Research Studies on Learning and Teaching of Mathematics: Dedicated to Edward A. Silver (Research in Mathematics Education) 3031354583, 9783031354588

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Research in Mathematics Education Series Editors: Jinfa Cai · James A. Middleton

Jinfa Cai Gabriel J. Stylianides Patricia Ann Kenney   Editors

Research Studies in Learning and Teaching of Mathematics Dedicated to Edward A. Silver

Research in Mathematics Education Series Editors Jinfa Cai, Newark, DE, USA James A. Middleton, Tempe, AZ, USA

This series is designed to produce thematic volumes, allowing researchers to access numerous studies on a theme in a single, peer-reviewed source. Our intent for this series is to publish the latest research in the field in a timely fashion. This design is particularly geared toward highlighting the work of promising graduate students and junior faculty working in conjunction with senior scholars. The audience for this monograph series consists of those in the intersection between researchers and mathematics education leaders—people who need the highest quality research, methodological rigor, and potentially transformative implications ready at hand to help them make decisions regarding the improvement of teaching, learning, policy, and practice. With this vision, our mission of this book series is: (1) To support the sharing of critical research findings among members of the mathematics education community; (2) To support graduate students and junior faculty and induct them into the research community by pairing them with senior faculty in the production of the highest quality peer-reviewed research papers; and (3) To support the usefulness and widespread adoption of research-based innovation.

Jinfa Cai  •  Gabriel J. Stylianides Patricia Ann Kenney Editors

Research Studies on Learning and Teaching of Mathematics Dedicated to Edward A. Silver

Editors Jinfa Cai Department of Mathematical Sciences University of Delaware Newark, DE, USA

Gabriel J. Stylianides University of Oxford Oxford, UK

Patricia Ann Kenney University of Michigan–Ann Arbor Ann Arbor, MI, USA

ISSN 2570-4729     ISSN 2570-4737 (electronic) Research in Mathematics Education ISBN 978-3-031-35458-8    ISBN 978-3-031-35459-5 (eBook) https://doi.org/10.1007/978-3-031-35459-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Edward A. Silver: A Celebration of His Professional Journey There is a Chinese saying: 饮水思源. In translation, the saying means that when we drink water, we should not forget where the water comes from, implying not to forget its origin. This saying is often used to recognize a person’s impact on us— that is, to be grateful for the source of one’s success. The purpose of this book is to embody this Chinese saying in honoring Edward A. Silver’s successful professional career as a mathematics educator and researcher. We planned the volume so that all chapters were written by at least one author who had the privilege of working with Silver as a graduate student or post-doctoral fellow. In this way, we exemplify Silver’s significant impact on the chapter authors and the many others who experienced his tutelage over the years. Our goals in producing this volume are to foster a continuing connection among those who fondly call themselves “Silverites” and to celebrate Silver’s long and successful professional career. Readers will find that the set of chapters are well-written and very insightful for the field of mathematics education research. However, the set of chapters in this volume represent only a part of Silver’s impact on mathematics education. Readers will find in each chapter the details of how he influenced numerous other researchers’ careers and research programs. Readers will also see that some of the chapters were written by multiple authors, some of whom did not work directly with him but instead were students or colleagues of one of the Silverites. This chain leading back to Silver shows just how far his influence extends—a kind of genealogy if you will—that branches out from Silver as the origin. We also recognize that one really cannot reflect on Silver’s long and successful career with only one book containing 12 chapters. For a summary of significant milestones in his career, we refer readers to the Biographical Sketch at the end of the book. In the remainder of this preface, we selectively highlight three major features of Silver’s career: keeping mathematics at the forefront, focusing on students’ mathematical thinking, and supporting professional learning in mathematics. v

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Keeping Mathematics at the Forefront The first feature of Edward A. Silver’s career that we highlight here is his focus on mathematics. He began his career as a mathematics teacher and developed an interest in fostering students’ understanding, in particular supporting mathematical understanding for all students. Silver began his journey as a mathematics educator after completing a Bachelor of Arts degree in mathematics at Iona College in New York. As a seventh-grade mathematics teacher, he became fascinated with the range of abilities among his students and was always interested in getting them excited about learning. This led him to focus on problem solving and inquiry because he found that these topics could capture his students’ imaginations and foster their thinking about mathematics. Silver also realized that contextualizing mathematics concepts within other subjects, such as science, could enhance understanding. His experience in classroom teaching motivated him to pursue a doctorate at Teachers College, Columbia University. After completing his EdD in mathematics education along with Master’s degrees in mathematics and mathematics education, Silver began teaching prospective mathematics teachers at Northern Illinois University, where he had an appointment in the mathematics department. His colleagues encouraged him to do research and to contribute to the field in this early stage of his career. This encouragement continued during his career transitions to San Diego State University in California (where again his appointment was in a mathematics department) and to the University of Pittsburgh (Pennsylvania), and finally to the University of Michigan. Silver has since been involved in several funded research projects that focused on rational numbers, division with remainders, interpretive reports of results from the National Assessment of Educational Progress (NAEP), Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR), and making research results accessible to practitioners. Silver’s interest in mathematical thinking and working with teachers to improve mathematics teaching and learning has been a consistent theme throughout his career. The interaction between students’ mathematical thinking and thoughtfulness has been the essence of his work. In addition to a focus on mathematics-based projects, Silver’s efforts to keep mathematics at the forefront of his work spread over into other areas of interest. In particular, in his Biographical Sketch, included later in this volume, he mentions mathematics in nearly all of his research interest statements: for example, the study of mathematical thinking, intellectually engaging and equitable mathematics instruction, and innovative methods of assessing and reporting mathematics achievement.

Focusing on Students’ Mathematical Thinking A second major focus of Edward A. Silver’s career has been on students’ mathematical thinking especially in the areas of problem solving and problem posing. He developed an interest in problem solving early in his classroom teaching career and

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doctoral studies. While pursuing his doctorate at Teachers College, he was influenced by his professors Jeremy Kilpatrick and Bruce Vogeli. Vogeli’s international perspective influenced Silver to take a global perspective on mathematics education. Kilpatrick’s interest in problem solving and inquiry had a lasting impression on Silver, who participated in problem-solving meetings with Kilpatrick and other former students. Silver was also influenced by George Polya’s ideas on mathematical problem solving and thinking, and he gained insight into the problem-solving process through his experiences with Polya’s (1962) book Mathematical Discovery. Additionally, during his graduate studies, Silver’s interest in cognitive psychology was sparked by taking a graduate seminar on Piaget. Silver’s interest in cognitive psychology and its relationship to mathematics education led him to write a grant proposal that was funded by the National Science Foundation. He organized and ran a conference that resulted in the publication of a book (Silver, 1985) entitled Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives. This project had a significant impact on Silver’s thinking about problem solving and the book (often called the “yellow book” because of the color of its binding) became widely cited and had a great impact in bridging the fields of cognitive psychology and mathematics education. In a seminal work entitled “On Mathematical Problem Posing” (1994), Silver advocated for problem posing as a valuable pedagogical tool in mathematics education and catalyzed the field of problem-posing research. In that paper, he discussed the importance of problem posing in mathematics education and argued that it is an essential aspect of mathematical thinking and thus can enhance students’ understanding of mathematics. Silver included a framework for problem posing and offered examples of how teachers can incorporate problem posing activities in their classrooms. Additionally, he emphasized the need for teachers to create a supportive environment that encourages students to take risks and explore different solutions to problems. Silver’s problem-posing work laid the foundation for later investigations that incorporated problem posing in students’ mathematical experience.

Supporting Professional Learning in Mathematics A third key focus of Edward A.  Silver’s career has been to support professional learning for teachers of mathematics and to mentor doctoral students and post-­ doctoral fellows in mathematics education. In the case of support for teachers, one of Silver’s crowning achievements was leading the QUASAR project at the University of Pittsburgh’s Learning Research and Development Center. QUASAR was an educational reform project that aimed to enhance mathematics instructional programs for economically disadvantaged middle school students (Silver & Stein, 1996). Given how conventional math teaching methods had failed to help many students in urban schools develop meaningful mathematical proficiency, the project sought to provide high-quality mathematics learning opportunities to students by enhancing the form and content of precollege mathematics

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instruction. QUASAR operated independently in six urban schools in the United States and served culturally and linguistically diverse students, including those for whom English was not the primary language spoken at home; the schools primarily served children living in poverty. The project aimed to promote a vision of mathematics instruction that blended attention to basic-level and high-level mathematical goals, emphasizing reasoning, problem-solving, communication, and conceptual understanding. Students in QUASAR classrooms constantly had opportunities to work on challenging tasks that involved multiple representations and multiple solution strategies and fostered the use of collaborative groups. The QUASAR project resulted in a number of publications, one of which was a casebook (Stein et al., 2000) that offered framework to help teachers and educators evaluate instructional decisions, materials, and learning outcomes according to a set of mathematics standards. Through nearly 500 classroom lessons, the book presents cases of mathematics instruction and provides readers with insights on how to create a challenging, cognitively rich, and exciting classroom environment that enhances students’ understanding of mathematics. The book has received high praise from professionals and is just another example of Silver’s impact in the area of professional learning. Although QUASAR was concluded more than 25 years ago, the project’s legacy continues. For example, much more is now known about the relationship between the nature of the mathematical tasks in which students engage and what students learn about mathematics. In particular, high-level tasks lead to greater student learning outcomes than regular exposure to low-level tasks that require memorization and application of known algorithms and procedures. This finding has been confirmed and extended by subsequent research, which has also provided more insight into the factors that impact the implementation of high-level tasks and how teachers can support students’ high-level engagement in ways that maintain the demands of the task. Research on mathematical tasks has influenced the practices of teacher educators and professional developers who seek to improve teaching and learning by designing learning opportunities for teachers that emphasize the cognitive demands of mathematical tasks. Finally, Silver’s work on the professional learning of mathematics is also reflected in his mentoring doctoral students and postdoctoral fellows. All three of this volume’s editors (two of whom were Silver’s doctoral students and one of whom served as his post-doctoral fellow) as well as a number of the chapter authors were beneficiaries of his wise counsel. During the interview with the editors, Silver reflected on his vision of working with his graduate students. In particular, his advice was to use time wisely in a doctoral program—that is, to choose the program that meets their needs and to take time to figure out what version of a mathematics educator they want to be (e.g., teacher educator, researcher, etc.). He also suggested that mathematics education doctoral students should remember what brought them to doctoral study in the first place and try not to lose that passion while they pursue their doctoral degree and thereafter.

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The Impact Continues As we attempt to demonstrate in this volume, one person can have a monumental impact on many others. Although 12 chapters cannot account for all of these impacts, they can highlight the impact one can have. In celebrating Edward A. Silver’s professional life, we also demonstrate how those who were impacted by working with him pass the torch to others, generation after generation. With this book, we honor Silver’s career as a mathematics educator and researcher both in terms of his efforts to excite students about learning mathematics through problem solving, problem posing, and inquiry as well as his support of teachers’ and others’ professional learning. Newark, DE, USA Oxford, UK Ann Arbor, MI, USA

Jinfa Cai Gabriel J. Stylianides Patricia Ann Kenney

References Polya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving. Wiley. Silver, E. A. (Ed.) (1985). Teaching and learning mathematical problem solving: Multiple perspectives. Lawrence Erlbaum Associates. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. http://www.jstor.org/stable/40248099 Silver, E.  A., & Stein, M.  K. (1996). The Quasar project: The “revolution of the possible” in mathematics instructional reform in urban middle schools. Urban Education, 30(4), 476–521. https://doi.org/10.1177/0042085996030004006 Stein, M., Smith, M., Henningsen, M., & Silver, E. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teacher College.

Edward A. Silver: Brief Biographical Sketch

Edward A. Silver joined the faculty at the University of Michigan in 2000, and he holds the William A.  Brownell Collegiate Professorship in the Marsal Family School of Education. In 2008 he was the Lappan-Phillips-Fitzgerald Visiting Professor at Michigan State University, and in 2015 he was the Fulbright Canada Visiting Research Chair in the Faculty of Education at the University of Alberta. Prior to joining the U-M faculty in 2000, he was Professor in the School of Education and Senior Scientist in the Learning Research & Development Center at the University of Pittsburgh (1987–2000). Before that he held faculty positions in the Department of Mathematical Sciences at Northern Illinois University (1977–1979) and in the Department of Mathematical Sciences at San Diego State University (1979–1987). He began his career teaching at the middle school and high school levels in New York. He received his BA from Iona College and multiple degrees (MA, MS, and EdD) from Teachers College, Columbia University. Silver’s scholarly interests include the study of mathematical thinking, especially mathematical problem solving and problem posing; the design and analysis of intellectually engaging and equitable mathematics instruction for students; innovative methods of assessing and reporting mathematics achievement; practice-based approached to enhancing teachers’ knowledge and effectiveness; and improving the interface between research and practice in education. He has directed numerous research projects and published more than 150 articles, chapters, and books on these and related topics. In addition to his scholarly pursuits, Silver has also held several leadership roles. He served as editor of the Journal for Research in Mathematics Education from 2000 to 2004 and as co-editor of The Elementary School Journal from 2008 to 2010. He was the Founding Director of Center for Research in Mathematics and Science Education at San Diego State University (1985–1987) and Director of the QUASAR project at the University of Pittsburgh (1989–1995). During his time at the University of Michigan, he has served as Chair of the Educational Studies Program (2003–2005), as Associate Dean for Academic Affairs (2005–2008), and as Senior Associate Dean for Research and Graduate Studies (2016–2021) in the

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Edward A. Silver: Brief Biographical Sketch

School of Education on the Ann Arbor campus. He also served as Dean of the School of Education at the University of Michigan-Dearborn from 2010 to 2013. Silver has received a number of recognitions for his work, including the 2004 Award for Outstanding Contributions of Educational Research to Practice from the American Educational Research Association; the 2007 Iris Carl Memorial Leadership and Equity Award from TODOS; the 2008 Judith Jacobs Lectureship from the Association of Mathematics Teacher Educators; the 2009 Distinguished Alumnus Award from the Department of Mathematics, Science and Technology at Teachers College, Columbia University; the 2009 Lifetime Achievement Award from the National Council of Teachers of Mathematics; and the 2011 Senior Scholar Award from the American Educational Research Association Special Interest Group for Research in Mathematics Education. In 2016, he was selected as a Fellow of the American Educational Research Association, and in 2017 he was elected to membership in the National Academy of Education.

Contents

Mathematical Problem-Posing Research: Thirty Years of Advances Building on the Publication of “On Mathematical Problem Posing” ��������������������������������������������������������    1 Jinfa Cai, Stephen Hwang, and Matthew Melville Professional Learning Tasks Through Job-­Embedded Teacher Professional Development����������������������������������������������������������������   27 Hala Ghousseini and Elham Kazemi To What Extent Are Open Problems Open? Interplay Between Problem Context and Structure����������������������������������������������������������������������   49 Roza Leikin, Sigal Klein, and Ilana Waisman The Researcher and the Practitioner: Stories About Establishing Pipelines, Building Bridges, and Crossing Borders��������������������������������������   71 Cathy G. Schloemer and Patricia Ann Kenney Understanding and Improving Mathematics Instruction Through a Cultural Lens��������������������������������������������������������������������������������   93 Meixia Ding, Rongjin Huang, Xiaobao Li, and Yeping Li From Mathematical Tasks to Research–Practice Partnerships: A Look at Edward Silver’s Influence on Our Efforts at Mathematics Instructional Improvement ����������������������������������������������������������������������������  115 Alison Castro Superfine and Benjamin M. Superfine Desirable Features of Mathematical Tasks: Views of Mathematics Teacher Educators ������������������������������������������������������������������������������������������  131 Cengiz Alacaci, Bulent Cetinkaya, and Ayhan Kursat Erbas Parental Involvement in Young Children’s Mathematical Learning: The Story of Family Math Tasks��������������������������������������������������������������������  157 Shuk-kwan S. Leung

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Engaging All Students in Challenging Mathematical Work: Working at the Intersection of Cognitively Challenging Tasks and Differentiation During Lesson Planning and Enactment ��������������������  179 Charalambos Y. Charalambous, Sofia Agathangelou, Seán Delaney, and Nicos Papadouris A Review of the Mathematical Tasks Framework and Levels of Cognitive Demand ��������������������������������������������������������������������������������������  219 Hui-Yu Hsu and Chen-Yu Yao A Critical Discursive Framework for Analyzing the Views About Mathematics Being Promoted by Mathematics Textbooks for Prospective Elementary Teachers������������������������������������������������������������  253 Leah N. Shilling-Stouffer and Gabriel J. Stylianides  Mathematical Tasks: The Lasting Legacy of the QUASAR Project������������  275 Margaret Smith and Mary Kay Stein Author Index����������������������������������������������������������������������������������������������������  299 Subject Index����������������������������������������������������������������������������������������������������  307

About the Contributors

Sofia  Agathangelou works at the Department of In-Service Teacher Training, Cyprus Pedagogical Institute (CPI). She holds a PhD and a Master’s degree in Curriculum Development and Instruction and a Bachelor’s degree in Educational Sciences. In the Department of In-Service Teacher Training, she designs, develops, and teaches modules for in-service teachers around topics of teacher knowledge in mathematics, teaching methodology (including differentiated instruction), and quality instruction. She also participates in relevant European and national research projects. Cengiz  Alacaci received his doctorate in Mathematics Education from the University of Pittsburgh (USA). He worked as a Graduate Research Assistant and later as a Post-doctoral Research Associate at the Learning Research and Development Center, University of Pittsburgh, with Dr. Edward A. Silver between 1995 and 1999. He later taught at Florida International University in the USA and Bilkent University in Turkey. He is currently a Professor of Mathematics Education at the Department of Mathematical Sciences, University of Agder, Norway. He has published and conducted research on mathematical problem solving, mathematics curriculum, and mathematics teacher education. Jinfa Cai is the Kathleen and David Hollowell Professor of Mathematics Education at the University of Delaware, and a Fellow of American Educational Research Association. He served as the Editor-in-Chief for the Journal for Research in Mathematics Education. He has also served as Program Director at the USA National Science Foundation. He was the Editor of the 2017 Compendium for Research in Mathematics Education. He has received several important awards, including the University of Delaware’s College of Arts and Sciences’ Outstanding Scholarship Award. He also received the Webber Award (honoring his significant contribution to the State of Delaware’s mathematics education). Recently, he has focused his research on mathematical problem posing.

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About the Contributors

Bulent Cetinkaya is an Associate Professor at the Department of Secondary Science and Mathematics Education, Middle East Technical University, Turkey. He earned his PhD in Mathematics Education from Syracuse University (USA). His research interests include mathematics teacher education, mathematical modeling and problem solving, teacher beliefs, and personal differences in teaching and learning. He has been involved as a principal investigator or researcher in several research and professional development projects supported by the National Science Foundation, General Electric, World Bank, and the Scientific and Technical Research Council of Turkey. Charalambos Y. Charalambous is an Associate Professor at the Department of Education, University of Cyprus, specializing in Educational Research and Evaluation. His research interests include teaching effectiveness with a particular focus on understanding the work of teaching and measuring teaching quality; contributors to teaching quality; and the effects of teaching on student learning. He is also interested in teacher initial training and teacher professional development through guided reflection on practice. Seán  Delaney is a Registrar and Vice-President (Academic Affairs) at Marino Institute of Education, an associated college of Trinity College Dublin, the University of Dublin. His research interests include the practice of teaching and teacher education, mathematical knowledge for teaching, and purpose and education. His book about teaching, Become the Primary Teacher Everyone Wants to Have, was published by Routledge and his podcast on education, Inside Education, has received over 150,000 downloads. Meixia  Ding is an Associate Professor of Mathematics Education at Temple University, USA. Her research focuses on how the instructional environment (e.g., teacher knowledge, curriculum) can be better structured to develop students’ sophisticated understanding of fundamental mathematical ideas (e.g., basic properties, relationships, and structures), which may lay a foundation for students’ future learning of more advanced topics like algebra. With the support of NSF’s CAREER award, she has recently explored the necessary knowledge for teaching early algebra in elementary school from a cross-cultural perspective. Ayhan Kursat Erbas received his doctorate in Mathematics Education from the University of Georgia (USA). He is currently a Professor of Mathematics Education at the Department of Mathematics and Science Education, Middle East Technical University, Turkey. He has published and conducted research on mathematical modeling, mathematical problem solving, mathematics curriculum and textbooks, international comparative studies, teaching and learning algebra, technology integration in mathematics education, and mathematics teacher education. Hala Ghousseini, PhD, is the John G. Harvey Professor of Mathematics Education at the University of Wisconsin. Her research focuses on teacher learning, classroom discourse, and the design of professional education that can support the learning of ambitious practice.

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Hui-Yu Hsu is an Associate Professor for the Graduate Institute of Mathematics and Science Education at National Tsing Hua University. She obtained a PhD degree at the University of Michigan under the supervision of Edward Silver from 2004-2010. In her dissertation, she adopted the Mathematical Tasks Framework and the construct of cognitive demand to examine the curricular and instructional materials, classroom teaching, and student learning of geometry in Taiwan. Her research interests include teaching and learning specific to geometry content, textbook analysis, and Event-Related Potential (ERP) examinations of students’ mathematics reactions. Rongjin  Huang is a Professor at Middle Tennessee State University, USA.  His research interests include mathematic classroom instruction, mathematics teacher education, and lesson study. He has published scholarly work extensively, including co-editing nine books such as Theory and Practice of Lesson Study in Mathematics (Springer, 2019) and Teacher Professional Learning Through Lesson Study in Virtual/Hybrid Environments (Routledge, 2023), and seven special issues for ZDM Mathematics Education and International Journal for Lesson and Learning Studies. Dr. Huang was a member of international program committee for ICMI STUDY 25 and is currently a council member of World Association of Lesson Study. Stephen Hwang is a Senior Research Associate at the Department of Mathematical Sciences, University of Delaware. His research interests include mathematical problem posing, the teaching and learning of mathematical justification and proof, the nature of practice in the discipline of mathematics, and teacher education. He is currently the Project Coordinator for Supporting Teachers to Teach Mathematics Through Problem Posing, an NSF-funded project that seeks to help middle-school mathematics teachers incorporate problem posing into their instruction. Elham Kazemi, PhD, is a Professor of Mathematics Education at the University of Washington. She studies children’s mathematical thinking and experiences, classroom discourse, and teachers’ collaborative learning. Patricia Ann Kenney had an atypical journey into the field of mathematics education. With a Bachelor’s degree in English and a Master’s degree in library (information) science, she started her career in 1974 as a university librarian. She returned to higher education to earn a second Bachelor’s degree in Mathematics (1981) and then a PhD in Mathematics Education (1988). She did a post-­doctoral fellowship under the tutelage of Dr. Edward A.  Silver at the University of Pittsburgh (Pennsylvania, USA) Learning Research and Development Center (1990-92), and then worked with him for nearly 20 years in a variety of capacities, most of which involved writing articles and book chapters as well as editorial work on publications such as the Journal for Research in Mathematics Education, the National Council of Teachers of Mathematics National Assessment of Educational Progress interpretive reports, and the More Lessons Learned from Research series. These experiences allowed her to utilize her background in English, research, and mathematics to the benefit of those with whom she worked.

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Sigal Klein is a PhD candidate, supervised by Prof. Roza Leikin, at the Department of Mathematics Education, University of Haifa (2023). In 1991, she graduated from the BSc program in Industrial Management at the Technion – the Israeli Institute of Technology. In 2019, she completed her MA degree in Mathematics Education at the University of Haifa. Prior to her Mathematics Education studies, she worked in industry and high-tech. In her teaching, she links mathematical knowledge and skills with the knowledge required in industry and high-tech with the aim of making the field of mathematics accessible to students. Her current PhD research focuses on the creativity required of students and teachers when solving open problems that have multiple solution outcomes and require producing multiple solution strategies. She heads the digitization of R&D mathematics education programs in the RANGE center. Sigal also teaches school mathematics at several levels (from elementary to high school) to students with a broad range of abilities. Roza Leikin is a Full Professor of Mathematics Education and Gifted Education at the Faculty of Education, University of Haifa, and since October 2020, she has served as the Faculty Dean. Dr. Leikin is the Establishing Director of the RANGE Center  – the Interdisciplinary Center for the Research and Advancement of Giftedness and Excellence. Prof. Leikin’s research focuses on mathematical creativity and ability, task design, teachers’ professional potential, and integration of neuro-cognitive research in the field of mathematics education. From 2012 to 2017, she served as the President of the International Group for Mathematical Creativity and Giftedness (http://igmcg.org/) and was the Head of the National Advisory Board in Mathematics Education of the Israeli Ministry of Education. Since 2013, she has served as a Senior Mathematics Education Editor of the International Journal of Science and Mathematics Education. She has edited 13 volumes related to research in mathematics and gifted education and has published more than 180 papers and chapters in peer-reviewed research journals, books, and refereed conference proceedings. Shuk-kwan S. Leung is a Professor and a former Director/Chair of the Institute of Education/Center for Teacher Education, National Sun Yat-sen University in Taiwan. She was also teacher and professor in Hong Kong. Her research interests focus on problem solving/posing, teacher education, and parental involvement. She serves on the editorial board for Mathematical Thinking and Learning and International Journal on Science and Math Education, has been active in PME and ICME, and published in journals and books. She won awards on research and teaching at university and national levels (National Science Council, Ministry of Education) and was a review panel member for NSC and MOE. Xiaobao  Li is an Associate Professor of Mathematics Education at Widener University. His research focuses on students’ learning difficulties in math, and how learning environment can be designed to make mathematics comprehensible and accessible for every child. The findings in this line of work were published in Cognition and Instruction and School Science and Mathematics.

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Yeping Li is a Professor and former Head of the Department of Teaching, Learning, and Culture, Texas A&M University, USA. His research interests include mathematics education, STEM education, international studies, and teacher education. He is the founding Editor-in-Chief of the International Journal of STEM Education and Journal for STEM Education Research, both published by Springer. In addition to publishing over 15 books and special journal issues, he has published numerous articles in topic areas that he is interested in. He received his PhD in Cognitive Studies in Education from the University of Pittsburgh, USA. Matthew Melville is an Assistant Professor at Purdue University Fort Wayne. His research interests include teacher professional learning, both for pre-service and inservice teachers, and international comparative studies. Currently, he is working on projects introducing a professional learning program, Japanese instructional circles, to different groups of high school and middle school mathematics teachers. Nicos Papadouris works with the Center for Educational Research and Evaluation, Cyprus Pedagogical Institute. He has a PhD in Science Education, an MEd in Science Education, a BA in Education, and a BSc in Physics. His research interests concentrate on teaching and learning in science (especially physics), science curriculum design and development, and the assessment of science learning. Cathy G. Schloemer taught mathematics at Indiana Area Senior High School in Indiana, Pennsylvania (USA), from 1977 to 2013. Cathy’s passion in the classroom was to strive to build students’ strong conceptual understanding, in addition to excellent procedural competence. She also mentored dozens of student teachers and wrote practical teaching articles that were published in professional journals. Cathy completed her doctoral degree under the supervision of Dr. Edward A. Silver at the University of Pittsburgh in 1994. She is fascinated by the interplay between research and practice in mathematics education and is intrigued by ways to make that interplay more seamless. Leah N. Shilling-Stouffer is an Associate Professor of Mathematics Education at Longwood University (USA), where she teaches a variety of mathematics courses, including those for future elementary and secondary teachers. She also serves as the Director of the Quantitative Reasoning (QR) Center, which provides students with academic support and enrichment services that help them to become confident and critical users of quantitative information and methods. Her primary research focuses on the importance, influences, and impact of mathematical beliefs on the learning and teaching of mathematics. Other research interests grounded in her work as QR Center director includes the exploration of barriers to success within and supports for students enrolled in introductory statistics courses. Margaret Smith is a Professor Emerita at the University of Pittsburgh (USA). Her work focuses on developing research-based materials for use in the professional development of mathematics teachers. She has authored or coauthored over 90 books and edited books or monographs, book chapters, and peer-reviewed articles

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About the Contributors

including Five Practices for Orchestrating Productive Discussions (co-authored with Mary Kay Stein). She is the recipient of the award for Excellence in Teaching in Mathematics Teacher Education from the Association of Mathematics Teacher Educators in 2009 and the Lifetime Achievement Award from the National Council of Teachers of Mathematics in 2019. Mary Kay Stein holds a joint appointment at the University of Pittsburgh (USA) as a Professor of Learning Sciences and Policy and a Senior Scientist at the Learning Research and Development Center. Her research focuses on classroom-based mathematics teaching and the ways in which policy and organizational conditions shape teachers’ practice. Dr. Stein was named an AERA Fellow in 2014. She has either co-authored or co-edited 9 books on mathematics education and school reform and has published over 50 articles in educational journals. Her invited talks comprise both domestic and international audiences, including two presentations at the International Congress on Mathematics Education (Seoul, South Korea, 2012; Hamburg, Germany, 2016). Gabriel J. Stylianides is a Professor of Mathematics Education at the University of Oxford and a Fellow of Oxford’s Worcester College (UK). His research focuses on issues related to the meaningful engagement of students in fundamental mathematical practices. He has published 4 books and over 90 other articles in refereed journals, conference proceedings, or edited volumes. His research received support from the US National Science Foundation, the US Department of Education, the Education Endowment Foundation, the Spencer Foundation, and the John Fell Fund. He is currently an Editorial Board member of the Elementary School Journal, the Journal of Mathematical Behavior, the International Journal of Educational Research, and Asian Journal for Mathematics Education. Alison Castro Superfine is a Professor of Mathematics Education and Learning Sciences, and the Director of the Learning Sciences Research Institute at the University of Illinois at Chicago (USA). Castro Superfine received her PhD in Mathematics Education from the University of Michigan. Her research focuses on teacher learning of both pre- and in-service teachers and on research-practice partnerships as vehicles for collaborative mathematics instructional improvement. Benjamin  M.  Superfine is a Professor of Educational Policy Studies and the Assistant Vice Provost for Faculty Relations at the University of Illinois at Chicago (USA). He also serves as a Faculty Affiliate in the UIC John Marshall Law School, the Learning Sciences Research Institute, and the Center for Urban Educational Leadership. Superfine received his JD and PhD in Education Foundations and Policy from the University of Michigan. Before joining UIC, Superfine practiced law at a firm in Washington, D.C. His research focuses on the history of education law and policy, school finance reform, standards-based reform and accountability, and teacher unions.

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Ilana  Waisman is a Research Fellow at the Neurocognitive Laboratory for the investigation of creativity, ability and giftedness, RANGE center, University of Haifa. She is a Senior Lecturer at the Shaanan Academic Teachers’ College. She received her MSc in Applied Mathematics from the Technion – Israel Institute of Technology. She completed her PhD at the Department of Mathematics Education, University of Haifa, under the supervision of Prof. Roza Leikin and Prof. Mark Leikin. She teaches mathematics and mathematics education courses for BEd and MA students. She also teaches a graduate course, “Looking at mathematics education through a neurocognitive lens,” at the University of Haifa. Her primary research focus is on the relationships between mathematical performance and associated neurocognitive activity, with a special focus on the neurocognitive characteristics of individuals with different levels of abilities. Chen-Yu Yao is a doctoral student in the Department of Educational Psychology and Counseling at National Tsing Hua University, Taiwan. He obtained a Master’s degree from the Graduate Institute of Mathematics and Science Education at the same university during 2018-2020. His research focuses on understanding students’ cognitive behaviors and development in mathematics by means of event-related potential (ERP) methodology.

Mathematical Problem-Posing Research: Thirty Years of Advances Building on the Publication of “On Mathematical Problem Posing” Jinfa Cai, Stephen Hwang, and Matthew Melville

In 1994, Ed Silver published a seminal paper entitled “On Mathematical Problem Posing.” This paper has been viewed as a “must-read” in mathematical problem-­ posing research because of the foundation it laid for this area of research. Indeed, according to Google Scholar, there are more than 1400 scholarly publications that have cited this paper since its publication, and the 10 most cited of those publications have themselves been cited over 7000 times.1 Moreover, 78 articles appearing in the top 10 high-quality journals in mathematics education (Williams & Leatham, 2017) have referenced this paper. For a relatively young area of study in mathematics education (concerted empirical research on problem posing has only been ongoing for the past 20–30 years), this is a rather extensive reach for a single paper. The great number of citations of this article is not surprising given that it is a seminal work on problem-posing research. In fact, it was the first paper to provide various perspectives on problem posing and problem-posing research. Silver (1994) raised six themes in “On Mathematical Problem Posing.” In this chapter, we describe these six themes and then examine the advances in

 Data retrieved from Google Scholar on March 17, 2023.

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During the preparation of this chapter, the authors were supported by a grant from the National Science Foundation (DRL 2101552). Any opinions expressed herein are those of the authors and do not necessarily represent the views of the National Science Foundation. J. Cai (*) · S. Hwang University of Delaware, Newark, DE, USA e-mail: [email protected]; [email protected] M. Melville Purdue University Fort Wayne, Fort Wayne, IN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Cai et al. (eds.), Research Studies on Learning and Teaching of Mathematics, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-35459-5_1

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problem-­posing research from the perspectives of problem posing as a cognitive enterprise, as something to be taught, and as something to teach through (Stanic & Kilpatrick, 1988). We decided to use these three perspectives to identify the advances that have been made in mathematics education research on problem posing for at least two reasons: (1) these three perspectives have demonstrated their usefulness for discussing problem solving and problem-solving research and, in parallel, for conceptualizing problem posing and problem-posing research (e.g., Cai & Leikin, 2020); and (2) the six themes Silver raised can be merged with these three perspectives. In this chapter, when we discuss advances, we have tried to note where themes in Silver’s paper have played out in state-of-the-art work, thus tracing the paper’s influence in the literature.

Conceptualization of Problem Posing Before we describe Silver’s six themes, we first describe a conceptualization of problem posing parallel to the conceptualization of problem solving by Stanic and Kilpatrick (1988). Stanic and Kilpatrick proposed three categories—problem solving as a cognitive enterprise, as something to be taught, and as something to teach through—into which research on problem solving, past and present, can be sorted. Similarly, Schroeder and Lester (1989) discussed three approaches to problem-­ solving instruction: teaching about, for, and via problem solving. These ways of thinking about problem solving suggest a parallel set of perspectives for research on problem posing: problem posing as a cognitive activity, problem posing as a learning goal unto itself, and problem posing as an instructional approach (Cai & Leikin, 2020). Problem posing as a cognitive activity refers to the process of posing mathematical problems—that is, “the process by which, on the basis of mathematical experience, students construct personal interpretations of concrete situations and formulate them as meaningful mathematical problems” (Stoyanova & Ellerton, 1996, p. 518). When considering problem posing as a cognitive activity, researchers want to understand how students construct personal interpretations of specific situations and then pose problems based on those situations. In examining this process, students’ thinking—particularly their mathematical cognition around problem posing—can be revealed. Thus, from this perspective, researchers can both examine the specific cognitive processes involved in posing problems as well as probe the wider scope of mathematical thinking and understanding that students exhibit as they pose problems in a particular mathematical situation (Silver & Cai, 1996). Problem posing as a learning goal refers to the development of problem-posing skills. In this perspective, the goal is to help individuals, including students and teachers, to increase their capacity to pose high-quality mathematical problems. By engaging in problem-posing tasks in mathematics instruction, learners can develop their problem-posing skills, and the development of problem posing skills is, in this

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perspective, viewed as one of the goals of mathematics instruction. Indeed, problem posing can be used as an outcome measure for students’ learning (Cai et al., 2013). Problem posing as an instructional approach refers to the idea of teaching mathematics through problem posing. Although developing problem-posing skills may be a goal of this kind of instruction, this perspective emphasizes engaging students in problem-posing tasks and activities to help them achieve both cognitive and noncognitive learning goals beyond developing problem-posing skills. For example, a mathematical topic may be introduced with a problem situation that embodies key aspects of the topic, and students could explore the topic and begin developing their understanding of it by posing mathematical problems based on that problem situation (Chen & Cai, 2020). Or, problem posing might be incorporated in instruction as a way to help students develop their identities as explorers of mathematics and to foster positive dispositions toward mathematics (Silver, 1994, 1997). In describing these three perspectives, we intend to use them to help organize an analysis of the impact of Silver’s paper on multiple aspects of mathematics education. For example, in reviewing the 20 most cited research articles that have built on the themes of “On Mathematical Problem Posing,” we noted how each article has situated itself with respect to the three perspectives. Based on our coding, 14 of the articles treat problem posing as a cognitive activity, 10 treat it as a learning goal, and 4 treat it as an instructional approach. It is important to note that the three are not entirely disjoint perspectives and, sometimes, problem-posing spans more than one perspective in actual research and practice. For example, when teachers use problem posing to teach a particular mathematical topic to their students, it seems clear that problem posing is being conceptualized as an instructional approach. But at the same time that teachers are using problem posing in this way, they may also be trying to understand their students’ mathematical thinking as they engage with the problem-posing task. In that sense, the teachers are treating problem posing as a cognitive activity—an activity that can provide a window into the students’ mathematical thinking. Moreover, once the students have posed some problems, the teacher may engage them in a discussion of the posed problems. As part of this discussion, the teacher may compare the posed problems, examine the mathematics embedded in them, and discuss the level of sophistication and complexity demonstrated in them. In doing so, the teacher is highlighting the problem-posing skills of the students, perhaps to help sharpen those skills for the whole class. Here, problem posing is, in part, itself the learning goal. Thus, research and practice involving problem posing may involve multiple perspectives even when the primary focus is clearly connected to one of the three perspectives.

Silver’s Six Themes in “On Mathematical Problem Posing” In writing “On Mathematical Problem Posing,” Silver was addressing a research community that was largely just starting to consider in depth the construct of problem posing and its myriad links to different aspects of mathematical thinking and

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learning. Thus, in addition to defining problem posing as “both the generation of new problems and the re-formulation, of given problems” (p. 19), Silver discussed various connections between problem posing and other constructs important to the field. In doing so, he was laying down six themes through which the field could recognize and build on the potential significance of investigating questions about problem posing. Silver’s six themes are: problem posing as (1) a feature of creativity or exceptional mathematical ability, (2) a type of inquiry-oriented instruction, (3) a prominent feature of mathematical activity, (4) a means to improving students’ problem solving, (5) a window into students’ mathematical understanding, and (6) a means to improve students’ disposition toward mathematics. These six themes, which Silver identified based on the state of the art of research in problem posing at the time, represent connections between problem posing and an extraordinarily wide swath of significant areas of mathematics education. However, we may organize these themes within the conceptualization of problem posing described above: Theme 1 is primarily related to problem posing as a learning goal; Themes 3 and 5 are primarily related to problem posing as a cognitive activity; and Themes 2, 4, and 6 are primarily related to problem posing as an instructional approach. In this section, we briefly summarize these connections following the categories that Silver used to present them.

 roblem Posing as a Feature of Creativity or Exceptional P Mathematical Ability Silver discussed the connection between problem posing and both mathematical ability and creativity. He explained that students who have a high level of mathematical ability might pose more complex problems than students who have a lower ability level. Students’ creativity in posing problems is also related to their ability to pose novel problems. Although students with high mathematical ability tend to pose more complex problems, this does not mean that problem-posing-based instruction should be limited to teaching only “gifted” students. Rather, problem posing can make mathematics more accessible for all. For example, students can use problem posing as an opening to reformulate a problem to make it more accessible to their own experience. Moreover, because problem posing is connected to creativity and mathematical ability, it is worthwhile to consider using problem posing to measure and to further develop those capacities in students. Silver would elaborate on this connection in another widely cited paper (Silver, 1997). Indeed, Silver (1997) not only provided a historical account of using problem posing to measure creativity but also proposed a relationship between engaging students in problem posing and their development of creative fluency, flexibility, and novelty, noting that mathematical creativity “is closely related to deep, flexible knowledge in [mathematics]; is often associated with long periods of work and reflection rather than rapid, exceptional insight; and is susceptible to instructional

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and experiential influences” (Silver, 1997, p. 75). Creative fluency refers to the ability to generate many ideas in response to a prompt (i.e., to generate many problems for a given problem-posing task). Flexibility in problem posing refers to the number of different types of problems that are posed. And, creative novelty refers to how uncommon an idea is (i.e., how unlikely a particular problem is to be posed). Silver (1997) argued that problem posing provides opportunities to develop all three of these features of creativity.

Problem Posing as a Type of Inquiry-Oriented Instruction Silver (1994) next showed how problem posing satisfies three different features associated with inquiry-based instruction: (a) to help students construct general rules, theories, or principles that are already known, (b) to help students construct genuinely novel theories or principles, and (c) to teach students how to solve problems through the use of self-questioning and self-regulatory techniques and metacognitive skills. (p. 21)

Noting that there are many different programs that offer inquiry-based instruction, but that they are often found among the economic elite, Silver argued that problem posing is a means to provide high-quality inquiry-based instruction to all students without the need for expensive curricular materials. In particular, this means that problem posing can benefit historically marginalized groups who have not often had access to inquiry-based instructional programs. Although Silver expressed the desire for all students to have access to inquiry-based instruction through problem posing, he called for further investigation of the effects of inquiry-based instruction on students. He also noted the need for more research on the connections between problem posing and students’ ability to solve problems.

 roblem Posing as a Prominent Feature P of Mathematical Activity Here, Silver expanded on Polya’s idea that problem posing is a common activity among mathematicians. For millennia, mathematicians have posed and solved problems for themselves and others. Thus, engaging learners in the activity of problem posing reflects a potentially strong link to the discipline of mathematics. Silver noted that this aspect of problem posing aligns with recommendations in the National Council of Teachers of Mathematics (NCTM, 1989) Curriculum and Evaluation Standards for School Mathematics for students in Grades 9–12 to be able to formulate their own problems. Moreover, Silver argued that problem posing would also be relevant to mathematics instruction that has the goal of preparing students for the types of ill-structured mathematical problems they will encounter in

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the real world—readying them to be “intelligent users of mathematics in order to solve problems of importance or interest to them” (p. 23). In contrast to focusing on the role of problem posing in the discipline of mathematics, this argument is founded in research that has shown that students who solve ill-structured problems embedded in a natural context (as in applied mathematics) engage in the problem-posing behaviors of formulation and reformulation that they would not ordinarily use to solve well-structured school mathematics problems (Lesh, 1981; Lesh et al., 1983). Thus, problem posing is a feature of both mathematical activity as it arises in the discipline of mathematics and as it arises in how problem solvers attack real-world problems.

 roblem Posing as a Means to Improving Students’ P Problem Solving Silver noted that one of the most cited reasons for interest in problem posing as an element of curriculum and instruction is its potential for helping students become better problem solvers. This echoed Kilpatrick’s (1987) claim that problem posing can be a good indicator of how well students can solve problems. Silver cited several examples of this application of problem posing, such as using problem posing to help students analyze problems more completely, to improve students’ ability to apply their knowledge and skills to solve problems, and to help students develop powerful problem-solving schemas. However, at the time Silver was writing, the research basis for connecting problem posing and problem solving was still relatively thin. He described some studies that provided mixed results regarding the connection, sometimes demonstrating an association between problem-solving ability and the quality of posed problems and sometimes failing to find such connections, and he called for further research to clarify the nature of the link.

 roblem Posing as a Window Into Students’ P Mathematical Understanding Another characteristic of problem posing that Silver described is its potential for revealing students’ mathematical thinking by, for example, encouraging them to explicitly mathematize problem situations. Silver noted that researchers have used problem posing in this way to better understand students’ conceptual understanding of mathematics (e.g., Hart, 1981) and to make inferences about their mathematical knowledge (e.g., Ellerton, 1986). In describing the findings of research in this area, Silver remarked that problem posing provides a window into students’ mathematical understanding, but it also presents a reflection of the character of the school mathematics students have experienced. In particular, Silver noted with some

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dismay that problem-posing-based research has found that students often lack any concern for making connections between the mathematics in a problem and its realworld referents (e.g., treating a fraction to represent a number of people). In this way, problem posing may elucidate students’ attitudes and dispositions toward mathematics as well as their problem-solving abilities.

 roblem Posing as a Means to Improve Students’ Disposition P Toward Mathematics Silver further expanded on the connection between problem posing and students’ dispositions toward mathematics. He observed that problem posing gives students the opportunity to generate engaging problems stemming from their own interests and that can also engage their peers’ interest. Moreover, the process of posing problems itself can also stimulate students’ interest. Given that mathematics anxiety is a nontrivial problem for mathematics education, Silver noted that problem posing appears to have the capacity to make mathematics less intimidating (e.g., Moses et  al., 1990) and can help students develop a positive affect toward mathematics (e.g., Healy, 1993). Although there is little evidence that students dislike problem posing and would form more negative dispositions toward mathematics because of it, Silver did caution that there are some cases, such as with students who are already successful at mathematics taught in the “normal” way, when it might be plausible that students could reject the new form of instruction where expectations are not always as clear.

Advances in Research on Problem Posing All of the six themes raised by Silver in “On Mathematical Problem Posing” have been taken up and investigated by researchers over the past 30  years. Here, we examine some of the progress that has been made, focusing on the three perspectives of problem posing: as a cognitive activity, as a learning goal, and as an instructional approach. Our survey here is necessarily somewhat abbreviated, but we direct the reader to several other reviews of the field that have been conducted since 1994. For example, Cai et al. (2015) conducted a comprehensive review of 10 areas of research concerning mathematical problem posing. Other researchers have also raised new questions and directions for problem-posing research (e.g., Singer et al., 2013). In more recent years, Cai and Leikin (2020) reviewed research specifically focused on affect in mathematical problem posing, and Cai and Hwang (2020) reviewed research on teachers learning to teach mathematics through problem posing. In addition, Baumanns and Rott (2021) conducted a systematic review of 241

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articles on problem posing with a specific focus on problem-posing situations. Overall, although we build on these prior reviews, we orient our discussion of advances in problem-posing research by focusing on advances related to the three perspectives about problem posing.

Advances in Research on Problem Posing as a Cognitive Activity Mathematical Understanding and the Capacity to Pose Problems An early focus of research rooted in the perspective of problem posing as a cognitive activity was to examine the kinds of problems that students and teachers are able to pose (see Cai et al., 2015, for a review). An overarching finding of this body of research is that teachers (both in-service and preservice) and students at multiple educational levels are capable of posing valid mathematical problems based on given problem situations, although typically a relatively small proportion of the posed problems in these studies are not mathematically valid problems. Even students who have traditionally struggled with mathematics, such as students enrolled in college remedial or developmental mathematical courses, are quite capable of posing important and complex mathematical problems (Silber & Cai, 2021). Indeed, Silber and Cai found that students who received a course grade of D or F posed mathematical problems of similar complexity to those posed by students who received an A in the course. As Silver (1994) observed, researchers have also made use of problem posing in cognition-focused studies to gain insights into the participants’ mathematical understanding. For example, Kotsopoulos and Cordy (2009) used problem posing as a formative assessment with seventh-grade students to monitor their progress with respect to the learning objectives in their experiment. Researchers have also used problem posing to assess preservice teachers’ (Tichá & Hošpesová, 2013; Yao et al., 2021) and in-service teachers’ (Ma, 1999) conceptual understanding of fractions. By analyzing the participants’ posed problems, these researchers could identify conceptual flaws and confusion about the meaning of fraction division. Moreover, examining posed problems can provide insights into the poser’s mathematical understanding that might not be revealed by other measures. For example, in a recent study, Yao et al. (2021) used computation tasks, a drawing problem-solving task, and a problem-posing task to assess 345 preservice mathematics teachers’ understandings of fraction division. Engaging in the problem-posing task seems to have encouraged preservice teachers to explore the fraction division situation more conceptually, thus giving them an opportunity to exhibit their conceptual understanding. These findings suggest that the activity of problem posing is conducive to preservice teachers’ development of their conceptual understanding or their inclination to make use of that understanding in ways that might not be observed outside of problem posing. In summary, problem posing has been effectively used as a window into students’ and teachers’ mathematical understanding.

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Relationships Between Problem Posing and Problem Solving Other research that draws on the perspective of problem posing as a cognitive activity has, as Silver suggested, examined the relationship between problem posing and problem solving in detail. Indeed, Silver (1997) located a connection between creativity and problem posing within the interplay between problem posing and problem solving. For example, Silver and Cai (1996) found that middle school students who were good problem solvers (on open-ended problem-solving tasks) tended to pose more, and more complex, mathematical problems than did less successful problem solvers. Cai (1998) later strengthened the evidence for this link with a cross-national sample of US and Chinese students and a set of paired problem-­ posing and problem-solving tasks. Subsequently, Cai and Hwang (2002) compared the problem posing and solving performance of US and Chinese sixth-grade students, focusing on the details of their problem-solving strategies and specific characteristics of the posed problems. They found a link between the Chinese students’ problem solving and their problem posing that appeared to be based on an abstract pattern-formation strategy, although a similar link was not found for the US students. A follow-up analysis with US seventh-grade students (Hwang & Cai, 2003), who were more likely than the sixth-­ grade students to use abstract problem-solving strategies, found a parallel link. Students who used more abstract problem-solving strategies tended to be more likely to pose problems that extended beyond the given information in the problem situation. This connection was further explored by Kar et al. (2010), who verified the significant correlation between problem solving and problem posing for preservice elementary teachers. More recently, Xie and Masingila (2017) explored the interplay between problem solving, problem posing, and visualization. They employed a series of tasks that alternated between posing problems for a given situation, then solving a problem involving that situation, and finally posing more problems with the same situation. The preservice teacher participants made associations between the concepts contained in their posed problems and in their solutions to the given problems. Moreover, the alternating activities appeared to develop the participants’ visualization ability, and the problem-solving process encouraged superior problem posing. In summary, researchers have used various ways to examine the links between problem posing and problem solving and have shown that there is a robust relationship between problem posing and problem posing. Problem-Posing Task Characteristics: Problem Situations and Prompts Another area of problem-posing research in this category, although not one discussed in Silver’s (1994) paper, is research probing the effects that problem-posing task characteristics have on problem-posing cognition. These task characteristics include the nature of the problem situation and the problem-posing prompt (Cai, 2022; Cai et al., 2022). Research on the effects of task characteristics is particularly

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important for informing the design of effective problem-posing tasks for instructional use. At the level of instructional task design, it is essential to identify the characteristics of effective problem-posing tasks—tasks that engage students and elicit useful posed problems. However, given that research has shown differences in students’ problem-posing development across grade levels (Cai, 2003; Guo et al., 2021), the important task characteristics may vary depending on the target student population. Moreover, small changes in task characteristics can produce large differences in problem-posing cognition. For example, Yao et al. (2021) found that the inclusion of a conceptual cue in a fraction-division-based problem-posing situation strongly influenced preservice teachers’ responses to exhibit conceptual understanding. With respect to problem-posing situations, researchers have identified specific features of interest, such as the use of a real-world or a purely mathematical context, the kinds of representations used, and the openness of the situation. In a problem-­ posing task, the problem situation provides context and data that students may use in formulating their own problems. Problem situations may involve real-world or purely mathematical contexts. In either case, students may make connections between their own experience and the problem situation in the process of posing a problem. A purely mathematical context may provide an opportunity for students to explore the abstract features of the mathematics and to relate it to their existing knowledge. And, a real-life context may provide an opportunity for students to abstract mathematical features from the context. Problem situations may also represent the given information in multiple ways, including words, numbers, tables, and pictures, thus drawing on students’ ability to interpret and use those representations. English (1998) found that students’ problem posing was improved when the problem situation was an informal context, such as a picture, that allowed more freedom to explore, rather than a formal symbolic context. Ultimately, the choice of problem situation provides the potential for students’ posed problems to connect to the desired mathematical learning goal (e.g., a particular concept), and it shapes the kinds of problems that students are likely to pose. Indeed, because the process of posing problems encourages the poser to deeply examine the problem situation, the choice of problem situation defines the territory that the students will explore as they pose. The openness of the problem-posing task can influence how the students make progress toward their learning goals. Stoyanova and Ellerton (1996) characterized three levels of openness: free, semistructured, and structured. These levels are not meant as a value-laden hierarchical structure; however, different levels of openness can provide varying levels of entry into problem-posing tasks. Free problem-posing situations are designed to provide no restrictions on the problem structure or solution structure, regardless of the context. Having a free problem-posing situation can provide students with a low entry point while maintaining a high ceiling for the complexity and sophistication of their posed problems. Semistructured problem-­ posing situations are designed to draw on students’ prior knowledge through either the problem structure or the solution structure. Structured problem-posing

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situations are designed to draw on students’ prior knowledge by providing both a specific problem structure and a specific solution structure. Structured situations can be further broken down into routine and nonroutine problems depending on whether the students have access to the specific solution structure or not. Routine and nonroutine problem situations provide different experiences for students (Baumanns & Rott, 2021). Leung and Silver (1997) specifically investigated the effect of situation characteristics on preservice teachers’ problem posing. In particular, they examined the effect on the posed problems of the presence of specific numerical information in the problem situation. Leung and Silver found that problem-posing performance was better when the problem situation contained specific numerical information than when it did not. Zhang et al. (2022) replicated and extended the study by Leung and Silver, focusing on elementary school students’ problem posing. Zhang et al. (2022) conceptualized three stages of problem posing: understanding the task, constructing the problem, and expressing the problem. They again found that the provision of specific numerical information in the problem-posing situation was associated with better problem-posing performance but only in the understanding the task and constructing the problem stages. Students’ performance in the expressing the problem stage did not show a significant difference with respect to provision of specific numerical information. Other research has attended to the problem-posing prompt—how one is asked to pose a problem—including whether or not providing a sample posed problem influences problem-posing cognition (Cai et al., 2022). The prompt in a problem-posing task is what lets students know what they are expected to do. As with the problem situation, the prompt influences both the kinds of problems students pose and the kinds of mathematical ideas they are likely to connect to (Cai et al., 2022). The prompt may leave the problem-posing task very open, or it may constrain what the poser is able to do. In an analysis of problem-posing tasks in Chinese and US textbooks, Jiang and Cai (2014) classified five types of tasks, each of which involved a different prompt for the student: reformulation of a given problem by posing a similar problem, posing additional problems after encountering a given problem, posing a problem that can be solved with a given operation, posing a problem that supplements given information, and describing a situation to match a given mathematical representation. Some features of the prompt can have very direct influences on problem-posing cognition. For example, the inclusion of a sample posed problem in the prompt can make it significantly more likely that a student will pose a problem parallel to the sample problem (Jiang & Cai, 2015). Jiang and Cai (2015) observed that, in addition to sample problems promoting the posing of similar problems, such sample problems also appeared to jump start the posing of complex problems. Moreover, Silber and Cai (2017) found that including a desired answer to a posed problem appeared to encourage preservice teachers to more closely attend to the mathematical concepts in problem-posing tasks.

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Advances in Research on Problem Posing as a Learning Goal There have been at least three areas of advancement in research that treats problem posing as a learning goal in the past 30 years: (1) research involving problem posing as a measure of learning outcomes, (2) research involving training teachers to pose better problems, and (3) research involving training students to pose better problems. Problem Posing as a Measure of Learning Outcomes Because researchers have documented that students’ success in problem solving is associated with their problem-posing abilities, problem posing has been used as a measure of this particular type of learning outcome (Cai & Hwang, 2002; Cai et al., 2013; Silver & Cai, 1996). For example, Cai et al. (2013) made use of problem-­ posing tasks to measure the effects on students’ algebra learning of a Standards-­ based middle school mathematics curriculum compared with the effects of more traditional curricula. They confirmed the association between students’ abilities to solve and pose problems for both types of curricula. Moreover, they assessed various characteristics of students’ responses to determine that students whose posed problems exhibited positive characteristics (such as reflecting the linearity of a given graph in their posed problem or embedding their posed problems in real-life contexts) were also strong problem solvers. Given its generative qualities, problem posing has long been used as a valid measure of students’ creativity. Several researchers have provided empirical data measuring creativity using problem posing (e.g., Leikin, 2009; Leikin & Sriraman, 2017; Leung & Silver, 1997; Lubinski & Benbow, 2006; Singer et al., 2017; Van Harpen & Sriraman, 2013; Voica & Singer, 2012). In their introductory paper of a special issue, Singer et al. (2017) provided a historical overview of the development of research and practice in creativity and giftedness with specific attention to creativity and giftedness in mathematics. Given the history of using problem posing to measure general creativity, it seems natural to use mathematical problem posing to measure creativity specifically in mathematics (e.g., Lewis & Colonnese, 2021). Training Teachers to Pose Better Problems Silver’s problem-posing research began with teachers as subjects (Silver, 1994; Silver et al., 1996), with a focus on teachers themselves posing mathematical problems based on given problem situations. Subsequent problem-posing research related to teachers has been extended to train teachers to pose better problems. Because the mathematical tasks students engage with in class shape the learning opportunities that are made available, the choice of such tasks is a critical aspect of teachers’ work. That is, teachers can increase the quality of students’ learning opportunities through their choice of worthwhile mathematical tasks, even when such tasks are built on what would ordinarily be routine mathematical activities

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(Butts, 1980; Crespo, 2003). Indeed, Crespo (2003) investigated how preservice teachers can learn to pose better problems for their students, finding that their posing practice can grow from posing single-step, computational problems to richer, open-ended problems that were more cognitively demanding of the students. Moreover, Crespo and Sinclair (2008) found that preservice elementary teachers’ problem posing was improved by exploring the problem situation before posing and by developing criteria for the quality of posed problems. Subsequently, Crespo and Harper (2020) found that preservice secondary teachers could make use of a framework on collaboration and equity to increase the “groupworthiness” of the tasks they posed. In another study, Koichu and Kontorovich (2013) found that the successful prospective teachers in their study posed the most interesting problems when blending exploration and problem solving with their problem posing. In a problem-posing project involving elementary school mathematics teachers, Cai and his collaborators (Cai et al., 2020, Li et al., 2020) found that teachers can both learn to be better problem posers and develop greater confidence in their ability to use problem posing to teach mathematics. In fact, after attending a series of problem-posing workshops, in-service elementary school mathematics teachers’ problem-posing performance and views about teaching mathematics through problem posing exhibited positive changes. Participating teachers were able to engage successfully with multiple problem-posing tasks. Training Students to Be Better Problem Posers Similar to training teachers to pose better problems, another line of problem-posing research has looked directly at developing students’ problem-posing abilities. English (1998) found that students were able to improve the breadth and level of challenge of the problems they posed when they had experience solving such problems as well as when they were prompted by informal contexts such as pictures. An exploratory study of elementary school students in China (Cai & Hwang, 2021) also focused on helping students become better problem posers (while their teachers were simultaneously learning about teaching through problem posing). Initial data showed that after 1 year, students in the problem-posing group performed significantly better on problem-posing tasks (Cai & Hwang, 2021). Meanwhile, Lewis and Colonnese (2021) used a “Three-Act Task” structure, combining generative questioning and exploration with manipulatives to help students grow in their ability to pose problems.

 dvances in Research on Problem Posing A as an Instructional Approach Shifting the locus of problem generation from being entirely centered on the teacher to being inclusive of students represents a significant change in the instructional approach of many teachers (Silver, 1994). The advances of research on problem

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posing as an instructional approach or Problem-Posing-Based Learning (P-PBL) have included the major issues of (a) describing and analyzing what teaching mathematics through problem posing looks like in practice and (b) understanding how teachers can learn to teach mathematics through problem posing. What Teaching Mathematics Through Problem Posing Looks Like Because teaching mathematics through problem posing is not currently common practice, there is a need for examples of this kind of teaching, both from a research perspective (to better understand its key features and characteristics) and from a practice perspective (to have models from which teachers can gain knowledge of this technique). There have been some efforts to describe how problem posing might be employed in the classroom to foster creativity (see, e.g., Silver, 1997), and there have also been attempts to improve preservice teachers’ knowledge and beliefs about mathematics through problem posing (Toluk-Uçar, 2009). More recent research in this area has focused on working with teachers to document cases of lessons that focus on problem posing (e.g., Cai, 2022; Chen & Cai, 2020; Xu & Cai, 2020; Zhang & Cai, 2021). These cases provide insight into several features of problem-posing instruction, including (1) how to design instructional tasks and activities that involve problem posing, (2) how lessons can be designed based on problem-posing activities and the learning opportunities they provide, (3) how problem posing can be incorporated into existing lessons to improve the quality of the learning opportunities or to create new learning opportunities, (4) how problem posing can be used to help teachers understand their students’ mathematical thinking and understanding, and (5) how the problems students pose can be taken up by teachers to move the lesson toward its learning goals. It should be noted that, although each of these features represents a major instructional event, they are not mutually exclusive but instead interrelated. To provide a glimpse of what teaching mathematics through problem posing might look like, we briefly consider the case study of Ms. Xu (Xu & Cai, 2020), a young elementary mathematics teacher learning to teach mathematics through problem posing. This case study describes her adaptation of a lesson, “Using Letters to Represent More Complex Quantitative Relationships,” that is in the unit on solving simple equations in her fifth-grade textbook. Ms. Xu had taught this lesson before, but she felt that its focus was too procedural. Moreover, the students had difficulty solving the problems that involved complex quantitative relationships. As a warm-up in the problem-posing-based lesson, Ms. Xu guided the students to pose problems or to describe situations that could be solved or represented by the expressions 5 × 8 and 120 – 30. Then, she showed the students the equation 1200 – 3x = ? and asked them to pose related mathematics problems. That is, she wanted the students to pose problems that could be represented by and solved using the given equation. The students attempted to pose various problems, and Ms. Xu selected and discussed one of the problems posed by the students: A large bottle of beverage has 1200 g, and 3 small cups are poured. Each small cup holds x g. How many g of

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Fig. 1  Problem-posing task used in Ms. Xu’s classroom

beverage are left in the big bottle at this time? The next problem-posing task Ms. Xu used is related to Fig. 1. She presented the figure and asked the students to pose mathematical problems based on the figure. After having students share and discuss their problems, Ms. Xu presented an extension problem based on a slightly more complex pattern for the students to continue thinking about. She then concluded by summarizing and highlighting the main points in the lesson. Overall, in this case study, Ms. Xu provided multiple opportunities for her students to pose problems that allowed them to connect their existing understandings with the symbolic representations in equations. Readers may refer to Cai and Hwang (2022) for a more detailed discussion of this lesson. How Teachers Learn to Teach Mathematics Through Problem Posing There is a rapidly growing body of research investigating how teachers can learn to teach mathematics through problem posing. One common resource for teacher professional learning is the set of curriculum materials itself, such as textbooks and teachers’ guides. For example, teachers participating in lesson study in Japan engage in a preparatory research process called kyouzai kenkyuu in which they investigate a range of instructional materials that includes textbooks and other curriculum materials (Doig & Groves, 2011; Melville & Corey, 2021). Similarly, Chinese teachers often study their textbooks as part of their professional learning (Fan et al., 2004). However, the current state of problem-posing inclusion in many textbooks is minimal. For example, Cai and Jiang (2017) compared how problem-posing tasks were included in popular Chinese and US elementary mathematics textbooks. They concluded that none of the textbook series incorporated problem posing in a substantial way, with only a small proportion (less than 4%) of instructional tasks involving problem posing. Moreover, the distribution of problem-posing tasks was inconsistent across grade levels and mathematical content areas. Ultimately, this state of affairs means that, for now, curriculum materials are unlikely to be robust

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supports for teacher learning about problem posing. Research on how teachers can learn to teach through problem posing has therefore needed to look well beyond existing curriculum materials. Some researchers have investigated how sustained professional development experiences can help teachers to develop their capacity to teach mathematics through problem posing. For example, Cai and his colleagues (Cai et al., 2020; Cai & Hwang, 2020, 2021; Li et al., 2020; Zhang & Cai, 2021) have reported on a longitudinal study of problem-posing professional development in multiple school districts across China. The program included multiday summer workshops over a few years. In these workshops, teachers learned about problem posing, engaged in both posing problems themselves and thinking about the kinds of problems their students might pose, designed and revised lessons that incorporated problem-posing tasks, and studied problem-posing lessons designed by other teachers (Cai & Hwang, 2021). Initial findings from this research support the general observation that professional development focused on teaching through problem posing appears to help teachers feel more confident about using problem posing in their own classrooms, to improve their ability to predict how their students will respond to problem-posing tasks, and to foster their construction of more sophisticated understandings of the advantages and challenges of teaching through problem posing (Cai et al., 2020; Cai & Hwang, 2020). Indeed, the study by Li et al. (2020) suggests that any effort to integrate problem-posing instruction in school mathematics must attend to teachers’ views about the advantages of teaching through problem posing and especially their views of the challenges of teaching in this way. In fact, for researchers and designers of professional development experiences, it will be important to find ways to help teachers overcome the challenges that they view as part of teaching with problem posing. Moreover, a number of teachers who have participated in the professional development workshops have collaborated with researchers to author problem-­ posing teaching cases—case studies of implementations of problem posing in specific lessons like the case of Ms. Xu described above (Xu & Cai, 2020)—that serve to share what they have learned about teaching mathematics through problem posing with other teachers (Zhang & Cai, 2021). Cai and Hwang (2021) have also argued that, for problem posing to truly penetrate the enacted school mathematics curriculum, teachers will need to leverage their roles as curriculum redesigners to modify and adapt their curriculum materials with problem posing. Indeed, it makes sense to build problem-posing practice on top of existing common practices. This allows teachers to take advantage of “low-­ hanging fruit” such as transforming the problem-solving tasks in their curriculum materials into problem-posing tasks by simply removing some information from the problem or by deleting the question from the problem statement (e.g., in a word problem). Some curriculum materials already include instructional routines along these lines. For example, the Illustrative Mathematics curriculum includes an instructional routine called “Co-craft questions” that have the teacher present students with a problem situation and ask them to pose problems based on the situation (Illustrative Mathematics, n.d.).

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 onclusions and an Eye to the Future of Research C on Problem Posing It is safe to say that “On Mathematical Problem Posing” propelled forward research on problem posing. Silver both helped to lay a foundation for problem-posing research and pointed out key directions that problem-posing research could explore. It has been encouraging to see in our brief review of the literature that there have been marked advances in problem-posing research since Silver’s (1994) paper, and we anticipate that there will continue to be advances for years to come. Our anticipation is supported by the observation that, internationally, there are still so few problem-posing activities in mathematics textbooks even though curriculum standards have long called for giving students opportunities to pose mathematical problems (Cai & Jiang, 2017). There is much room for continued progress to incorporate problem posing in students’ mathematical experiences. Over the years, a number of researchers have continued to synthesize research on problem posing and suggest future directions for research (e.g., Cai et al., 2015; Cai & Hwang, 2020; Cai & Leikin, 2020; Silver, 2013; Singer et al., 2013). We do not intend to repeat here the discussions provided in these prior syntheses. Instead, we narrow our gaze to focus on three specific areas of research on mathematical problem posing that have built on these prior writings because we believe that, among the many areas of research related to problem posing, these areas are ripe for progress that could significantly move the entire field forward.

Affective and Cognitive Processes of Problem Posing The first specific area of research we wish to highlight again concerns the processes of problem posing. However, we believe it is important to expand beyond cognitive processes to attend to both affective and cognitive processes of problem posing (Cai & Leikin, 2020). It is clear that there are important affective aspects of problem-­ posing processes that require attention. For example, Akay and Boz (2010) established that problem posing can have a positive influence on self-efficacy in mathematical understanding. In problem-solving research, the field of mathematics education has a well-developed knowledge base about the pertinent cognitive and affective processes (e.g., McLeod & Adams, 1989; Schoenfeld, 1985; Silver, 1985). However, in problem-posing research, even though researchers have identified general strategies or heuristics students may use to pose problems (e.g., Brown & Walter, 1983; Cai & Cifarelli, 2005; Christou et al., 2005; Cifarelli & Cai, 2005; English, 1998; Koichu, 2020; Leung & Silver, 1997; Rott et al., 2021; Silver & Cai, 1996) and have begun to document the multiple affective processes of mathematical problem posing (collectively, what Schindler & Bakker, 2020, describe as the “affective field”), there is not yet a general problem-posing analogue to

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well-­established frameworks for problem solving (Garofalo & Lester, 1985; Polya, 1957; Schoenfeld, 1985). Some researchers (Cai et  al., 2022) have called for research on affective and cognitive processes of problem posing with a particular focus on task variables, processes, and products. They specifically highlight the need for research on the following questions: How do different characteristics of problem situations and prompts influence subjects’ problem posing? What affective and cognitive processes are involved in subjects’ problem posing based on problem situations and prompts with different characteristics? Moreover, they recommend that these research questions be explored at different levels, such as the individual level, the small-group level, and the classroom level.

Teaching Mathematics Through Problem Posing One of the major goals of educational research is to improve students’ learning. Instructional improvement can happen at various levels, such as national, state, district, or school levels. However, one of the most effective arenas for instructional improvement might be at the level of the lesson. At this level, there appears to be strong potential for problem posing to provide more high-quality learning opportunities for students through integrating problem posing into the lesson. If teachers do incorporate worthwhile problem-posing tasks into their mathematics lessons, at least two questions arise: (1) How can students’ responses to the problem-posing tasks provide their teachers with useful insights about the students’ mathematical thinking? and (2) How can teachers effectively make use of those insights along with the students’ posed problems in the lesson? Certainly, the more information that teachers obtain about how students are thinking and what they know, the more they will be able to use that information to create effective learning opportunities for all of their students. As noted above, problem-posing researchers have used problem-posing tasks to understand and assess students’ thinking (e.g., Cai & Hwang, 2002; English, 1998; Silver & Cai, 1996). For example, in Kotsopoulos and Cordy’s (2009) study with seventh-grade students, the researchers were able to use the students’ problem-posing journals to gain the needed window into their students’ understanding of the learning objectives. And, once teachers have gained information about students’ mathematical understandings through problem posing, how are they to use the students’ posed problems effectively to shape classroom discourse and help the students learn? Students may pose different kinds of problems, including ones that are irrelevant to the mathematical goals of the lesson and ones that are so difficult to solve that, even if they are relevant, attending to them could derail the lesson. Teachers must make important instructional decisions about which problems to discuss in the moment, how to discuss those problems, and which problems to save for later. A key issue here is deciding whether a posed problem has the potential to help the class make progress toward the specific learning goals of the lesson. Zhang and Cai (2021) analyzed 22

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problem-posing teaching cases to understand how teachers in those cases made use of student-posed problems. They found a consistent pattern of teachers making intentional decisions about which problems to use based on the relevance of the problems to their instructional goals. However, this requires a great deal from the teacher—to know which problems can highlight the key mathematical concept at hand—that is, to see the potential mathematics in the posed problems. Although some of this thinking can be done in advance when planning the lesson and anticipating the kinds of problems students might pose, the field does not yet have a comprehensive understanding of how teachers can handle student-posed problems. Further analysis of authentic problem-posing lessons, whether through the medium of teaching cases or through videotaped exemplars, may be one route through which the field addresses this area (Cai, 2022).

 eachers’ Professional Learning to Teach Mathematics Through T Problem Posing Because teaching through problem posing requires significant instructional decision making, both in the planning and implementation of problem-posing lessons, teachers will need adequate preparation. As it stands, although teachers are quite capable of posing problems themselves, learning how to engage their students in productive problem posing that serves the learning goals of a mathematics lesson will require significant teacher professional learning. Teacher learning involves both developing teachers’ knowledge of problem posing and how it may be used in mathematics classrooms and developing productive beliefs about problem posing. Ultimately, the objective is to parlay teachers’ learning into substantive changes in their classroom instruction to include problem posing with the goal of improving students’ learning. Research has consistently shown the importance of teachers’ beliefs on teachers’ professional learning and classroom instruction (Philipp, 2007; Richardson, 1996; Thompson, 1992), and this is no less true for learning to teach mathematics through problem posing. In particular, teachers’ beliefs about the advantages and challenges of problem posing as an instructional tool will shape their adoption and implementation (Cai et al., 2020). Further research is needed to articulate the specifics of what knowledge and beliefs teachers should develop to effectively use problem posing to teach mathematics as well as what kinds of professional learning experiences best support that development. One promising route for developing and sharing this kind of professional knowledge has already been mentioned above. Problem-posing teaching cases—carefully documented cases of teaching mathematics through problem posing—may be an effective, tangible artifact for both supporting professional learning and disseminating what has been learned (Cai et  al., 2023). In particular, this is possible when teachers themselves play a major role in the development and writing of such cases. Case-based learning has been used as a powerful tool for teacher development (e.g.,

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Smith & Friel, 2015; Stein et al., 2000). For example, in China, Cai and his colleagues (Cai et al., 2021; Zhang & Cai, 2021) have been engaged in problem-posing research in which elementary mathematics teachers who are learning to teach mathematics through problem posing engage in professional development workshops and collaborate with teacher researchers and teams within their schools to develop problem-posing teaching cases. The cases these teachers are developing are more than simply lesson plans or a report on the implementation of a lesson (Zhang & Cai, 2021). They include multiple components that capture a snapshot of the teaching as well as the theory and experience underlying the instructional decisions therein. The first component explains the mathematical learning goals for the lesson, including a description of what it means to understand the content topic and a mathematical analysis that situates the content within the mathematical framework of the curriculum. The second component is a cognitive analysis of the learning goals and content, focusing on potential difficulties for students and the prior understanding and knowledge students need to succeed in the lesson. The third component is a description of the major components of the lesson, focusing on each instructional task, along with a rationale for each problem-posing task that explains the purpose of the task and what students should take away from it and details on implementation, including potential student responses (e.g., posed problems), ideas about how the teacher could deal with those responses, and specific reflections from experiences with implementing the lesson. The teaching case concludes with a reflection that summarizes how the lesson fostered students’ mathematical understanding and provides some guidance to other teachers about what to pay attention to when teaching a lesson of this type. The teaching cases (and all four components) can be iteratively and continuously improved as the lessons are repeatedly implemented so that they embody the best of what the teachers and researchers learn as they work toward refining lessons and units. Even if resources such as teaching cases can support significant teacher learning about teaching through problem posing, the research on teacher professional learning has not produced robust evidence that teacher learning generally has the desired effect on students’ learning. In fact, researchers have reviewed over 1300 studies that addressed the effect of professional development on student learning outcomes, finding it challenging to detect the translation of professional learning into student achievement gains (Guskey & Yoon, 2009; Yoon et  al., 2007). Some researchers have critiqued the rigor of design in studies examining the relationship between teacher professional learning and improvements in student learning. For example, Yoon et al. (2007) found that only nine investigations of the over 1300 studies they reviewed met the What Works Clearinghouse (WWC) standards for rigor in design. Indeed, one reason the complex relationship between teacher professional learning and improvements in student learning is challenging to study is because the relationship is embedded partly within the phenomenon of classroom instruction, which is itself a dynamic, complex phenomenon. In the case of teaching through problem posing, we highlight the importance of attending to intentional changes in classroom instruction that can mediate the relationship between teachers’

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professional learning and students’ achievement gains. In particular, we propose not only that the development and refinement of teaching cases can serve as a mechanism for teacher learning but also that the teaching cases themselves can act as tangible, dynamic objects that store information on quality lessons (and their continuous improvement) that can be shared within and across communities. In this sense, teachers engage in professional learning as they contribute to  and modify teaching cases, and the teaching cases store and make accessible sufficient detail about an improved lesson to convey what is needed to support the enactment and further improvement of the lesson. This level of detailed information must include the rationales for the changes and improvements that make problem posing effective in the lesson. Currently, we are engaged in a project to build a system of professional learning that includes the development and improvement of problem-posing teaching cases with middle school mathematics teachers (Cai et al., 2021). Our goal is to longitudinally investigate the promise of supporting teachers to teach through problem posing for enriching teachers’ instructional practice and improving students’ learning. It is our hope that more researchers will engage in such hands-on work to improve instruction and foster students’ deep learning of mathematics.

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McLeod, D. B., & Adams, V. M. (1989). Affect and mathematical problem solving: A new perspective. Lawrence Erlbaum Associates. Melville, M., & Corey, D.  L. (2021). Kyouzaikenkyuu: An exploration of Japanese mathematics teachers’ daily planning practices. Journal of Mathematics Teacher Education. https://doi. org/10.1007/s10857-­021-­09493-­5 Moses, B., Bjork, E., & Goldenberg, E.  P. (1990). Beyond problem solving: Problem posing. In T.  J. Cooney & C.  R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s (pp. 82–91). National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Journal for Research in Mathematics Education. https://doi. org/10.2307/749544 Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–315). Information Age. Polya, G. (1957). How to solve it: A new aspect of mathematical method. Princeton University Press. Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula (Ed.), Handbook of research on teacher education (2nd ed., pp. 102–119). Macmillan. Rott, B., Specht, B., & Knipping, C. (2021). A descriptive phase model of problem-solving processes. ZDM. https://doi.org/10.1007/s11858-­021-­01244-­3 Schindler, M., & Bakker, A. (2020). Affective field during collaborative problem posing and problem solving: A case study. Educational Studies in Mathematics, 105, 303–324. https://doi. org/10.1007/s10649-­020-­09973-­0 Schoenfeld, A. H. (1985). Making sense of “out loud” problem-solving protocols. The Journal of Mathematical Behavior, 4(2), 171–191. Schroeder, T. L., & Lester, F. K. (1989). Developing understanding in mathematics via problem solving. New Directions for Elementary School Mathematics, 31, 42. Silber, S., & Cai, J. (2017). Pre-service teachers’ free and structured mathematical problem posing. International Journal of Mathematical Education in Science and Technology, 48(2), 163–184. Silber, S., & Cai, J. (2021). Exploring underprepared undergraduate students’ mathematical problem posing. ZDM–Mathematics Education, 53, 1–13. Silver, E.  A. (1985). Research on teaching mathematical problem solving: Some underrepresented themes and needed directions. In Teaching and learning mathematical problem solving: Multiple research perspectives. Lawrence Erlbaum Associates. Silver, E.  A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM–Mathematics Education, 97(3), 75–80. Silver, E.  A. (Ed.). (2013). Teaching and learning mathematical problem solving: Multiple research perspectives. Routledge. Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27(5), 521–539. Silver, E.  A., Mamona-Downs, J., Leung, S.  S., & Kenney, P.  A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27, 293–309. Singer, F. M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7. Singer, F. M., Sheffield, L. J., & Leikin, R. (2017). Advancements in research on creativity and giftedness in mathematics education: Introduction to the special issue. ZDM, 49(1), 5–12. Smith, M. S., & Friel, S. N. (Eds.). (2015). Cases in mathematics teacher education: Tools for developing knowledge needed for teaching. Information Age. Stanic, G.  M. A., & Kilpatrick, J. (1988). Historical perspectives on problem solving in mathematics curriculum. In R. I. Charles & E. A. Silver (Eds.), Research agenda for mathematics education: The teaching and assessing of mathematical problem solving (pp. 1–22). National Council of Teachers of Mathematics. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-­ based mathematics instruction: A casebook for professional development. Teachers College Press.

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Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing in school mathematics. In P.  C. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Mathematics Education Research Group of Australasia. Thompson, A.  G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D.  A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). Macmillan. Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143. Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166–175. Van Harpen, X. Y., & Sriraman, B. (2013). Creativity and mathematical problem posing: An analysis of high school students’ mathematical problem posing in China and the USA. Educational Studies in Mathematics, 82(2), 201–221. Voica, C., & Singer, F. M. (2012). Creative contexts as ways to strengthen mathematics learning. Procedia-Social and Behavioral Sciences, 33, 538–542. Williams, S. R., & Leatham, K. R. (2017). Journal quality in mathematics education. Journal for Research in Mathematics Education, 48(4), 369–396. Xie, J., & Masingila, J. O. (2017). Examining interactions between problem posing and problem solving with prospective primary teachers: A case of using fractions. Educational Studies in Mathematics, 96(1), 101–118. Xu, H., & Cai, J. (2020). Fostering the development of students’ thinking in teaching “Using letters to represent quantitative relationship” through collaborative learning and problem posing. Elementary Mathematics Teachers, 2020(2), 14–16. [in Chinese]. Yao, Y., Hwang, S., & Cai, J. (2021). Preservice teachers’ mathematical understanding exhibited in problem posing and problem solving. ZDM–Mathematics Education, 53, 1–13. Yoon, K. S., Duncan, T., Lee, S. W.-Y., Scarloss, B., & Shapley, K. (2007). Reviewing the evidence on how teacher professional development affects student achievement (Issues & Answers Report, REL 2007–No. 033). U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance, Regional Educational Laboratory Southwest. Zhang, H., & Cai, J. (2021). Teaching mathematics through problem posing: Insights from an analysis of teaching cases. ZDM–Mathematics Education, 53, 1–13. Zhang, L., Cai, J., Song, N., Zhang, H., Chen, T., Zhang, Z., & Guo, F. (2022). Mathematical problem posing of elementary school students: The impact of task format and its relationship to problem solving. ZDM–Mathematics Education, 54, 497–512.

Professional Learning Tasks Through Job-­Embedded Teacher Professional Development Hala Ghousseini and Elham Kazemi

Mathematics instruction that is both socially and intellectually ambitious requires teachers to support students’ engagement in tasks that are cognitively demanding and authentic to students’ own experiences (Silver, 1985; Stein et  al., 2009). Through his scholarship, Edward Silver contributed to our understanding of the nature and import of students solving complex tasks. He argued that inquiry-­ oriented mathematics instruction that regularly uses cognitively demanding problem-­solving and problem-posing tasks and activities can enhance students’ creative approaches to mathematics and lead to the development of student understanding and problem solving and reasoning (Silver, 1997; Silver et  al., 2009). Also central to Silver’s contributions to understanding inquiry-oriented mathematics instruction is his portrayal of its cognitive and pedagogical demands for teachers. His studies of teacher professional development highlight, for example, how teachers can deepen their understanding of mathematics through opportunities to make connections “within the mathematics, and between the mathematics and matters of pedagogy and student thinking” (Silver et al., 2007, p. 275). They also reveal teachers’ perceived obstacles to their use of multiple solutions in the classroom, which include limitations of instructional time and the demand of making choices between multiple solutions. Far from an obstacle to learning, Silver’s scholarship portrays the idea of “demand,” both cognitive and pedagogical, as an opportunity for learning: Cognitively demanding tasks provide a window into students’ sensemaking and practices and the nature of learning activities that support this sensemaking. H. Ghousseini (*) University of Wisconsin-Madison, Madison, WI, USA e-mail: [email protected] E. Kazemi University of Washington, Seattle, WA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Cai et al. (eds.), Research Studies on Learning and Teaching of Mathematics, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-35459-5_2

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Similarly, Silver argued, teachers would benefit from opportunities to consider pedagogical demands—challenges and opportunities—associated with facilitating student engagement with cognitively demanding tasks (Silver et al., 2007). Ed Silver explained the nature of cognitive and pedagogical demand in teacher learning, particularly, in his description of the role of Professional Learning Tasks (PLTs) in teacher professional development. He posited that PLTs can be designed to prompt teachers to consider the challenging aspects of instructional situations and consequently invest their knowledge in the cognitive and pedagogical demands of supporting student learning. PLTs, he argued, are particularly positioned to bring teacher learning closer to practice because they emulate the complexity of the work of teaching within the particularities of an instructional context (Silver, 2009). In this chapter, we draw on Silver’s ideas of how PLTs can engage teachers in the cognitive and pedagogical demands of practice by focusing on a model of practice-­ based professional development that contextualizes PLTs inside joint enactments of teaching. Joint enactments of teaching engage teachers as a collective in knowing and doing the work of teaching while students are present—in contrast to only working jointly on a lesson before or after its enactment. Joint enactments are characterized by pauses in which participants narrate what they are thinking or considering and by invitations for others’ pedagogical reasoning about problems of practice. Using an example from our current research, we elaborate the affordances of situating PLTs inside joint enactments of practice for teachers’ collaborative sensemaking about responsive mathematics teaching.

Learning Complex Performance as a Social Practice Studies of inquiry-oriented instruction reveal its complex nature where teachers not only support students’ engagement with disciplinary ideas in the context of real-­ world situations (Barron & Darling-Hammond, 2008; Lampert, 1990) but also manage students’ collaborative interactions and the way they negotiate authority, identity, and positioning in the classroom (Hand, 2012; Langer-Osuna et al., 2020). This form of ambitious teaching rests on commitments and approaches to learning to teach that reflect the complexity of this work. It is also a type of performance that is not simply devised in the moment but learned as a social practice, which Silver and his colleagues (Stein et al., 1999) described as a shift away from learning techniques toward learning opportunities that bring teachers together in the “reexamination, ongoing experimentation, and critical reflection that are required to develop the beliefs, knowledge and habits of practice” (p. 239). An extensive body of literature suggests how transformative teacher learning in mathematics can be supported through sustained, connected experiences and contexts in which teachers have collective opportunities to make sense of new knowledge and instructional practice (e.g., Ball & Cohen, 1999; Lefstein et al., 2020; Silver et al., 2009). Transformative learning is not an individual, isolated process; it is propelled by practitioners interacting with one another around purposive activities using a shared repertoire of

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resources they call upon to create “mutual, collective, inter-knowing” (Yanow, 2001, p. 59). Viewed from a social practice perspective, learning complex performance can best develop when participants engage in mutual activities where they experiment with new possibilities for action and arrangements for participation in practice (Dreier, 2008). Change and learning happen through the adjustments participants make to their contributions to shared activities. Practice-based approaches to teacher learning have flourished over the past two decades based on this vision of professional learning as a social practice. Practice-­ based models of teacher professional development aim to position teachers as agents and active collaborative contributors to their learning through particular social and organizational contexts (Lampert et al., 2015). Ed Silver’s research on teacher learning helps articulate the spirit of this approach where he maintains that practice-­ based models can “coordinate and link different facets of teacher knowledge to each other and to the settings in which the knowledge is used” (Silver et al., 2007, p. 261). He also clarified the way Professional Learning Tasks (PLTs) can be designed around specific goals for teacher learning in ways that build on and connect teachers’ prior knowledge and experiences. Silver asserted that PLTs can provide a context for teachers to grapple with the complexity of teaching where they “treat a particular situation as problematic. Helping teachers discern the problematic component of the task and consequently own it and see it as meaningful is a critical aspect of the work that goes in the design and facilitation of PLTs” (Silver, 2009, p. 245). Through inquiry into the problematic components of PLTs, teachers can grapple with the pedagogical demand of instructional situations and leverage their collective knowledge. Recent history has seen a range of rich innovations in practice-based professional development models centered around PLTs. These models include the use of student work, classroom video, and narrative cases. Samples of student work have been used in PLTs aimed at developing teachers’ deeper understanding of student thinking (Kazemi & Franke, 2004) and mathematics (Suzuka et al., 2009). Video footage of classroom instruction has also been purposefully selected and used to design PLTs that invite teachers’ analytical thinking (Borko et  al., 2011). In his scholarly work, Ed Silver also demonstrated how narrative cases of teaching can be used as anchors for PLTs to stimulate teachers’ reflection and analysis aimed at one or more aspects of teaching practice (Silver et al., 2007, 2009). Narrative cases, he argued, provide a rich context for PLTs through which teachers can experience the relationships among mathematics, pedagogy, and student learning in a particular mathematics classroom while illustrating core issues and persistent challenges embedded in the use of cognitively demanding tasks. Through case analysis and discussion teachers then can build collective proficiency in learning to treat a lesson as a unit of analysis, making claims about the contingent aspects of teaching based on evidence rather than opinion (Silver et al., 2006).

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The Role of PLTs as Levers to Teachers’ Sensemaking Silver (2009) maintained that the highly contextual nature of [PLTs] is intended to allow teachers to propose, debate, and consider solutions to pedagogical dilemmas and explore pedagogical possibilities as they move back and forth between past and current teaching experiences and the activity space of the professional development experience. (p. 245)

Silver’s representation in Fig. 1 of the role played by PLTs in bridging teachers’ lived experiences in their classrooms and their professional development experiences highlights the promise and strength of practice-based approaches in creating spaces for teachers to connect learning about teaching with their own practice. In fact, Silver et al. (2005) presented evidence of meaningful shifts in teachers’ practice motivated by insights that teachers gleaned from the analysis and discussion of PLTs such as video and narrative cases. Similarly, Kazemi and Franke (2004) found that teachers who engaged in PLTs that deepened their understanding of their own students’ mathematical thinking were able to appreciate the details of students’ ideas and envision instructional trajectories to support them. Silver’s representation, however, also points to a tension in common practice-­ based approaches: The PLTs, although connecting the domains of teacher practice and practice-based professional development, are implemented in contexts that are outside teachers’ actual enactment of practice where they make moment-to-moment decision making in response to students (e.g., drawing on samples of student work rather than responding to students in the moment). Ball and Cohen (1999) alluded to this tension in their argument that practice-based learning requires experience with tasks that “must be immediate enough to be compelling and vivid” while at the same time “sufficiently distanced to be open to careful scrutiny, unpacking, reconstruction, and the like.” (p.  12). Ball and Cohen explained that situating PLTs in practice, in the immediacy of teachers’ decisions, could interfere with opportunities to learn and may distract from the teaching that is underway. However, distancing teachers’ professional learning experiences from actual enactment of practice also leaves teachers with the task of individually connecting their learning to their own teaching and grappling independently with the complexities that may arise. This stands at odds, in our view, with fundamental conceptions of professional learning as a social practice, which is grounded in the idea of shared goals and collective responsibility (James et al., 2007; Lee & Louis, 2019). Learning complex practice is not accomplished only through collaboration around artifacts of practice—planning lessons prior to enactment, debriefing a classroom video—but also through the

Fig. 1  The role of professional learning tasks (Silver, 2009)

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Fig. 2 Reconceptualizing the role of PLTs in relation to teacher practice and professional development

“mutual, collective, inter-knowing” that Yanow (2001) referred to wherein teachers build on each other’s reading of a situation and their interpretations of students’ sensemaking and participation. In this respect, we argue that PLTs in practice-based teacher education can be conceptualized to be more than bridges that link teachers’ practice to professional development experiences they engage with outside their actual teaching. We posit that PLTs can also act as contexts for their collaborative sensemaking in practice— inside shared enactments of teaching that are part of the professional development. This proposal would shift Silver’s (2009) representation to show how PLTs can be at the center of practice-based professional development that is embedded in teachers’ own classrooms (see Fig. 2). The PLT in this configuration becomes nested both in practice and collaborative professional development rather than the bridge between them as shown in Fig. 2. A PLT in this model can be a space of inquiry for teachers while teaching, geared toward the students in this particular class, their current understandings, and who they are as a group of learners. We elaborate how this can operate in what follows.

How May PLTs Act as Spaces of Inquiry While Teaching? Through our work, first with preservice teachers and then with practicing teachers, we have experimented with embedding PLTs in the enactment of practice where in-the-moment teaching decisions can be available for teachers’ collective consideration. Our work with preservice teachers led us to create the interactive pedagogy of rehearsals (Lampert et al., 2013) where novices collectively plan and approximate the enactment of an activity in front of their peers and then implement it with students in actual classroom settings. In the context of rehearsals, preservice teachers could (re)try, reconsider, and receive feedback from other participants about aspects of practice they had studied and prepared for. Teacher educators and rehearsing teachers would also bring into the rehearsal particular aspects of the students and classrooms where they would subsequently teach the activity. They could pause the

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rehearsal to open spaces for deliberation and discussion about pertinent problems of practice or pedagogical considerations that could support student learning and participation (Ghousseini et al., 2015; Kazemi et al., 2016). We further developed this work with practicing teachers through professional learning experiences called Learning Labs that engage teachers, alongside their colleagues, in ambitious teaching inside the complex social and material ecology of teachers’ classrooms (Kazemi et  al., 2018). Learning Labs are designed around learning cycles through which teachers investigate and enact instructional activities that are deliberately planned to be responsive to the curriculum and students in classroom contexts. Typically, the teachers, along with university- and school-based teacher educators, go through four phases of inquiry (see Fig. 3): (1) new learning, (2) coplanning, (3) coenactment, and (4) debriefing. In the new learning phase, they start by participating in activities that engage them in disciplinary content, aspects of ambitious teaching, or children’s thinking. The team then coplans an instructional activity to be enacted in one of their classrooms. During the planning, the team identifies content and participation goals for students, potential student responses, and strategies for supporting student engagement. The group also anticipates how they would work together as a collective during the lesson—where different team members would sit, who would lead the lesson at different times, and when team members would pause instruction to reflect or ask a question about something they noticed or even engage directly with students. Participants might even step into the teacher role to test the suggested course of action and then collectively assess its consequences (Ghousseini et al., 2022; Kazemi et al., 2021). The cycle ends with teachers debriefing the enacted lesson. Those familiar with Lesson Study will note its similarity to Learning Labs, especially the collaborative nature of coplanning and debriefing the lesson. However, Learning Labs differ from Lesson Study (see, for example, Lewis et  al., 2009) in important ways. The lesson that is planned through Learning Labs is not developed through extensive planning as is the

Fig. 3  The Learning Labs cycle as a PLT

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research lesson in Lesson Study. Rather, the Learning Lab lesson is planned within a relatively short period of time, typically 30–45 min, and is rather loose in its structure to enable participants to make modifications to it during enactment. Unlike Lesson Study, during enactment, participants can pause instruction to confer with one another and change the direction of the lesson. Another important difference between Lesson Study and Learning Labs is that teachers can share or take turns facilitating the planned lesson. In both rehearsals and Learning Labs, participants know that they can draw on shared practices and tools during lesson enactments for the purpose of collective experimentation while teaching. In Learning Labs with in-service teachers, we work on an overarching PLT across the full cycle of the four phases in Fig. 3. The task is for teachers to understand and prepare to enact the process of mathematical argumentation and to learn to support broad participation from students in this process. This main PLT is supported by a number of smaller activities that are built around each phase of the Learning Lab. Consistent with Silver’s (2009) conceptualization of PLTs, the overarching task is launched through purposive learning opportunities for teachers that involve the examination of mathematics problems, video, or narrative cases of teaching anchored in a mathematical focus. The planning phase focuses on an instructional activity where teachers identify the mathematical goals and consider possible student strategies, generating in the process questions to elicit students’ ideas and represent their thinking. The enactment or collaborative teaching phase involves inquiry into practice where teachers both do the work of teaching in response to students’ performance and have opportunities to pause to consider problematic aspects of an instructional situation. Teachers and teacher educators collectively contribute to discerning the “problematic component of the task and consequently own it and see it as meaningful” (Silver, 2009, p. 245). During the debriefing phase, teachers further elaborate on their perception of the cognitive and pedagogical demands of the lesson. Accordingly, the PLT that is stretched across the four phases of the cycle affords different opportunities to learn for teachers as they consider how to facilitate cognitively and socially demanding mathematical problems before, during, and after teaching. Working collaboratively on discerning and managing the complexity of teaching has especially been missing from the literature on teacher collaborative learning opportunities. Throughout our studies of what happens across these cycles of investigation and enactment into teaching, we have repeatedly seen teachers developing contextualized and nuanced insights into how to plan for, facilitate, and reflect on cognitively demanding tasks that center students’ sensemaking. Next, we turn to a brief example to show how a PLT stretched across the Learning Lab cycle represents the way PLT can be embedded both in practice and collaborative professional development. We highlight the affordances of such a model through the deliberate connected activities in each phase of the cycle that together address the goals of the overarching PLT.

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Case Example In the case we present here, we illustrate how we conceptualize the PLT in our professional development model as the opportunities for a group of educators to further their understanding of supporting upper elementary students in the practice of mathematical argumentation. As stated above, the PLT is stretched across a full cycle of the Learning Labs professional experience: new learning, collaborative planning, collaborative teaching or enactment, and debriefing.

 ew Learning and Planning: Engaging with Complexities N Before Teaching The literature on argumentation repeatedly asserts that argumentation is more than an activity—it rests on establishing a classroom culture for argumentation over time through intentional norm setting, activity selection, and guidance (e.g., Knudsen et al., 2018). As with all ambitious teaching, developing a culture for mathematical argumentation is complex work for teachers and creates many intellectual and pedagogical demands because teachers have to learn how to intentionally (1) choose and structure mathematical problems or routines, (2) develop norms for what counts as acceptable arguments, and (3) provide language supports and use discourse structures to engage students in the practices of argumentation (Krummheuer, 2007; Makar et al., 2015). The upper elementary teachers at Hilltop Elementary School,1 together with school- and university-based instructional coaches and their principal, were exploring these demands in the context of a 2-year professional learning2 focus on the practice of mathematical argumentation (Kazemi et al., 2021). One aspect of the group’s work was problem selection and problem posing. In the case we illustrate here, teachers were experimenting with a growing patterns problem which is well suited for supporting students’ functional thinking (see, for example, National Council of Teachers of Mathematics [NCTM], 2020). During this focal Learning Lab, the group’s learning goals entailed exploring the affordances of this growing patterns problem for mathematical argumentation and inquiring together into how to launch a particular growing pattern problem (see Fig. 4) so that the intellectual work of discerning its structure would rest with students. The focal Learning Lab we describe below was at the end of the first year of work on mathematical argumentation. It was the first day that the group explored whether and how the growing patterns problem could help them with their overall goals of cultivating students’ facility with mathematical argumentation. The group  All names but university-based teacher educators are pseudonyms.  The school had been using Learning Labs as an aspect of their professional learning support for 8 years at the time of data collection. 1 2

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Fig. 4  The growing pattern problem used by teachers in the Learning Lab as drawn on chart paper

of educators was interested in supporting students to observe and notice the structure of a growing pattern and use their observations to make claims about the arrangement of a future term in the growing pattern. The group gathered for this particular Learning Lab consisted of two university-based teacher educators, two school-based instructional coaches, two fourth-grade teachers, two fifth-grade teachers, a special education teacher, and the school principal. In the new learning phase of the Learning Lab (the first activity of the PLT), the group learned about the growing patterns problem by trying out the problem themselves and reading and watching several video accounts of similar problems in use with students. As they did so, the participants began to explore what mathematical ideas might emerge and enumerated the myriad decisions they could make as teachers that might influence learners’ intellectual engagement: • • • •

Whether or not to ask students to identify how many squares are in each term; Whether or not to ask students to describe the way the term is structured; How to use color markers to display students’ noticing or counting strategies; The pros and cons of using a table to keep track of the term number and total number of squares; • The language that students might use to refer to various parts of the term such as “layers,” “tower,” and “floors”; • The possible models that students could build and their uses; and • The various ways students might discern the structure of the growing pattern either recursively or explicitly relating term number and total number of squares. This list reflects the group’s considerable investment in relating their instructional decisions to students’ opportunities for sensemaking. They considered how these decisions would influence students’ access to the problem, broad participation, and the cognitive demands of the problem. The group drew on multiple domains of knowledge as they considered how their pedagogical actions might relate to student engagement and access. For example, the teachers were attuned to knowledge of students and content when they tried to predict the language that students might use to describe various parts of the figure and how listeners in a conversation would be able to follow each other’s contributions to a discussion. In the coplanning phase of the Learning Lab (the second activity of the PLT), the group considered whether to show one term at a time or to show multiple terms at a time as well as how their word choice to prompt student noticing might affect their

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thinking. They wondered, “Should they use the term ‘pattern’ or should they prompt, ‘What is staying the same? What is changing?’” With each new idea, participants considered how students would react and what they might do alternatively. Consider, for instance, the following extended exchange when one of the fifth-grade teachers, Leslie, proposed potentially beginning the lesson with a discussion of the meaning of the word “pattern.” Notice how the educators proposed possible ways that students might enter a conversation about the meaning of the term pattern and how they considered together what might happen as their students engaged with the problem. We comment on the italicized and bolded contributions after the excerpt. Leslie (fifth-grade teacher): And I’m wondering if it would be worthwhile at first to talk about the word pattern and, you know, elicit from them what is…when we’re discussing math, what is a pattern to you? What is the meaning of that word? And it could be, you know, just to elicit it’s something that happens regularly that, you know, whatever it is that they say, Sonya (fourth-grade teacher): That’s predictable Leslie:  That’s predictable and you have evidence to show why that is what you’re predicting. Brooke (fifth-grade teacher): Do we just start out with that or wait until the third step to reveal it? Leslie: I think that we should kind of start out with it because everybody has access to the word pattern like this is what I think the pattern is and then say, “Well, we’re going to be doing work with patterns today, but let’s start with this figure.” Tina (fourth-grade teacher): Yeah, I think if we do that though, I think we do need to clarify, I think like ABAB, I see how kindergarteners see it, like a star circle, star circle like it repeats. And this doesn’t. Eva (school-based coach): This grows Tina: It grows this way. It’s not like, which is why think I saw it three 2 2, three 2 2. To me, that’s a pattern. So I think we’re going to have kids that see pattern as ABCABCABC. Lynn: And if we’re doing it, that’s in the fifth-grade classroom, they would be doing it differently than fourth graders Tina:  Our kids don’t have exposure. Like, I know, they do a few patterns in K-1, yes, but it’s like ABABAB

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Eva: In choral counts (a common instructional activity used in the school), we do patterns all the time. How is this a different kind of pattern? Sonya: Like a visual pattern, not a numerical one? Elham (univ. teacher educator): Yeah. We can also talk about like, how is this changing or growing, which is not even using the word pattern really, until… The way pattern comes in here is like, do you see any regularity? Do you see anything that is staying the same in the way we’re counting because that’s what helps you get to a generalized expression is that we’re counting it in the same way each time. So, it’s that repetition in repeated reasoning. Alison (univ. teacher educator): So, kind of like those fraction comparison ones we did (referring to a prior Learning Lab problem). Elham: Yeah. What’s the same in our thinking here in the way we counted it and what’s different so. So, if we do say, here’s term one. How many are here? Here’s term two. What happened, what changed? What stayed the same? How would you count this one? Here’s term three. What changed? What’s staying the same? How did you count this one? You could record the different ways that they saw and counted it. And then you could say, what do you think the next one will be if it keeps growing in this way? Would that work? As one facet of the collaborative planning they were doing, this exchange provides a window into how the group’s planning considerations were deeply connected to their own understanding of how patterns appear across the grades and how that relates to what a particular group of fourth-grade students might do in the enactment that is about to take place. Although the teachers did not yet know how the fourth-grade class will make sense of this growing patterns problem, they were raising questions for one another about how patterns that grow relates to the other ways students in their own school typically encounter patterns in the school curriculum as repeating patterns (ABABAB) or numerical patterns (emphasized in bold text). Recall that this is the first time teachers tried a growing patterns problem in their classrooms so there was some trepidation about what students may or may not do: “Our kids don’t have exposure.” The exchange was also marked by various propositions (shown in italics) about how teachers could frame and start the problem. Linguistic markers like, “I’m wondering if it would be worthwhile…,” “If we do that…,” or “Would that work?” reflect how the group was thinking aloud together

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with their goals of engaging students in noticing the structure of the growing pattern without doing the intellectual work for them. Thinking aloud together matters in this activity of the PLT because the teachers will teach this lesson together. Thinking aloud about possible scenarios (including the prompt that a teacher might say) brought every teacher into the problem space they will be walking into where their ideas will move from the realm of hypotheticals. By the end of their coplanning, the group decided against beginning the launch of the problem by discussing the term “pattern” but, clearly, they had intuitions regarding exploring the meaning of pattern with their students. Additionally, the complexities the teachers considered during planning are propositional; they planned with their students in mind, anticipating what might happen given their best understanding of their intended goal. But they cannot make the actual instructional decisions or consider the merits of their instructional decisions until they make them and see what their students do in response. That is what the next activity of the PLT will afford for these participants.

Joint Enactment: Engaging Complexities During Teaching Next, in the joint enactment phase of the Learning Lab (the third activity of the PLT), the group taught their planned lesson together in Sonya’s fourth-grade classroom. They walked into this enactment with the understanding that some teachers would facilitate students’ engagement with the problem while other participants would sit among students. The group also shared the routine of being able to pause instruction at moments where they felt the need to confer about their next instructional moves. These pauses, referred to as Teacher Time Outs, signify times when participants, perceiving challenging or problematic aspects of instruction, share the complex demands of instructional decision making in a particular moment of practice (Gibbons et al., 2017). Teacher Time Outs can take the form of participants interacting with the students or deliberating among themselves about a particular aspect of instruction. The improvisational decisions that are made during this joint enactment also rely on the participants’ extended planning. Their puzzling over the language to use to invite students to make sense of the structure of growing terms and the representation of these terms become shared resources that they draw on in this joint enactment of teaching. During the enactment, the students shared many ways to describe the terms, fitting with what the teachers had anticipated and bringing additional language to describe the squares, such as “stacked 2, 2, 3” and “two columns of 3 and 1 more.” Some students also noted a numerical pattern of successive odd numbers in the total number of squares used: Term 1 had three squares, Term 2 had five squares, and Term 3 had seven squares. After discussing the structure of the first three terms, the teachers distributed linking cubes and decided to ask students to build the fourth term. However, to the teachers’ surprise, the students built several different configurations seemingly at odds with the ways that the class had previously decomposed the structure of each term. The teachers gathered these different models and displayed them by holding them up in the front of the room (see Fig. 5). They asked

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Teachers hold up the range of models of Term Recreations of Term 4 built by students and 4 built by the students

displayed in photograph, from left to right

Fig. 5  The fourth term as built by students

students to explain why they built the fourth term as they did. Through those exchanges they learned that some students were not extending the pattern they observed in the first three terms but changing the structure. For example, the student who made the fourth term in the shape of a square said that she knew the fourth term would have nine cubes and she arranged those nine into the shape of a square. Another student who placed two cubes in the “toe” of the shape thought that the shape would grow in that horizontal direction as well as the vertical. Some students stated that the growing pattern would be too “obvious” or “normal.” The fact that teachers were all together during this joint enactment gave them the opportunity to collectively explore why students built these particular models. As they listened to students’ rationales, they realized that some students built the fourth term in ways that diverged from how teachers anticipated students’ understanding of patterns. Consequently, one teacher signaled a Teacher Time Out to talk directly to the students about what they meant by “obvious and normal.” In the following excerpt, notice that while Leslie initiated the Teacher Time Out, other educators are positioned with authority to join in the conversation and respond with curiosities they have in the moment (see bolded text for teacher questions and italicized text for important student responses). Leslie (fifth-grade teacher): So, several of you used a couple of words, and I really want to know what you mean by that. So one person or more than one person said that if we made this term four (holding up model of what Term 4 should be) that, it would be too obvious. And so they thought Term 4 was this way (showing a square by moving the top two cubes to the side to form a 3 × 3 square as shown in Fig. 2). And I want to know what obvious means. What do you mean, “too obvious”? You used that phrase (looking at Student 1). You said that it would be too obvious. I’m really wondering what you mean by that.

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Student 1 (who had built a 3 × 3 square for Term 4): I mean by that because Term 2 and Term 3, so Term 2 was five and then ‘cuz it said, I do it, three added two and then you stacked it up on the other two and Term 3 you did the same thing. And then I think Term 4 couldn’t just be the same thing, just adding two more on top because if you did that because if you already did that then you can already see the Term 2 and Term 3 are the same except that you added two more on top. Leslie: And if I just added two more on top for four, that would be too obvious. Student 1: Yeah. Leslie: And then what would [Term] 5 be? Student 1: Probably the same thing adding two more, and if you kept doing that, people might think, oh, I don’t know, it’s going to be too obvious. Tara (school-based coach): So, I think it’s so obvious that it’s like no mental challenge. Student 1: No Elham (univ. teacher educator): It’s not complicated enough. Tara: It’s not complicated enough. Student 1: No Brooke (fifth-grade teacher): And I heard you say it can’t be the same thing each time. It being the same thing would be too obvious. Student 1: Yeah Leslie: Yeah, I see. Did anybody have any other ideas about obvious? (to a different student) Is that what you meant too when you said obvious? Student 2: Yeah. I kind of agree because it then becomes easier and easier as you just keep doing the same thing. So then I would like to revise my thinking so now I think the 9 (Term 4) would be in the shape of a square instead of just stacking up. Leslie: Because you think this is just too obvious, like we wouldn’t give you something that simple [Educators quietly chuckle.] Heather (principal): That makes my heart so big that you know us so well that we know how challenged you like to be Leslie: So, then my other question is you said that would be normal. Is that what you meant by normal too? That it would just be like, now we’re just having two on top each time.

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Tara: Could I ask a question? You know the word, predict, you know, when you’re predicting. Would the word predictable, would that describe what you’re thinking, that it would be predictable? Student: Yeah Tara: Easy to predict Elham: Well, I think at first he was actually saying that it could also be that way, it’s okay if it was obvious. Eva: Yeah, right, that’s what I heard too Leslie: Oh oh, you say it’d be OK if it was obvious. Elham: Yeah, that could be normal too, he was sort of saying that. Leslie: Wow! This joint enactment of the growing patterns problem allowed the group of educators to see firsthand how students engaged with the problem and to further elicit their thinking and see the consequences of their pedagogical decisions on students’ engagement. Additionally, the use of the Teacher Time Out allowed educators to engage in the instructional demands of a moment of instruction in a way that was consequential to its unfolding rather than a postreflection on practice. For example, with five participants chiming in, the group was trying to unpack what students might have meant when they used the words “obvious” and “normal.” Through this exchange, the participants supported each other in understanding student thinking as they learned that students came up with a new geometric configuration for Term 4 because otherwise it would just be “easier and easier.” As the educators asked clarifying questions during this Teacher Time Out, they came to understand that some students interpreted nine cubes as being the important mathematical feature to maintain in Term 4 rather than the geometric structure they had described when they examined Terms 1 through 3. Through the Teacher Time Out, the educators tried to restate what they heard students say, as exemplified in Tara’s attempt in the excerpt “Because you think this is just too obvious, like we wouldn’t give you something that simple.” This collective in-the-moment sensemaking, in interaction with the students, then helped the group of educators to understand what students meant by “too obvious and normal”: Students expected some challenge as part of this pattern, and when they perceived a lack of challenge due to the consistency in the pattern’s structure (“it was doing the same thing!”), they created a new structure. We think Ed Silver would see this as an interesting twist on cognitive demand, where some students, perceiving not enough cognitive demand, adjusted the problem structure.

Debriefing: Engaging the Complexities After Teaching The educators were buzzing when they left the classroom to engage in the debriefing phase of the Learning Lab (the fourth activity of the PLT), eager to make sense of what the students had said and done. We drop in as they began to puzzle through

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what the students meant when they said the patterns were obvious. Note here how they recall what students said to them (in italicized text) and how they continued to try to interpret what they heard and consider their next instructional steps as a result (in bold text). Tara (school-based coach): She [Student 1] was fixated on what the numbers should be. The creative part of her wanted to just say you could add the two next to anywhere, and the pattern would continue. Multiple educators: Yeah, right. Tara: She wasn’t looking at the structure of the shape, she was looking at the pattern of the numbers. Eva (school-based coach): That’s right. Tara: And she thought ‘obvious,’ like why would we do something obvious? Tina (fourth-grade teacher): Simple. Tina: Just add another two to the top. Tara: Yeah, add two, add two. Tina:  Cuz (another student) was like, it’s 9. And I was like, “how do you know” and he said, “cuz 3, 5, 7, 9” I asked him, how do you? “it goes 3, 5, 7, 9”? So you can put it anywhere Leslie (fifth-grade teacher): So that makes me think “what a pattern is.” Because a pattern is obvious. Elham (univ. teacher educator): Oh yeah, the numerical pattern. Tina: He was looking at the numbers and saying I can structure it anyway I want. Tara: What if we talked about the pattern of a shape? In their processing of students’ responses, the teachers attended to the purpose of the growing patterns problem: the idea that it was really asking students to coordinate what they noticed about numerical patterns reflected in the total number of cubes in each term with its geometry. Through their deliberations about students’ thinking, they concluded that students could keep those ideas separate from each other. Tara’s question, “What if we talked about the pattern of the shape?” led the teachers to consider what it would mean to help students discern such patterns— how each term was growing from the previous one. As the discussion continued, the educators were clearly appreciative of the complexities and nuances in their students’ ideas. They made clear their fundamental belief that there was always logic in how students approached problems, and making sense of those ideas was necessary for steering a lesson toward particular instructional goals. They agreed that the students were correct that the geometry could change at any time and accordingly wanted to figure out how to frame the problem and the discussions around it so that

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students wouldn’t disregard the structure of the shape. This led to an extended discussion about what the words “pattern” and “structure” might mean to students. For example, might some students think structure refers to the final built figure and not its internal design or build? This further complicated for the group what they would need to consider in returning to this problem. They entertained ideas about conveying to students that patterns in the numbers can be coordinated with the geometry of the structure. For example, they wondered if posing this question would be useful for students’ developing understanding of the growing patterns task: “What does it mean that all these terms have the same structure? And that the same structure is growing?” Teachers’ deliberations and reflections during this activity of the PLT extend the collective sensemaking they engaged in during the other phases of the Learning Lab cycle (new learning, coplanning, and enactment). Their mutual engagement in this debrief was motivated by curiosity, surprise, and wonder enabled by the opportunity to experience a lesson collectively, where they felt mutual accountability for supporting student learning. Their considerations led them to make decisions about revisions they would make to the launching and monitoring student understanding and mathematical argumentation in their next enactment of a growing patterns problem.

Discussion Our work in this chapter has been greatly inspired by Ed Silver’s representation of the role of PLTs in practice-based professional development. Our own work on designing practice-based professional development for teachers has been aligned with his perspectives. Silver posited that PLTs can bring teacher learning closer to practice through their highly contextual nature. When deliberately designed, he added, PLTs can engage teachers’ inquiry into complex practice where they coordinate and connect different facets of their knowledge. We have connected these characteristics of PLTs to our design of the Learning Labs activities that aimed to support teacher learning about facilitating their students’ sensemaking through mathematical argumentation. Most importantly, our argument in this chapter was influenced by Ed Silver’s depiction of PLTs as bridging teachers’ moving “back and forth between past and current teaching experiences and the activity space of the professional development experience” (Silver, 2009, p. 245). Given our work with preservice and in-service teachers on models of professional learning that bring investigations of practice closer to the enactment of teaching, we explored in this chapter the implications of such models for the role and position of PLTs. Embedding PLTs in practice through the Learning Labs model conceptualizes PLTs as larger than bridges: The overarching PLT is, first, stretched across a cycle of practice-based activities that involves new learning, coplanning, joint enactments of teaching, and collective debriefs. Second, across all these phases, the PLT remains nested inside the professional development and actual teaching practice in the presence of students. We highlighted through the case narrative the affordances of this

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conceptualization of PLT where participants participated, across all these experiences, in inquiry about student thinking and in deliberations about the actions they would take to support it. The opportunities for learning in this conceptualization emerge in the various spaces the PLT provides teachers to connect their knowledge and experiences, not only between their own teaching and the Learning Lab context but also from each phase of the cycle to another, all within an integrated whole. We showed, for instance, in the case narrative how teachers’ deliberations during planning had the potential to cross beyond hypotheticals to the realm of joint enactments where teachers were able to test some of the collective pedagogical envisioning they made in anticipation of student thinking. In each phase of the Learning Lab as a context for working on the PLT, teachers are able to tackle a real challenge to ambitious instruction by investing their knowledge in the cognitive and pedagogical demands of instructional situations in a way that motivates continued work rather than abandonment of new ideas. Being in the company of others and inquiring into practice together can also address the oft-repeated observation that teachers can take up superficial aspects of reform recommendations and not make any substantive change to the way they relate to the discipline or to their students. The narrative case also illustrates the way the embeddedness of the PLT in practice, across the Learning Lab phases, affords a different type of cognitive and pedagogical demand for teachers as compared to when professional development experiences are situated outside teachers’ practice. The Learning Lab cycle provides a space for teachers across all four phases to engage collaboratively in proposing, debating, and considering solutions to pedagogical dilemmas and exploring pedagogical possibilities (Silver, 2009) to problems of practice that emerge in the moment. In attending to these dilemmas collaboratively, teachers were contextualizing their individual and shared knowledge of content, students, and pedagogy to consider how to give students access to the mathematical task in a way that centered their sensemaking around clear mathematical goals. This aspect of purposive, communal pedagogical problem solving provides space for participants to engage their self-knowledge and shared resources—a tenet of social practice theories of learning (Penuel et al., 2016; Yanow, 2001). This theory stipulates that when teachers are able to collaborate across activity structures (new learning, planning, enacting, and debriefing) around challenging and uncertain aspects of practice, they are positioned to be able to adjust their contributions to one another (O’Connor & Allen, 2010) and to various contexts of practice. Our work on embedding PLTs inside joint enactments of practice entails a particular positioning for students. This model of collaborative teaching in classrooms positions students as colearners with the Learning Lab participants wherein they hear the educators’ deliberations about their thinking and participation. Sometimes in our collaborative lessons, we find that students weigh in during the interactive pauses, clarifying an idea or offering their view of what teachers might try. For example, in the case narrative in this chapter, one can see how students were helping teachers understand what they had meant by “obvious,” responding to their in-the-­ moment pondering and inquiries. For these reasons, we have found that in embedding PLTs inside joint enactments of practice, both adults and students can see

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uncertainty and asking questions as central aspects of what it means to learn with others. The students’ presence also provides a stimulus for adjusting one’s thinking based on students’ participation and ideas (Ghousseini et al., 2022). Additionally, because the Teacher Time Outs that happen during the lesson (either in interaction with students or among the teachers) are grounded in a lesson that has been coplanned, the shifts between pauses and enactment occur in a coherent instructional experience designed with particular students in mind. In this sense, we find that our model of PLTs nested in practice counters views that posit that the bustle of immediacy would interfere with opportunities to learn or “restricts attention to the sort of teaching underway in that particular class” (Ball & Cohen, 1999, p. 14). As Ed Silver (2009) maintained, PLTs can be deliberately designed to prompt teachers to consider the challenging aspects of instructional situations and consequently invest their knowledge in the cognitive and pedagogical demands of supporting student learning. The Learning Lab cycle is an example of such deliberate design that anchors teachers’ inquiry into practice inside an integrated intentional activity that allows them to pursue learning goals inside the immediacy and authenticity of practice.

Limitations and Future Research Our conceptualization of the role of PLTs in supporting teacher learning as tasks that are embedded in practice, involving collective teaching, planning, and reflection, has some organizational and cultural limitations. First, designing and facilitating Learning Labs require changes in the organizational conditions of schools to allow for collective learning that is embedded in teachers’ workday. Most schools, at least within the United States, do not have schedules and the myriad resources available to regularly carry out Learning Labs. For example, time for teachers to see each other teach is often lacking as is the availability of skillful facilitators who can establish a culture of collaborative learning, trust, and shared accountability. Stigler and Hiebert (2009) noted this issue in the context of the United States, where they argue that professional development usually involves professional developers as presumed experts who present “workshops for teachers during specially designated days during the school year” (p. 36). Stigler and Hiebert (2009) also remind us that although professional development practices, like Lesson Study (or Learning Labs), can be important alternatives to traditional teacher learning opportunities, it is clear that numerous subtle but “powerful cultural forces” (p. 36) can stand in the way of bringing teacher learning closer to practice. Reorganizing schools so generative teacher learning can happen on the job through deliberately designed PLTs that are embedded in practice requires cultural changes. Deprivatizing teaching is necessary for developing PLTs that include joint enactments of teaching. Teachers do need to be able to see each other teach. But we also contend that deprivatization of teaching—making teaching public—is not sufficient for the kind of professional learning that we are trying to foster.

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It is not sufficient to simply see each other teach. To be actively involved in an adult learning endeavor, collaboratively teaching and wrestling with new ideas, also means that teacher learning needs to be public. Teachers’ partial ideas, revisions, and growth are available and visible to one another when joint enactments go beyond demonstrating teaching while others simply observe. In our Learning Labs, joint enactments create shared consequences for teachers because everyone in that space is accountable for enacting and making sense of the journey that the lesson took. Finally, embedding PLTs in practice in the context of Learning Labs should not assume that, like any collective learning endeavor, what each person takes from the experience is the same. Organizational supports are still needed to help teachers contextualize their learning in their own practice with their own students. School coaches can provide such supports generatively when coaches can be participants in the Learning Lab experience. In this way, coaches can connect the PLT beyond the specific Learning Lab cycle to teachers’ daily work. Hence, the model of PLTs we are advancing in our work creates significant demands for teacher educators to monitor both collective and individual learning and to consider how learning cycles build coherence over time. Further research on how school-based teacher educators can do this work can provide insights into the way organizational changes can support teacher learning that is closer to their daily practice. Additionally, the Learning Lab cycle is one example of this broadened conceptualization of PLT. Further research can inform how other models of professional learning can embed PLTs in teachers’ practice and the sort of affordances they bring for teacher learning.

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To What Extent Are Open Problems Open? Interplay Between Problem Context and Structure Roza Leikin, Sigal Klein, and Ilana Waisman

Introduction There is a consensus among researchers, mathematics educators, and mathematicians that problem solving is central to the teaching and learning of mathematics and the development of mathematics as a scientific field (Polya, 1973; Schoenfeld, 1985; Silver, 1985; Verschaffel et  al., 2020; Liljedahl & Cai, 2021). Defining a problem as a situation that requires a solution entailing mental effort implies that solving mathematical problems is an inherently challenging process. Solving mathematical problems is based, to a large extent, on individual experience, including successful use of strategies in the past, as well as on intuition and imagination, which are significant in solving unfamiliar problems. This observation implies that the level of challenge embedded in a problem is relative and is a complex function of the problem-solving expertise and creativity of the individual solver. One of the central questions in mathematics instruction is how to organize the learning process such that it will challenge students with different levels of mathematical abilities (for the concept of ability, see Krutetskii, 1976), mathematical proficiency, and creativity (Leikin, 2023). One of the methods recommended by mathematics educators for developing mathematical proficiency and mathematical creativity in all students is the implementation of open mathematical problems and tasks (Haylock, 1987, 1997; Leikin, 2018; Leikin et al., 2023; Pehkonen, 1995, 1997; Silver, 1995, 1997). This chapter focuses on the relationship between the mathematical challenge embedded in open mathematical problems and the types of openness. Special attention is given to open-end problems, particularly to the complexity of open problems and its link to “realistic” considerations when solving the problems. We start with a R. Leikin (*) · S. Klein · I. Waisman Department of Mathematics Education, RANGE Center, University of Haifa, Haifa, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Cai et al. (eds.), Research Studies on Learning and Teaching of Mathematics, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-35459-5_3

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background section in which we analyze different interpretations of openness of mathematical problems as they appear in Silver (1995, 1997), Pehkonen (1995, 1997), Krutetskii (1976), and Haylock (1987, 1997). The analysis is framed by the following distinctions: • Open-start versus open-end problems following Pehkonen’s (1997) definition of open problems as related to starting situation versus goal situation. • Multiple Solution Strategies Tasks (MSTs) versus Multiple Solution Outcomes Tasks (MOTs): MSTs explicitly require solvers to solve a problem in as many different ways as they can (Leikin, 2007). In MOTs, solvers are expected to find several solution outcomes (Leikin et al., 2023). Following the analysis of classes of open problems, we explore the relationships between creativity and openness as in Silver (1997) and Haylock (1987, 1997). We then zoom in on problems with realistic contexts (Greer, 1993; Reusser & Stecler, 1997; Silver et al., 1993; Verschaffel et al., 1994, 1997, 2000), which are considered a special class of open problems. Studies have found that students’ success when solving realistic problems is significantly lower than when solving standard problems due to students’ inclination to suspend real-world knowledge and realistic considerations when solving word problems (Reusser & Stebler, 1997; Verschaffel et al., 1999; 1997). To explore the complexity of this class of problems from a new angle, a distinction is drawn between the mathematical structure of the problems and their context (either mathematical or realistic). We examine the hypothesis that the complexity of this class of problems is related to their structure and to their place in the MOT class rather than to the phenomenon of suspension of sense making. The concluding discussion focuses on the connections between problems’ openness, multiplicity of solutions, and the related insight component of mathematical reasoning that determine the mathematical challenge embedded in mathematical tasks. We argue that mathematical instruction should systematically integrate discussion of the connections between multiple solution strategies for a particular problem and the completeness of the problem’s solution space.

Different Classes of Open Problems The notion of open mathematical problems as it appears in the mathematics educational literature includes a variety of situations and questions to be answered. The goal of teaching mathematics is not only performing calculations and algorithms but enhancing understanding and developing reasoning and creativity (Pehkonen et al., 2013). The Discussion Group led by Pehkonen at PME-17 in Japan produced a special issue of ZDM in which four articles analyzed the use of open problems in different countries (Nohda, 1995; Pehkonen, 1995; Silver, 1995; Stacey, 1995). Silver (1995) analyzed the nature and use of open problems in mathematics education and identified several different meanings of the term open problems: • Problems open in mathematics • Problems allowing multiple interpretations

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• Problems having multiple solution methods • Problems that lead to other problems or generalizations Problems open in mathematics are previously identified problems which remain unsolved. These problems remain open until a mathematician or group of mathematicians solve them. In 1900, Hilbert, at the mathematics conference in Paris, formulated 23 problems that meaningfully influenced the research agenda for mathematicians in the twentieth century. Some of these problems remain open today (Simons Foundation, 2022). Problems with multiple interpretations (e.g., “what is the height of a child of 1 year old?”) “do not have a single exact answer, rather a range of plausible solutions can be justified” (Silver, 1995, p. 68). Note here that many mathematical problems can be solved using multiple different solution methods. However, their openness with respect to the performance of different solution methods by a particular solver depends on how the task is presented to the solver—for example, whether the instructional setting encourages a group of students to find different ways to solve a problem. Silver (1995) provided an example of “chickens and rabbits heads and legs problem” that can be solved in several ways. Problems that lead to generalization and other problems are a type of problem-­ posing problems, the openness of which is ultimate because the outcomes depend on the solver. Figure 1 depicts the Divisibility-of-the-Product problem, which is a variation on a generalization problem that leads to other problems, borrowed from Silver (1995, p.  68). We invite readers to solve this and the following problems included in this paper. The DOP problem is an open-start problem due to its explorative nature and the possibility of “extension to other problems.” It is also open with respect to the ways in which the problem can be introduced to students as an MST.  One can use a generic example and draw conclusions leading to a formal proof that n(n + 1)(n + 2) (n + 3)⋮24, for example, by consideration of different values of n : n = 4k, n = 4k + 1, n = 4k + 2, n = 4k + 3 or by induction. It can be also proven that 24 is the biggest number that divides the product of any four consecutive integers. The problem has a particular solution outcome and cannot be considered an open-end problem. The openness of the task depends to a large extent on the instructional setting—the teacher’s trust in students’ capacity to tackle the problem successfully and decision to allow a self-regulated problem-solving procedure, which is essential for maintaining the explorative nature of the problem. Pehkonen (1997) defined the openness of a problem based on the openness versus closedness of its “starting situation” and “goal situation.” If the starting situation is not clearly explained (e.g., givens are missing) or the goal is not defined (e.g., a statement, without a question), the problem is defined as an open problem. Pehkonen (1995, 1997, with reference to Avital, 1992) argued that open problems are of an

Show that the product of any four consecutive integers is divisible by 24. Is there any larger number that divides the product of any four consecutive integers?

Fig. 1  Divisibility-of-the-Product (DOP) problem

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exploratory nature (for elaboration see also Sullivan et  al., 1997). In Pehkonen’s (1995, 1997) categorizations, open problems require: • • • •

Mathematical investigations, Problem posing, Variations on provided problems (what-if and what-if-not questions), and Relation to real-life situations.

Pehkonen (1997) clarified that in many real-life situations, the starting situation or goal is not exactly described. Figure  2 depicts a Matches-Polygons problem, which is a variation of a problem provided by Pehkonen (1997). Similar to the DOP problem, the MP-1 problem (Fig. 2) is open due to its explorative nature. In contrast to the DOP problem (Fig. 1), the MP-1 problem is also an open-end MOT. Questions MP-2—MP-6 can be considered a guide to exploration. When considering polygons in an “orthogonal grid,” questions MP-2, MP-3, MP-4, and MP-5 each have a complete solution (Fig. 2): Questions MP-2 and MP-3 require finding the corresponding sets of numbers whereas question MP-4 asks about one number. Task MP-4 demands that the solver draw a complete set of different polygons. As such, Tasks MP-2, MP-3, and MP-5 are MOTs that have complete sets of solution outcomes. Figure 3 depicts solutions to the MP problem in an orthogonal grid. Interestingly, the set of all the polygons attained in the orthogonal grid can be described as a set containing 1 square, 3 rectangles (one of which is a square), 6 hexagons, 12 octagons, 2 decagons, and 1 dodecagon. Thus, the MP problem in the orthogonal grid has a complete solution and is not open ended. When the assumption that the grid is orthogonal is reduced, the set of polygons increases infinitely as a function of angles between the adjacent sides (matches). Many additional problems can be posed and, thus, the MP problem is open according to Silver’s criteria, “problems that lead to other problems or generalizations.” Still, the answer to problem MP-2 has a complete solution, 0  m : n