework that we might use when teaching.

To which Luz later responded, “So in other words, to check [PSTs’] potential deficit labeling around students and consider how it is playing out in classroom teaching and interactions?” Sylvia and Luz’s comments remind us, as MTEs, of the power of labels on students and if they serve to open access to resources or limit students’ potential to be seen as successful (Pollock, 2017). When PSTs read autobiographies like Theodore’s, Luz’s, and Sylvia’s, MTEs can open spaces with their PSTs to unpack the problematic impact of labels assigned to students of color and why they exist in education. (And we note that reading Crystal’s autobiography might serve as a learning opportunity to unpack notions of white privilege and open up ways to discuss mathematics as a gendered, sexist context that girls and women continue to navigate when becoming teachers of mathematics.) As PSTs learn to adopt equity-­ based mathematics instructional practices, they also need an entry point for having honest discussions about deficit language that oppresses and marginalizes students; the mathematics autobiography can be one such entry point.

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 raming the Mathematics Autobiography as a Tool to Support F PSTs’ Curiosity About Each Other Finally, we discussed how the process of writing and sharing our mathematics autobiography does not and should not stop when the last person shares their story. When we consider that we are constantly writing and rewriting the stories of our lives as educators, we can use the process of a mathematics autobiography to reflect on our work and to plan future changes to our practice. Because we only have our PSTs for a short time in their teacher preparation program, we considered how the mathematics autobiography can be a living document that encourages iterative growth and reflection through praxis. Theodore described a PST in his methods class who wrote about her emerging understanding of critical race theory in her teaching. After having graduated, she has a deeper stance connected to global inequity and war-driven systems of oppression that affect the experiences of her students. As Theodore discussed his experience with this PST, he considered whether “Maybe the autobiography stories are like seeds, initial lenses into PSTs’ evolving intentions and practice?” To unpack Theodore’s comment, we considered several questions for ourselves and for our work as MTEs: • How might the process of writing and sharing our autobiographies have a lasting effect on our PSTs long after they have left our classroom? • How might the autobiographies help PSTs to continue to question whether a particular instructional practice is benefitting the needs of all students in the class or just a select few? • If we know that PSTs leverage their prior experiences when learning how to teach mathematics, what would happen if we positioned the mathematics autobiography in a way that created a new, more critical space for PSTs to adopt a stance of teaching mathematics that promotes equity, a space that they could return to when they have classrooms of their own? When we consider the mathematics autobiographies in this way, they become something much more than an assignment in a methods class; the autobiography can become a powerful vehicle for developing teachers who adopt a humanizing stance to their practice.

Considerations and Challenges In this chapter, we presented an introduction to the Mathematics Autobiography Activity and how we have come to learn about the power of our stories in mathematics classrooms. But not every space is safe and not every story is one that feels safe to share. Given the current political climate, we see active steps being taken to diminish the contributions, rights, and safety of people and communities who are incorrectly blamed for the social ills of society. Immigrant PSTs might not feel

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as though their stories about learning mathematics in their home country are safe stories to share; they have been told by their federal government that they are not welcome here. Black and Brown Americans may not want to share their stories of discrimination and police brutality as it relates to the ways they have experienced a violent world (See Gutstein’s (2012) project titled “Driving While Black/Driving While Brown” for more information on the intersection of police profiling and mathematics). Some experiences are traumatic, and writing a mathematics autobiography can reopen those wounds. MTEs, especially those who may not have experienced the trauma of racism, sexism, bigotry, or discrimination, need to be cognizant of the existence of these stories and lived experiences and the potential they hold for re-traumatization before asking PSTs or practicing teachers to trust that their stories (and the emotions tied to these stories) will be treated respectfully and carefully (Pollock, 2017). The Mathematics Autobiography Activity is meant to be an opening to learn about and learn from the experiences that we bring to the classroom, but not with the purpose of creating more divisiveness among ourselves. Therefore, prior to engaging in the Mathematics Autobiography Activity (or engaging in any of the TEACH Math Modules with students for that matter), PSTs need to develop a trust among themselves and with their MTEs that deficit language and perspectives are not to be used. In turn, MTEs need to create a space for PSTs to share their stories when they feel safe and willing to share those experiences. In this chapter, we also discuss the mathematics autobiography and present specific ways that we contextualize and situate our stories of mathematics as a means of disrupting and dismantling the notions of inequity in our mathematics classrooms.

Conclusion and Implications Our chapter drew from existing literature about using personal narratives to help PSTs and teachers develop new, culturally responsive perspectives about teaching and learning (Aguirre, 2009; Celedón-Pattichis & Ramirez, 2012; He & Cooper, 2009; Kalinec-Craig & Bonner, 2016; LoPresto & Drake, 2004). He and Cooper (2009) use the mathematics autobiography as a first step for PSTs to begin to learn about the lives of children from backgrounds different than their own by sharing written autobiographies, which draw on a variety of perspectives and experiences. But as MTEs, we are cautious to not let the autobiographies transform into another assignment in our class and into a place that promotes assumptions or stereotypes. Instead, we view the mathematics autobiography as a journey of self-discovery and reflection while also acknowledging that we still have more to learn about one another and each other’s prior experiences. The mathematics autobiography can serve multiple purposes. First, it can be used to examine our PSTs’ prior experiences through various lenses (e.g., Critical Race Theory, LatCrit, Black Feminism, Queer Theory, curriculum ideologies and theories) and to unpack the intersections we all have in our life stories (Cho,

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Crenshaw, & McCall, 2013; Hearn, 2012). It is essential for MTEs to learn more about the students in their teacher preparation program because it models for them our expectations of culturally responsive teaching. To acknowledge that we live and identify in multiple spaces is to acknowledge that each of these spaces play a role in how we learn mathematics, use mathematics, and see ourselves as mathematicians. When teachers acknowledge the multi-dimensional spaces that they and their students reside within, they may be more responsive to the needs and experiences of their students. The mathematics autobiography also opens up spaces to unpack and make sense of issues of inequity and marginalization in mathematics classrooms. For PSTs who have never been labeled as an ELL or as a special education student, they may not know the destructive implications of those labels on a young child. Many white, middle class, native English-speaking PSTs may not have experienced the injustice of systemic racism and oppression compared to others of their peers because American education and schooling were designed for them to succeed. Therefore, MTEs need to find more ways to elevate the voices of PSTs and children that have been silenced or stereotyped in the past (Gutiérrez, 2015; Prieto & Villenas, 2012), which include experiences from teachers such as Theodore, Sylvia, and Luz that you read about earlier. Future MTEs also need to be prepared to self-reflect through praxis so that they can help their PSTs to do the same; acknowledging and interrupting established privilege is not work that should only be the burden of PSTs but of all who play a role in education and schooling (Kalinec-Craig & Bonner, 2016; Wager, 2014). The mathematics autobiography places a human face on the topics, concepts, and skills that PSTs typically learn during a mathematics methods class. Testimonials of students who learned mathematics in a second language can serve as concrete reminders to PSTs of the importance of considering students’ cultural and linguistic backgrounds as assets and serving as advocates for them. Testimonials of students and teachers who have experienced “othering” (Foote & Bartell, 2011; Kumashiro, 2000) encourage more teachers to adopt a culturally responsive pedagogy that values students’ funds of knowledge and perspectives. Finally, the mathematics autobiography can serve as an entry point for our colleagues in departments and colleges outside of education and teacher preparation. Our PSTs have interacted with professors in many departments prior to beginning their teacher preparation coursework. We acknowledge that the experiences of our PSTs in their content coursework are a part of their journey to become a mathematics teacher. The (mathematics) autobiography could be used to help professors in the sciences, social sciences, humanities, and fine arts departments to consider their own educational journeys as a means of exploring how their teaching philosophy and practices do (or do not) value the cultural and linguistic resources that their students bring to the post-secondary classroom. The mathematics autobiography can be one tool that can support more teachers to adopt humanizing stances toward their practice by sharing their experiences and positionalities.

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References Aguirre, J.  (2009). Privileging mathematics and equity in teacher education: Framework, counter-­resistance strategies and reflections from a Latina mathematics educator. In B. Greer, S. Mukhopadhyay, A. B. Powell, & S. Nelson-Barber (Eds.), Culturally responsive mathematics education (pp. 295–319). New York: Taylor & Francis. Aguirre, J. M., Mayfield-Ingram, K., & Martin, D. B. (2013). The impact of identity in K-8 mathematics learning and teaching: Rethinking equity-based practices. Reston, VA: The National Council of Teachers of Mathematics. Celedón-Pattichis, S., & Ramirez, N. G. (2012). Beyond good teaching: Advancing mathematics education for ELLs. Reston, VA: National Council of Teachers of Mathematics. Cho, S., Crenshaw, K., & McCall, L. (2013). Toward a field of intersectionality studies: Theory, applications, and praxis. Signs, 38(4), 785–810. Drake, C., Spillane, J. P., & Hufferd-Ackles, K. (2001). Storied identities: Teacher learning and subject-matter context. Journal of Curriculum Studies, 33(1), 1–23. Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany: State University of New York Press. Foote, M.  Q., & Bartell, T.  G. (2011). Pathways to equity in mathematics education: How life experiences impact researcher positionality. Educational Studies in Mathematics, 78, 45–68. Freire, P. (1970). Pedagogy of the oppressed (30th anniversary ed). New York: Continuum. Gutiérrez, R. (2013). Why (urban) mathematics teachers need political knowledge. Journal of Urban Mathematics Education, 6(2), 7–19. Gutiérrez, R. (2015). HOLA: Hunt for opportunities–learn–act. The Mathematics Teacher, 109(4), 270–277. Gutstein, E. (2012). Reading and writing the world with mathematics: Toward a pedagogy for social justice. Hoboken, NJ: Taylor and Francis. Hartlep, N.  D. (2013). The model minority stereotype: Demystifying Asian American success. Charlotte, NC: Information Age Press. He, Y., & Cooper, J. E. (2009). The ABCs for pre-service teacher cultural competency development. Teaching Education, 20(3), 305–322. Hearn, M. C. (2012). Positionality, intersectionality, and power: Socially locating the higher education teacher in multicultural education. Multicultural Education Review, 4(2), 38–59. hooks, B. (1994). Teaching to transgress: Education as the practice of freedom. New  York: Routledge. Kalinec-Craig, C., & Bonner, E. (2016). Seeing the world with a new set of eyes: (Re)examining our identities as White mathematics education researchers of equity and social justice. In N. Joseph, C. Haynes, & F. Cobb (Eds.), Interrogating whiteness and relinquishing power (pp. 91–112). New York: Peter Lang. Kumashiro, K. K. (2000). Toward a theory of anti-oppressive education. Review of Educational Research, 70(1), 25–53. Lee, S. J. (1996). Unraveling the “model minority” stereotype: Listening to Asian American youth. New York: Teachers College Press. LoPresto, K.  D., & Drake, C. (2004). What’s your (mathematics) story? Teaching Children Mathematics, 11(5), 266–272. Lortie, D. C. (1975). Schoolteacher: A sociological study (2nd ed.). Chicago: The University of Chicago Press. Martin, D.  B. (2015). The collective black and “principles to actions”. Journal of Urban Mathematics Education, 8(1), 17–23. Pollock, M. (2017). Schooltalk: Rethinking what we say about—And to—Students every day. New York: The New Press.

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Prieto, L., & Villenas, S. (2012). Pedagogies from Nepantla: Testimonio, Chicana/Latina femi- nisms, and teacher education classrooms. Equity & Excellence in Education, 45(3), 411–429. 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. White, D. Y., Crespo, S., & Civil, M. (2016). Cases for mathematics teacher educators: Facilitating conversations about inequities in mathematics classrooms. Charlotte, NC: Information Age Publishing.

Chapter 11

Preparing Pre-Service Elementary Mathematics Teachers to Critically Engage in Elementary Mathematics Methods Theodore Chao, Luz A. Maldonado, Crystal Kalinec-Craig, and Sylvia Celedón-Pattichis

Keywords  Mathematics education · Teacher education · Autobiography · Critical reflection · Instagram · Photovoice · Rights of the Learner This chapter focuses on activities that prepare pre-service teachers (PSTs) to engage in the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project modules. Using Freire’s (1970) construct of praxis as critical reflection coupled with transformative action, we present activities we use to start critical reflection discussions with PSTs so that the subsequent TEACH Math modules support them in enacting transformative action in their practice. We first present our specific contexts in working with PSTs, introduce our activities and how we utilize them for connecting to specific TEACH Math modules, and then reflect upon our successes and struggles with these activities. On the surface, the TEACH Math project can be viewed as a collection of three modules designed to engage PSTs to teach mathematics using research-based practices focused upon equity. But as four mathematics teacher educators (MTEs) who have used these modules in various ways, we view the overall enactment of the TEACH Math modules as more than just a combination of the three modules. Our T. Chao (*) The Ohio State University, Columbus, OH, USA e-mail: [email protected] L. A. Maldonado Texas State University, San Marcos, TX, USA e-mail: [email protected] C. Kalinec-Craig University of Texas at San Antonio, San Antonio, TX, USA e-mail: [email protected] S. Celedón-Pattichis University of New Mexico, Albuquerque, NM, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_11

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experiences allow us to view these modules as specific extensions of Freire’s (1970) theory of critical consciousness that, when enacted thoughtfully, create equitable mathematical educational experiences for all PSTs and, thereby, their students. Here, we speak about (a) what it means to prepare our PSTs for engaging with the TEACH Math modules, (b) how we approach the TEACH Math modules so that PSTs view them as transformative, rather than as an assignment they complete for a grade, and (c) the power of the TEACH Math modules when PSTs have been properly prepared for them through critical reflection. Our experiences, not just with TEACH Math but with teacher education in general, tell us that PSTs rarely see issues of equity, agency, and access as pertinent to their teaching situation, particularly in the context of mathematics (Turner et  al., 2014). Additionally, although we engage our PSTs in critical self-reflection, we have seen that this does not always lead to changes in actual teaching dispositions or identities. Each of us can recall PSTs who speak proudly about orienting their teaching around equity when in our course, but then abandon these principles a few years later when they enter the classroom, loading our Facebook feeds with deficit language and dispositions about their own students. We acknowledge that when our PSTs graduate, they often work within an institutional and structural system that maintains the status quo of power and inequity; seeing graduates use deficit language might be a product of the system and not reflective of them as teachers. This happens when critical reflection is uncoupled from transformative action. And critical reflection comes from “reading” the world with mathematics in a way that the mathematics of our PSTs’ everyday worlds is connected to the mathematics they teach (Gutstein, 2006). We see the TEACH Math modules as promoting Freire’s transformative action, what Gutstein (2006) refers to as “writing” the world with mathematics, so we couple these modules with the following activities meant to spur critical reflection. We should warn readers that, even as veteran users of these modules, we still struggle and are constantly evolving in our enactment of the TEACH Math modules. Rather than offering takeaway solutions here, we use this chapter to discuss our experiences using these activities to engage PSTs in the critical reflection necessary to fully engage in the modules as transformative action.

The Spaces in Which We Live We speak with four unified voices about how we implement the TEACH Math modules in our unique situations. Our instruction of PSTs shares many facets; most notably our alignment with critical issues of equity, agency, and empowerment and our experiences in helping to pilot the TEACH Math modules (see Chap. 2). We all teach in what are commonly referred to as “Elementary Mathematics Methods” courses serving future elementary school teachers in a university setting. Our PSTs take our methods course after already completing at least one prerequisite mathematics content course and most of our students take our methods course while also engaging in student teaching at a local elementary school.

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Our situations are different in a few ways, however. Theodore prepares PSTs within a Prekindergarten (PreK) to third-grade teacher education program at a large public university in Ohio. His social justice and urban education-oriented teacher education program involves either undergraduates in their second of fourth semesters or master’s students in their first of two semesters. While PSTs take his methods course, they also intern as student teachers 2 days a week in a PreK-third-grade setting. Before his PSTs take his methods course, PSTs have taken 10 credit hours of mathematics coursework designed specifically for pre-service elementary teachers. Theodore has used all three modules in his teaching, but consistently focuses on the Mathematics Learning Case Study Module and the Community Exploration Module. Luz prepares elementary PSTs who will be certified as either English as a Second Language (ESL) or Bilingual Education Early Childhood (EC)-6th generalists at a large, public, Hispanic Serving Institution (HSI) in Texas. Her elementary mathematics methods course, required for both undergraduate and graduate certification programs, is a “floater” class and does not need to be taken at any specific time in the program. PSTs do, however, have to have passed three preliminary mathematics courses taught by the mathematics department prior to enrollment. Because of this “floater” status, her PSTs are not necessarily in a school placement at the time they take their mathematics methods course. Luz consistently implements the Mathematics Autobiography Activity, the Mathematics Learning Case Study Module, and the Community Exploration Module in her elementary mathematics methods courses. Crystal also prepares PSTs seeking an EC-6th certification at a large, public HSI in Texas. When PSTs take Crystal’s methods course, they are simultaneously enrolled in reading comprehension, science methods, and assessment courses. Crystal’s PSTs are also typically in the second of fourth semesters for their teacher preparation program and might be undergraduates or post-baccalaureate students. The PSTs complete 80 hours of fieldwork across 10 weeks during the math methods semester with children who reflect her city’s cultural and linguistic diversity. Finally, Crystal’s PSTs also engage in a 10-week afterschool program called Support and Enrichment Experiences in Mathematics (SEE Math) where they work with a child to solve interdisciplinary problem-solving tasks that connect to the child’s home and community knowledge. Crystal uses the Mathematics Autobiography Activity, the Mathematics Learning Case Study Module, and the Community Mathematics Exploration Module in her elementary mathematics methods course. Sylvia prepares elementary PSTs who are in the Bilingual/TESOL Cohort in a public HSI located in New Mexico. Before taking her elementary mathematics methods course, PSTs enroll in 9 credit hours of mathematics content courses designed for elementary teachers. Student teaching is comprised of three semesters with the mathematics methods course offered during the first semester of the three. At the same time PSTs are enrolled in the mathematics methods course, they teach in a K-8 public school classroom 2  days a week, focusing on working with ­culturally and linguistically diverse students and reflecting the bilingual or TESOL practices they are developing. Although Sylvia does not teach the methods course in Spanish

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because there is a mix of students who are also in general elementary education, she supports PSTs’ Spanish language development to teach mathematics by providing readings in Spanish and by having PSTs write their assignments in Spanish. Sylvia uses the Mathematics Autobiography Activity, the Mathematics Learning Case Study Module, and the Classroom Practices Module in her elementary mathematics methods course.

The Activities While the TEACH Math modules have been carefully crafted and piloted by experienced MTEs and researchers, we do not use them as independent curriculum modules but rather integrate them throughout our methods courses. The activities we detail below come from our experience in creating spaces for critical reflection within our specific contexts before our PSTs engage with the modules. We find that this critical reflection should be approached with caution, as exploring issues of power, privilege, and societal inequity can be fraught with emotional triggers and the recalling of uncomfortable moments (D’Angelo, 2011; Tatum, 2003). We see this critical reflection as necessary before introducing the modules, however, so that we do not reify deficit perspectives our students (or we) might bring into our classrooms. We know that in order to open up spaces for our students to engage in praxis, critical reflection coupled with action (Freire, 1970), we, as mathematics teacher educators (MTEs), must be sensitive yet vigilant against oppression in our approach. Using our own voices, we speak about five activities (see Table 11.1) we have honed in our own mathematical methods teaching to use prior to engaging our PSTs in one or more of the TEACH Math modules: (a) a Photovoice Interview used to explore PSTs’ notions of identity for the Mathematics Learning Case Study Module, (b) Numbers about Me posters to explore personal mathematical connections for the Mathematics Learning Case Study Module, (c) Instagram Math Trails to prime the Community Walk Activity in the Community Mathematics Exploration Module, (d) connecting TEACH Math with Complex Instruction (Cohen, Lotan, Scarloss, & Arellano, 1999) and Rights of the Learner (Kalinec-Craig, 2017) to explore ideas about status and equitable participation in the Classroom Practices Module, and (e) reflective mathematics autobiography stories to prepare students for the Mathematics Learning Case Study module.

The Photovoice Interview Theodore: I utilize Photovoice Interviews in order to create critically reflective discussion around identity as connected to mathematics learning. Photovoice involves participants introducing personal photographs into an interview context and then

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Table 11.1  Five activities to support PST critical reflection prior to engagement with the TEACH Math modules TEACH Math Module Case study Module Problem-­ solving Interviews Numbers about PSTs create posters of Case Study Me the numbers that make Module Getting to up their personal Know You worlds Interview Community Instagram Math PSTs post Instagram Exploration Trails photographs of Module mathematics in their Community community Walk Activity Classroom Notion of status/ PSTs analyze math Practices autobiographies and Rights of the Module videos with lenses of Learner Video Lens status and Rights of Activity the Learner PSTs share and reflect Case Study Mathematics Module on own mathematics autobiography Getting to auto biographies in reflections connection to Sylvia’s Know You Interview journey Activity Photovoice Interviews

Description PSTs share and talk about personal photographs with each other

Rationale Personal photographs reveal inequities in mathematics learning experiences

To help PSTs recognize mathematics outside of classroom settings to create narratives of mathematics in their lives To visually document mathematics in their own worlds

To connect issues of status and inequitable participation to classroom practice

To reflect on the many cultural, linguistic, and community connections that can be acknowledged or ignored when listening to a child’s mathematics story and to use personal backgrounds and interests in the interviews they conduct with their own students

using the photographs to frame the stories they tell (Wang & Burris, 1997). To model this technique, I introduce myself to my class on the very first day by sharing personal photographs of myself and my family. I do this to make myself vulnerable and to model how photographs expose aspects of my personal life in ways that speech or words cannot. For instance, my photographs reveal that my partner and I both identify as Chinese American, that I am a father of three children, and that my father is a Christian pastor, which means I grew up with a heavy emphasis on Christian values and positioning. While it is easy to say these things about myself or list them on a slide, these photographs open up windows into which my PSTs peek into my world and therefore get to know me in ways that make me vulnerable, while still allowing me power in how I position myself in these photographs (Holland, Lachicotte, Skinner, & Cain, 1998; Wang & Burris, 1997). After I share my Photovoice story, my PSTs read Chapter 2 “Identities, Agency, and Mathematical Proficiency: What Teachers Need to Know to Support Student

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Learning” and Chapter 3 “Know Thyself: What Shapes Mathematical Teacher Identities?” from the book The Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices (Aguirre, Mayfield-Ingram, & Martin, 2013). These ­chapters focus on how to support students’ emerging mathematical identities and how to reflect upon one’s own emerging identity as a mathematics teacher. In the next class, my PSTs present two personal photographs that “represent your mathematics identity when you were a student” to share in their own Photovoice Interviews. The PSTs partner up and share photographs with each other for 20  minutes (10  minutes each), with one PST playing the role of “photograph sharer” and the other playing the role of “interviewer.” The photograph sharer talks about whatever they choose, not limited only to what is shown in the photograph. While the photograph serves as an anchor to start initial stories, tangential (and often more revealing) stories often erupt from the initial photograph or story (Chao, 2014). The interviewer must follow strict guidelines on what they can ask: (a) Only questions pertaining to the detail of the photograph (e.g., “Who are the people in this photograph?”) as opposed to judgmental or interpretive questions (e.g., “Why didn’t you tell your parents that you hated math?”) and (b) Questions asking for more detail during a story (e.g., “Tell me more about that.”) (see Appendix A for the full Photovoice Interview activity description). On the surface, the Photovoice Interview activity reveals stories about how the PSTs envision themselves as mathematical learners. It forms a platform for PSTs to address personal anxieties about mathematics or worries about teaching mathematics, much like the autobiographies written about by Drake and colleagues (Drake, 2006; Drake, Spillane, & Hufferd-Ackles, 2001; LoPresto & Drake, 2004). In this case, the use of photographs allows PSTs to unveil visually the intersections of their various identities as they connect to their evolving mathematical and teacher identities, allowing PSTs to confront the ways their mathematics learning histories echo issues of gender discrimination, racial segregation, and what Freire (1970) calls the “banking” model of education. Through the elicited stories, PSTs note how, in their own experiences, mathematics education was sometimes a site of oppression. They reflect upon their mathematical experiences, visualize the mathematics teacher they want to be, and contemplate how they will create an inclusive community within their future mathematics classrooms. After the Photovoice Interview activity, we are armed with visual stories that help us disarm the myth of mathematics as a politically neutral space. We can confront dangerous (and subconscious) habits of comparing ourselves to each other based upon our mathematics knowledge. The Photovoice Interviews serve as a catalyst to quickly show how the sociopolitical dimensions of power and privilege involved with teaching mathematics have been present in our own education. And now, PSTs are ready to engage in the Mathematics Learning Case Study module and focus on listening to the complex mathematical thinking of their Case Study student and possibly not fall trap to deficit-oriented generalizations.

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Numbers about Me Posters and Instagram Math Trails Luz: One idea I want PSTs to begin to engage with is to view mathematics as more than just what happens within the four walls of a classroom. Much like Theodore described with the Photovoice Interviews, helping PSTs engage in critical reflection about mathematics often begins with reflecting on their own identities and their relationships to mathematics. Many of my PSTs come from traditional schooling experiences where mathematics was a speedy, rule-driven subject you are either good at or not. To that end, I use two assignments in my elementary mathematics methods course aiming to help PSTs engage in critical reflection of the world around them, what Gutstein refers to as “reading” the world with mathematics (Gutstein, 2006). For many who have only encountered mathematics through worksheets or paper-pencil driven assignments, “reading” the world with mathematics can begin with observation of their own lives and experiences. The first activity, the Numbers about Me poster, stems from an idea presented in Peterson’s (2013) chapter, “Mathematics Across the Curriculum” in Rethinking Mathematics: Teaching Social Justice by the Numbers (Gutstein & Peterson, 2013), which encourages students to think about the many numerical facets of their lives. A similar activity can be found in the Teaching Children Mathematics article, “That’s My Number” (Chao, 2016). The second activity I use to support PSTs in beginning to “read the world” is an Instagram-based Mathematics Trail. For the Numbers about Me activity, I start by presenting the following number to my PSTs when introducing myself: 582. I ask my students to read this number, asking if anyone can read it in another language (all attempts are validated and I have heard this number in Cantonese, Portuguese, German, Korean, Russian, among others, and seen it with American Sign Language). I also ask what the value of the 8 is, and what number is in the hundreds place. Then, I emphasize that despite answering all my questions, no one has yet suggested why this number might be important to me. So my PSTs offer a few guesses such as a birthday (which is flattering), an area code, or even the total number of As I have given out as a final grade for this course. Finally, I reveal that this number represents the total number of miles I have to travel to El Paso, Texas, my hometown, from my current city, to eat some of my mom’s cooking. I then present PSTs with examples of past Numbers about Me posters (see Fig.  11.1) as well as child-generated examples (see Appendix B). PSTs are then instructed to bring back their own Numbers about Me poster in the third or fourth week of the course, each with at least six numerical connections. PSTs are encouraged to be creative and add pictures or words to their poster. In the past, my PSTs have used this assignment to create timelines, 3D geometric shapes, paper bag booklets, and digital posters that focus on numerical aspects of their personal ­identities. This activity is then connected to the Mathematics Learning Case Study Module. The first interview in the module, which is geared toward understanding the mathematical knowledges and experiences of a student, often reveals to the PSTs that many children also have a hard time seeing the world around them through mathematics. After this interview, the PSTs share their Numbers about Me posters with their case study students and continue to have conversations about mathematical connections in everyday life.

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Fig. 11.1  Example Numbers about Me Poster

My second activity, an Instagram-based Math Trail, is connected to the Community Mathematics Exploration Module. I based my concept of Math Trails on Kay Toliver’s (1993) work. Toliver, a veteran New York City educator, guides her students on walks through their Harlem neighborhood to discover and document the mathematics in their world. To update this activity for a smartphone world, we use the social photo-sharing application, Instagram, for my PSTs to document the mathematics in the communities surrounding the schools that they student teach in. Throughout the semester, PSTs post photographs of mathematical happenings in their neighborhoods to a class Instagram account. PSTs are encouraged to look for two different types of pictures to take: (a) those in which mathematics is occurring (e.g., determining the tip at a restaurant) and (b) pictures in which mathematical “problems” could be generated (e.g., estimating how many cars travel through an intersection for an hour by counting how many travel through in 5 minutes). This Instagram-based Math Trail serves as an introduction to the necessary work of exploring, documenting, and learning about the mathematics within the communities their students live in for the Community Walk Activity found in the Community Mathematics Exploration Module. Although intended to introduce PSTs to two different TEACH Math modules, both of these activities together begin to change the conversation around what counts as mathematics. For those PSTs who find it a bit more challenging to see the world around them through a mathematical lens, the sharing of the artifacts from these two activities generates conversations and allows PSTs to re-think what they understood and understand about mathematics.

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Status, Equitable Participation, and Rights of the Learner Crystal: In my elementary mathematics methods class, I utilize the TEACH Math research and modules in conjunction with the research of Complex Instruction (Cohen et al., 1999) and an emerging framework that I learned from a mathematics teacher educator (Olga Torres) called the Rights of the Learner (RotL, as will be described in more detail below). My PSTs know from the first day of class that this is a place where we need to reimagine a mathematics classroom as a room where the ideas and knowledge of all students are always front and center. We need more teachers who seek opportunities to learn from children and to utilize equity-based teaching practices. The Mathematics Autobiography Activity is a crucial first step to helping my PSTs adopt equity-based mathematics teaching practices. With their autobiographies, we begin the semester by talking about moments when the PSTs felt like they were identified or named as the “smart” student or the student whom the teacher rarely called upon to contribute. I ask them to share moments when they felt they did not have an equitable opportunity to participate in the classroom (Featherstone et  al., 2011). I transition to talking with them about the notion of status, a perceived social ranking that assumes some students can contribute more than others (Cohen et al., 1999; Cohen, Lotan, & Catanzarite, 1988) and how this can be a way to analyze our experiences and imagine a different scenario where everyone can contribute and feel valued. We talk about what a classroom might look like if instead of children following the rules in mathematics classrooms (e.g., following a prescribed set of procedures, using only the words posted on a word wall), they have rights to exercise and employ at their discretion. When children have rights to exercise and not necessarily rules to follow, teachers know more about how students think, not just how well students can replicate what the teacher wants them to say or do. At this point, my PSTs have already read my blog posts about the RotL: (a) to be confused; (b) to claim a mistake and revise their thinking; (c) to speak, listen, and be heard; and (d) to write, do, and represent what makes sense to them (Kalinec-­ Craig, 2017). PSTs then reconsider their mathematics autobiographies using the lens of status and these four RotL in order to reflect critically on their own mathematical experiences as students. For example, the PST may talk about how they felt like they had low status because they were afraid to make a mistake or did not necessarily use the teacher’s terminology when describing their mathematical thinking. Status and the RotL are thereafter coupled as a framework for praxis: reflection coupled with action to create change (Freire, 1970). To contextualize the ideas of status and the RotL, I show a video that illustrates an issue of status playing out among three students, one of whom is Black and named Reggie (Cohen & Stanford University, 1994). In the video, the PSTs watch how Reggie attempts to contribute meaningfully to the task at hand, but Matt and Chris (his two classmates) dismiss Reggie, or sometimes outright ignore him. This positioning means that Reggie is perceived to be less capable of contributing to the task at hand. Reggie is not in a place to exercise his RotL.

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We then rely on the Power and Participation Lens from the Video Lens activity described in Chap. 2 of this volume to unpack the video in three ways. First, we briefly analyze the transcript from the video to offer evidence for claims as to who appears to have high or low status. Second, we examine how the arrangement of the desks might prevent students like Reggie from having access to the materials, a tenet of Complex Instruction that argues for spaces that are physically equitable for all students to contribute. Finally, we reflect on our prior experiences based on what we saw in the video and think about Reggie’s RotL. Did he have an opportunity to speak, listen, and be heard? Could he write, do, and represent what made sense to him? The whole group discussion that results from pairing the Reggie video and the mathematics autobiography not only helps PSTs see similarities between themselves and the experiences of the students in the videos but also to reflect on what they will or might do as teachers to interrupt similar issues of inequitable participation and status in their own classrooms.

Mathematics Autobiography Reflections Sylvia: In my elementary mathematics methods course, I start by asking my PSTs to reflect on their own experiences with learning mathematics in schools from elementary grades to the present. The questions that PSTs reflect on have to do with the following areas: (a) How mathematics was taught to them (what activities they engaged with in the mathematics classroom), (b) If there was a high or low point in learning mathematics and who was involved, (c) Whether or not their mathematics teachers drew from their cultural and community knowledge to teach mathematics, (d) How they were supported to learn mathematics at home, (e) What their experience was if mathematics was taught in a language other than English, and (f) How they were alike or different than other students in their classes. After they reflect on these questions, PSTs discuss how these experiences impact the kind of mathematics teacher they will be or want to be. I introduce this assignment during the first week of classes, telling PSTs to come ready to discuss these experiences during the second week of class. This mathematics autobiography serves as an entry point for all students to share how they learned mathematics and to discuss issues of access and equity with their future students. I also share my own story as a second language learner of English and connect these experiences to the first chapter of Beyond Good Teaching: Advancing Mathematics Education for ELLs (Celedón-Pattichis & Ramirez, 2012). When I share my own mathematics journey (See Chap. 10 in this volume), I emphasize the important role two of my teachers, an English teacher and a mathematics teacher, played in shifting my placement from an English as a Second Language (ESL) track to a college preparatory track when I was a high school freshman. This shift in placement determined the rest of my path to the University of Texas at Austin. This

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critical intervention and advocacy by my teachers to challenge administrators in that placement, to support my parents in navigating the educational system, and to prepare the financial aid application was crucial to my preparation to apply to universities and to have the needed courses to be successful in college. By reflecting on their experiences learning mathematics and discussing equity and social justice issues, my PSTs are then prepared to enter the Mathematics Learning Case Study Module. The first part of this module includes Getting to Know Students with an interview. I have my PSTs develop their own questions to get to know their students and to find out what their students are interested in so that they can later use this information to construct word problems for problem-solving interviews. I use this mathematics autobiography reflection to support my PSTs in understanding their own students’ cultural and linguistic backgrounds, personal interests, and how to use this information to plan problem-solving lessons and interviews.

Our Takeaways from This Work We share these activities as examples of how we prepare our PSTs to engage in critical reflection before engaging in the transformational action of the TEACH Math modules. Our experiences show us that, when we situate ourselves as also on the same journey as our PSTs, we are able to keep the focus on empowering students mathematically. For instance, Luz shares her own personal number first, to open up how distant her mother’s cooking, and therefore a part of her community, is from her, and Sylvia brings up her own mathematics autobiography and journey before asking her PSTs to share their own. All of us have experienced the power of these modules in our own teaching, watching our PSTs grow in their critique of their own mathematics education and their development of a vision of equitable teaching in their future classroom. For instance, Sylvia’s personal educational journey opens up many issues for our PSTs to reflect upon that they might not have associated with mathematics learning. Luz’s Numbers about Me poster helps her PSTs think critically about the mathematics in their lives they did not previously recognize was there. Theodore’s use of the Photovoice Interviews immediately opens up an aura of vulnerability in his classroom among his PSTs. Crystal’s use of the Complex Instruction video and the RotL framework helps her PSTs focus on how students can be easily “othered” within a mathematics class based upon subconscious prejudice. We have all heard from our PSTs how these activities, coupled with the TEACH Math modules, helped them realize the importance of how they must serve as advocates for their future students and how to approach their mathematics teaching with an eye for equity and social justice.

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Doing This in Your Mathematics Methods Course Our experience shows us that enacting this work is not easy. Sharing our own ­personal stories, experiences, and cultural and linguistic backgrounds involves taking a risk and entering a “brave space” (Spencer, Russell, & Johnson, 2016). We also run the risk of reinforcing certain stereotypes that we want PSTs to move away from. For instance, Theodore has been told that when he shares about being Chinese American, which leads to a story about the negative implications of the Model Minority Myth (Hartlep, 2013; Museus & Iftikar, 2013), it is the first time his PSTs have ever thought about the negative implications of the “Asians are good at math” stereotype. While this is illuminating to PSTs, how many of them will now generalize this story for all of the Asian and Pacific Islander students they will teach? We must be careful that, through this work, we deconstruct the many stereotypes and generalizations of parents, students, and communities that our PSTs might encounter. Additionally, this work is especially more difficult for scholars of color because we experience more resistance than white scholars (Rockquemore & Laszloffy, 2008). All of us have cultural backgrounds as well as personal and professional experiences (positive or negative) that we can draw from to construct our own mathematics autobiographies and testimonies. We are the authors of our own stories that we choose to construct and share with our PSTs. Furthermore, we can use all experiences to reflect critically on ways that we, as educators, can advocate for all students, especially those who may need to navigate new school systems and so forth. Our main takeaway from utilizing these activities is that the very structure of story and narrative, in sharing our own or the stories of students and other PSTs, is a powerful technique to help PSTs internalize the deep issues of equity, power, and oppression that the TEACH Math modules help them address. We note that almost all of the activities we utilize involve PSTs telling or creating stories about mathematics in their own lives. Perhaps the first part of praxis for emerging mathematics teachers involves creating and recreating personal narratives about mathematics and then critically reflecting on these narratives.

 ppendix A: Mathematics Teacher Identity Photovoice A Interviews This project is about exploring your emerging mathematics teacher identity as described in the two Aguirre et  al. (2013) chapters, Identities, Agency, and Mathematical Proficiency: What Teachers Need to Know to Support Student Learning and Know Thyself: What Shapes Mathematics Teacher Identities? 1. For your homework, digitally prepare two photographs to share in class (i.e., have them available on your phone or tablet). The photographs should represent your mathematics identity when you were a student. Upload them to the course website.

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2. In class, you will partner up and share your photographs with each other in a 20-minute Photovoice interview (10  min each), which is an interview mechanism that allows the sharer to use the photograph to augment his/her voice: (a) During these photovoice interviews, the sharer can talk about whatever s/he wants to. The sharer does not have to only stick with what the photograph shows. (b) The interviewer is only allowed to ask two things. Otherwise, the interviewer is to listen. (i) Specific details of the photograph (e.g., Where was this photograph taken?, Who is in this photograph?) (ii) “Tell me more about that…” to extend or get more detail about what the sharer might be talking about. Caution: A photovoice interview can bring up very vulnerable and personal stories. Be respectful and ready to listen. Remember, as a listener, not to judge or critique. Simply listen. After the interview, you will reflect on the experience by writing down your mathematics teacher identity story. This will be less than a page long and encapsulate the story you told in the interview as well as reflections on the interview itself.

Appendix B: Students’ Examples of Numbers about Me [1] I have 1 cusin (cousin) [2] I have 2 dogs. [Their] names are Reina and Corazon. [5] I livt (lived) in 5 hosis (houses). [7] Yo tengo siete años. (I am 7 years old.) [7] is my favrit (favorite) dogs age. [30] retronts (restaurants) [2] I have 1 brother and 1 sister that makes 2 [7] pepol (people) in my haous (house) [1000] movis (movies claimed to have watched). [14] I have 14 cusins (cousins)

References Aguirre, J. M., Mayfield-Ingram, K., & Martin, D. B. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. Reston, VA: National Council of Teachers of Mathematics. Celedón-Pattichis, S., & Ramirez, N. G. (2012). Beyond good teaching: Advancing mathematics education for ELLs. Reston, VA: National Council of Teachers of Mathematics.

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Chao, T. (2014). Photo-Elicitation/Photovoice interviews to study mathematics teacher identity. In J. Cai, J. Middleton, & L. Van Zoest (Eds.), Current research in mathematics teacher education: Contributions by PME-NA researchers. New York: Springer. Chao, T. (2016). That’s my number. Teaching Children Mathematics, 22(9), 576–576. Cohen, E.  G., & Stanford University. (1994). Status treatments for the classrooms [CD-Rom]. New York: Teacher’s College Press. Cohen, E. G., Lotan, R. A., & Catanzarite, L. (1988). Can expectations for competence be altered in the classroom? In M. Webster & M. Foschi (Eds.), Status generalization: New theory and research (pp. 27–54). Stanford, CA: Stanford University Press. Cohen, E. G., Lotan, R. A., Scarloss, B. A., & Arellano, A. R. (1999). Complex Instruction: Equity in cooperative learning classrooms. Theory Into Practice, 38(2), 80–86. D’Angelo, R. (2011). White fragility. International Journal of Critical Pedagogy, 3(3), 54–70. Drake, C. (2006). Turning points: Using teachers’ mathematics life stories to understand the implementation of mathematics education reform. Journal of Mathematics Teacher Education, 9(6), 579–608. Drake, C., Spillane, J. P., & Hufferd-Ackles, K. (2001). Storied identities: Teacher learning and subject-matter context. Journal of Curriculum Studies, 33(1), 1–23. Featherstone, H., Crespo, S., Jilk, L. M., Oslund, J. A., Parks, A. N., & Wood, M. (2011). Smarter together!: Collaboration and equity in elementary math classroom. Reston, VA: National Council of Teachers of Mathematics. Freire, P. (1970). Pedagogy of the Oppressed. 30th anniversary ed. New York: Continuum. Gutstein, R. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York: Routledge. Gutstein, R., & Peterson, B. (2013). Rethinking mathematics: Teaching social justice by the numbers (2nd ed.). Milwaukee, WI: Rethinking Schools. Hartlep, N.  D. (2013). The model minority stereotype: Demystifying Asian American success. Charlotte, NC: Information Age Publishers. Holland, D., Lachicotte, J., Skinner, D., & Cain, C. (1998). Identity and agency in cultural worlds. Cambridge, MA: Harvard University Press. Kalinec-Craig, C. (2017). Rights of the Learner Blog Posts. Retrieved January 12, 2018, from https://embracinglifewithmajorrevisions.wordpress.com/rights-of-the-learner-blogs/ LoPresto, K.  D., & Drake, C. (2004). What’s your (mathematics) story? Teaching Children Mathematics, 11(5), 266–271. Museus, S.  D., & Iftikar, J.  (2013). An Asian critical theory (AsianCrit) framework. In M.  Y. Danico & J.  G. Golson (Eds.), Asian American students in higher education (pp.  18–29). New York: Routledge. Peterson, B. (2013). Teaching math across the curriculum. In E. Gutstein & B. Peterson (Eds.), Rethinking mathematics: Teaching social justice by the numbers (pp. 9–12). Milwaukee, WI: Rethinking Schools. Rockquemore, K. A., & Laszloffy, T. (2008). The Black academic’s guide to winning tenure without losing your soul. Boulder, CO: Rienner Publishers. Spencer, J., Russell, N., & Johnson, K. (2016). Exploring racial consciousness and faculty behavior in STEM classrooms. Paper presented at the Annual Meeting of the Association of Mathematics Teacher Educators (AMTE). Irvine, CA. Tatum, B. D. (2003). Why are all the black kids sitting together in the cafeteria: And other con versations about race (5th ed.). New York: Basic Books. Toliver, K. (1993). The Kay Toliver mathematics program. The Journal of Negro Education, 62(1), 35–46. Turner, E., Aguirre, J., Bartell, T., Drake, C., Foote, M., & Roth McDuffie, A. (2014). Making meaningful connections with mathematics and the community: Lessons from prospective teachers. TODOS research monograph, 3, 60–100. Wang, C., & Burris, M. A. (1997). Photovoice: Concepts, methodology, and use for participatory needs assessment. Health Education & Behavior, 24(3), 369–387.

Chapter 12

Our Linguistic and Cultural Resources: The Experiences of Bilingual Prospective Teachers with Mathematics Autobiographies Gladys H. Krause and Luz A. Maldonado

Keywords  Elementary teacher education · Bilingual education · Mathematics education · Mathematics · Prospective teachers · Autobiographies · Culturally diverse students · Linguistically diverse students In one class of 28 bilingual prospective teachers (BPSTs), the phrases “I have never been good at math” or “Math is a scary subject” or “Mis sentimientos por las matemáticas son como una montaña rusa” (“My feelings about mathematics are like a roller coaster”) were lamentably common. Such feelings are certainly not confined to this particular group of BPSTs. As we work with these students, we find—as hinted at in the last quote above—that such feelings come and go: in some stories BPSTs indeed reflect on their enjoyment learning mathematics and feeling knowledgeable, capable of solving mathematics problems, and motivated to continue learning. We have worked with BPSTs for more than a decade and the commonality of their comments strikes us every time. They all loved mathematics, obtained high grades, and took advanced mathematics courses in their high schools. But this sample is not indicative of Hispanic mathematics achievement nationwide: the statistics tell a different story. According to the National Assessment of Educational Progress (NAEP, 2009), Hispanic students performed 21 points below White students in mathematics from 2007 to 2009. This achievement gap was similar to that of previous years for both ethnicities. The stories narrated here are intended to be indicators of the need to better understand the broader mathematical experiences that Latinx students in the United States face. At the same time, we highlight the need for bilingual teachers in US schools G. H. Krause (*) William and Mary, Williamsburg, VA, USA e-mail: [email protected] L. A. Maldonado Texas State University, San Marcos, TX, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 T. G. Bartell et al. (eds.), Transforming Mathematics Teacher Education, https://doi.org/10.1007/978-3-030-21017-5_12

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since we are in “crisis” mode as the increasing number of bilingual learners will lead to a (further) shortage of bilingual teachers (Sheets, Flores, & Clark, 2011). Moreover, mathematics has been identified as one of the fields (together with special education and science) that have the highest rates of teacher turnover (Cuban, 2010; Grissmer & Kirby, 1991; Ingersoll, 2001; Ingersoll & May, 2012; Krause, 2014; Rumberger, 1987). What, then, might the modules for elementary mathematics methods courses from the Teachers Empowered to Advance Change in Mathematics (TEACH Math) project help us understand about the preparation of bilingual mathematics teachers? In this chapter we describe the experiences of five BPSTs, as retold in their mathematics autobiographies. Twenty-eight BPSTs provided open-ended, written reflections on formative events in their mathematical education (see Appendix A for the assignment description), an activity we have found to elicit heightened engagement in teacher preparation classes and to deepen their abilities to put themselves in their students’ shoes. The assignment is sometimes used as a preliminary activity by TEACH Math researchers (see http://teachmath.info) and was given to the students in the first week of their mathematics methods class at a university in southern United States. All of the students in the class were English–Spanish bilinguals and received instruction for this particular class in Spanish. In this class, 26 students identified as Latinx and two as White. The BPSTs wrote a “math life story” recounting their experiences with mathematics, considering how these experiences impacted their attitudes towards, and understanding of, mathematics. We identify in their stories how their feelings and views about their own position as students in the classroom, their languages, their cultures, their families, and their socio-economic status influenced their academic development in mathematics and their decisions to become bilingual teachers. We then analyze how the BPSTs’ experiences can inform our practice as instructors of mathematics methods. From this we draw conclusions for providing quality mathematics education in bilingual classrooms. Although we do not attempt to say that the experiences narrated here form the common denominator among all BPSTs, we draw attention to what appeared to be common within this particular group of BPSTs as they were growing up and learning mathematics. The five stories shared here are representative of the common themes we found across the 28 individual stories. Although past research has examined experienced teachers’ stories (Lloyd, 2006), little research has focused on stories written and lived by BPSTs about their mathematics classrooms and mathematics learning experiences. We value each one of these experiences shared by the BPSTs, not only because of their authenticity but also because of how these same experiences shape the way these future teachers see education and how they shape their identities as teachers of mathematics. Their stories draw attention to the notion that the nature and substance of prospective teachers’ learning is influenced by, and part of, their emerging identities as mathematics teachers (Skott, 2001, 2004; Spillane, 2000).

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A Word on Terminology We will refer to “bilingual” students, classrooms, prospective teachers, and teachers throughout this chapter. By using the term “bilingual,” we specifically refer to those teachers and students who speak English and Spanish fluently. We will refer to Spanish–English bilingual classrooms as those that intentionally serve the needs of Spanish-speaking bilingual students (García & Sylvan, 2011; Martin-Jones, 2000). In specific situations we employ the term “emergent bilinguals” when referring to students who are actively engaged in learning English as a second language or when we cite the work of other authors who use the term English Language Learners or English as a Second Language. We adopted this terminology from García (2008), who argues that by identifying students as “emergent bilinguals” we are referring to the children’s potential to develop their bilingualism instead of suggesting a limitation. We will use the term “Latinx” instead of “Latino” in order to be more inclusive of diverse gender identities. We use Latinx to refer to any person of Latin American descent residing in the United States. On some occasions we will use the term Hispanic, the word adopted by the US government since the 1970s to give people from Latin America a common identity (MacDonald, 2004), when we cite the work of state or federal entities supporting data related to education in the United States or when we quote people who have specifically used this word.

Background The work of this chapter builds on the body of life-history research found in education (Casey, 1993; Connelly & Clandinin, 1999; Dhunpath & Samuel, 2009; Drake, Spillane, & Hufferd-Ackles, 2001). Drake et al. (2001) describe what they call “life-­ story interviews” as a useful method for learning about the identities of the individuals telling the stories. According to this method, stories, as lived and told by teachers, can serve as a lens through which they understand themselves personally and professionally and through which they view the content and context of their work. According to Dhunpath and Samuel (2009), telling stories about one’s life is a process of recording how the narrator positions themselves in relation to the topic being discussed. Through stories about one’s life, the narrator chooses a position relative to the narrative that varies according to the personal impact of the events in the story. This research is important, particularly in the case of the BPSTs, because telling a story manifests a conscious process of documenting and representing their emerging views about teaching and learning mathematics. For decades researchers have investigated how teachers’ own experiences affect how they teach (Ball, 1997; Lortie, 1975; Smith, 1996). These investigations are based on how teachers’ beliefs are formed and what past experiences they had as students. LoPresto and Drake (2004) continued this line of investigation with a specific focus on teaching mathematics. They used stories of prospective and in-service

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teachers’ past and present relationship with mathematics in order to understand teachers’ relationship with this subject, and to use this understanding when designing methods courses and professional development. Moreover, extensive work exists that studies how teachers position themselves in their personal narratives, and how that positioning informs the decisions they make during mathematics instruction (Celedón-Pattichis & Ramirez, 2012; D’Ambrosio & D’Ambrosio, 2013; Drake et al., 2001). Drake et al. (2001) found that teachers’ attitudes on mathematics and literacy were directly informed by their prior experiences as students. In particular, some teachers used their prior negative experiences as a means for not replicating similar negative experiences for their students as they learned mathematics. This chapter focuses exclusively on BPSTs’ mathematics autobiographies. By analyzing this particular population of prospective teachers, we seek to emphasize how culture and language influence different aspects of their mathematics learning. We identified scenarios, recounted in the BPSTs’ own voices, where the meeting of cultures hindered or stimulated their own learning of mathematics, in order to highlight elements supporting mathematical content during mathematics instruction in bilingual contexts.

BPSTs’ Voices: Reflecting on the Past Aguirre, Mayfield-Ingram, and Martin (2013) point out that teachers’ experiences while learning mathematics influence their identities as mathematics teachers. They explain how teachers’ mathematical identities are partly rooted in their own experiences as students of mathematics and will ultimately influence their instructional decisions in the mathematics classroom. Perhaps unexpectedly, we have found that BPSTs’ own negative mathematical experiences have provided a decisive motivation for turning to mathematics education and finding their teaching identities in providing quality mathematics instruction to the bilingual Latinx community.

Their Stories As noted above, the stories described in this chapter come from the mathematics autobiographies that 28 BPSTs submitted as an assignment for the Bilingual Mathematics Methods class at a university in southern United States. We have chosen five stories as they are representative of common experiences told and lived by the majority of the students in the class. As described earlier in this chapter, the purpose of mathematics autobiographies was to capture the BPSTs’ own experiences while they were students learning mathematics, as a means to gain deeper insight into BPSTs’ ways of making sense of mathematics teaching and as a means to learn about specific issues pertaining to

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bilingual students’ learning of mathematics. In addition to reflecting on their classroom and instructional experiences as students, the assignment also required them to reflect on their experiences learning mathematics outside the classroom and to describe aspects of their cultural background that they thought might have influenced the way they learned mathematics. This particular aspect of the mathematics autobiography assignment is meant to capture the BPSTs’ experiences that are usually invisible to educators, but that are relevant to informing teaching practice; for example, we have seen how parents and communities contribute to the work students do in the classroom (Colegrove & Krause, 2017). This assignment is given at the beginning of the semester and we use the BPSTs’ experiences and stories to shape the direction of the class and to foster meaningful discussions throughout the rest of our implementation of the TEACH Math modules. Johana’s experience  I got accepted to the School of Engineering at [X] University and all of a sudden I felt like I could accomplish anything. I was in love with my major and I thought since I was at a new school I could start over and just excel at math. I was completely wrong. I failed my first round of exams and I was in shock. I thought a lot about why my life had gotten to that point. I became angry because I thought a lot about my high school experience and how I never learned anything while I was there. The only thing that I ever did there was worksheets and meaningless projects. Everyone in my college classes seemed prepared. They knew those triangle tricks that I never understood. They knew words I had never heard before. I compared myself to them. I was this Latina girl from the east side of Houston (imagine the statistics against me). They were white boys from the north side of Houston. I had never been on a plane in my life and they already had their pilot’s licenses. They got 5’s on all of their AP tests and in my high school it was a miracle if you got a 3. I felt like an alien beside them. I was so mad at my high school for lacking the resources to prepare me for college. It didn’t help to notice that everyone else from my high school that came to [the university] was also struggling. I was mad at the school system and I wanted to change it. I decided to change my major to bilingual education because I didn’t want this same story to happen to anyone else. I thought back to when I was in bilingual education classes and remembered happy times. My teachers were excellent at implementing both languages in the classroom. Math was always exciting because we would work in groups and do activities to help us understand. I remember talking about fractions and making arroz de leche in class. I remember my teacher asking us about “the big fight” that was going to happen that night when she was explaining how to round numbers. The big numbers knocked out the competitor and moved on to the next round. We could all relate because our dads would be watching the big fight that night. Learning felt natural then and it didn’t seem like a burdensome task. I remember my teacher pulling me aside in the hallway and telling me that I was the only one to get a perfect score on both the reading and math tests. I was not surprised too much because math came natural to me then. My parents never had to help me and as for high school they couldn’t help me because they never even finished school. As a future teacher, I want my students to feel the way I did about math when I was young. Not

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only that but as a teacher and as an employee in the district I really want to push STEM [Science, Technology, Engineering, and Mathematics] for Latina girls. Not many resources are offered but I want to be the one to change that in my district. I want future aspiring engineers to be able to have that support to excel in their colleges and universities. I can give my students the strong foundation they need in math and give them a positive math experience so that they can develop positive math identities. It is so important for me that bilingual Latino students be able to do things they never thought possible. This is my biggest passion these days and I hope that my math experience will help me create an excellent math experience for the next generation! Johana self-identifies as a Latina girl and describes herself as coming from a low socio-economic background, struggling to adjust to a place where women and racial minorities are typically underrepresented (US Department of Commerce, 2013). Her story marks an example of the common BPST perception of a lack of rigorous mathematics preparation. Her story also provides details of her views on her position in academia: she lacks the tricks others seem to know, she doesn’t understand words they seem to grasp, and she feels her socio-economic background sets her at a disadvantage. Similar experiences recur throughout the BPSTs’ stories. Lending support to this finding, data from the US Department of Commerce (2013) reported that by 2011, Hispanics occupied only 7% of the workforce in STEM. Daniela’s experience  In the eighth grade, I was placed in pre-algebra. This meant that I had skipped a whole year of math. I was intimidated by the students in my pre-algebra class because they had an even greater knowledge in math than I had. These students had been in math honors class for the past two years, so they were able to fully understand what the teacher was talking about. The students in the class were middle class Caucasian and that played an even stronger role in intimidation. I was from a low-income Hispanic family, so automatically I felt intimidated by those “superior” to me and separated. No one was like me. I remember the first day of class I wanted to cry. I had spent the last two years being the smartest student in my math class, with people who were from the same economic status as I was and had my skin color, and now I had no idea what was going on and felt like an outsider. By the end of the week, seeing that I had not learned anything, I had decided to go back into the regular math class, but when I spoke to Mr. K, my eighth grade teacher, he asked me to give it another chance. He told me that he knew that I could make it and that if he didn’t think I had the capability to be in his class, he wouldn’t have accepted me in. His words gave me hope. For the next two months, I stayed after school to receive tutoring classes from him. He was able to catch me up with what they were learning. At the end of the year, I no longer felt as intimidated as I was and had rekindled my love for math. I think back on how different it would’ve been if Mr. K wouldn’t have cared to have me in his class. I know I wouldn’t have been as successful in mathematics if it weren’t for him. My Algebra teacher was also my Pre-Calculus teacher. What I enjoyed about those classes was that Mr. A was Hispanic and spoke Spanish with me. He did not speak it to teach math specifically, but he used it with me when discussing other

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issues. The use of my native language in the class, specifically from a teacher, made me feel valued and accepted. His use of Spanish even caused my Caucasian classmates to become closer to me. For example, some wanted to learn Spanish and others just wanted to speak to me because they knew Mr. A and I had a strong relationship. As with Johana, Daniela felt that her status as Hispanic or Latina was a factor that had determined her (unfortunately poor) academic position in the classroom. They also shared a sense that Spanish made them feel safe and happy. It helped them to feel valued and accepted. Angela’s experience  I don’t remember having a hard time with any subject area in high school; in fact I graduated valedictorian and received a scholarship to attend a university of my choice. Later, I arrived at [X] University in the fall of 2011 with an intention of following through with pre-pharmacy. I was a [Y] Scholar and I knew I had to take several mathematics courses for my career choice. That semester, I took 12 hours: calculus, biology, chemistry and psychology. I look back at it now and I still cringe at the fact that I didn’t have anyone to talk me out of such a heavy course load. Oh and I forgot to mention that I also worked 30 hours a week. The real incident that affected my attitude towards math, however, happened before I even started my classes. That year, the college of natural sciences implemented a change that said that all pre-pharmacy or pre-med students, had to take a placement exam and score a 70 or above, by the 5th class day in order to secure their spots in the calculus and chemistry courses. To this day it has been one of the hardest and most stressful exams I’ve ever had to take. It was just so tedious and ill structured and it would time-out at random times. It makes me cry when I remember how hard I struggled to make a 70 on that exam, and more so how hard it was for me to even have the time to take it. I come from a very humble family, my mother is disabled due to cancer, my older brother is also disabled due to Cerebral Palsy and my dad and I are who work and we’ve struggled a lot to make ends meet. At my house we’ve never had internet because to us, internet is a luxury we can’t afford. In order to take the exam, I needed internet, so I would go to the library to take it. One time I stayed at the library until about 4 a.m. using the wifi, because the exam kept timing out because it took me so long to solve a problem. I spent almost an entire 3 weeks taking both the calculus and the chemistry exams until finally passing them with the bare minimum. Angela’s story, too, describes experiences of academic success. She likewise struggled to survive in the rigorous academic life that college required from her. But Angela’s story has an additional detail often lacking in our BPST accounts: she worked nearly full-time while taking a full course load; and in an era where everything is digital, internet access is still a luxury in her home. Hector’s experience  Once I got into middle school I remember math becoming a lot more complicated. I always struggled in mathematics especially once we started learning algebra in my 6th grade class. I remember feeling lost once we started doing more complex problems that had multiple steps. I didn’t know how to ask for

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help and that really ended up hurting me. I remember also in 7th grade I had just moved to Texas and was in math class learning about percentages. I sat on the couch one night and asked my mom to help me and she didn’t know how to help me. It was frustrating because I wanted to work really hard on my homework but didn’t even know where to start. It had to take me going to extra tutoring hours and looking up more problems on the computer to practice and learn how to work on the problems by myself. Now I feel like math is something that is needed for kids to fully grasp and understand what is going on around them. I could understand it but I could never fully justify my work when it came time to explain. The mathematics that was taught in school for me was never really connected to my home life experiences. I feel like that was one of the reasons that math was never really something that I could relate to. Reflecting now upon not having that makes me feel more strongly towards math and trying my best in my pre-service learning to know more about how to make math something that is culturally relevant for my kids. Having that background as to how it can make you feel isolated and un-relatable will hopefully help me understand and know when my kids are viewing it in the same manner. Luckily though for me when I was growing up math was supported in my home because I would always be helping my mom do phone calls with bills and letting her know what we had to pay at the store. She always told me that she didn’t get the chance to finish middle school back in Mexico so I was lucky to be having an education here in the United States. I would always come home and she would be waiting for me at the kitchen table ready to see what was in my folder so I could get started on it. Hector’s story describes his difficulties explaining mathematical concepts, his feelings about mathematics, his home values, and the acknowledgement of how his experiences motivate and inform his instructional practices. Another aspect of Hector’s story relays his mom’s inability to help with the homework, not because she “did not care” but because of her lack of formal education. Nevertheless his story tells how supported he felt at home about his education. This complements Colegrove and Krause (2017) and Acosta-Iriqui, Civil, Diez-Palomar, Marshall, and Quintos-Alonso (2011) consistent findings of Latinx parents’ commitment to their children’s success in school that is hindered by English as a barrier to participating in their children’s schooling more fully. Marta’s experience  I was in ESL [English as a Second Language] when I was in elementary school, but only for kindergarten and first grade. I don’t really remember much about that time thus, most of the math that I remember is from when I was speaking English mostly. I remember that when I was in middle school the teachers would always put me in the group with the ELL [English Language Learners] students so that I could help them. This was as much support as my teachers would give the ELL students in our classroom. Honestly, I didn’t mind doing this. This was something that I had been doing my whole life since my mom would make me translate for her.

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I feel like my middle school was not very good because I went to a school that had a lot of students from lower socio-economic communities. Because of our student population, a lot of our teachers had preconceived notions about the type of students we were. This made it hard for the minority students to get into the honors classes, so I didn’t see a lot [of] Hispanic students in my classrooms. This might have been another reason why a lot of my math teachers didn’t find the need to make math relevant to me. Marta’s description includes not only her position in the classroom but also her view of how racial minorities were seen growing up. We also see how from very early on she started receiving instruction in English and does not recall instruction of mathematics in Spanish. Importantly, while the school environment let her bilingual ability languish in those early years, it was later used as a classroom resource in middle school.

Discussion Throughout the stories in this chapter we identified five common themes in the BPSTs’ experiences: 1. Their position in the mathematics classroom: they saw themselves as “inferior” or “not having any chances” because they were women and Hispanic/Latina, and this caused them to confront stereotypes about their academic performance and status. 2. Their overall academic performance: they all were good students and performed to high standards in their schools. Some of them graduated as valedictorians and received scholarships to go to the schools of their choice. They recognized the importance of education and the impact it would have on their lives, valued it, and cultivated it. 3. Their mathematics performance: despite their academic effort and scholastic recognition, their preparation in mathematics did not provide a sufficient foundation to attain their goals of pursuing careers in mathematics-related disciplines. 4. Their academic instruction in a language other than Spanish: they framed their experiences in terms of a lack of instruction in their native language and of a concomitant feeling of happiness during instruction in their native language. They experienced a sense of belonging when a teacher used and acknowledged their native language. 5. Home values, support, and responsibilities: we saw how Angela described her responsibilities at home as an almost full-time job while she was also a full-time student. Hector mentioned how his mother could not help with some of his homework due to her level of education, though he still felt supported and motivated at home to work hard in the classroom. We believe that these individual experiences are likely shared by other individuals, but nevertheless arise only infrequently in a given classroom because of their

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personal nature. In the five stories shared by the BPSTs here, culture and language played a significant role in the descriptions of the settings and situations of the stories. Interestingly, the more these stories focused on descriptions of the students’ culture and linguistic background, the more the students felt constrained when learning mathematics.

Implications for Classroom Practices For Prospective Teachers The stories above reflect the heartfelt retellings of the lives and academic experiences of five BPSTs, and their outlook for developing their teaching practices based on those experiences. As we saw in their stories, what teachers do in the classroom can influence the role future teachers’ play in their own classrooms, and influence the decisions university students will make in pursuing their professional goals. Their stories not only offered rich information about them and their classroom experiences but also brought into focus how these experiences served as catalysts for change. In the context of the mathematics classroom for our methods course, the writing of their experiences challenges BPSTs to make sense of mathematics education reform, particularly in light of their own experiences as bilingual mathematics learners. In addition to shifting their perspective from student to teacher, BPSTs must grapple with issues and questions unique to the teaching and learning of mathematics. Reflection on their experiences as learners of mathematics can help prospective teachers perceive the importance for their teaching practice of acknowledging, valuing, and incorporating the diverse backgrounds of their students in the mathematics classroom. Reflecting on their experiences provides an opportunity for BPSTs to test the assumption that mathematics, and its teaching and learning, is universal. Latinx children often come to school with different ways of making meaning, ways often not recognized by schools (García & Otheguy, 2017). By reflecting on their own experiences, BPSTs can start to identify the value of these ways of knowing in the teaching and learning of mathematics. This mirrors Sheets et  al.’s (2011) call to “educar para transformar” (educate to transform) as a model of what it means to prepare bilingual teachers. Specifically, they detail a preparation program that requires BPSTs to “experience a personal evolution that questions existing beliefs, enhances ethnic identity, initiates teacher identity, and promotes efficacy” (p. 15). For example, Johana’s story describes how she is motivated and inspired to “change” the opportunities that are given to Latina students in the district that she comes from. In Daniela’s story we see how she recognizes the role played by Mr. A in motivating her to work up to her capacity to develop an understanding of mathematics. In Angela’s story we see how she recognizes today that her course load was higher than it should have been, given all the other responsibilities that came with

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also being a provider for her family. She recognizes that she has the capacity to be challenged academically, and that she can succeed, but she needed guidance on how to navigate her courses. Hector’s story illustrates how he recognizes that mathematics was never related to his culture, and he reflects on the importance of finding ways to connect students’ experiences outside of the mathematics classroom to their experiences in the classroom. He says that he never found a connection with mathematics and his culture and feels that, even though he understood mathematics, it was always difficult to explain why or how things worked the way they did. Lastly, we see in Marta’s story that she reflects on how students’ socio-economic status affected the quality of mathematics instruction received. By writing about their school mathematics experiences, BPSTs took the time to reflect on their school experiences from the perspective of the mathematics instruction they received, their culture, and their language—and to think about how all these events may have impacted their lives and the decisions they made.

For Teacher Educators The themes described above informed our practice in the bilingual mathematics methods class by focusing our instruction on addressing aspects of (a) language, culture, and community in the context of mathematics instruction; (b) the ability to perform complex mathematics tasks; and (c) the ability to provide high-quality mathematics instruction at any elementary grade level. We detail each of these below. Addressing issues of language, culture, and community opens the door to valuing and stimulating the use of students’ native language in mathematics instruction instead of suppressing it. We, as educators, can take advantage of both language and culture as resources for providing quality mathematics instruction. For instance, mathematics instruction based on children’s mathematical thinking (Carpenter, Fennema, Peterson, & Carey, 1988) has proven to be effective in helping students achieve higher levels of mathematics understanding. This type of instruction requires a classroom setting where children are allowed to share their ideas in a way that is natural to them, and where the teacher listens, values, and uses these ideas to build on their understanding of mathematics (Carpenter, Fennema, Franke, Levi, & Empson, 2015). Classroom practices in the bilingual classroom must go beyond the traditionally recommended instructional strategies for working with emergent bilinguals such as using pictures, diagrams, or models; adjusting the speed of the teacher’s speech; creating a vocabulary list; and providing different assessment forms (Moschkovich, 1999). These traditionally recommended strategies provide little guidance on developing the mathematics in emergent bilinguals’ mathematics instruction. Addressing issues of BPSTs’ ability to perform complex mathematics tasks has helped us target mathematics instruction in a way that helps BPSTs regain their confidence in solving complex mathematics tasks and learn to build on students’

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thinking to advance important mathematical ideas. We recognize that to be an effective mathematics teacher, teachers need to be sensitive to their students’ mathematical thinking and able to make sense of that thinking, which in turn requires deep understanding of the mathematics itself (Ball, Thames, & Phelps, 2008). We know that our BPSTs possess that knowledge, given how their stories relate their early facility with mathematics in school; however, circumstances related to their culture, language, and socio-economic status seemed to affect the way in which BPSTs view their capacity to perform in school mathematics. By working with our BPSTs on regaining their ability to perform complex mathematics tasks, we can also address issues of their ability to provide high-quality mathematics instruction at any elementary grade level. Because elementary teachers are responsible for establishing the foundation for the mathematical education of children, prospective teachers’ mathematical preparation and ability to teach high quality mathematics is an important concern (Ball, 1990). Prospective teachers not only need to understand place value and number composition, make sense of the standard algorithms and nonstandard strategies for performing operations, and develop the ability to reason flexibly about numbers and operations (Whitacre, 2015); they also need to be aware of the implications of teaching mathematics in a multicultural setting in order to be able to effectively teach mathematics content. For example, prospective teachers need to be aware that, when an immigrant student is confused or has a question, the students’ native traditions may discourage questioning the teacher; or the prospective teachers may encounter algorithms as taught in other countries which may open their eyes to the variety of possible ways to compute answers (Perkins & Flores, 2002). This realization can help teachers become more accepting when students deviate from the teachers’ expected procedures or algorithms. These are small examples or realizations that change the way in which prospective teachers see students’ participation and engagement in the class, allowing them be more curious about their own students’ mathematical thinking and ways of doing mathematics. This ultimately motivates both the prospective teacher and elementary school student to engage in meaningful mathematics tasks. We have used BPSTs’ stories in the classroom to alter what BPSTs view as the traditional roles for students and teachers in the mathematics classroom: for instance, their mathematics instruction generally becomes less teacher-dominated. And their stories also help alter BPSTs’ views of who is “good at math”: the so-called achievement gap between Latinx and White students is located in the schools and not in the students themselves or in their families.

Conclusion Each individual’s experiences are unique. Though this group’s experiences may not generalize to all BPSTs, we draw attention to the commonalities of experiences growing up and learning mathematics represented here by our 28 participants and

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further posit that it mirrors the experiences of many other BPSTs. These experiences derive value not only from their authenticity but also from how these same experiences shape the BPSTs’ identities as mathematics teachers. In contrast to deficit views of BPSTs, these narratives show BPSTs’ depth of educational insight and provide a framework within which they evaluate the utility of the instruction they encounter in their preparation as teachers. Moreover, according to the report Achievement Gaps: How Hispanic and White Students in Public Schools Perform in Mathematics and Reading on the National Assessment of Educational Progress (NAEP, 2009), from 1990 to 2009, the proportion of Hispanic students in the overall US student population increased at grade 4 from 6% to 22%, and at grade 8 from 7% to 21%. As the population of Latinx students grows in this country, the potential impact of the academic gap generated by the schools between Latinx and White students also increases. With this we see the urgent need to sharpen our efforts to provide quality education to all. In the particular case of Latinx bilingual students, failure to use their native language and recognize their importance and the contributions of their cultural background in their mathematics instruction imposes a barrier to content communication. As educators, if we focus our efforts on recognizing and using the cultural and linguistic background of our students as resources, we cultivate the emergence of prospective teachers already equipped with the ability to maximize the linguistic and cultural resources of their future students. This new teaching force will be able both to recognize their student’s cultural contribution to the classroom and to help them create a positive identity in the mathematics classroom. This provides an important affirmation for the student that facilitates learning. If we let these rich linguistic and cultural resources languish throughout students’ education, we run the risk of losing the very bilingual students who go on to become teachers.

 ppendix A: Elementary Mathematics Methods Mathematics A Autobiography Activity You will write a “math life story” to reflect on your own experiences with mathematics as a student, and in life, and to think about how those experiences impacted your attitude towards mathematics as well as your understanding of mathematics. You will also reflect on how your own experiences may impact your work as a teacher. Begin by reflecting on the following questions: • What do you remember most about learning math in elementary and middle school? • How do you feel about math? How have your feelings changed over time? • How do you think your school math experiences impacted your attitude towards math? • How do you think your school math experiences impacted your understanding of mathematics? What experiences made it easier/harder for you to learn math?

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• What did your teachers do or not do to connect mathematics to your home/cultural/community experiences? How do you think this impacted your experience? • How was your math learning supported at home and in your community? Did your parents or other family members engage in activities involving math? Did you do any activities that involved or applied math outside of school (e.g., sports, hobbies, games)? • If you received mathematics instruction in a language other than your home language, what was your experience like? What did teacher do or not do to support your learning? • In what ways were you alike or different from the other students in your math classes? Consider math backgrounds, ethnicity, race, gender, linguistic, and/or socio-economic backgrounds. Please be specific in your own identification(s) and those of others. • How do you think your experiences, feelings and beliefs might impact the kind of mathematics teacher that you will be, or the kind of teacher that you want to be? For each question, think about specific experiences and events that you remember, instead of just generalities. Make sure to address all questions.

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Celedón-Pattichis, S., & Ramirez, N. (2012). Beyond good teaching: Advancing mathematics education for ELLs. Reston, VA: National Council of Teachers of Mathematics. Colegrove, K., & Krause, G. (2017). “Complicando algo tan sencillo”: Bridging mathematical understanding of Latino immigrant parents. Bilingual Research Journal, 40(2), 187–204. Connelly, F.  M., & Clandinin, D.  J. (1999). Storied identities: Storied landscapes. New  York: Teachers College Press. Cuban, L. (2010). As good as it gets. Cambridge: Harvard University Press. D’Ambrosio, U., & D’Ambrosio, B. S. (2013). The role of ethnomathematics in curricular leadership in mathematics education. Journal of Mathematics Education at Teachers College, 4(1), 19–25. Dhunpath, R., & Samuel, M. (2009). Life history research: Epistemology, methodology, and representation. Rotterdam, The Netherlands: Sense Publishers. Drake, C., Spillane, J., & Hufferd-Ackles, K. (2001). Storied identities: Teacher learning and subject-­matter context. Journal of Curriculum Studies, 33(1), 1–23. García, O. (2008). Teaching Spanish and Spanish in teaching in the U.S.: Integrating bilingual perspectives. In A. M. de Mejía & C. Helot (Eds.), Integrated perspectives towards bilingual education: Bridging the gap between prestigious bilingualism and the bilingualism of minorities (pp. 31–57). Clevedon, UK: Multilingual Matters. García, O., & Otheguy, R. (2017). Interrogating the language gap of young bilingual and bidialectal students. International Multilingual Research Journal, 11(1), 52–65. García, O., & Sylvan, C. E. (2011). Pedagogies and practices in classrooms: Singularities in pluralities. Modern Language Journal, 95(3), 385–400. Grissmer, D., & Kirby, S. (1991). Patterns of attrition among Indiana teachers, 1965–1987 (Research Report). Retrieved from Rand Corporation website: https://www.rand.org/content/ dam/rand/pubs/reports/2007/R4076.pdf Ingersoll, R. (2001). Teacher turnover and teacher shortages: An organizational analysis. American Educational Research, 38(3), 499–534. Ingersoll, R., & May, H. (2012). The magnitude, destinations, and determinants of mathematics and science teacher turnover. Educational Evaluation and Policy Analysis, 2(1), 2–31. Krause, G. (2014). An exploratory study of teacher retention using data mining (Doctoral dissertation). Retrieved from https://repositories.lib.utexas.edu/handle/2152/24742 Lloyd, G. (2006). Preservice teachers’ stories of mathematics classrooms: Explorations of practice through fictional accounts. Educational Studies in Mathematics, 63(1), 57–87. LoPresto, K.  D., & Drake, C. (2004). What’s your (mathematics) story? Teaching Children Mathematics, 11(5), 266–271. Lortie, D. (1975). Schoolteacher: A sociological study. Chicago: University of Chicago Press. MacDonald, M. V. (2004). Latino education in the United States: A narrated history from 1513– 2000. Houndmills: Palgrave Macmillan. Martin-Jones, M. (2000). Bilingual classroom interaction: A review of recent research. Language Teaching, 33(1), 1–9. Moschkovich, J.  N. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11–19. National Center for Education Statistics. (2009). National Assessment of Educational Progress (NAEP) Mathematics Assessment. Retrieved from https://nces.ed.gov/nationsreportcard/ mathematics/ Perkins, I., & Flores, A. (2002). Mathematical notations and procedures of recent immigrant stu dents. Mathematics Teaching in the Middle School, 7(6), 346–351. Rumberger, R. (1987). The impact of salary differentials on teacher shortages and turnover: The case of mathematics and science teachers. Economics of Education Review, 6(4), 389–399. Sheets, R., Flores, B., & Clark, E. (2011). Educar para transformar: a bilingual education teacher preparation model. In B. Flores, R. Sheets, & E. Clark (Eds.), Teacher preparation for bilingual student populations: Educar para transformar (pp. 9–24). New York: Routledge.

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Index

A Adaptation local context, 27, 28 sample timelines, 27 Autobiographies, 44, 137, 149, 150, 155–157, 162, 164, 165 Crystal’s, 44 personal histories, 44 mathematics, 137 as entry point for PSTs, 141 Luz’s mathematics, 138 Maria, personal histories, 45 stories, 142 Sylvia’s mathematics, 139 teacher preparation program, 142 Theodore’s mathematics, 138 B “Banking” model of education, 152 Bilingual, 94, 163 Bilingual education, 165 Bilingual prospective teachers (BPSTs) academic instruction, language, 169 academic performance, 169 academic success, 167 complex mathematics tasks, 171, 172 elementary teachers, 172 Hispanic/Latina, 167 home values, 169 instruction, English, 169 instructional practices, 168 issues, language, culture and community, 171 lack, rigorous mathematics preparation, 166 life-story interviews, 163 mathematical education, 162

mathematics, 162 autobiographies, 164 classroom, 169 education, 170 performance, 169 motivation, 168 reflecting, experiences, 170, 171 support and responsibilities, 169 traditional roles, 172 C Centers, 114–116 Child-centered approach, 108, 113 Childhood, 108 Children’s community knowledge, 93 Children’s funds of knowledge, 45 deficit-based assumption, 46 knowledge and experiences, 45 mini-explorations, 48–49 MTEs, 46 reading and video clips, 47 Children’s mathematical thinking, 3, 6, 7 Children’s multiple mathematics knowledge bases, 44 Classroom Practices Module, 25, 33 activities, 20 curriculum spaces, 20 goals, 19 lesson plans, 20 myriad classroom practices, 19 video cases, 20 Community, 137, 142 Community-based mathematics practices CME, Cohort analysis, 62, 63 PSTs, 62, 63

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178 Community-based math teaching, 87 Community connections, 18–19, 77, 80–81 Community explorations, 49–51 Community Mathematics Exploration (CME) Module, 24, 25, 28, 34 activities, 18 challenges, 53 community walk, 18 goals, 18, 58 with PSTs reading and video clips, 47–48 scaffolds, 47 scaffolding, 46 Community ties, 95, 97, 102 Community Walk Activity, 49, 53 Crystal’s mathematics autobiography, 137 Crystal’s PSTs, 149 Culturally diverse students, 173 Culturally relevant mathematics teaching (CRMT), 61, 65, 66 See also Prospective urban teachers (PSTs) Culturally relevant pedagogy (CRP), 60, 67, 70 Culturally relevant teaching (CRT), 72 Culture, 135, 143, 144 Curricular analysis process, 121 Curricular noticing, 121, 123, 127, 128 Curriculum case study analysis community and curricular resources, 122, 123 CSAT, 122 elementary mathematics methods course, 122 community connections, 127–128 description, 119 funds of knowledge, children’s, 119 materials, 120 “opening” curriculum spaces, 120 Curriculum adaptation, 120, 123, 126–128 Curriculum analysis, 15 flag activity, 124 real-world connections, 124 unrealized opportunities, 125–126 Curriculum materials, 8 Curriculum Spaces Analysis Table, 28 Curriculum Spaces Analysis Tool (CSAT), 119, 129–130 curricular noticing, 121 description, 120 features, 121 MMKB, 120 PSTs, 120

Index D Deficit language, 29, 30 Design–implement–analyze cycle, 70 Diversity, 85 E Early childhood analyzing videos, 108, 109 classrooms, 114 contexts, 107 teaching and learning mathematics, 114 Video Lenses tool, 109, 116 Elementary education, 92 Elementary mathematics methods activities critical reflection, 150, 151, 153, 157 cultural backgrounds, 158 ideas of status, 155 Instagram-based Math Trail, 154 mathematics autobiography reflections, 156, 157 mathematics classrooms, rules, 155 MTEs, 150 Numbers about Me, 153, 154 photovoice interviews, 150–152 Power and Participation Lens, 156 researchers, 150 RotL, 155 takeaway, 157, 158 Bilingual/TESOL Cohort, 149 courses, 23, 26 crystal’s methods, 149 ESL, 149 interviewer and photograph sharer, 152 SEE Math, 149 third-grade teacher education program, 149 Elementary teacher preparation, 137–139 Emergent bilinguals, 163 Engagement, students, 82 “beamed with pride”, 83 building classroom community, 82–83 math, 82 problem-solving lesson, 82 English as a Second Language (ESL), 149, 156 English Language Learner (ELL), 141 Equity, 85–86 Everyday life, 119, 120, 122 External pressures, 59 F Facilitation, 23, 30, 37

Index Flag activity, 123–128 Funds of knowledge, 3, 7–9, 144 knowledge and experiences, 43 mathematics instruction, 44 I Instagram-based Math Trail, 154 Instructional modules, 15 Classroom Practices Module, 19–21 Community Mathematics Exploration Module, 18–19 Mathematics Learning Case Study, 16–18 Instructor-initiated scaffolds, 61 K K-6 mathematics curriculum, 122 L Latinx, 163 Lesson planning, 19, 20 Life-story interviews, 163 Linguistically diverse students, 173 Luz’s mathematics autobiography, 138 M Math centers, 114, 116 Math education, 86 Mathematical content and early childhood practices, 109, 116 Mathematics autobiographies, see Autobiographies Mathematics autobiography reflections, 156, 157 Mathematics classrooms, 43 Mathematics education, 6, 152, 157, 162, 164, 170 connections, modules and big ideas, 25 social justice (see Social justice) (see TEACH Math project) Video Lenses tool (see Video Lenses tool) Mathematics Learning Case Study Module, 25, 26 activities, 16, 18 final write-up and reflection, 19 “Getting to Know You” interview, 17 goal, 16 mathematics lesson development, 19 problem-solving interviews, 17 shadowing experience, 17 Mathematics methods courses, 17, 20

179 Mathematics module adaptations, 32–38 Mathematics teacher educators (MTEs), 4, 5, 10, 43–44, 92, 93, 95, 98, 150 challenge, 77 elementary education certification program, 46 histories and social identities, 44 prior experiences and positionality inform, 136 (see also Social justice) Mathematizing, 92, 93, 96, 97, 99, 100 Mathematizing a photograph, 48 Mediation, 60 Multiple mathematical knowledge bases (MMKB), 9–11, 26, 31 clinical experiences, 60, 61 CME Module, 59 connections with children, 59 CRMT, 61 definition, 92 ideologies, 58 language and culture, 58 mathematics instruction, 59 PSTs (see Prospective urban teachers (PSTs)) TEACH Math modules, 60 teacher/curriculum developers, 59 teaching practices, 58 urban PSTs, 58 urban teacher education program, 60 N Narratives, 44–45 National Assessment of Educational Progress (NAEP), 161 Numbers about Me activity, 153, 154 P Photovoice Interviews, 150–152 Place value, 114–117 Play, children early mathematics learning, 114 free play, Counting Bugs, 112 game Hi Ho Cherry-O, 110 games, 111 and interventions, 112 Video Lenses tool, 110 Positionality, 31 Post-baccalaureate credential program, 94 Power dynamics, 31 Prekindergarten (PreK), 149 Privilege, 31

180 Prospective urban teachers (PSTs), 3 barriers, 64 BPSTs (see Bilingual prospective teachers (BPSTs)) clinical experiences, 73 CME Module, 66, 78, 79 community visits, 78, 79 CRMT, 65 analyzing students, 70 CME Module, 68, 69, 71 community-based math lessons, 68 confidence developed, 67 field experience, 68 mathematics instruction, 69, 71 non-routine lesson, 68 pedagogical tendencies, 70 third-grade students, 69 culturally relevant mathematics teacher, 65 development process, 70, 71 educational activities, 6 elementary mathematics methods (see Elementary mathematics methods) family and cultural practices, 8 mathematics content courses, 52 mathematics vs. CRMT, 67 MTEs’ methods courses, 5 personal and CME experiences, 67 reflections, 78, 79 research, 6 students and communities future classroom, 81 interests, 80 linking, mathematics teaching, 81 local community, 80 mathematics connection, 80, 81 teaching mathematics, 64 traditional math, 64 understandings and practices, 6 R Readings, 47, 48, 54 “Real world,” vs. “school math”, 83 Reflections, PSTs, 77–79, 81, 88 Resources curricular and community, integration, 123 curriculum, 119 mathematical, 120 Rights of the Learner (RotL), 155 S School mathematics vs. real mathematics, 83–84

Index School–university partnership model, 61 Social justice children’s community knowledge, 93 community ties, 97–98 courses, primary routes, 92 in curriculum, 92 mathematizing, 92 tasks, connections, 95–96 Student case studies Classroom Practices Module, 19–21 Community Mathematics Exploration Module, 18–19 Mathematics Learning Case Study Module, 16–18 Support and Enrichment Experiences in Mathematics (SEE Math), 149 Sylvia’s mathematics autobiography, 139 T TEACH Math modules, 24, 60 TEACH Math project, 136 conceptualization, 4–5 conference, 3 co-PIs, 4 design principles, 5 elementary mathematics methods (see Elementary mathematics methods) goals, 15 MMKB, 9, 10 PSTs’ understandings and practices, 6 teacher learning, 5 website, 15 Teacher education, 15, 20, 23, 24, 148 programs, 6, 11, 57, 58, 60, 92–95 Teacher educators, 23 Teacher learning, 92, 99 Teacher preparation program, 77, 87 Teachers beliefs, 136 Teachers Empowered to Advance Change in Mathematics (TEACH Math) project, see TEACH Math project Teaching practices community-based math lesson, 86 teaching, 87 curriculum, 87 future classrooms, 86 math teaching philosophy, 88 time investment, 87 Tensions, 29, 31 Testimonials/testimonios, 140 Theodore’s mathematics autobiography, 138

Index Time investment, 87 Traditional math, 64 U Urban PSTs process, 60 Urban teacher education program, 60 V Video case study, 20

181 Video clips, 47–49, 54 Video Lenses tool, 107–108 continuing use, 117 Counting Bugs, 112, 113, 117 early childhood instructional practices, 114–116 early mathematics content, 109, 110 Hi Ho Cherry-O, 110, 111, 117 on young children, 108, 109 Videos, 108, 109