Problem Posing and Problem Solving in Mathematics Education: International Research and Practice Trends 9819972043, 9789819972043

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Tin Lam Toh · Manuel Santos-Trigo · Puay Huat Chua · Nor Azura Abdullah · Dan Zhang   Editors

Problem Posing and Problem Solving in Mathematics Education International Research and Practice Trends

Problem Posing and Problem Solving in Mathematics Education

Tin Lam Toh · Manuel Santos-Trigo · Puay Huat Chua · Nor Azura Abdullah · Dan Zhang Editors

Problem Posing and Problem Solving in Mathematics Education International Research and Practice Trends

Editors Tin Lam Toh National Institute of Education Nanyang Technological University Singapore, Singapore Puay Huat Chua National Institute of Education Nanyang Technological University Singapore, Singapore

Manuel Santos-Trigo Mathematics Education Department Centre for Research and Advanced Studies CDMX, Mexico Nor Azura Abdullah Universiti Brunei Darussalam Brunei Darussalam, Brunei Darussalam

Dan Zhang Beijing Academy of Educational Sciences Beijing, China

ISBN 978-981-99-7204-3 ISBN 978-981-99-7205-0 (eBook) https://doi.org/10.1007/978-981-99-7205-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Contents

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Problem Posing and Problem-Solving in Mathematics Education: International Research and Practice Trends . . . . . . . . . . . Tin Lam Toh, Manuel Santos-Trigo, Puay Huat Chua, Nor Azura Abdullah, and Dan Zhang Trends and Developments of Mathematical Problem-Solving Research to Update and Support the Use of Digital Technologies in Post-confinement Learning Spaces . . . . . . . . . . . . . . . Manuel Santos-Trigo

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Problem Posing and Modeling: Confronting the Dilemma of Rigor or Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corey Brady, Paola Ramírez, and Richard Lesh

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Types of Mathematical Reasoning Promoted in the Context of Problem-Solving Instruction in Geneva . . . . . . . . . . . . . . . . . . . . . . . Maud Chanudet

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Prospective Secondary School Mathematics Teachers’ Use of Digital Technologies to Represent, Explore and Solve Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexánder Hernández, Josefa Perdomo-Díaz, and Matías Camacho-Machín

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Primary School Teachers’ Behaviors, Beliefs, and Their Interplay in Teaching for Problem-Solving . . . . . . . . . . . . . . . . . . . . . . . Benjamin Rott

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Movie Clips in the Enactment of Problem Solving in the Mathematics Classroom Within the Framework of Communication Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Tin Lam Toh and Eng Guan Tay

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On Teaching of Word Problems in the Context of Early Algebra . . . 121 Nicolina A. Malara and Agnese I. Telloni

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Problem Posing by Mathematics Teachers: The Problems They Pose and the Challenges They Face in the Classroom . . . . . . . . 151 Alina Galvão Spinillo, Síntria Labres Lautert, Neila Tonin Agranionih, Rute Elizabete de Souza Rosa Borba, Ernani Martins dos Santos, and Juliana Ferreira Gomes da Silva

10 Problem Posing Among Preservice and Inservice Mathematics Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Ma. Nympha Beltran Joaquin 11 An Approach to Developing the Problem-Posing Skills of Prospective Mathematics Teachers: Focus on the “What if not” Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Zoltán Kovács 12 Regulation of Cognition During Problem Posing: A Case Study . . . . 217 Puay Huat Chua 13 Problem Posing in Pósa Problem Threads . . . . . . . . . . . . . . . . . . . . . . . 233 Lajos Pósa, Péter Juhász, Ryota Matsuura, and Réka Szász 14 Conclusion: Mathematics Problem Posing and Problem Solving: Some Reflections on Recent Advances and New Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Edward A. Silver

Editors and Contributors

About the Editors Tin Lam Toh is Associate Professor and Head of the Mathematics and Mathematics Education Academic Group of the Singapore National Institute of Education, Nanyang Technological University, Singapore. He obtained his Ph.D. in Mathematics from the National University of Singapore with research topic in Henstock Integration Theory. He has been teaching in Singapore school and was Head of Mathematics Department of the school before he joined the National Institute of Education as a teacher educator. He continued his research in Pure Mathematics and picked up his research in mathematics education, focusing on mathematical problem solving, calculus education, and comics for mathematics instruction. He has been the Principal Investigator of several large research grants in mathematics education and continues to publish in international refereed journals in both mathematics and mathematics education. Manuel Santos-Trigo completed his doctorate in mathematics education at the University of British Columbia, Canada. He is a full professor at the Centre for Research and Advanced Studies, Cinvestav-IPN, in Mexico City. He teaches graduate courses in the mathematics education department and does research in mathematical problem solving and the use of digital technologies. His current research project involves characterizing teachers and students’ problem-solving reasoning in scenarios that include and foster the students’ systematic and coordinated use of digital technologies and online developments to understand concepts and to develop problem-solving competencies. Puay Huat Chua’s research interests are in mathematics problem posing and in harnessing data analytics for decision making in education. He received his Doctor of Philosophy from the National Institute of Education (NIE), Nanyang Technological University, Singapore. He was Senior Research Analyst with Research Evaluation in the Planning Division in the Ministry of Education and was involved in work on

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international comparative studies. He was also a vice-principal of three secondary schools. He is currently Senior Teaching Fellow in NIE Office of Education Research, actively working on conceptualizing and operationalizing knowledge mobilization and on research impact studies. Nor Azura Abdullah is an assistant professor and teacher educator at the Sultan Hassanal Bolkiah Institute of Education, Universiti Brunei Darussalam. She earned her Ph.D. in mathematics education from the University of Hong Kong in 2021. Initially focusing on teachers’ pedagogical content knowledge and development, her research interests have since expanded to teachers’ values in mathematics teaching and learning, as well as professional development through practices such as Lesson Study. With knowledge and experience in mathematics education at primary and secondary levels, Nor Azura also teaches and supervises graduate courses within the faculty. Dan Zhang is a professor of Beijing Academy of Educational Sciences and head of the coach team of Beijing primary school mathematics. After her bachelor and master degree on mathematics from Beijing Normal University, she obtained her Ph.D. from Northeast Normal University, majored in curriculum and teaching methodology. She has hosted several research grants including National “Eleventh Five-Year” educational research plan and Beijing Municipal Educational Science “Thirteenth FiveYear Plan.” During the past ten years, she has researched on problem posing and published a series of journal articles and books. Her work on problem posing has got the first prize of Beijing Basic Education Teaching Achievement Award and the second prize of National Basic Education Teaching Achievement Award.

Contributors Nor Azura Abdullah Universiti Brunei Darrusalam, Bandar Seri Begawan, Brunei Neila Tonin Agranionih Federal University of Paraná (UFPR), Curitiba, Brazil Corey Brady Simmons School of Education and Human Development, Southern Methodist University, Dallas, USA Matías Camacho-Machín Universidad de La Laguna, San Cristóbal de La Laguna, Spain Maud Chanudet University of Geneva, Geneva, Switzerland Puay Huat Chua National Institute of Education, Nanyang Technological University, Singapore, Singapore

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Juliana Ferreira Gomes da Silva Federal University of Pernambuco (UFPE), Recife, Brazil Rute Elizabete de Souza Rosa Borba Federal University of Pernambuco (UFPE), Recife, Brazil Ernani Martins dos Santos Pernambuco University (UPE), Recife, Brazil Alexánder Hernández Universidad de La Laguna, San Cristóbal de La Laguna, Spain Ma. Nympha Beltran Joaquin University of the Philippines—Diliman, Quezon City, The Philippines Péter Juhász Alfréd Rényi Institute of Mathematics, Budapest, Hungary Zoltán Kovács Eszterházy Károly Catholic University, Eger, Hungary Síntria Labres Lautert Federal University of Pernambuco (UFPE), Recife, Brazil Richard Lesh School of Education (Emeritus), Indiana University, Bloomington, USA Nicolina A. Malara Department of Physics, Mathematics and Computer Science, University of Modena & Reggio E., Modena, Modena, Italy Ryota Matsuura St. Olaf College and Budapest Semesters in Mathematics Education, Northfield, MN, USA Josefa Perdomo-Díaz Universidad de La Laguna, San Cristóbal de La Laguna, Spain Lajos Pósa Alfréd Rényi Institute of Mathematics, Budapest, Hungary Paola Ramírez Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile Benjamin Rott Institute of Mathematics Education, University of Cologne, Cologne, Germany Manuel Santos-Trigo Centre for Research and Advanced Studies, Cinvestav-IPN, Mexico City, Mexico Edward A. Silver University of Michigan, Ann Arbor, USA Alina Galvão Spinillo Federal University of Pernambuco (UFPE), Recife, Brazil Réka Szász Budapest Semesters in Mathematics Education, Northfield, MN, USA Eng Guan Tay National Institute of Education, Nanyang Technological University, Singapore, Singapore

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Agnese I. Telloni Department of Education, Cultural Heritage and Tourism, University of Macerata, Macerata, Italy Tin Lam Toh National Institute of Education, Nanyang Technological University, Singapore, Singapore Dan Zhang Beijing Academy of Educational Sciences, Beijing, China

Chapter 1

Problem Posing and Problem-Solving in Mathematics Education: International Research and Practice Trends Tin Lam Toh, Manuel Santos-Trigo, Puay Huat Chua, Nor Azura Abdullah, and Dan Zhang

This book presents both theoretical and empirical contributions on problem solving and posing in relation to the framing of teaching and learning scenarios of mathematics in schools. Mathematical problem solving has been the focus in mathematics education traceable to at least seven decades since the publishing of George Pólya’s seminal book How to Solve It in 1945. Problem solving has since then become not only an important part of the mathematics curriculum, but also a teaching and learning approach in many countries around the world. Problem posing as an essential activity in mathematical practices and in mathematics education can be seen as beginning as a natural consequence of Pólya’s problem solving model, Looking Back as the fourth stage. It has evolved and grown rapidly in the recent decades, gaining prominence within mathematics education. The topic of mathematics problem posing has expanded significantly, with traces of its presence in the research and practice agenda. It can also be seen that the mathematics curriculum frameworks in many parts of

T. L. Toh (B) · P. H. Chua National Institute of Education, Singapore, Singapore e-mail: [email protected] P. H. Chua e-mail: [email protected] M. Santos-Trigo Centre for Research and Advanced Studies, Cinvestav-IPN, Mexico City, Mexico e-mail: [email protected] N. A. Abdullah Universiti Brunei Darrusalam, Bandar Seri Begawan, Brunei e-mail: [email protected] D. Zhang Beijing Academy of Educational Sciences, Beijing, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_1

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the world have incorporated problem posing as an instructional focus, building on problem solving as its foundation. The juxtaposition of problem solving and problem posing in mathematics presented in the contributions of this book aims to address the needs and perspectives of the mathematics education research and practice communities at the present day. In particular, this book aims to address the three key points: 1. To present an overview of research and development regarding students’ mathematical problem solving and posing; 2. To discuss new trends and developments in research and practice on these topics; and 3. To provide insight into the future trends of mathematical problem solving and posing. The three objectives were the targeted aims of the Topical Study Group (TSG) 17 on Mathematical Problem Solving and Posing in ICME14 held in Shanghai, China, in July 2021. Thus, the chapters included in this book address issues and themes involved in the three above objectives. Authors presented initial versions of the chapters during the sessions of the Topic Study Group at the ICME conference. Further, some authors were invited to refine and extend their contributions based on their alignment to the group’s three aims. Chapter 2 begins with an overview of mathematical problem solving in the context of digital tools and remote learning scenarios by Santos-Trigo. Recognizing problem solving as a distinctive human activity that shapes and influences what individuals do to face and deal with social, labour, professional or academic situations; and reflecting on the process of identifying, formulating, and solving problems, how people develop resources, strategies, and ways of reasoning to solve problems becomes feasible. This chapter examines the questions that have inspired mathematicians and mathematics educators for years: How are mathematical problems formulated and what the process of approaching and solving problems involves? How do students develop problemsolving competencies? How do teachers and students’ use of digital tools shape the ways they reason and solve mathematical problems? In Chap. 3, Brady et al. discuss a perspective on problem posing research grounded in their research in modeling. They identify honoring and amplifying students’ and teachers’ mathematical agency as a shared commitment between researchers in modeling and problem posing. However, a tendency to assume a tight coupling between problem types and individual mathematics topics is a deep-rooted issue. The authors advocate a shift towards activities that highlight problematic situations to engage students in mathematical modeling and interpretations. The authors argue that to achieve relevance, mathematics education needs to engage with a broader range of problem situations, to utilize a wider array of representational tools, and to embrace a more encompassing view of mathematical problem posing, problem solving, and modeling, as interpretive human activities. In Chap. 4, Chanudet presents a study of the types of mathematical reasoning that are involved in the challenges faced by school students in the context of problemsolving instruction, to achieve a robust understanding of both teachers’ practices

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and potential students’ learning. After operationalizing the definition of the term “mathematical reasoning”, the chapter reports a research study conducted in Geneva on teachers’ selection of problems from textbooks for a mathematical problemsolving course at the secondary level. The chapter concludes with a characterization of teachers’ practices and a formulation of some hypotheses about students’ potential learning related to problem solving. In Chap. 5, Hernández et al. report a research study which aims to analyze the work done by a group of prospective secondary school mathematics teachers as they solve mathematics problems using digital technology. The study focused on analyzing the mathematical activity as they progressed in solving one problem in terms of different problem-solving episodes. It was shown that the use of digital technology facilitated the students’ process to generate and pursue new paths to represent mathematical objects, transform representations, formulate conjectures, and observe and justify relationships and conjectures. In Chap. 6, Rott reports a study on how teachers in Germany primary schools organize their teaching for problem solving. The lessons were videotaped and the teachers were interviewed. The result shows a strong correlation between the teachers’ belief about teaching and focusing for problem solving. In Chap. 7, Toh and Tay examine the characteristics of four video clips which are extracted segments from popular movies, and which can potentially be used for mathematics instruction on problem posing and solving. Together with an extensive literature review, a framework of video development/adaptation for mathematics instruction focusing on mathematical problem solving based on communication theory is presented. In Chap. 8, Nicolina and Telloni report a project on the teaching of verbal problems for primary and lower secondary school, based on verbal problems focusing on connected activities intertwining problem solving, argumentation and problem posing. In the research, the students, through classroom discussions and under the careful guide of the teacher, were brought to (a) see a verbal problem as representative of a class of problems and as a generator of (first numerical, then algebraic) sentences representing its solving process(es); (b) compare these sentences and, through the variation of the numerical data of the problem, conceive them as mathematical objects; (c) grasp the regularities of particular sets of such sentences, up to discover and reify specific arithmetical properties; (d) construct the texts of problems in realistic contexts, related to a specific numerical sentence. Due to the intervention, the students showed the development of linguistic and reasoning capabilities, the emergence of pre-algebraic and metacognitive habits of mind and a positive disposition towards problem solving. The chapter concludes with implications for teacher education and further research. In Chap. 9, Spinillo et al. present two research studies on how mathematics teachers deal with problem posing. The first study analyzed the characteristics of the problems they posed on formulating eight mathematical word problems for which the solution involved multiplication and/or division. The second study analyzed the classroom practice of teachers who used problem posing in their classroom to teach mathematics to elementary school students. The data obtained in the first study

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allowed the authors to know the characteristics of mathematical problems posed by teachers, analyzing aspects related to the verbal statement and the types of problems they formulate. The data obtained in the second study was used to document teachers’ opinions about their students and their teaching practice regarding problem posing in the classroom. In Chap. 10, Beltrand-Joaquin explores the differences in word problems posed by pre-service mathematics teachers and in-service mathematics teachers. Forty word problems constructed by the two groups of teachers were examined. The results of the data analysis showed that the problems posed by pre-service teachers dwelled more on varied contexts, although the contents included mostly basic mathematics concepts. In contrast, the problems generated by in-service teachers were generally theoretical in nature but involved higher mathematical content and processes. It was notable that most of the problems constructed by both groups were of the non-routine application type of tasks and that some were unrealistic, ambiguous, or had errors. In Chap. 11, Kovács attempts to answer and document whether problem posing can be mastered by examining the performance of novice and expert group of student teachers. The author concluded that problem-posing is a learnable activity, and argues that the skills required for problem-posing go beyond the boundaries of the skills and abilities involved in problem-solving. In Chap. 12, Chua presents the regulatory phases of cognition during problem posing through a case study of a grade 9 student who worked on a geometric problemposing task. The study paints the different phases in the regulation of cognition during problem posing, namely, property noticing, problem construction, checking solution, and looking back. The looking back phase was not strongly exhibited. Discussion of these phase descriptors in classroom problem-posing instructions are also discussed in the chapter. In Chap. 13, Pósa et al. present how an instructional method of mathematical guided discovery through specifically designed problem threads, encourages students to pose problems, and hence create their own problem strands. To illustrate this pedagogical approach, they describe a sample problem thread involving geometric transformations from easy to challenging problems. The solutions of selected problems were also included. The editors of this book are truly honored to have Professor Ed Silver to write the concluding chapter for this book. In addition to stressing the importance of problem posing and problem solving, Silver commented on the diversity of the geographical locations of the authors contributing to this book. The team of authors not only consists of “seasoned contributors”, but also new entrants to the field. The inclusion of new perspectives of problem solving and problem posing by building on the earlier works makes the field of research on problem posing and problem solving promising. Silver commented that both curriculum perspective and cognitive activity on problem posing and problem solving are evident in the book, and investigation related to teaching mathematics through problem posing and problem solving have become more visible in addition to research focusing on the nature of problem posing and problem-solving processes. Silver further noted the importance of research on problem posing and problem solving to be conducted within classroom context

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studying the mathematical and pedagogical challenges. He acknowledged that most contributions in this book were studies made outside the classroom context, but their studies were authentic and relevant to the classroom instructions. In addition to this volume documenting the development of problem posing and problem solving from ICME 14, a special issue on the same topic based on the development from ICME14 was also published in the Hiroshima Journal of Mathematics Education in 2022 (Volume 15 No. 1). The content discussed in the papers included in the special issue include researchers’ advocate of strategies to develop students’ problem-solving competency and problem posing (Bos & Bogaart, 2022; Ramirez et al., 2022), characterization of the processes exhibited by students in problem solving within the classroom context (Favier, 2022). Empirical studies include the use of Graphic Organizers for teaching problem solving (Abdullah & Abbas, 2022), and an exploratory study on students’ problem-posing based on free-posing and semistructured tasks (Chua & Toh, 2022). The special issue also includes an international comparative study between students in China and the United States on the problems they posed on division (Luo et al., 2022), and a textbook analysis on problem-solving tasks in mathematics textbooks of China (Wang & Wang, 2022). The editorial team offers this book together with the special issue in the Hiroshima Journal of Mathematics Education as a contribution to the contemporary knowledge of problem posing and problem solving. We look forward to further work that advances from the studies that stem from the studies reported in the two contributions that arose from ICME14 TSG 17.

References Abdullah, N. A., & Abbas, N. H. (2022). Teachers’ exploration using graphic organizer for problem solving in primary mathematics. Hiroshima Journal of Mathematics Education, 15(1), 19–33. Bos, R., & van den Bogaart, T. (2022). Teachers’ design of heuristic trees. Hiroshima Journal of Mathematics Education, 15(1), 5–17. Chua, P. H., & Toh, T. L. (2022). Developing problem posing in a mathematics classroom. Hiroshima Journal of Mathematics Education, 15(1), 99–112. Favier, S. (2022). A characterization of the problem solving processes used by students in classroom: Proposition of a descriptive model. Hiroshima Journal of Mathematics Education, 15(1), 35–53. Luo, F., Yu, Y., Meyerink, M., & Burgal, C. (2022). Fifth grade Chinese and United States students’ division problem posing: A small-scale study. Hiroshima Journal of Mathematics Education, 15(1), 85–97. Ramírez, M. C., Pérez, M. M. A., & Hernández, N. G. (2022). A strategy for enhancing mathematical problem posing. Hiroshima Journal of Mathematics Education, 15(1), 55–70. Wang, R., & Wang, C. (2022). Historical comparison of problems and problem-posing tasks in Chinese secondary school mathematics textbooks. Hiroshima Journal of Mathematics Education, 15(1), 71–84.

Chapter 2

Trends and Developments of Mathematical Problem-Solving Research to Update and Support the Use of Digital Technologies in Post-confinement Learning Spaces Manuel Santos-Trigo

Abstract Problem solving is a distinctive human activity that shapes and influences what individuals do when facing social, professional, or academic situations. Reflecting on what the process of identifying, formulating, and solving problems involves becomes a relevant task to understand how people develop resources, strategies, and ways of reasoning to solve problems in different domains. How are mathematical problems formulated? And what does the process of approaching and solving problems involve? How do students develop problem-solving competencies? How do teachers and students’ use of digital tools shape the ways they reason and solve mathematical problems? These questions have inspired mathematicians and mathematics educators to investigate what the process of formulating and solving mathematical problems entails and ways for students to understand mathematical concepts and to solve problems. In this chapter, some seminal conceptual frameworks are reviewed to shed light on principles and tenets to support and frame learning scenarios that foster students’ problem-solving competencies. Further, the consistent and systematic use of digital technologies becomes relevant for learners to enhance their ways of reasoning to work on mathematical tasks and to engage in and extend mathematical discussions beyond classrooms. Thus, digital tools provide a set of affordances for students to dynamically model tasks and rely on heuristic strategies such as orderly dragging objects, quantifying parameters, tracing loci, using sliders, etc. to work and solve the problems. Keywords Mathematical problem-solving trends · Conceptual frameworks · Digital and semiotic tools · Mathematical reasoning

M. Santos-Trigo (B) Centre for Research and Advanced Studies, Cinvestav-IPN, Mexico City 07360, Mexico e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_2

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1 Introduction and Background Problem posing and looking for different ways to solve problems are distinctive activities that permeate and distinguish human behaviours. People encounter and deal with different types of tasks in their daily activities and in professional or educational contexts. Some tasks or problem situations might only require that individuals activate routine procedures or rules to approach and solve them; but others, such as those found in disciplinary domains, demand that people design a plan in which resources and strategies become relevant to represent, explore, and solve those problems. Cropley (2019) stated that “…our capacity for creativity and problem solving is a strong and persistent characteristic of anatomically modern humans, constantly rising to new challenges and embracing the technological possibilities of any given era” (p. 153). In mathematics education, an essential part of the research agenda has been to understand and characterize what the teachers and students’ process to comprehend concepts and to formulate and solve problem entails (Santos-Trigo, 2020a, 2020b). Thus, research programs in mathematical problem solving have contributed significantly to characterize what thinking mathematically involves and to explain teachers and students’ behaviours to solve problems (Schoenfeld, 1985). Furthermore, findings in mathematical problem-solving research have provided relevant information regarding ways to frame and support learning environments to foster students’ development of problem-solving competencies (Santos-Trigo, 2022). Recently, the COVID-19 pandemic crisis and the long social confinement led educational systems worldwide to transform and adjust learning environment to include both remote and face-to-face activities. In this perspective, the imposed social worldwide lockdown has modified not only the ways that people interact and face some daily tasks or job’s undertakings, but it has also challenged the development and implementation of professional and school tasks. Hence, extant research results and problem-solving developments need to be examined and interpreted in terms of what changes and adjustments are required to structure post-confinement mathematics teaching/learning scenarios (Santos-Trigo et al., 2022a). Specifically, it becomes important to analyse the role and importance of using digital technologies and online developments for teachers and students to work on mathematical tasks (Foster et al., 2022). During the lockdown, teachers and students relied on digital apps to continue educational goals and to carry out school activities. Trouche et al. (2020) pointed out that the use of diverse resources in learning environments produces constant adjustments to teaching practices: “Teachers search for resources, select resources and modify them; they use them in class, and this can lead to further modifications” (p. 1244). In this perspective, teachers adjusted their instructional model to present contents and to foster their students’ participation through communication apps and looked for various ways to monitor and assess the students’ work and problem-solving competencies (Santos-Trigo, 2022). In some cases, class sessions were video-recorded, and students had an opportunity to review what they had studied during the sessions. Likewise, some students

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consulted online materials and platforms to revise or extend their understanding of mathematical concepts or looked for examples of solved problems to work on proposed tasks. In addition, teachers and students engaged in synchronous or asynchronous discussions to address class issues and to share ideas and problem solutions. Hence, teachers and students extended their ways and strategies to approach mathematical tasks. For instance, they relied on platforms or online resources and digital apps to understand concepts, share ideas, and discuss mathematical tasks. In general terms, learning spaces, in post pandemic confinement, are being extended to integrate both students’ face-to-face activities and remote or online work. Atweh et al. (2022) pointed out that “mathematics teacher education faces the challenge of taking the lead in re-imagining alternative possible futures, commencing with re-engaging with the question of the purposes of mathematics education” (p. 3). What digital technologies and online developments are important for teachers and students to learn mathematics and to develop problem-solving competencies? How should learning environments be structured to provide students with an opportunity to work on face-to-face and online activities to construct and understand mathematical concepts or knowledge and to solve mathematical problems? What type of assignments should students work on in a remote or online environment and what type of tasks they should discuss in face-to-face activities? To address and discuss these types of questions in terms of problem-solving research and results, four interrelated themes are examined: (A) The importance of tools (concrete and abstract or semiotic tools) in the development of mathematics. I argue that the use of tools shapes the formulation of mathematical problems and the ways or methods to reason and solve those problems; (B) a review of seminal problem-solving conceptual frameworks that explain and characterize students’ processes to work on mathematical problems; (C) problem-solving principles and basis to frame learning environments for students to understand concepts and to solve problems including the systematic use of digital tools and online developments to work on mathematical tasks; (D) ways to assess and monitor students’ understanding of concepts and problem-solving competencies.

2 The Role of Tools, Methods, and Results in Mathematics Developments The purpose of this section is to argue about the importance of characterizing the extent to which the individuals or group’s use of tools permeates their ways of reasoning and solving mathematical problems. That is, Mathematics developments and results go hand-in-hand with the ways in which the mathematics community formulates and represent, explore, and work on mathematical problems. Cai and Hwang (2019) pointed out that advancements in mathematics and sciences involve the formulation of significant and interesting problems. Thus, mathematical contents and results can be traced and explained in terms of what problems were posed and pursued throughout the history of the discipline and the methods used to approach and solve

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them (Santos-Trigo, 2020a, 2020b, 2022). Furthermore, how individuals or groups formulate and solve mathematical problems have been an area of interest and research in the field of mathematics education. Hence, to elaborate on ways of reasoning involved in mathematical practices, it becomes relevant to address and discuss what tools1 and how individuals or groups used them to approach mathematical problems. The social context, tools, and methods used to deal with mathematical problems shape and influence the development of the discipline. Trouche (2016) pointed out that “…the process of creating artefacts and the process of creating mathematics feed one another” (p. 133). Arcavi (2020) recognizes that the use of tools extends and increases the power of human cognition to approach and solve problems: “Since pre-historical times, a central occupation of humans was the design, creation, use and improvement of tools in order to transcend inherent limitations” (p. 421). For instance, tool affordances can facilitate data analysis, the representation and exploration of mathematical objects and relationships, carrying out complex calculations or operating symbols, and visualizing concepts and relations. Monaghan and Trouche (2016) pointed out a distinction between an artefact and a tool: An artefact is a material object, usually something that is made by humans for a specific purpose, e.g. a pencil. An artefact becomes a tool when it is used by an agent, usually a person, to do something. …The materiality of an artefact is not just that open to touch. An algorithm, e.g. for adding two natural numbers, is an artefact and it is material in as much as it written and can be programmed into a computer. (p. 6)

Hence, artefacts also include abstract or semiotic developments such as the Cartesian system or a formula to find roots of quadratic equations. The difference between an artefact and a tool is relevant to explain how and when an artefact is transformed by the user to become a tool or an instrument for solving mathematical problems. That is, the user’s tool appropriation is achieved in terms of the user construction of a cognitive scheme to activate tools affordances to solve mathematical problems. Trouche (2016) discussed mathematical developments of Mesopotamian schools in terms of tools such as tablets, tokens, tables, etc. to do and learn mathematics (Scribal schools). He recognized that Mesopotamian mathematical practices relied on the use of clay tablets and tokens to develop computational methods. Empirical methods to find numerical patterns to solve practical problems (finding land areas or tax payments) appear in clay tablets. …manipulating numbers through tokens, memorising tables and intermediate results, developing and using highly structured algorithms dedicated to specific mathematical tasks, expressed in a very few lines for saving place on clay tablets. (Trouche, 2016, p. 134)

Hence, ways of reasoning associated with the development of Mesopotamian mathematics privileged the use of concrete objects and tools to find arithmetic and geometric patterns that were validated and supported with corresponding calculations. 1

The term tool is used to refer to artefacts (concrete or symbolic ones) that individuals transform into an instrument to represent, explore, and solve mathematical problems.

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The Greek mathematics relied on the use of straightedge and compass to work on geometry problems which were central to Euclid’s development of the axiomatic method to present, support, and validate mathematical results. Arzarello et al. (2021) stated that “In Euclid’s masterpiece, the Elements, no real, material tools are envisaged; rather their use is objectified into the geometrical objects defined by definitions and axioms” (p. 99). They go on to mention that “From the perspective of classical geometry, drawing tools, despite their empirical manifestation, may also be conceived as theoretical tools defining a particular geometry” (p. 100). The Greeks also focused their attention to the three classical geometric problems: squaring the circle, trisecting an angle, and doubling a cube. Attempts to solve these problems led the mathematics community to develop new contents and results in different areas such as theory of equations and algebra. Further, Descartes (1596–1650) introduced the coordinate system to algebraically represent and solve geometry problems (analytic geometry). The use of the Cartesian system opened a new window to represent, explore, and analyse mathematical relationships and became a powerful tool to model and work on problems that involve optimization phenomena. That is, the tools used to represent, explore, find, and analyse mathematical relationships not only shape ways of reasoning to work on mathematical tasks, but are also important to visualize and present problems solutions. Descartes’ contributions also include his attempt to find a universal method to solve problems. In Rules for the direction of the mind, he outlined rules to work on complex problems in areas such as mathematics, science and philosophy. In general, tools include material artifacts such as straightedge and compass, or abstract and semiotic tools or objects like the Cartesian system or formulae to solve equations or digital apps (GeoGebra) whose affordances are important to represent and solve mathematical problems. What is relevant in using the tools to deal with mathematical problems? The subject’s tool appropriation is important, and it involves analysing ways in which subjects represent, reason, and solve mathematical tasks. Hence, it becomes central to analyse ways in which learners rely on different tools to understand concepts and to develop mathematical problem-solving competencies. “The introduction of digital tools into the everyday practice of school mathematics brings perturbation to, and adaptations of, established relations between teacher, students, subject and tools…” (Ruthven, 2022, p. 8). Indeed, he proposed a framework to analyse the integration and use of digital technologies in teaching practices in terms of three interrelated dimensions: (a) the ergonomic perspective that accounts for the interaction between humans, tools and learning environments; (b) the epistemological dimension refers to the influence of tool affordances to both subject and didactical knowledge; and (c) the existential dimension refers to the student self-conception and about the subject to use the tool.

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3 Conceptual Frameworks in Mathematical Problem-Solving Approaches The idea of explaining how the mathematics community formulates and solves mathematical problems and characterizing ways in which teachers and students develop problem-solving competencies has been an area of interest in mathematics and mathematics education (Polya, 1945; Schoenfeld, 1985, 2020, 2022). To this end, both the mathematics and mathematics education communities have contributed significantly to categorize and relate the process involved in solving mathematical problems and ways to learn the discipline. Polya (1945) proposed, based on his own experience as a mathematician, four interrelated phases to organize the process to understand concepts and to solve mathematical problems: (i) Understanding the problem, that is, it is important for teachers/students to identify what data or information is given, what is the goal, and how data are related. (ii) Devising a plan, understanding the problem provides elements for learners to make a solution plan. Here, it is relevant to look for similar or familiar problems to think of a solution method or to explore particular or simpler cases to find patterns or introduce auxiliary elements to visualize mathematical relations. (iii) Carrying out the plan implies checking steps and involved operations including consistency of units of involved objects’ attributes (area, speed, lengths, angles, etc.). (iv) Looking back includes analysing and examining the result, revising arguments to support the solution, analysing the extent to which the method used to achieve the solution can be applied to solve other problems and to extend the initial problem and to formulate new ones. In addition, Polya emphasizes the importance for learners to rely on an inquisitive approach to delve into concepts and to solve mathematical problems. Specifically, he introduces a set of heuristic strategies that are useful for students to think of and work on problems throughout the four problem-solving phases. Polya (1945) illustrates, with several examples, the questions that learners can ask to understand and make sense of problem statements, to design and implement a solution plan and to look back at the solution process. Some questions involve the use of heuristics to both approach the problem and to overcome difficulties that might arise during the students’ attempt to advance and solve the problem. Thus, problematizing the subject means that students constantly pose questions to work on mathematical tasks and to develop mathematical thinking. Even the study of basic mathematical objects might become a source to formulate questions and explore mathematical concepts and relations. For instance, the study of triangles (properties and attributes) appears in elementary school. Focusing on ways to construct some triangles based on the consideration of some basic elements of the figure might lead

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learners to think of a family of problems in which they will have an opportunity to activate different concepts and resources to solve them. The next task that involves the construction of a triangle (Polya, 1945), is used to illustrate problem-solving phases in terms of questions that are relevant to understand and analyse the given information and to think of a simpler case to solve the problem. The problem statement also provides directions on how students can formulate problems based on problematizing elements of a mathematical object (triangle) and its construction. The questions posed throughout the problem-solving phases provide directions for students to identify relevant data and strategies to work on the problem. Construct a triangle, being given a side a, the altitude h perpendicular to a, and the angle ∝ opposite to a (Polya, 1945, p. 80). (i) Understanding the problem and data representation. What data are given? How are they related? Can you sketch possible ways to arrange the given data to construct a triangle? Can you start the construction with the given angle/side? Is it possible to draw a segment AB in such a way that point A is situated on ray FE and point B on ray FG of angle EFG? (Fig. 1). (ii) Solution plan and its implementation. Can you start the triangle construction with one of the given elements, for example the angle? The idea is start with a simpler case to draw the possible triangle. Figure 2 shows the given angle EFG. How to add the other elements or conditions? Point P is any point on ray FE, a circle with centre P and radius side a is drawn. This circle intersects ray FG at Q. Thus, triangle PQF holds two conditions of the triangle, the given angle ∝ and the side a. Thus, to introduce the height datum in this representation, a perpendicular line to line PQ that passes by vertex F is drawn. And segment HR corresponds to the given heigh of the triangle. Then, to get the solution is to find a position for vertex F in which triangle PQF has its height from F to side PQ the length h. Fig. 1 Representing the given data

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Fig. 2 Reducing the triangle construction to finding a position for vertex F

How to move vertex F in such a way that the measure of angle EFG remains constant? Is there any relation between the circle that circumscribes triangle PQF and the position of vertex F that keeps a constant arc? How to draw a circle that passes through the three vertices PQF? Fig. 3 shows the circle with centre the intersection of side perpendicular bisectors that passes through the three vertices. This circle intersects the parallel line to PQ that passes by point R at S and T. Then triangles PQS and PQT hold all the asked condition (solution). (iii) Another approach. Polya (1945) states that it is always important to look for different ways to solve a problem. Again, thinking of a simpler case that involves fewer condition to draw the triangle becomes relevant to explore other approaches. In the previous construction the given angle was the initial element to draw the triangle, what about if it starts with the side a? The use of GeoGebra provides affordances to construct a dynamic model of the problem. Figure 4 shows side a, line BL is any line that can be moved by moving point L around the circle with centre at B. BJ is drawn by rotating line BL around point B an angle ∝ of 38° (the given angle). From point A, a parallel line to BL is drawn and this line intersects line BJ at P. Here, triangle APB holds two of the given conditions, the angle, and the side. What is the locus of point P when point L is moved along the circle with centre at B? Fig. 5 shows that the locus is a circle. To introduce the altitude condition, a perpendicular line k to AB that passes by point A is drawn and segment AK on the perpendicular is congruent with the altitude

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Fig. 3 Inscribing triangle PQF to find triangles PQS and PQT that hold all given conditions

Fig. 4 Drawing triangle ABP that holds two given conditions

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Fig. 5 The locus of point P when point L is moved along the circle is a circle

h. The parallel line to AB that passes through K intersects red circle at points M and N. The triangles ABN and ABM have all the required conditions (Fig. 6). To achieve this solution, a set of heuristics associated with the use of GeoGebra such as dragging objects orderly, tracing loci, and moving lines on the plane became crucial to construct the triangle. (iv) Looking back. It is observed that the problem statement does not include specific values for each given element, but in the graphic representation, each one has specific value. For what values of side a, angle ∝, and height h is it possible to guarantee that the triangle can always be constructed? By changing the length of the altitude CD, and keeping the side AB and angle ∝ fixed, it is observed that the possible values of the altitude to construct the triangle should not be greater than the length of segment QR, where QR is the perpendicular bisector of AB (Fig. 7). This task illustrates that there are different ways for students to engage in problemposing activities. The case involved focusing on a side, an altitude, and an angle as given data to draw the triangle; but what about if the given data includes the permitter of a triangle isosceles and one of its altitudes, how to construct that triangle? English (2019) stated that “It seems that the crux of learning to teach through problem posing is how to effectively design or adapt tasks (with associated questions) that are of

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Fig. 6 Triangles ABN and ABM hold all given conditions

Fig. 7 Exploring the domain for the altitude h to guarantee the triangle construction

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high cognitive demand and foster students’ mathematical thinking and development” (p. 1). In this case, the situation involves three elements of a triangle, and the problem is to construct it with those elements. Thus, a family of construction problems might emerge by considering different elements of figures and asking for ways to form or draw such figure. This idea can also be applied to other figures, for example, can you construct a square ABCD if you are given a point R on one side of the square and its centre of symmetry? Or what about if two opposite vertices of a square ABCD are given, can you construct that square? etc. A Dynamic Geometry System (GeoGebra) was used to represent and to explore relations between the given elements in the triangle task. The use of GeoGebra allows students to model accurately the problem and to move elements within the model to find and empirically test properties and relations of model elements. Dragging objects within the model becomes an important heuristic to identify and explore patterns and relations to solve the problem. Summary section: The four phases that Polya proposes to approach and solve mathematical problems emphasize the importance for learners to problematize the solution process. That is, questions are the vehicle for students to analyse and make sense of problem statements, to think of a plan and its implementation, to review the solution process and to extend the initial domain of the problem and solution methods. Within the complexity involved in characterizing the essence of the discipline and what is relevant to understand concepts and to solve mathematical problems, Polya’s work shed lights on providing a framework to orient and guide the students’ development of problem-solving competencies. Polya’s contributions have inspired the development of research programs in mathematical problem solving in mathematics education worldwide (Schoenfeld, 2015; Liljedahl & Santos-Trigo, 2019). Schoenfeld (1985) designed and implemented a research program that involved a mathematical problem-solving approach to foster and guide university students to use heuristic methods to work on nonroutine problems. During the problemsolving sessions, students worked on tasks individually, in small groups and shared and discussed their approaches within the entire group. The analysis of students’ work led Schoenfeld to propose a conceptual framework that explains students’ problem-solving performances in terms of four intertwined dimensions or categories: (i) The students’ ways to identify, access, and use knowledge base (facts, procedures, and skills) and resources to work on the problem; (ii) the use heuristic methods or cognitive strategies such as analysing simpler or specific cases, looking for similar problems, drawing figures, working backward, etc. to understand and make sense of problem statements and to design and implement a solution plan; (iii) the students’ use of metacognitive or self-control and monitoring strategies to decide what plan and solution actions to take including when to abandon certain direction during the solution process; (iv) Students’ belief systems that reflect their conceptions and mathematical views that shape their way to approach mathematical problems. That is, students’ beliefs set up the context and ways to activate their knowledge base, the heuristic, and control strategies to solve the problems.

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In terms of Schoenfeld’ framework, the resources that are relevant to work on the triangle construction task include knowledge regarding ways to draw the triangle height, perpendicular bisector of sides, and angle bisector, how to circumscribe a triangle and properties of inscribed angles. Reducing the problem condition, considering only two instead of three conditions and focusing on finding a position of one point (vertex), and dragging elements within the model are key heuristics to approach the triangle construction. Subject’s belief system shapes decisions regarding what path to explore and when to abandon or to reconsider a different approach including different ways to solve the task. For instance, if students believe that any mathematical problem can be solved by using a formula or rule, then their approach to the problem will involve looking for such formula to solve it. Or if they believe that any problem can be solve in five or ten minutes, then after that time they abandon the problem, etc. Lesh and Zawojewski (2007) introduced model-eliciting activities to emphasize that learning mathematics takes place through modelling processes in which learners get engaged in interactive cycles to interpret, represent, explore, and refine models in approaching a realistic complex situation. Students begin their learning experience by developing conceptual systems (i.e., models) for making sense of real-life situations where it is necessary to create, revise, or adapt a mathematical way of thinking (i.e., a mathematical model). …students are expected to bring their own personal meaning to bear on a problem, and to test and revise their interpretation over a series of modelling cycles. (p. 783)

Lesh and Zawojewski (2007) identify steps or actions associated with the students’ interactive model cycles to work on model-eliciting problems. Description involves the initial construction of a model of a problem situation in which learners relate the real world and the modeled world. Another step consists of manipulating and operating the mathematical model within the context of real situation to interpret and assess the pertinence of the model. Prediction is relevant for learners to connect the model solution with the real world and to verify or assess the consistency and usefulness of the prediction within the real-world situation. Lesh and Zawokewski recognize that when students work on model-eliciting activities, they refine their initial interpretation of the problem based on small group discussions to contrast model solutions that are expressed through written or oral language, graphs, tables, etc. An important goal for students to work on modeleliciting activities is to identify what ideas, knowledge base and ways of thinking they bring to bear on the problem and that lead them to engage into modelling cycles to refine, extend, their ways of thinking and solving those problems. What bases, principles and common grounds and differences distinguish or are relevant in Polya, Schoenfeld and Lesh’s frameworks to support learning scenarios that foster students’ problem-solving competencies? Polya’s framework sheds lights on ways to approach mathematical problems based on his own experience to work and develop mathematics. Schoenfeld implemented a research program to document the extent to which a problem-solving approach explicitly addresses the students’ use of heuristic strategies to solve

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mathematical problems. Thus, while Polya supported and validated main phases to structure problem-solving process through his own experience in developing and solving mathematical problems (retrospective and introspective approaches); Schoenfeld collected and analysed data regarding how university students behaved and approached problems during a problem-solving course that enhance the use of heuristic strategies. Hence, students’ problem-solving performances are explained in terms of what resources, cognitive and metacognitive strategies and belief systems bring to bear on problem solutions. Likewise, Lesh’s work is supported through multiple research studies that involve the design and implementation of modeleliciting activities with teachers, students, and researchers. Lesh privileges the use of contextualized or complex tasks to engage students in modeling-cycles activities to develop mathematical concepts and to constantly refine models to work and solve the tasks. The three problem-solving frameworks or developments have been seminal and useful to organize curriculum proposals and to support and implement problemsolving approaches. Schoenfeld (2020) has updated his research framework to support mathematical instruction. He outlined five dimensions of powerful mathematical classrooms, identified as Teaching for Robust Understanding TRU framework: (1) [the mathematics or content, that refers to]…the content and practices in which the students engage are mathematically rich; (2) [Cognitive demand] students engage in sensemaking and “productive struggle”; (3) [Equitable access] all students engage equitably with the core content and practices; (4) [Agency] the environment provides students opportunities to develop a sense of agency, take ownership of the mathematical content, and build positive mathematical identities; and, (5) formative assessment shapes ongoing classroom practices in ways that adjust to “meet the students where they are”. (p. 1171)

Recently, Foster et al. (2022) argued that the TRU framework dimensions are students-centred and require a new didactical contract to make explicit students and teachers’ responsibilities and commitments to understand and solve problems. Thus, the students’ role involves monitoring their own learning, explaining, and sharing their ideas with peers and teachers, and listening to other students and posing questions to the group and their teachers. Indeed, this didactical contract is consistent with the activities that students engage in learning spaces that combine remote and face-to-face instruction. Li et al. (2014) pointed out that “…the emphasis on cognitive challenges of mathematical tasks should help readers to think about the use of specific approaches and practices” (p. 6).

4 Problem-Solving Principles and the Use of Digital Technologies What has the mathematics education community learned from the social confinement experience in organizing and structuring learning environments that integrate both students’ remote work and face-to face mathematical discussions? How does the

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teachers and students’ systematic and coordinated use of diverse digital technologies and online developments provide affordances to reconcile or complement students’ face-to-face problem-solving interaction and online work and assignments? SantosTrigo (2022) argues that in a hybrid learning environment, strategic and tactical plan and decisions are relevant for teachers to structure and orient teaching practices and for students to engage in problem-solving activities. A teaching strategic plan involves teachers’ decisions to assign and follow up students’ online work and what activities they should work on during their face-toface interaction. That is, students are guided to consult specific online platforms or encyclopedias to revise concepts or contextualize problem statements (Santos-Trigo et al., 2021). While a tactical instructional plan includes decisions regarding the type of online assignments and tasks that students will work, ways to address and discuss mathematical tasks in face-to-face interaction and making explicit how students’ work will be assessed. In terms of students’ decisions and actions: …a strategic plan helps them identify what resources or online developments to consult, what digital tools are important to use; how and with whom they should interact to understand concepts and to solve problems; what material needs to be revised, etc. While a tactic plan involves the actual actions that students take and perform to understand concepts and to develop problem-solving competencies. It involves the activation of technology affordances to model, explore, solve problems and to communicate results. It also includes the use of technology apps to discuss ideas and to share mathematical results. (Santos-Trigo, 2022, p. 44)

Mathematical tasks are central for students to engage in mathematical discussions and to stimulate cognitive conflicts to foster questioning, reformulations, generalization, and to build multiple connections (Santos-Trigo et al., 2022b). In this process, students rely on the use of digital tools and online developments to represent, explore, solve, and extend mathematical tasks and to present, share, and discuss their task approaches and solutions with peers and the group. Monaghan and Trouche (2016) pointed out that “The increasing presence of digital technology in everyday life and work opens up new opportunities (and problems) for linking in-school to outof-school mathematical activities” (pp: 265–266). Santos-Trigo (2022) argues that even routine tasks can be a departure point for students to engage in mathematical thinking. For instance, with the use of GeoGebra, students can construct dynamic models of those problems and explore mathematical relationships associated with the behaviour of some elements within the model. Hence, to transform and extend the initial nature of the tasks, students rely on questions to delve into concepts, resources, and strategies to make sense of the problem statement, to represent and to look for mathematical relations to solve and extend the initial problem. Likewise, looking for different ways to represent and solve a problem provides an opportunity for students not only to think of different concepts and strategies to approach the problem, but also to contrast strengths and limitations associated with the solution paths. Here, students should also discuss the extent to which the methods used to solve the problem could be applied to solve other problems. McKnight et al. (2016) reported that the use of technology is transforming teachers’ practices in terms of “transforming learning routines, which includes accessing advanced learning resources and content, igniting

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cognitive processes that enhance learning (e.g., active inquiry vs. memorization), and changing teacher roles from delivery of content to facilitator or learning coach” (p. 206). A pathway for teachers to integrate the systematic use of technology affordances in their teaching practices is that they directly get involved in mathematical problem-solving experiences to work on and discuss the extent to which the use of digital technologies opens up novel ways to reason, solve, and extend the problem domains (Santos-Trigo, 2022). Next, a task that involves the construction of a rectangle is discussed to illustrate the importance of problematizing the learning process and to look always for different ways to solve tasks during both students’ remote and face-to-face work. The goal is to sketch a route for teachers and students to rely on digital technology affordances to work on mathematical problems and to understand concepts. What is a rectangle? How to find its area/perimeter? Is a parallelogram with equal diagonal a rectangle? Etc. In general, students are encouraged look for information to contextualize the task, in this case, they might consult online platforms such as Wikipedia to review properties and theorems associated with this figure. The idea is that students share and discuss with peers what they have explore about concepts involved in problem statements previous to address the task in face-to-face work (in classrooms). Question: Can you draw a rectangle if you are given its perimeter and its diagonal? How can the given data be represented? What is the goal? How is the rectangle diagonal related to the rectangle sides? Is the sum of two rectangle sides always larger than the diagonal length? (i) a dynamic model. In Fig. 8, segment AB is half of the given perimeter, CD is the given diagonal. Point P is any point on segment AB. Points E and F are the interception points of the circle with centre at C and radius AP and the circle with centre at D and radius PB.

Fig. 8 Geometric representation of given data

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When point P is moved along segment AB the path generated by point E and F is shown in Fig. 9. The path or locus of F and E is an ellipse since the sum of CF and DF is constant (segment AB). A family of triangles CDF appears when point P is moved along AB. The idea is to find a position for point P in which triangle CFD or CED becomes a right triangle. To achieve this, a circle with centre at the midpoint of segment CD and radius MC is drawn, then the intersection points of this circle with the ellipse are the vertices to construct two rectangles that fulfil the given conditions (Fig. 10).

Fig. 9 The locus of point E and F when point P is moved along AB is an ellipse

Fig. 10 The use of an ellipse to solve the task

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Fig. 11 Rectangles ALEJ and AKGM fulfil the given conditions

(ii) The construction of a family of rectangles with a fixed perimeter. In Fig. 11, segment AB represents half of the given perimeter (semi-perimeter) of the rectangle and point P is any point on segment AB. A rectangle APQR is drawn with side PQ congruent with PB, then the perimeter of rectangle APQR corresponds to the given perimeter. The locus of point Q when point P is moved along segment AB is the segment BF. To introduce the diagonal length, a circle with centre A and radius the diagonal value is drawn. Points E and G are the intersection of the circle and segment BF, the required rectangles are ALEJ and AKGM (Fig. 11). It is observed that when point P is moved along segment AB a family of rectangles with a constant perimeter is generated. To delve into the family of generated rectangles implies examining how some attributes (area, diagonal, ratio of sides, etc.) of those rectangles behave. Point H has its x-coordinate the length of side AP and y-coordinate the area of rectangle APQR. Here, by moving point P on segment AB, the area associated with the corresponding rectangle varies and a question might be posed: From the family of rectangles with fixed perimeter, which one has the maximum area? To graph the area behaviour associated with the family of rectangles, the locus of point H when point P is moved along segment AB is drawn and it represents the graphic representation of the area variation of the family of rectangle with the given perimeter. With the tool affordances, it is possible to examine properties of the graph. A tangent line to the graph at point H is drawn and the slope of this tangent line is determined (Fig. 12). The tangent values are important to analyse when the curve is

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Fig. 12 Representing the area variation of the family of rectangles with fixed perimeter

increasing/decreasing or changes its concavity. It is observed that the slope of the tangent line varies when point P is moved along segment AB. Figure 13 shows that when the value of the slope of the tangent line to the graph is zero, the area reaches the maximum value and corresponds to a square with sides half of segment AB. (iii) The algebraic approach. To model the problem algebraically, if x and y are the sides of the rectangle, then: P 2x + 2y = P or x + y = 2 √ 2 2 d= x +y Solving these equations for P = 8 and d = 3.28 and representing the solution graphically (Fig. 14). Comment: The question addressed in this task involves problematizing the study of mathematical objects and their properties and attributes. As in the triangle task, the idea is to focus on elements of a particular mathematical object and to ask about ways to construct it in terms of the given elements. The use of GeoGebra offers affordances to dynamically model the task in terms of embedded concepts. For instance, the first approach that starts with the construction of a triangle (in which the construction of the rectangle is reduce to the construction of a right triangle) with one side (the given

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Fig. 13 The area maximum area of the family of rectangles is obtained when the slope of the tangent line at H is zero and the rectangle becomes a square

Fig. 14 Graphic representation of the solution of the system of equation for specific values of the perimeter and diagonal

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diagonal) and the sum of the other two sides equals to the semi-perimeter leads to generate a family of triangles with one vertex on an ellipse which is the locus of such vertex when point P is moved along segment AB. To decide where point F (Fig. 10) should be located for angle AFB to measure 90°, a circle with centre the midpoint of AB and radius MA or MB was drawn and the intersection of this circle with the ellipse provided the position for the vertices of the triangle to becomes a right triangle. Clearly, the tools affordances plays an important role to connect concepts such as conic sections, drawing right triangles in a circle, and inscribed angles to solve the task. The second approach involves drawing a family of rectangles with a fixed perimeter and to connect a circle with radius the diagonal of the rectangle to find the task solution. The dynamic model of the family of rectangles with fixed perimeter led to explore a variation task to find what element of that family of rectangles reaches the maximum area. These dynamic approaches are relevant to represent and making sense of involved concepts geometrically and to connect concepts that traditionally are studied in different courses, conic sections in analytic geometry and optimization problems in calculus. In general terms, GeoGebra affordances open new paths for learners to engage in mathematical thinking. Hence, students think of mathematical objects in terms of properties that can be represented and explore through the activation of GeoGebra commands and affordances. In this process, canonical heuristics such as thinking of simpler problem or exploring particular or special cases are enhanced with the use of the tool. Furthermore, new heuristics that include, dragging objects within the model, tracing loci of objects, measuring objects attributes, etc. become crucial to find and explore mathematical relationships that are relevant to solve the task. The solution process shown in addressing the triangle and rectangle tasks emphasizes the importance for teachers and students to engage in an inquiring approach to understand mathematical contents and to solve problems. To contextualize and review the study of triangles, students might consult online developments to discuss themes related to elements and types of triangles, construction or existence of triangles (conditions), points, lines, inscribing or circumscribing a triangle, etc. Similarly, to review and extend concepts that appear in the rectangle task, students also can consult available platforms that offer short videos that address and explain those concepts and ways to solve related problems (Santos-Trigo et al., 2021). The idea is that teachers identify what online developments students should consult to review the involved concepts and to monitor their students’ learning. Indeed, some platforms include quizzes and problems for students to solve. Engelbrecht and Oates (2021) stated that “Online resources that are now at the disposal of our students enable them to direct their own learning, collaborating with teachers and fellow students all over the world” (p. 39).

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5 Assessing and Monitoring Students’ Problem-Solving Competencies Lesh and colleagues’ model-eliciting activities explicitly address the importance for students to reveal what they know and the resources that they access and activate to understand concepts and to deal with different types of problems. Thus, it becomes relevant for students to register what they think and do during their learning process and problem-solving approaches. Both individual and collective work become essential to monitor students’ understanding and ways of reasoning to solve problems. Schoenfeld (2020) pointed out that “what students reveal about their thinking allows teachers to adjust the level of cognitive demand, keeping students engaged in productive struggle” (p. 19). In terms of the teachers and students’ didactic contract, the idea is that students acknowledge and accept the commitment and responsibility of making explicit and sharing their work with peers and the teacher. Likewise, the teacher encourages and fosters the students’ individual and group participation and provide feedback throughout the problem-solving sessions. The idea is to keep students working productively, not simply to find errors and correct them – that is, for classroom dialogue to continue to provide students with opportunities to build upon the understandings they have developed, and to address emerging misunderstandings. (Schoenfeld, 2020, p. 19)

One way to organize and exhibit the students’ work is to provide a space for students to record and share their ideas, understanding, and learning experiences. Santos-Trigo (2022) proposed a digital wall or a problem-solving digital notebook for students to monitor and register their problem approaches and learning experiences. Digital apps such as Teams and GeoGebra provide channels or a book template for students to record their problem-solving work and experiences. The idea is that students continuously register and share their ideas, comments, difficulties, and problem-solving approaches. To this end, the digital wall, then, becomes the space for students to share their work, and is also a reflective tool in which students discuss peers’ ideas and problem solutions. Thus, students are guided to record and share their learning and problem-solving experiences around issues that include: (i) The type of questions they pose and pursue to make sense of problem statements and ways to represent the problems. (ii) Different methods used to approach and solve the problem, including a discussion of the concepts and ways in which they were used to achieve the problem solution. (iii) The construction of dynamic models of tasks and strategies that were used to identify relations to support and validate problem solutions. (iv) Online resources and digital apps that were used to work on the tasks. Including what videos were reviewed to understand concepts and to revise ways to solve similar problems. (v) Examples of problems that can be solved with the methods used to approach the initial problem.

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(vi) Posing new problems by varying initial data, extending the problem domain and ways to generalize solutions or results. (vii) Students’ short videos in which they present their problem solutions. (viii) Ways in which they considered and were influenced by peers’ ideas and work to understand concepts and to solve problems. (ix) Quizzes or tests and self-assessment they worked to monitor their own learning. GeoGebra also provides a book template for students to record their work in terms of chapters, the questions they posed, their problem solutions, concepts and strategies used to approach the problem, their self-assessment, online material, and feedback received from the teacher and peers.

6 Looking Back and Remarks Problem posing and solving mathematical problems are essential activities in the development of mathematics. The use of tools, both material or semiotic ones, shapes what problems to formulate and the types of reasoning involved to achieve problem solutions. The development of conceptual frameworks has significantly contributed to understand what is essential to characterize the process of solving mathematical problems and ways in which learners construct mathematical concepts and problem-solving competencies. A relevant issue to extend the scope of these conceptual frameworks is to make explicit what the students’ use of digital technologies brings to the way they understand concepts and solve mathematical problems. Clearly, the students’ consistent use of digital technologies extends their problemsolving approaches in terms of broadening the ways of representing, exploring, and reasoning to include dynamic modeling representations and also to share, discuss, and communicate solutions. What society lived and experienced in accomplishing tasks worldwide during the pandemic crisis altered and transformed ways to communicate, interact, move, and carry out multiples daily and professional tasks. This pandemic crisis has also brought and provided an opportunity to re-examine and update educational practices based on accumulated problem-solving research findings to support learning environments that include both students’ remote or online work and face-to-face learning activities. Santos-Trigo (2022) pointed out that: The irruption of digital technologies to organize and carry out educational tasks not only shaped and marked the implementation of school activities during the critical period of the pandemic, but also, contributed to identify resources and new ways to interact with students to foster, monitor, and assess their learning. (p. 46)

Thus, digital technology apps and online developments or platforms become a tool for teachers and students to deal with educational tasks (Santos-Trigo, 2019). What type of digital technologies should learners use to represent, explore, and understand concepts and to solve mathematical problems? What platforms and online

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developments should teachers and students consult to review and extend their concept understanding and to assess their own learning and problem-solving competencies? The selection and ways to use the digital tools implies addressing explicitly the users’ appropriation of tools affordances to understand concepts and to solve mathematical problems (Santos-Trigo, 2020a). The two tasks, discussed previously, show that GeoGebra offers teachers and students a set of affordances to dynamically model concepts and problems that not only extend the ways to observe and identify objects mathematical properties and relations, but also, the implementation of problemsolving strategies. A common principle that supports and permeates a problem-solving approach to learn mathematics is that teachers and students conceptualize the discipline as a set of dilemmas that they need to solve in terms of mathematical concepts, resources, and strategies. Thinking of dilemmas or problematizing the study of the discipline implies that students formulate and pursue questions to understand concepts and problem statements, to think of and implement a solution plan, and to extend and formulate new problems (Berger, 2014). With the use of technology, students pose questions that expand their ways of reasoning about problems and solutions. For instance, to model concepts and problems dynamically requires that students think of data and problem statements in terms of properties of embedded concepts that can be represented geometrically. Furthermore, the scope or domain of elements of dynamic models can be changed directly by varying initial values (length, area, perimeter, angle, etc.) and this variation leads to analyze the behavior of family of cases associated with the model (Santos-Trigo & Reyes-Martínez, 2019). Extending confined learning classrooms to include both remote students’ work and teacher-students’ face-to-face interaction implies that teachers not only assign the tasks that students will work in that online environment, but also, what apps and platforms the students will activate and consult to solve the tasks. In addition, course material should incorporate short lesson videos in which experts or teachers model problem-solving approaches or concept explanations. In this material, there will be examples of solved problems to illustrate the importance for students to always look for multiple ways or methods to solve a problem. Here, students are encouraged to discuss concepts and resources used to reach the solution. It could also include links to platforms in which students can consult, review, and extend their understanding of involved concepts. The students’ online work becomes part of the material that the teacher and students address during the face-to-face interaction. In this context, the work that students register in the digital wall is crucial to monitor the students understanding of concepts and problem-solving performances, including self-assessments or quizzes answers. Arcavi (2020) pointed out that “the exploitation of the opportunities and affordances of the digital media enables teachers to adapt e-textbooks (or parts of them) to different students or different classroom situations” (p. 434). Finally, the framing of a hybrid learning environment requires the design of a support system in which teachers and students can receive continual technical support to use apps and to orient students in their problem-solving approaches or answering

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specific questions. This system might also include opportunities for students to engage in synchronous or asynchronous discussions with peers and the teacher.

References Arcavi, A. (2020). From tools to resources in the professional development of mathematics teachers. In S. Llinares & O. Chapman (Eds.), International handbook of mathematics teacher education: Volume 2, tools and processes in mathematics teacher education (pp. 421–440). Koninklijke Brill NV. https://doi.org/10.1163/9789004418967_016 Arzarello, F., et al. (2021). Experimental approaches to theoretical thinking: Artefacts and proofs. In G. Hanna & M. de Villiers (Eds.), New ICMI Study Series. Proof and proving in mathematics education. https://doi.org/10.1007/978-94-007-2129-6_5 Atweh, B., Kaur, B., Nivera, G., Abadi, A., & Thinwiangthong, S. (2022). Futures for post-pandemic mathematics teacher education: Responsiveness and responsibility in the face of a crisis. ZDM Mathematics Education. https://doi.org/10.1007/s11858-022-01394-y Berger, W. (2014). A more beautiful question. Bloomsbury Publishing. Kindle Edition. Cai, J., & Hwang, S. (2019). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research. https://doi.org/10.1016/j.ijer.2019.01.001 Cropley, D. H. (2019). Homo Problematis Solvendis—Problem-solving Man. Springer. https://doi. org/10.1007/978-981-13-3101-5_13 Engelbrecht, J., & Oates, G. (2021). Student collaboration in blending digital technology in the learning of mathematics. In M. Danesi (Ed.), Handbook of cognitive mathematics (pp. 1–39). https://doi.org/10.1007/978-3-030-44982-7_37-1 English, L. (2019). Teaching and learning through mathematical problem posing: Commentary. International Journal of Educational Research. https://doi.org/10.1016/j.ijer.2019.06.014 Foster, C., Burkhardt, H., & Schoenfeld, A. (2022). Crisis-ready educational design: The case of mathematics. The Curriculum Journal, 00, 1–17. https://doi.org/10.1002/curj.159 Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Information Age Publishing. Li, Y., Silver, E. A., & Li, S. (2014). Transforming mathematics instruction: What do we know and what can we learn from multiple approaches and practices? In Y. Li et al. (Eds.), Advances in Mathematics Education. Transforming mathematics instruction: Multiple approaches and practices (pp: 1–12). Springer. https://doi.org/10.1007/978-3-319-04993-9_1 Liljedahl, P., & Santos-Trigo, M. (Eds.). (2019). Mathematical problem solving, current themes, trends, and research. ICME-13 monographs. Springer.https://doi.org/10.1007/978-3-030-104 72-6_4 McKnight, K., O’Malley, K., Ruzic, R., Horsley, M. K., Franey, J. J., & Bassett, K. (2016). Teaching in a digital age: How educators use technology to improve student learning. Journal of Research on Technology in Education, 48(3), 194–211. https://doi.org/10.1080/15391523.2016.1175856 Monaghan, J., & Trouche, L. (2016). Introduction to the book. In J. Monaghan, et al. (Eds.), Mathematics Education Library. Tools and mathematics (pp. 3–12). Springer. https://doi.org/ 10.1007/978-3-319-02396-0_1 Polya, G. (1945). How to solve it. Princeton University. Ruthven, K. (2022). Ergonomic, epistemological and existential challenges of integrating digital tools into school mathematics. Asian Journal for Mathematics Education, I(I), 7–18. https:// doi.org/10.1177/27527263221077314 Santos-Trigo, M. (2019). Mathematical problem solving and the use of digital technologies. In P. Liljedahl & M. Santos-Trigo (Eds.), Mathematical Problem Solving. ICME 13 monographs

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(pp. 63–89). Springer Nature ISBN 978-3-030-10471-9, ISBN 978-3-030-10472-6. https://doi. org/10.1007/978-3-030-10472-6_4 Santos-Trigo, M. (2020a). Problem-solving in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 686–693). Springer. Santos-Trigo, M. (2020b). Prospective and practicing teachers and the use of digital technologies in mathematical problem-solving approaches. In S. Llinares & O. Chapman (Eds.), International handbook of mathematics teacher education: volume 2, tools and processes in mathematics teacher education (pp. 163–195). Koninklijke Brill NV. https://doi.org/10.1163/978900441896 7_007 Santos-Trigo, M. (2022). Intertwining cumulated and current mathematical problem-solving developments to frame and support teaching practices. In C. Fernández, S. Llinares, A. Gutiérrez & N. Planas (Eds.), Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 35–50). PME. Santos-Trigo, M., Barrera-Mora, F., & Camacho-Machín, M. (2021). Teachers’ use of technology affordances to contextualize and dynamically enrich and extend mathematical problem-solving strategies. Mathematics, 9(8), 793. https://doi.org/10.3390/math9080793 Santos-Trigo, M., & Reyes-Martínez, I. (2019). High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies. International Journal of Mathematical Education in Science and Technology, 50(2), 182–201. https://doi.org/10.1080/ 0020739X.2018.1489075 Santos-Trigo, M., Reyes-Martínez, I., & Gómez-Arciga, A. (2022a). A conceptual framework to structure remote learning scenarios: A digital wall as a reflective tool for students to develop mathematics problem-solving competencies. International Journal of Learning Technology, 17(1), 27–52. Santos-Trigo, M., Reyes-Martínez, I., & Gómez-Arciga, A. (2022b). The importance of tasks and the use of digital technologies affordances in mathematical problem-solving approaches. In L. Uden & D. Liberona (Eds.), Learning technology for educational challenges, CCIS 1595 (pp. 113–124). https://doi.org/10.1007/978-3-031-08890-2_9 Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press. Schoenfeld, A. H. (2015). How we think: A theory of human decision-making, with a focus on teaching. In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education (pp. 229–243). https://doi.org/10.1007/978-3-319-12688-3_16 Schoenfeld, A. H. (2020). Mathematical practices, in theory and practice. ZDM Mathematics Education, 52, 1163–1175. https://doi.org/10.1007/s11858-020-01162-w Schoenfeld, A. H. (2022). Why are learning and teaching mathematics so difficult? In M. Danesi (Ed.), Handbook of cognitive mathematics (pp. 1–35). Springer. https://doi.org/10.1007/978-3030-44982-7_10-1 Trouche, L. (2016). The development of mathematics practices in the Mesopotamian scribal schools. In J. Monaghan et al. (Eds.), Mathematics Education Library. Tools and mathematics (pp. 117– 138). Springer. https://doi.org/10.1007/978-3-319-02396-0_5 Trouche, L., Rocha, K., Gueudet, G., & Pepin, B. (2020). Transition to digital resources as a critical process in teachers’ trajectories: The case of Anna’s documentation work. ZDM Mathematics Education, 52, 1243–1257. https://doi.org/10.1007/s11858-020-01164-8

Chapter 3

Problem Posing and Modeling: Confronting the Dilemma of Rigor or Relevance Corey Brady, Paola Ramírez, and Richard Lesh

Abstract In this chapter, we articulate a perspective on problem posing research grounded in our own research in modeling. We begin by identifying a shared commitment between researchers in modeling and problem posing: to honor and amplify students’ (and teachers’) mathematical agency. We believe this shared commitment can serve as the basis for a shared agenda of modeling and problem posing, going forward. However, we next identify a potential issue, which is rooted in a tendency to assume a tight coupling between problem types and individual topics in the mathematics curriculum. To illuminate the nature and significance of this issue, we consider problem solving and posing through the lens of Donald Schön (1983) study of epistemological crises across professions, and we consider how Schön’s dilemma of “rigor or relevance” applies to mathematics education. We urge a shift toward activities that highlight problematic situations and that ask students to engage with such situations by interpreting them mathematically. We argue that to achieve relevance, mathematics education needs to engage with a broader range of problem situations, to utilize a wider array of representational tools, and to embrace a more encompassing view of mathematical problem posing, problem solving, and modeling, as interpretive human activities. Keywords Problem posing · Problem solving · Modeling · Mathematical agency · Complexity

C. Brady (B) Simmons School of Education and Human Development, Southern Methodist University, Dallas, USA e-mail: [email protected] P. Ramírez Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile e-mail: [email protected] R. Lesh School of Education (Emeritus), Indiana University, Bloomington, USA © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_3

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1 Introduction With deep roots in cognitive psychology (e.g., Newell & Simon, 1972), early problem-solving research was initially focused on the response processes provoked in individuals by problems, and on the variations in these processes that corresponded to manipulation of “task variables” (Goldin & McClintock, 1984). Over time, however, this paradigm began to break down, as it became increasingly clear that problemsolving was more complex and situated than initially understood (Hegarty et al., 1995; Lester & Kehle, 2003; Schoenfeld, 1992; Silver, 1985; Stanic & Kilpatrick, 1989). More generally, shifts in the theoretical frameworks of mathematics education researchers favored a widening of the view on problem solving from informationprocessing theories toward sociocultural theories that encouraged a conception of problem-solving as situated cognition unfolding within a community of practice (Lesh & Zawojewski, 2007). Along with this widening of the view, important ideas that had first surfaced in problem solving research flourished through connections with problem posing (Brown & Walter, 1983; Kilpatrick, 1987; Silver, 1985, 1994) and with modeling (Lesh & Doerr, 2003; Lesh & English, 2005; Lesh & Harel, 2003; Lester & Kehle, 2003; Mousoulides et al., 2008; Zawojewski, 2013), illustrating how these domains are quite closely intertwined. In particular, problem-solving and problem-posing research has increasingly attended to the need to invite students to mathematize (Freudenthal, 1968, 2005) authentic situations that create the need for students to respond with original constructions that involve “important and worthwhile mathematics” (Cai & Hwang, 2020) and that are “thought-revealing” (English, 2020; Lesh et al., 2000). This trend has strengthened potential connections with modeling. Moreover, problem-posing has shown the ability to bridge between student learning (e.g., English & Watson, 2015) and both pre-service and in-service teacher learning (e.g., Crespo, 2015; Ellerton, 2015), opening a range of fresh new lines of work (Singer et al., 2015). These are exciting developments. From the range of definitions for problem-posing found in the literature (cf. Baumanns & Rott, 2022), we select that of Stoyanova and Ellerton (1996): a “process by which, on the basis of mathematical experience, students construct personal interpretations of concrete situations and formulate them as meaningful mathematical problems” (p. 518). While this definition is broad, we appreciate its emphasis on interpretation and meaningfulness. We are particularly interested in activities that create the need for students to construct mathematical interpretations, and where these mathematizations reflect the understanding and meaning they have made of the “concrete situations.” In this chapter, we draw further connections between the two distinctive traditions of problem-posing and modeling, considering their views on the nature of problems and their potential role in the learning and teaching of powerful mathematical ideas. In particular, we aim to identify ways in which research in problem posing and modeling might cross-pollinate for the good of both. We begin by identifying a

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common motivation across the two traditions, around a goal to foster students’ mathematical agency, to use mathematics to interpret their world in meaningful ways. However, we then show how this goal can be inhibited when problems are seen as “carriers” of individual topics in the mathematics curriculum, tailored to elicit specific ideas or structures univocally. This leads us to ask “What is a problem?” for mathematics learning. We consider this question through the lens of Schön’s (1983) analyses of reflective practice across professions. Using Schön’s terms, we propose that by supporting students in “taming the messes” that present themselves in realworld settings, mathematics education can respond to Schön’s dilemma of “rigor or relevance.” We close with reflections on the kinds of real-world settings that it is most important for students to engage with, to prepare themselves as creative STEM professionals or even as literate citizens of a connected and complex world.

2 Positioning Students (and Teachers) with Mathematical Agency Kilpatrick (1987) was an early advocate for encouraging students to take greater agency in problem solving, through problem posing. He not only urged researchers to attend to “problem formulating” but argued that this “should be viewed not only as a goal of instruction but also as a means of instruction” (Kilpatrick, 1987, p. 123). In doing so, he noted that “many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (124). Similarly, the 1991 Standards of the National Council of Teachers of Mathematics made room for this kind of work, identifying the importance of activities where “students design their own explorations and create their own mathematics” (NCTM, 1991). And Silver (2013) noted that the 1991 National Statement on Mathematics for Australian Schools similarly specified that students “should be expected to…pose and attempt to answer their own mathematical questions” (Australian Education Council, 1991, p. 39). Thus, problem-posing (within one’s own investigations) was from its beginning motivated in part by finding ways to increase students’ (and teachers’) mathematical agency, by connecting them with the richness of authentic mathematical work. But the use of this term in the literature illustrates that degrees of agency observed in classrooms can in fact vary along a dimension separate from, but related to, the level of authenticity of the mathematical enterprise that the class engages in. In particular, students within either “didactic” or “discussion based” classrooms (Boaler & Greeno, 2000) can experience varying degrees of “exercised agency” (Gresalfi et al., 2009). And this is true even though in both types of classrooms that Boaler and Greeno (2000) described, the authenticity of the mathematical enterprise was limited to absorbing and applying concepts on a specified curricular trajectory. A key opportunity in connecting problem posing and modeling thus arises with the goal of increasing the authenticity of mathematical activity, so that students develop new mathematical constructs in response to situations that create the need

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for a general class of construct (Lesh & Harel, 2003; Lesh et al., 2008). In such contexts, problem posing for oneself can take center stage, as an intrinsic aspect of collaborative groups’ work in mathematizing problematic situations in the world. The research on model-eliciting activities, or MEAs (Lesh et al., 2000) has focused on such settings. In particular, two illuminating examples described extensively in the literature are the Volleyball (Lesh, 2003) and Shadows (Lesh & Harel, 2003) problems. In both cases, students engage in advising realistic but imaginary Clients, who face real-world dilemmas. In Volleyball, the Clients are organizers of a volleyball camp, who want to divide camp participants into fair teams, based on a variety of tryout information. These include both quantitative data (such as height and vertical leap) and qualitative data (comments from their regular-season coaches and nondichotomous descriptions of outcomes of a spiking drill). In working to address the organizers’ needs, students often explicitly or implicitly pose problems like, “How can we quantify volleyball-playing skill, so as to rank players?” In pursuing such problems, students encounter the data-modeling challenges of constructing a measure or a statistic. In Shadows, the Clients are designers of an optics exhibit at a children’s museum. This MEA begins with a definite problem (determining which of a set of figures can be oriented to cast a square-shaped shadow). However, this question is situated within the broader goal of helping the exhibit designers, and this encourages students to frame their own generalizing problems. For example, they may ask themselves, “Under what conditions can one shape cast a shadow that is another shape?” or “How can you be certain that a given shape cannot cast a square shadow, no matter how it is oriented?” Because these student-generated problems reflect groups’ conceptions and explorations of the domain of projection transformations, we view them as reflecting a high degree of mathematical agency. A signature question for MEA research is: How can school-based modeling activities be designed to support mathematical activity as it experienced beyond school (Lesh et al., 2007)? Similarly, connecting to authentic practices in mathematical work has often been inspirational for problem-posing research as well. Moore-Russo and Weiss (2011) considered processes observed in problem-posing studies by analogy with the work of mathematicians. Cai and Cifarelli (2005) introduced a microworld (cf. Papert, 1980) as a setting where processes of exploration, problem posing, and problem solving could be interwoven. Stickles (2011) and Crespo and Sinclair (2008) found that for students to engage in effective problem-posing, it was essential for them to feel familiar with and deeply engaged in problem settings that were personally meaningful to them. Indeed, problem-posing researchers have increasingly noted that phenomenological and “insider” perspectives on problem-posing activities are useful for guiding research. Researcher responses to these perspectives include supporting participants in articulating their own aesthetic criteria for good problems (Crespo & Sinclair, 2008), investigating whether a problem is “interesting” to the problemposers themselves (Koichu & Kontorovich, 2013), and attending to how problemposing pedagogy empowers learners and transforms their relationship to mathematics by positioning them as co-designers of their educational experiences (Crespo, 2015; Koichu, 2020).

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3 Problem-Posing in Everyday Instruction As part of a Special Issue of Educational Studies in Mathematics, Ellerton (2013) articulated a rationale for a specific approach to problem posing, enabling the practice to fit into everyday instruction as a substantive extension of problem-solving. In her “Active Learning Framework,” she articulated a model for increasing students’ mathematical agency by creating connections between students’ and teachers’ mathematical work (Fig. 1). As this model of active learning progresses from left to right, mathematical practices and concepts are increasingly appropriated by students. Their “activity” and agency increase, as they are increasingly positioned as the owners of mathematical ideas. Ellerton also demonstrates how the agency of the teacher is complementary to that of the student here. There is, in effect, a transfer of agency from the teacher (at first, holding all agency) to the student (at last, holding maximum agency). This transfer is completed in the last two (problem-posing) blocks of the sequence, where students create problems that have the same structure as those that they have been given by authority figures (the teacher, the textbook, and/or the internet), and where their classmates work through and discuss these problems. The logic of this framework is clear. The framework provides a valuable lens on both the strengths and the limitations of problem-posing research. The strengths include the respect for students’ agency, as they gradually take on authority approximating that of the teacher and create problems for an authentic peer audience. The limitations in fact lie in the restrictions on mathematical agency accorded to both students and teachers within the broader enterprise implied in the model. Just as

Fig. 1 Active learning framework. Reproduced from Ellerton (2013), Fig. 5, page 99, with permission from Springer Nature

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students are constrained to follow the structural template of example problems, so teachers are also constrained to present example problems that capture the single, isolated content topics that define curriculum units. Problems of this kind are generally limited to application problems, whereas more authentic problems typically demand that students draw upon multiple mathematical constructs from different topic areas to construct viable solutions. As we will see, this limitation in the enterprise described in the Active Learning framework is inherent in the way that problems are themselves framed with respect to mathematics learning. Here, problems appear to be cast as the “carriers” of mathematical content—each “new content topic” in a course ushers in a new problem type, whose structure is the common factor across the “model example” of block 1, the “textbook/internet” examples of blocks 2 and 3, the problems based on the “model problem” in block 4, and the student-posed problems of blocks 5 and 6. For another perspective on the limitations imposed by a restricted definition and role for “problems,” consider Silver’s (2013) reflection on problem-posing research, from the same ESM Special Issue. He expressed concern about the loss of precision in research if the same term is applied to (a) teachers posing problems for their students, or (b) for themselves; and (c) students posing problems for their classmates (Silver, 2013, p. 159). However, in discussing Ellerton’s framework, we have seen the potential value in drawing parallels between the mathematical work of teachers and students. Doing so also highlights the asymmetry in Silver’s account: he does not consider “students posing problems for themselves” as a fourth category of interest.

4 How Do Problems Relate to Topics in the Curriculum? Silver’s (2013) omission of students’ posing problems for themselves limits the potential of problem-posing to be experienced as an organic component of authentic mathematical work, as described above (AEC, 1991; Kilpatrick, 1987; NCTM, 1991). We suggest that this is related to how problems are implicitly conceptualized as “carriers” of key constructs in the curriculum, leading students to identify individual curricular contexts with particular problem types (see also, Ramirez, 2019). This can become a limitation, because the relationship between “concepts” and “problems” is bidirectional. Restricting attention to problems and problem-types that have the purpose of illustrating the concepts of the curriculum implicitly suggests that the curriculum has the purpose of preparing students to solve a fixed list of established problem-types. This sets a potential trap for mathematics education, which we will examine through the lens of Schön’s (1983) dilemma of “rigor or relevance.”

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5 Interlude: Taming Messes and the Dilemma of Rigor or Relevance In his study of disciplines and professions in the twentieth century, Donald Schön described radical shifts in an array of fields, including many with theoretical bases in the health and social sciences. In these fields, an old paradigm of “Technical Rationality” (in which professional practice was strongly grounded in disciplinary theories), was undermined by an increasing sense that these theories were failing to guide practitioners as they faced the complexity of practice. In the earlier order, theoretical work was the foundation and warrant for determining which questions were valid and researchable problems in the field. In contrast, in the new order, questions of practical implementation contributed materially to structuring research and determining which problems were valid and worthy of pursuit. In other words, across all of these fields of professional and disciplinary knowledge, there was a shift toward Praxis as a source of problem-identification and definition. There were fundamental changes in what counted as an important problem and how problems were structured. We argue that similar dynamics and tensions are operating now in mathematics education, between mathematics content (in the role of Theory) and problem posing and modeling (in the role of Praxis).

5.1 Taming Messes Under the old perspective of Technical Rationality, practice consisted of “instrumental problem solving made rigorous by the application of scientific theory and technique” (Schön, 1983; p. 21). Applying this to the context of mathematics education, this would correspond to framing problem solving and modeling in terms of “application” problems. A first challenge to Technical Rationality arises when practitioners notice that “[i]n real-world practice, problems do not present themselves to the practitioner as givens. They must be constructed from the materials of problematic situations which are puzzling, troubling, and uncertain.” (p. 40, emphasis added). To adapt to this view of practice, mathematics education would need to foreground the act of “constructing” problems—i.e., a form of problem posing. Indeed, Schön’s description of “problem setting”—the work needed to interpret a “mess” (a problematic situation) as an approachable problem—is also consistent with our definition of problem posing as an interpretive and meaning-producing activity: It is this sort of situation that professionals are coming increasingly to see as central to their practice. ...When we set the problem, we select what we will treat as the ‘things’ of the situation, we set the boundaries of our attention to it, and we impose upon it a coherence which allows us to say what is wrong and in what direction the situation needs to be changed.

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5.2 Rigor or Relevance A second challenge to the perspective of Technical Rationality is based in the limited set of conceptual tools that are fully covered by rigorous theory. On one hand, a practice can only be rigorous if it restricts itself to tools covered by the theory. On the other hand, it can only be relevant if it engages with the phenomena of the world most pertinent to praxis. An example of this challenge within mathematics education (specifically Calculus Reform), is the field’s resistance to numerical methods and its affinity for analytic functions. Kaput (1994) quoted Thomas Tucker in asking, rhetorically, “Are all functions encountered in real life given by closed algebraic formulas? Are any?” (Tucker, 1988, p. 16). In other words (in the context of Calculus), if we limit ourselves to closed-form functions whose derivatives and integrals we know analytically, there are very few (if any) real-world situations that we can even approach in a way congruent with real-world praxis. Schön called this second challenge the dilemma of “rigor or relevance.” He described it in terms of conceptual geographies for the professions: In the varied topography of professional practice, there is a high, hard ground where practitioners can make effective use of research-based theory and technique, and there is a swampy lowland where situations are confusing “messes” incapable of technical solution. The difficulty is that the problems of the high ground, however great their technical interest, are often relatively unimportant to clients or to the larger society, while in the swamp are the problems of greatest human concern. (Schön, 1983, p. 42)

The mathematics that forms the foundation for creative future STEM work (and even for participating as a literate citizen) involves “taming messes” in the “swampy lowland.” Returning to Calculus Reform, by expanding to numerical methods, Pat Thompson’s Project DIRACC (2019) has demonstrated how an attention to relevance can be applied at a whole-course level. In a complementary way, Kaput’s SimCalc projects illustrated how piecewise definitions, direct manipulation, imported data, and executable simulation representations could expand the modeling expressivity of functions and democratize access to the ideas of Calculus (see, e.g., Kaput, 1996; Kaput & Roschelle, 1997).

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6 What is a Problem? The problem-posing and modeling communities have the potential to collaborate to address both of these challenges—taming messes and engaging with problematic situations from the “swampy lowland” of relevant mathematics. To do so, however, it is necessary to reconsider the nature and role of problems. In describing “creative problem solving,” Liljedahl noted that inspired by Schoenfeld (1982), the community has recognized and become interested in “problems for which the solver does not have ‘access to a solution schema’” (Liljedahl et al., 2016, p. 15). In such settings, even experts are put off balance and begin to exhibit what Lesh and Harel (2003) refer to as “local conceptual development.” Through iterative efforts (posing problems and trying on possible interpretations), they get their bearings mathematically and construct coherent interpretations. Designing and analyzing such contexts demands a more open definition of “problem,” such as the one provided by Lesh and Zawojewski (2007): A task, or goal-directed activity, becomes a problem (or problematic) when the “problem solver” (which may be a collaborating group of specialists) needs to develop a more productive way of thinking about the given situation. (Lesh & Zawojewski, 2007, p. 782)

Under this definition, no task or activity is inherently a problem. Instead, it becomes problematic when it creates the need for the kind of response described by Liljedahl and colleagues. Following this approach, we do not define problem solving as a search for a procedure (cf. Newell & Simon, 1972) that will allow us to go from the “givens” to the “goals” of a problem. Instead, problem solving consists of iterative cycles, like those described by Schön, in which the problem solver seeks to construct a more productive interpretation of both givens and goals within a problematic situation. Thus, mathematical problem solving is about interpretive mathematizing—conceptualizing, describing, or explaining situations mathematically—and not simply about executing rules or procedures in a manner that approximates the performance of an “expert.” Moreover, these iterative and experimental re-framings of the “problematic situation” are in fact “problem-posing.” In this sense, posing problems for oneself is an integral component of all authentic problem solving.

7 Mathematics is Interpretive: What is a Model? Shifting the construct of the “problem” in this way offers a different means of casting some core questions of problem-solving research. It opens up the perspective that problem posing and problem solving can be occasions for students to engage consciously in developing “problem solving personae” (Lesh & Zawojewski, 2007) whose style, stance, and strategies they document, as part of becoming reflective practitioners (Schön, 1983) themselves.

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Why personae (plural) versus persona (singular)? Because taking a stance on a “mess,” to turn it into a problem, is a matter of constructing and selecting a productive way of interpreting the situation. It can involve perspective-taking, acts of “seeing-as” (Wittgenstein, 1958), and a variety of tentative “frame experiments” (Schön, 1983) that use mathematics to create alternative lenses to look at the problematic situation. Moreover, when working in a group, it can mean adopting different roles at different moments in collaborative work. Flexibility is essential. Thus, a successful problemsolver’s history does not involve the development of a single “expert” persona or identity; rather, it more closely follows the model of “adaptive expertise” (Bransford et al., 2010; Hatano & Inagaki, 1986). In this approach, the conceptual tools and frames that problem-solvers provisionally construct, adapt, or adopt, to interpret problematic situations play a central role. In the Models and Modeling Perspective (Lesh & Doerr, 2003), these constructions are called models: Models are conceptual systems (consisting of elements, relations, operations, and rules governing interactions) that are expressed using external notation systems, and that are used to construct, describe, or explain the behaviors of other system(s)—perhaps so that the other system can be manipulated or predicted intelligently. (Lesh & Doerr, 2003, p. 10)

The “other systems” in this definition are embedded in the world: problematic situations (“messes”). In the process of developing “more productive ways of thinking,” a model is judged by whether it yields a viable interpretation of a mess, making it a tractable problem.

8 The Reflective Role of Heuristics in This Approach As students gain individual and collective experiences with constructing mathematical interpretation systems (i.e., models) that make problematic situations tractable, it is important that they reflect on and develop a descriptive language for the approaches they have taken. It is here (in supporting students to make their approaches to “taming messes” a shared topic of reflection and documentation), that heuristics can play a vital role. English et al. (2008) note that Pólya’s (1945) work was understood originally in this descriptive sense, and only later were attempts made to use it prescriptively to teach “general” problem solving. Liljedahl et al. (2016) note the etymological connection between Archimedes’s famous bathtub “eureka” moment and the study of heuristics. Liljedahl then problematizes this connection: eureka moments are moments of sudden insight and reinterpretation of the world, in which people construct new strategies. They thus represent moments of moving beyond one’s pre-existing problem-solving “toolkit.” Supporting students in creative problem solving does benefit from careful thought about successful strategies. However, rather than learning such strategies in the abstract, documenting heuristics is important reflective and retrospective work, serving to externalize one’s insights and make them shareable with others. In this vein,

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various researchers have identified supports for learners to reflect upon, describe, and share their own strategies and innovations (e.g., Hamilton et al., 2007; Prediger, 2005). Under this approach, discussions of problem-solving strategies would become essential after authentic problem-solving activities, to give students opportunities to share their successes and struggles. And accounts of others’ successes and struggles can be used as supports for identifying one’s own efforts as significant and seeing how these experiences can be generalized. For instance, Brady (2023) described how teachers in a comparative Geometry course engaged in over half a semester of their own exploration, conjecture-generation, and informal proving before reading Lakatos’s (1976) Proofs and Refutations. Later, as they read the exchanges among Lakatos’s dramatis personae, they recognized the patterns of thinking that they themselves had engaged in, attributed to famous European mathematicians. This gave the teachers evidence that the interpretive frames and ways of thinking that they had devised were powerful. The distinctive personalities in Lakatos’s text mapped to mathematics problem-solving personae that had shown up in their own classroom. In this way, the heuristics work of Pólya (1945) and others can support the construction of a shared descriptive language in classrooms, which fosters the development of empowered communities of practice. Statements from heuristics do not play into this model merely as prescriptive “guidelines” for general problem solving. Instead, they provide tools for students to describe notable victories in their shared problem-solving past.

9 The Answer, Certainty, and Closure in This Approach When classroom groups of students are invited to interpret problematic situations as an essential part of their work in posing and solving problems, they produce a diversity of ways of mathematizing those situations. This means that, across different groups, students are not working on precisely “the same” problem. This can undermine beliefs of some teachers and students (as well as some researchers) about the existence of a single “correct” or optimal answer. Moreover, reflecting on the assumptions they themselves have made in “taming the mess” of the situation, students may recognize that their solution is only as valid and stable as those assumptions. Finally, when students have gone through multiple iterations of problem-framing and problem-solving to come to a satisfactory solution, they can see that this process could continue further, which undermines the idea of closure in problem solving. While these ambiguities can be unsettling for students who have associated mathematics with clarity, certainty, and closure, ambiguity and interpretation are essential components of life in Schön’s (1983) “swampy lowland.” These interpretive dimensions undermine traditional images of mathematics as a “purely deductive activity in which perfectly rigorous formal proofs are used to produce theorem after theorem” (a critique in UNESCO, 2012, p. 10).

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10 Connections Between Teachers’ and Students’ Work At this point, it should be clear that in Silver’s (2013) listing, the problem-posing done by teachers for themselves does in fact have an important analogue in students’ work. When students engage authentically problematic situations with agency, they can reframe those situations through problem-posing actions for their own benefit. Moreover, this perspective shows that it is necessary and relevant to teachers’ instructional practice to have experiences of posing problems for themselves. Far from being only an enriching luxury, such experiences pave the way for teachers to recognize the value of authentic problem-solving and to understand the nature of that work, first-hand (cf, Ball et al., 2008). Finally, the effort of reflecting on their own work to identify the problem-solving personae that they have developed can help teachers to recognize patterns in their students’ thinking and to support students in discussing their own problem-solving processes and personae.

11 The Dilemma of Rigor or Relevance in Mathematics Education When problems no longer have the role of being “carriers” for single mathematical topics, it is possible to venture into the “swampland” of problematic situations that characterize authentic mathematical work in the world outside of school. Such settings can often create the need (Lesh & Harel, 2003) for solutions that: 1. Optimize, maximize, or minimize quantities, reasoning about tradeoffs and constraints. 2. Quantify qualitative information (e.g., to operationalize constructs or to create composites). 3. Identify or posit operational systems in the world, with objects, relations and actions that enable interpretations and predictions of system-level behavior. The extensive literature on Model-Eliciting Activities (Lesh et al., 2000), provides examples of such problems being used with students of all ages, from elementary school (English & Watters, 2005; Sevinc & Brady, 2019) to university and beyond (Diefes-Dux et al., 2004). In particular, the Volleyball and Shadows problems mentioned in Sect. 2, illustrate features 1–3 above. Collaborating with problemposing researchers in constructing and analyzing such activities could help to develop a mathematics education that gives students experiences with “taming messes”—the first of the challenges we identified through Schön’s analysis. In that connection, it should be noted that such modeling activities can position students as authentic constructors of model-based solutions, using concepts from the standard curriculum, and working in traditional representational media. The mathematical innovations they create arise from combining components in novel ways and making use of interactions among them. However, to fully address the dilemma

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of “rigor or relevance” in mathematics education, we may have to reconsider the mathematical tools that we embrace.

11.1 Complicated or Complex? According to Davis et al. (2000, p. 55), the behavior of a complicated mechanical system, such as a refrigerator, is planned, directed, and determined by its internal architecture. In contrast, many social, engineered, and natural systems exhibit complexity. In complex systems, the simple behaviors of the system’s components, or “agents,” (e.g., cars in a city, animals in an ecosystem) can interact to produce emergent properties at the aggregate level (e.g., traffic jams, nonlinear population dynamics). These emergent properties can be counter-intuitive to observers and they illustrate why a complex system cannot not adequately be described by reducing it to its components (cf. Gershenson & Heylighen, 2005). Complicated situations may require students to employ advanced algebraic techniques or concepts; and they may demand subtlety in their application of these conceptual tools. Nonetheless, these complicated situations can generally be described adequately for modeling using static media and using the tools of algebra. In contrast, complex situations may involve feedback loops, tipping points, or other emergent behaviors. They thus call for representations that are dynamic and executable. And they often require mathematical tools beyond closed-form, one-way, algebraic functions. An elementary illustration of the difference can be found in the oscillating behavior of pendula. On the basis of a simple pendulum, an extraordinarily complicated but deterministic watchwork can be built; indeed, such a timepiece (built by a “divine watchmaker”) was a ruling metaphor for the ordered universe, before the advent of complexity science. However, if we merely add a second pendulum, extending from the “bob” of the first, we get a double pendulum. This is a complex system, whose future behavior is so sensitive to the “initial conditions” of its present state that precise prediction long into the future is effectively futile. All of the conceptual tools of physics and calculus needed to mathematize the simple pendulum are essential in building an understanding of the complex system of the double pendulum, but these tools are not sufficient on their own.

11.2 New Representational Tools and Approaches If we consider a shift from complicated to complex systems to be part of our response to the demand for relevance in mathematics education, we may need to engage with representational media that can adequately describe and simulate complex systems. Incorporating complexity modeling tools such as NetLogo (Wilensky, 1999) in mathematics classrooms will demand substantial curricular changes; but the expressive

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benefits are also substantial, allowing students to create executable representations of complex systems. NetLogo is an accessible complexity modeling tool, due to its “agent-based” representational infrastructure (Wilensky & Papert, 2010). In particular, rather than using differential equations to model complex systems (e.g., traffic or ecosystems), NetLogo uses “agent-based modeling,” in which students identify the “agents” that make up a system (e.g., cars and stoplights; wolves and sheep). Each agent is then given simple logic that can be expressed with computational rules. The computer then allows the modeler to create many such agents in a virtual world and “run” their interactions, causing systems-level properties to emerge (e.g., good traffic flow or traffic jams; stable ecosystems or extinctions). Mathematics education research must adjust its theoretical lenses to do justice to students’ thinking as they construct and reason with such computational models (cf, Amado et al., 2019). However, responding to this challenge will enrich our understanding of mathematical thinking, problem posing, problem solving, and modeling.

12 Discussion and Conclusion There is an increasing realization that in learning mathematics, students should engage with problems that are meaningful to them (Hernandez-Martinez & Vos, 2018); that take their own models and lived experiences into account (Jung & Magiera, 2021); and that embed mathematics learning in conceptual practices (Hall & Jurow, 2015). The idea that fostering connections between mathematics and students’ lives outside of school “should help students improve all aspects of their mathematics learning” (McNair, 2000, p. 551) has wide appeal. Indeed, research on problem posing is motivated in large part by a desire to position students (and teachers) with the mathematical agency to use mathematics to make sense of their world. However, when we emphasize teaching problem solving as a domain-general, procedural skill, and when “problems” are viewed as “carriers” of atomic topics in the curriculum, the agency of teachers and students can be diminished, and the “relevance” of problem-solving can be lost in a drive for “rigor.” It should be noted that a precisely analogous tendency also poses a threat to relevance in the field of mathematical modeling. In the name of making the messy process of modeling more teachable, the interpretive and meaning-making practices of modeling can also be collapsed, making “modeling” look more like “applications” of single textbook topics. For both problem-posing and modeling, the result of this “collapse” can be that authentic processes (of problem solving or modeling) are caricatured, as the field sacrifices “relevance” in favor of “rigor”—both at the level of the mathematics (focusing exclusively on algebraic functions, one-way causal relations, etc.) and at the level of the human processes and practices (privileging context-free experimental studies with manipulable “task variables”). However, there are hopeful lines of research, grounded in desires for problem solving, problem posing, and modeling to be “relevant.” This work strives to support

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and study the creative mathematical work that learners of all ages are capable of. Learning in such settings promises to prepare students for success in a connected world, where complexity and emergence are the rule, and where “wicked problems” (Rittel & Webber, 1973) demand hard conceptual, social, and political reframing work. In pursuing such a research agenda, one needs all the allies one can get. This chapter has identified potential connections between lines of work within both problem-posing and modeling research, in the hope of fostering new collaborations and exchanges of ideas among them.

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Chapter 4

Types of Mathematical Reasoning Promoted in the Context of Problem-Solving Instruction in Geneva Maud Chanudet

Abstract In this chapter I present a study aiming to learn what kinds of mathematical reasoning are involved in the problems students have to face in the context of problem-solving instruction with the goal of better understanding both teachers’ practices and potential students’ learning. Indeed, problem solving is usually seen as a means to develop and assess students’ mathematical learning but can also be considered as an independent learning goal. In the context of this second purpose, the potential or at least the intended mathematical learning related to problem solving usually remained unclear. Nevertheless, some researchers highlight that problem solving can, among other things, help students develop mathematical reasoning skills. This leads me, in a first part, to define the mathematical reasoning and to characterize the different types of mathematical reasoning students can mobilized to solve mathematical problems at secondary level. Then, I focus the research on a course devoted exclusively to problem solving at secondary level, in the canton of Geneva (Switzerland). I analyze the problems of the official resources to identify the types of mathematical reasoning at stake. It helps to clarify the institutional purpose and to highlight and better understand the choices made by teachers. Then, I analyze the problems that teachers propose to students over a school year and how they articulate these problems. It leads to characterize teachers’ practices and to formulate some hypotheses about students’ potential learning related to problem solving. Finally, I give some leads to help teachers organize a teaching of problem solving linked to the mathematical reasoning at stake. Keywords Problem solving · Mathematical reasoning · Resources · Teachers’ practices · Students’ learning

M. Chanudet (B) University of Geneva, Geneva, Switzerland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_4

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1 Introduction The interest given to problem solving in mathematics education is not recent. For many years, problem solving has been seen as a major aspect of mathematics, its teaching and its learning (Liljedahl et al., 2016, p. 1). It can be explained by the place played by problem solving in the discipline itself as Halmos summarize. The mathematician’s main reason for existence is to solve problems, and that, therefore, what mathematics really consists of is problems and solutions. (Halmos, 1980, p. 519)

Two functions are usually promoted for problem solving in mathematics education. On the one hand, it can be used as a tool to develop and evaluate students’ learning of specific mathematical notions or concepts. Problem solving is then seen as a way to develop specific mathematical learning. On the other hand, it can be a learning object. Then the aim is to develop skills and knowledge centered on this aspect of mathematical activity. In that sense, curricula in many countries encourage both teaching content through problem solving and teaching of problem solving. Contrary to NCTM standards “which structurally puts problem solving on an equal footing with learning to work with representations, communicating results etc., rather than as the goal for these processes” (Stacey, 2005, p. 344), the unified curriculum for mathematics in French-speaking Switzerland put problem solving as the main goal for compulsory education, with these two roles, problem solving as a learning means and problem solving as a learning goal. The priority aims of the mathematics and natural science domains emphasize the role of problem solving to construct and mobilize reasoning specific to mathematics (CIIP, 2010). Nevertheless, the term of “reasoning” is not defined and only used in two configurations: deductive reasoning and probabilistic reasoning. Although mathematical reasoning is considered as a central aspect of the practice of the mathematics, its teaching and its learning, and as a central learning goal of many mathematics curricula, its definition is usually missing, as if it were known and shared by all. However, the research conducted by Jeannotte et al. (2020) shows that Canadian primary teachers define it in various ways, with different meanings (justify, identify, extrapolate, apply, explain, prove, make choices, analyze, etc.). The review of the literature in the field conducted by Jeannotte (2015) shows that researchers themselves have different conceptions of mathematical reasoning. This is in line with the observation made by Yackel and Hanna “Writing about reasoning in mathematics is complicated by the fact that the term reasoning, like understanding, is widely used with the implicit assumption that there is universal agreement on its meaning” (2003, p. 223). These elements highlight that, even if mathematical reasoning is widespread in mathematics education, there is a need for clarification of what it really is. On top of that, it seems interesting to question the articulation between problem solving and mathematical reasoning. Some studies in a French speaking context focused on mathematical reasoning and problem solving but mainly at primary level (ChoquetPineau, 2014; Hersant, 2010; Houdement, 2009).

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These elements lead me to investigate the types of mathematical reasoning involved in problems students have to face in the context of problem-solving instruction at lower secondary, and thus, what they can learn from problem solving when it constitutes a goal in itself. More specifically, the questions I want to address are: What types of mathematical reasoning are promoted in official resources dedicated to problem solving? What types of mathematical reasoning are involved in the problems that teachers propose to students in the context of problem-solving instruction? The remaining part of this chapter is structured as follows. First, I present my theoretical framework based on results about problem solving and mathematical reasoning. I then expose the research design, with a focus on the context and on the methodology used to answer my research questions. After that, I show the results of the analysis of official textbooks and of the practices of mathematics teachers about problem solving. Finally, I present a collaborative work that aims to develop a resource for teaching problem solving organized around the mathematical reasoning involved in problem-solving.

2 Theoretical Framework 2.1 Problem Solving The term of problem is polysemous and largely used in everyday life. In the context of mathematics education, a problem can be defined in a relationship between the subject and the situation (Brun, 1990; Fagnant & Demonty, 2016; Richard, 2004). It highlights that a problem is not a problem in itself but according to the conditions in which it is proposed. Thus, in the scholar context, the same problem, according to the knowledge of the student for whom it is intended, according to the moment when it is proposed during schooling, according to the way it is managed by the teacher in class, according to its formulation, etc., can lead to a resolution in which the student will put into play personal and new procedures or to an automated procedure. In the latter case, some authors (Julo, 1995; Schoenfeld, 1985) use the term of exercise. Faced with a problem, a student must engage in a form of activity called problem solving that can be described as a cognitive process aimed at a goal, without any known method to implement (Mayer & Wittrock, 2006, p. 287). In others terms, problem solving can be considered as “a response to a question for which one does not already know a method by which it can be answered” (Monaghan et al., 2009, p. 24). A lot of research has been conducted on problem solving for many years. Some, since the main work of Pólya (1945), focus on describing the problem-solving process (Favier, 2022; Richard, 2004; Rott, 2012; Schoenfeld, 1985). Others try to identify attributes of the problem solver that contribute to problem-solving success (Carlson & Bloom, 2005; Lester et al., 1989; Schoenfeld, 1985, 1992). But, as Hersant and

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Choquet emphasize (2019), the questions concerning knowledge at stake in inquiry based learning situations, and more generally problem solving, is still a crucial issue for mathematics education. That is why I want to investigate this question: what can students learn from problem solving when it constitutes the learning goal? Indeed, intended learning outcomes about problem solving are usually unclear and are not linked to mathematical knowledge (Hersant, 2010; Houdement, 2009). However, according to Houdement (2009), solving problems can help students reinvest their knowledge, but also develop proving and reasoning skills. The reinvestment of knowledge is central when problem solving is seen as a means of constructing and reinvesting mathematical concepts set as objectives in the curricula. When problem solving is the object of instruction, the knowledge involved is known and mastered by the students, even if they do not always know a priori which ones to use to solve the problem. These notions or concepts are supports to work on problem solving. The difficulty in defining what can be learned in problem solving does not lie here. If this reinvestment of knowledge is important in problem solving, it is not the main issue. Therefore, I am interested in characterizing learning related to mathematical reasoning. Some research has relied on different types of mathematical reasoning to describe possible learning in problem solving (Choquet-Pineau, 2014; Georget, 2009; Hersant, 2010; Houdement, 2009), but without describing precisely how these categories were constructed or selected. To do that, it is necessary to define properly mathematical reasoning.

2.2 Mathematical Reasoning Mathematical reasoning and, in a correlated way, proof, are interesting leads to identify possible learning in problem solving. These two aspects, problem solving and mathematical reasoning are the heart of the two main international large-scale assessments. The Trends in International Mathematics and Science Study (TIMSS) 2019 mathematics framework is organized according to two dimensions: the content domains (subject matter) and the cognitive domains (thinking processes) which is divided in three: applying, knowing and reasoning. Reasoning is seen as a skill that students develop with the age “the eighth grade has less emphasis on the knowing domain and greater emphasis on the reasoning domain.” (Lindquist et al., 2019, p. 15). This cognitive domain is associated with problem solving in that “The third domain, reasoning, goes beyond the solution of routine problems to encompass unfamiliar situations, complex contexts, and multistep problems.” (Lindquist et al., 2019, p. 22). This domain is then associated with cognitive skills as analyze, integrate/synthesize, evaluate, draw conclusions, generalize, justify. It is even more the case for the Program for International Student Assessment (PISA). Indeed, two central aspects characterize “mathematical literacy” as defined by the last PISA 2022 framework: reasoning and problem solving. “Mathematical

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literacy is an individual’s capacity to reason mathematically and to formulate, employ, and interpret mathematics to solve problems in a variety of real-world contexts.” (OECD, 2018, p. 8). The authors precise that reasoning encompasses both deductive and inductive and highlight the strong relationship between mathematical reasoning and problem solving. They define mathematical reasoning as: Mathematical reasoning (both deductive and inductive) involves evaluating situations, selecting strategies, drawing logical conclusions, developing and describing solutions, and recognising how those solutions can be applied. Students reason mathematically when they: - Identify, recognise, organise, connect, and represent, - Construct, abstract, evaluate, deduce, justify, explain, and defend; and - Interpret, make judgements, critique, refute, and qualify. (p. 14–15)

With this assumption, reasoning mathematically encompasses many types of activities, of actions and refers to special structures of reasoning (deductive and inductive). Jeannotte’s review of the literature of the field leads her to qualify mathematical reasoning from a theoretical perspective and to propose a conceptual model for mathematics at school. In this model, two complementary aspects of mathematical reasoning are emphasized: a structural aspect and a procedural aspect (Jeannotte, 2015; Jeannotte & Kieran, 2017; Jeannotte et al., 2020). If the structural aspect of mathematical reasoning focuses on its logical structure, as discussed by Toulmin (2007) or Pierce (n. d.), its processual aspect highlights that when someone carries out a reasoning, he or she makes actions which are oriented, directed towards a goal.

2.2.1

Structural Aspect of Mathematical Reasoning

The structural aspect “refers to the way in which the discursive elements combine in an ordered system that describes both the elements and their relation with each other.” (Jeannotte & Kieran, 2017, p. 7). It brings to consider specific reasoning steps, abductive, deductive and inductive, dependent on the nature of the conclusion and on their epistemic value. Inductive and deductive are usual explicit learning goals in mathematics curricula. I focus below on mathematical reasoning which could be mobilized by students at lower secondary. Deductive form of reasoning plays a major role in the processes of proof. Indeed, it is usually considered as the only form of reasoning that infers a true conclusion from true data and warrant. Then, it is interesting to distinguish two cases according to the nature of the warrant. If the warrant refers to mathematical knowledge, in the sense of learned knowledge in didactic transposition (Chevallard, 1985), which are elements identified in the curricula (theorems, properties, etc.), I call it hypotheticodeductive reasoning. On the other side, if the warrant is based on experience or does not refer directly to scholarly knowledge, which include warrants related to logic notions (for example, for any proposal “A”, “A” is true or “no A” is true), I call it logico-deductive reasoning.

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It is also possible to distinguish specific forms of deductive reasoning depending on the articulation of deductive steps. Specifically, students can use an exhaustive case study. It implies that they test all possible solutions until they find the good one (Battie, 2003, p. 48). Such reasoning involves enumeration, which is a knowledge identified by didacticians and schoolable (Chevallard, 1985), and that Rivière defines as strong when the following two conditions are met (weak if only one of the two is) “each item must be dealt with at least once and only once”1 and “each item must be dealt with no more than once and only once” (see footnote 1) (Riviere, 2017, p. 594). To summarize, it is possible to characterize exhaustive case study as a deductive reasoning which involves enumeration, associated with at least one of the following two characters: the possibility of reducing the set of cases to be enumerated and the necessity of testing the potential solutions which have been enumerated. Close to this, it is possible to refer to case separation study, which consists in reducing the resolution of a problem to the study of a finite number of cases, handled separately (Battie, 2003). Other types of reasoning are also mobilizable to solve problems like the reasoning by the absurd, the reasoning by recurrence, the reasoning by contraposition, but with older students. Inductive form of reasoning is usually described as a way to go from the particular to the general. In that sense, Grinstein and Lipsey highlight that “Inductive reasoning includes thinking that proceeds from examining multiple, well-chosen examples in the problem solving process to making a conjecture that is then tested in other, more general cases.” (Grinstein & Lipsey, 2001, p. 297). But considering only the structural aspect of reasoning is not sufficient to define some specific types of reasoning. The processual aspect of reasoning is then necessary.

2.2.2

Process Aspect of Mathematical Reasoning

Jeannotte described the process aspect of mathematical reasoning as “commognitive processes that are meta-discursive, that is, that derive narratives about objects or relations by exploring the relations between objects.” (Jeannotte & Kieran, 2017, p. 9). Jeannotte (2015) identified nine processes of the several overlapping mathematical reasoning processes found within the literature, which she grouped into two types: those related to the search for similarities and differences (generalizing, conjecturing, identifying a pattern, comparing, and classifying), and those related to validating (justifying, proving, proving formally). On top of that, the process of exemplifying can support each of the other processes. These different processes are close to those described by Stylianides (2008). Considering these processes help to describe a central approach to solve problems: the experimental approach, which can be characterized as a dialectic between phases of experimentation, conjecture, test and proof. Such an approach articulates theory and emporia (Gardes, 2013, p. 34) and is mainly inductive. Hersant asserts 1

My translation.

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Fig. 1 Synthesis of the different types of mathematical reasoning and approaches

that the scientific nature of the experimental approach consists in this articulation between induction and deduction (2010). This approach is close to inquiry. Indeed, in the Encyclopedia of Mathematics Education, Dorier and Mass explain the focus on inquiry in mathematics education thank to “the increasingly shared view that mathematics and sciences education are closely connected, that mathematics is not purely deductive and that mathematical concepts may be grasped through some experimental practice.” (Dorier & Maass, 2014, p. 302). Another major approach is trial and error. Facing a problem, students make tests and adjust them until they reach the solution. Problems which lead to use this approach are usually proposed to young students, as if it is easy to adjust its tests. However, Favier (2022) shows that it is far to be the case. His research highlights that students encounter difficulties to respect at the same time all the data of the problem, to adjust their tests by changing parts while keeping the whole, or to exploit the link between the test and the feedback, whether this involves using the deviation from the goal to characterize a relevant new trial or anticipating the effects of an adjustment. His study leads to believe that managing such problems in class constitutes a real challenge for generalist teachers. In summary, different types of mathematical reasoning and approaches can be used by students at lower secondary to solve problems, characterized by their structural and/or processual aspect. This can be illustrated as follows (Fig. 1).

3 Research Design 3.1 Context This research is part of a broader doctoral research project, which studies teachers’ assessment practices in the context of mathematics problem-solving instruction (Chanudet, 2017, 2019a). It focuses on a problem-solving centered course given in the canton of Geneva in French-speaking Switzerland and called mathematical and scientific approaches (MSA). The curriculum then at stake for MSA course at the time I conducted the research is given in the appendix (Annex 1).

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This annual course is designed for a 45-min period per week and delivered to 13–14-year-old students (grade 8). Its main goal is to improve students’ problemsolving competencies. Nevertheless, the institutional requirements are formulated in general terms and the intended learning outcomes are unclear. Teachers are supposed to make students solve different open-ended problems (Arsac et al., 1991) in order to foster their problem-solving competencies and especially an example-conjecturetest-proof approach, which can be considered as an experimental approach. They are invited to select such relevant problems within two official resources gathering about 70 problems. The first resource is the textbook shared with ordinary mathematics courses. It includes a chapter dedicated to problem solving and called “Research & Strategies” which constitutes the part of the resource I analyzed. The other one is a specific document (called “problem pool”), elaborated by some former teachers of MSA and devoted exclusively to this context. It presents different problems related to different mathematical domains and different types of strategies. On top of that, MSA course leads to an independent summative assessment. As assessing problem solving remains difficult especially because it “requires access to evidence of process” (Monaghan et al., 2009, p. 25), this assessment focuses on students’ problem-solving processes. But, as Goos claims “new approaches to classroom assessment are needed in order to give teachers access to students’ thinking” (2014). That is why the research narrative (Bonafé, 1993; Bonafé et al., 2002; Chevallier, 1992; Sauter, 1998) has been chosen as a means to assess students’ problem solving competencies and to give teachers an access to students’ processes of thinking. A research narrative can be defined as a specific contract between teacher and students in which students have to explain as precisely as possible, in natural language, how they processed to solve the problem, including mistakes, wrong ways, dead-ends, help received. Reciprocally, the teacher has to assess students on these and only these elements, and not the results obtained and especially without considering whether the students found the solution or not. As with open problems, the emphasis is laid on the development of a scientific attitude, but the authors also stress what is specifically offered by these narratives in natural language and how they make accessible to the teacher the richness and singularity of the solving processes. (Artigue & Houdement, 2007, p. 373–374)

In this context, I address the following research questions: what types of mathematical reasoning are promoted in courses centered on problem solving at secondary school? More specifically, I wonder: what types of mathematical reasoning do institutional resources emphasize? What types of mathematical reasoning are involved in the problems teachers choose to develop students’ problem-solving skills? How do they articulate these problems during a school year?

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3.2 Methodology To answer these questions, at first, I analyzed problems proposed in the official resources of MSA course by taking the point of view of the mathematical reasoning involved during their resolution. To do this, I solved each problem from the point of view of the students, i.e., with the knowledge they are supposed to have at their level of schooling and not from the point of view of a teacher for example. Then, I identified the main type(s) of reasoning, the one(s) that seemed to be the most prevalent in this resolution. This does not mean that, from a mathematical point of view, these identified and retained types of reasoning are the only ones that can be in play. Some students can use other types of reasoning than those a priori identified, especially students who used inefficient procedures. But I consider that these types of reasoning are sufficiently important in the problem to be highlighted from a didactic point of view; in other words, these problems present an interest to work with students on the selected type(s) of reasoning. To ensure reliability in the process of identifying mathematical reasoning at stake during the resolution of the problem, my colleague Stéphane Favier proceeded the same way. Then, we compared our results, and in the few cases of disagree, we solved and analyzed together problems in order to determine the main type(s) of reasoning at stake. These disagrees were mainly due to different ways of solving the problems. This led us to enrich our a priori analysis and to integrate different possible types of reasoning for a same problem. Secondly, I studied the practices of two teachers (Sarah and Paul) teaching the MSA course.2 Sarah is teaching the MSA course for the first time, while Paul has a long experience with this course. They work in two different schools. The school where Paul works is in a more affluent neighborhood than the school where Sarah works. I focused here on the problems they submitted to their students during a school year. I proceeded in the same way as described above and paid attention to the articulation between these problems. I completed and confronted these results with Sarah’s and Paul’s comments collected during the three interviews I had with each of them, at different moments.

3.2.1

Examples of Analysis of Problems in Terms of Mathematical Reasoning

I would like to give two examples about the way I analyze problems in terms of mathematical reasoning involved during the resolution. It is important to precise that solving a problem can involve the mobilization of one type of mathematical reasoning or more. On top of that, it’s clear that, even if each problem gets an “optimal” way to solve it, other types of mathematical reasoning may sometimes be used and led to success. 2

For more details on the profiles of these teachers and the conditions under which they were recruited to participate to the study, see (Chanudet, 2019b).

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Jaffar kidnapped Princess Jasmin and is holding her prisoner in one of three cells in his palace. Aladdin, who runs to save her, finds himself in front of the three cell doors, each bearing a sign, only one of which tells the truth. Aladdin knows that he can only open one door before the guards arrive. Which door must he open to save Jasmin?

Fig. 2 Statement of the problem “The kidnapping of Jasmin” (The diagram in this figure is drawn by the author)

Let’s examine the following problem intitled “The kidnapping of Jasmin”3 (CIIP, 2012, p. 219) (Fig. 2). To be sure which door Aladdin should choose, there are three cases to study: each corresponding to the fact that the indication written on the door is the only one that tells the truth. • Study of the 1st case: the indication in cell 1 is true, i.e., the statement “Jasmin is in this cell” is true, so Jasmin would be in cell 1. In this case, according to the data of the statement, the two other indications must be false and in particular that of cell 2 “Jasmin is not in this cell”. However, as Jasmin is not in cell 2, so this indication is true. This contradicts the data (the only door that tells the truth is the one Jasmin is in) of the problem and allows us to conclude that it cannot be the indication of cell 1 that is true. • Study of the 2nd case: the indication of cell 2 is true, i.e., the statement “Jasmin is not in this cell” is true, so Jasmin would be either in cell 1 or in cell 3. The indications given by the other two doors must therefore be false. Thus, for cell 1 “Jasmin is in this cell” leads us to deduce that Jasmin would then be in cell 3. For cell 3, “Jasmin is not in cell 1” being false allows to deduce that Jasmin would be in this cell 1. These last two deductions are in contradiction which allows us to conclude that the indication given by cell 2 cannot be the one that is true. • Study of the 3rd case: the indication given by cell 3 is true, i.e., the statement “Jasmin is not in cell 1” is true. We therefore deduce that Jasmin can be in cell 2 3

This problem is from “Research & Strategies” chapter of official textbook for this grade, described in the previous part of this text dedicated to the context of the study.

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Determine all positive integers of up to 7 digits, divisible by 6 and whose digits are 0 or 1. Fig. 3 Statement of the problem “0 and 1”

or cell 3, and that the indications of the two other cells must be false. Cell 1 says “Jasmin is in this cell”, which is indeed false because cell 3 says true. The cell 2 “Jasmin is not in this cell” must also be false, which leads to the conclusion that Jasmin is in this cell 2. The reasoning involved to solve this problem is structured on the different cases to be studied, by implementing a reasoning by exhaustive case study. The study of each of these cases is based on a deductive reasoning whose rules are related to the notions of logic. It leads to logico-deductive reasoning. The study of the second and third cases (and possibly of the first) also requires an exhaustive review of the other doors without which it is not possible to conclude with certainty. Therefore, this problem is interesting to work on these two types of reasoning: exhaustive case study and logico-deductive reasoning. Once again, it does not mean that all students will use only these types of reasoning, but they are necessary to solve the problem and, in that sense, interesting for illustrating these specific mathematical ways of solving problems. The second example is related to the problem called “0 and 1”4 (Fig. 3). The resolution of this problem implies to mobilize both exhaustive case study and hypothetico-deductive reasoning to limit the set of possible cases. Indeed, the set of positive integers of 7 digits at most are all candidate solutions. However, treating them one by one to test them does not seem feasible or at least reasonable. This should lead students to rely on mathematical properties, especially on criteria of divisibility by 2 and 3 to reduce the set of numbers that satisfy the conditions of the problem. It reduces it to the numbers with 3 or 6 digits 1 and which end in 0. The problem is now to determine all positive integers of up to 7 digits, 3 or 6 digits 1 and which end in 0. It is possible thanks to an exhaustive case study or by disjunction of cases, according to the number of digits composing the number then to play on the permutation of the digits (numbers with 1, 2, 3 digits: no solution; 4-digits numbers: one solution 1110; 5-digits numbers: 3 solutions 11100/11010/10110; 6digit numbers: 6 solutions 111000/110100/110010/101100/101010/100110; 7-digit numbers: 11 solutions 1111110/1110000/1101000/1100100/1100010/1011000/ 1010100/1010010/1001100/1001010/1000110). It implies to get organized to make sure you don’t forget any of the 21 solutions. I now present the results of my analyses in both contexts: the types of mathematical reasoning involved in the official MSA course resources and those promoted by teachers via their choice of problems for students.

4

This problem is from the second resource which is specific to MSA course.

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4 Results and Discussion 4.1 Types of Mathematical Reasoning Promoted in Official Resources for MSA Course Within the 76 problems proposed in the two official resources for MSA course (“Research & Strategies” chapter and the “problems pool”, see Methodology), I identified 97 possible uses for the different types of reasoning identified above, although not in the same proportion (Fig. 4). To illustrate how I obtain these results, consider the problems analyzed above which are two of the 76 problems in the official resources. For “The kidnapping of Jasmin”, I consider both exhaustive case study and logico-deductive reasoning to be relevant types of reasoning for solving the problem. Therefore, they count as two types of reasoning for a single problem. For “0 and 1”, I consider exhaustive case study and hypothetico-deductive reasoning. I do the same for each one of the 76 problems. Then I calculate the number of occurrences of each type of reasoning (for example 11 occurrences for exhaustive case study) divided by the total number of reasoning involved (97) which led to the percentage express in the pie chart. The most represented type of reasoning is hypothetic-deductive reasoning. It covers problems involving geometric properties and theorems as well as problems involving algebraic or arithmetic properties. The trial-and-error approach also appears very frequently. It is interesting to note that although the official instructions of MSA course highlight the mobilization of example-conjecture-test-proof approach as a main learning goal, this type of reasoning (experimental approach) is at stake in only 17% of cases. Therefore, there is a gap between the intentions announced in the official instructions and the types of reasoning potentially used in the problems proposed in the official resources. Without such an analysis, teachers can randomly choose problems they submit to students within the official resources, thinking that they are representative to the different types of reasoning and in the same proportion as their importance in the curriculum, but it is clearly not the case. 17% 33%

25% 5% 9%

11%

Fig. 4 Distribution of the types of reasoning involved in problems from official resources

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The specific analysis of the problems from the chapter “Research & Strategies” of the textbook shared with ordinary mathematics courses and dedicated to problem solving gives interesting information. It appears that when problems are similar from a global point of view (form of the question, mathematical domain at stake, form of the statement, etc.), the way to solve them is similar. For example, the problems that take the form of games all lead to the mobilization of an experimental approach and a reasoning by exhaustive case study. Indeed, although the choices concerning the implementation of these games in class should vary from one teacher to another, the students may be first invited to play some games. The fact that the question asked concerns the winning strategy invites them to systematically organize the trials carried out, and thus to reason by exhaustive case study, with the aim of conjecturing winning positions or configurations. The students can then prove these conjectures by reasoning, again, by exhaustive case study. Similarly, the problems that are formulated in the geometric framework all lead to the mobilization of hypothetico-deductive reasoning, with the support of properties, formulas of areas and perimeters of polygons or circles/ disks. Only one other problem involving hypothetico-deductive reasoning is based on different properties, in this case related to fractions. It seems interesting to discuss the possible effects of this similarity between the form of the problem and the type of reasoning involved. It can certainly make it possible to work on similar problems in order to allow the students to reinvest the reasoning already encountered. But it also can lead to didactic contract effects. Indeed, after several similar problems, the context or more generally the similar form of the new problem can lead the students to resort almost de facto to the type of reasoning already used, and thus limit their initiative and their reflection about what approach to adopt. On the other hand, logico-deductive reasoning is used differently in several problems. It allows, and is sufficient, to solve some of them, including those for which the use of a table is efficient. Other problems need to be articulated to a reasoning by exhaustive case study. Some problems are apparently very similar although they are not solved in the same way. In particular, the use of a table is only relevant for one of them. Conversely, two problems are different in form although they mobilize the same type of reasoning. These subtleties, which are not perceptible without a detailed analysis of the type of reasoning involved, seem to be important from a teaching perspective. Indeed, the help given to the students during the research but also the elements that can be put forward at the end of the research will be different from one problem to another: the organization for a problem mobilizing an exhaustive case study, the choice of the trials and their articulation with the conjecture emitted in the case of an experimental approach or the choice and the validity of the rule used in view of the data during a hypothetico-deductive reasoning to name a few. Moreover, the four games proposed in this resource lead to the mobilization of an experimental approach. Only two other problems invite to use such an approach, with an almost immediate conjecture concerning one of them. As it stands, the resource does not propose a truly consistent problem in terms of the conjectures to be developed. However, the formulation of conjectures is central to the experimental

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approach, which is itself often considered as the main learning goal of problem solving. Finally, it appears that the problems leading to hypothetico-deductive reasoning and which are strongly focused on mathematical notions in the mathematics curricula are like some of those present in the associated chapters. The challenge for students solving these problems is to select and correctly mobilize the relevant mathematical properties. Other problems are certainly based on mathematical notions without making them an object of study but more a context to train problem solving (in this case the experimental approach for example).

4.2 Types of Mathematical Reasoning Promoted by Teachers Once this study of institutional resources carried out, let us look at the actual practices of teachers and at their choices in terms of the problems proposed to students. To understand the choices teachers made, it is interesting to look first at the resources from which they selected problems for students. Of the 22 problems she submitted to students during the school year, Sarah selected 13 problems from the official resources analyzed earlier. For Paul, there were 8 out of 23. The other problems came from colleagues, websites, current or previous official textbooks, or ongoing training. Let’s see if the freedom they take in selecting problems out of the official textbooks leads to more variety and a better fit with the learning objectives of the MSA course. Figure 5 shows the types of reasoning involved in problems teachers submitted to their students for MSA course and in what proportion. To obtain these pie charts, I proceeded as described in the methodology part and the previous section of this chapter. Even if Sarah and Paul refer to the different types of mathematical reasoning in various proportions, it is interesting to notice that, for both, it is very different from Sarah's class 7% 4%

Paul's class 26%

22%

14% 10%

3%

14%

21%

15% 26%

28%

10%

Fig. 5 Distribution of the types of reasoning involved in problems chosen by teachers

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that observed in official resources. Especially, hypothetico-deductive reasoning is more rarely involved contrary to logico-deductive reasoning which is more frequently at stake in problems they select. Sarah declares that she relies on the strategies mobilized during the resolution of problems to select those she proposes. However, she identifies and mainly promotes three types of reasoning: the experimental approach, the logico-deductive reasoning and the exhaustive case study. Even though Paul principally relies on his experience to select problems, without any problem-centered criteria, the problems he chooses are various in terms of types of reasoning mobilized. By analyzing the organization of problems between them, it appears that, in Sarah and Paul’s classes, logico-deductive reasoning is the subject of targeted work at the beginning of the year, with problems proposed consecutively over five or six sessions, while other types of reasoning are mobilized in an occasional way throughout the year. I make the hypothesis that is because this type of reasoning is particularly easy to identify. Sarah also relies on strategies involved during resolution to structure her teaching. The types of reasoning are worked alternately over the year. She seeks to confront students with different types of problems, aiming both at the diversity of the types of problems encountered and at long-term work. She expects students to be able to use already encountered strategies when solving new problems. At the beginning of the year, Paul proposes some problems aimed at developing general skills related to the organization to highlight the importance of an exhaustive data processing methodology. However, he does not have criteria for choosing and organizing the problems he selects. This lack of a common thread seems related to his difficulties to clearly identify the learning objectives targeted in this course and to recognize the strategies mobilized during the process of resolution.

5 Conclusion This study shows that even in the case of a course focused on problem solving and especially on the experimental approach, the most representative type of mathematical reasoning at stake in problems in official textbooks is hypothetico-deductive reasoning. Even though it is necessary to help students develop such reasoning skills, this type of mathematical reasoning is already worked on in all the other chapters associated with the different mathematical themes (algebra and function, geometry, numbers and operations, etc.). Thus, it seems necessary that problems proposed in official textbooks lead to a wider range of mathematical reasoning in order to foster students’ various reasoning competencies. Nevertheless, regarding official resources, teachers orient their choices towards a larger set of problems, which lead to different kinds of mathematical reasoning. The curriculum for MSA course leaves the intended learning outcomes about problem solving unclear. My doctoral research (Chanudet, 2019b) has shown that teachers encounter difficulties in the context of MSA course to select and organize problems to propose to students with a mathematical and didactic approach. These results led me and my colleague Stéphane Favier to explore a way to give teachers

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tracks to organize problem solving instruction in the context of MSA course and define more precise learning objectives. Indeed, it seems interesting to consider the types of mathematical reasoning at stake when solving a problem as a way to organize a teaching of problem solving. I give an example of this operationalization in the context of a collaborative work with teachers, which lead to review the specific resource proposed to teachers for teaching problem solving in the context of MSA course. The former specific resource offered to MSA teachers (the “problems pool” analyzed above, see Context section) is a set of problems, organized according to the mathematics domain involved. The first page of the resource presents a table with the correspondence between the main “strategy of resolution” and the problem. But for many problems, these strategies are not relevant in the sense that they are not enough precise to help teachers chose a problem according to his or her teaching goal. For example, a type of strategy is “examples, counter-examples”. In the same category, there are problems which need to prove by using a counter example, problems in which examples lead to trial-and-error approach and problems in which examples help identifying a conjecture within an experimental approach. These different roles given to examples can create confusion for teachers who want to organize a teaching of problem solving. That is why we propose a different way to characterize and organize problems. The new resource elaborated with teachers also offers a set of problems, but more relevant with the official learning goals. It means that we ensure that the different types of mathematical reasoning were well represented. On top of that, problems are organized according to the type of mathematical reasoning involved during the resolution. Indeed, in addition to the choice of each problem, the question arises of how to plan and articulate problems over the long term, using mathematical criteria. In line with the work of Julo (1990, 1995) and with the notion of patterns of problems, we hypothesize that thinking in terms of mathematical reasoning can help organize the teaching and develop planning. Indeed, Julo defines “patterns of problems” as “traces left in memory by previously encountered situations and organized into structured objects with a certain number of characteristic properties5 ” (Julo, 1995, p. 90). The idea is therefore to multiply the research of various problems to allow students to both enrich their memory of problems and develop patterns of problem. The different types of mathematical reasoning should constitute these “characteristic properties”. The emergence of these different characteristics can however only be done by a wellorchestrated confrontation of a certain number of similar problems, i.e., mobilizing the same type of mathematical reasoning. This confrontation implies, moreover, that the problems the students are going to encounter are not too distant in time so that they can refer to them more easily. To conclude, I think that the question of what can be institutionalized as a result of such classes, in relation to reasoning, also remains crucial and that more researches 5

My translation.

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are necessary to analyze effective learning of students in terms of mathematical reasoning.

Annex Annex 1 Programme of “Mathematical and Scientific Approaches” Course (MSA) I Organization […] This weekly period is intended to support a teaching that contributes to the strengthening and development of problem solving strategies and mathematical situations activities. II Programme • The suggested activities are linked with three main topics: – Numbers and Operation – Space and measure – Function and algebra • The problem solving strategies are: – – – – –

Analogy Trial and error—Example/counter-example Inductive and deductive reasoning Organized study of all cases and exhaustion of solutions Introduction to proofs

• These strategies contribute to the development of: – Scientific procedures – The rules of scientific debate III Mathematics development course: introduction […] The allocation of an additional period in the curriculum for grade 8 students with scientific profile, aims to enable these students to learn and become familiar with this important part of mathematical activity. The aim is not simply to solve problems “one by one”, but also to discover and systematize problemsolving methods. In particular, the aim is to place the student in a learning situation where she/he will have to implement a “scientific approach”, that leads her/him to the following scheme: Try-conjecture-test-prove This part of mathematical activity is required when students are confronted with the so-called open-ended problems. This places the pupil in the most typical situation of mathematical activity, that of confronting a problem which enables

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her/him to work as a mathematician who is confronted with a problem to which she/he does not know the solution. IV Open ended problem (a) According to a definition proposed by a group of researchers at the IREM of Lyon, an “open-ended problem” has the following characteristics: • the wording is short • the wording does not introduce the method or the solution, the solution must not be reduced to the use or an immediate application of recent coursework • the problem must be situated in a conceptual field that students are familiar enough with, so that they can easily “take possession” of the situation and engage in trials, conjectures, draft resolutions, or counterexamples. (b) Solving a problem consists of a series of steps outlined in the official textbook: 1. Appropriation of the wording: “understanding the problem to identify its purpose”. At this stage, the teacher must ensure that all students are involved in the problem. That is to say that they are able to construct a correct representation of the data, understand the constraints and the goal to achieve. If necessary, the teacher answers questions, rephrases or makes the student rephrase the problem. 2. Data processing: “design a plan”, then “put the plan into action” and “get back to the solution”. This stage corresponds to the research and resolution of the problem itself. A relatively short time slot can be allocated for individual research, followed by a second group work time. During the individual research phase, the teacher can verify that each student has actually read the problem, has at least partially assimilated it, and that, during the group work, she/he will not only follow the ideas of the one who speaks first. Group work helps to avoid the discouragement of certain pupils, to stimulate the exchange of ideas among students, to learn how to collaborate, to listen to each other, to defend their point of view, to respect each other. 3. Communication of research procedures and results: “Write the results in a form that anyone can understand and follow the work done”. At this stage, the student must account for all the resolution of the problem, individual phase and the group work included. Such a written report gives the teacher the first insight into the student’s research work and provides an occasion for evaluation. The writing of this report is a basis for the evaluation and is therefore an important competence for the student. This is why the practice of “research narrative” has been chosen as a thread for this course. According to the textbook, a research

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narrative is “a comprehensive account of research, including trials and errors that didn’t lead to a satisfactory result, or wrong conjectures, and the reasons which lead them to abandon them.” V Research narrative 1. Presentation of research narrative […] The objectives of this pedagogical practice can evolve throughout the year. They may initially be: • to develop students’ curiosity and critical thinking • to provide a communication tool that facilitates students’ writing • to put in place the rules of mathematical debate, in particular the following ones: a counterexample is sufficient to invalidate a statement, examples that verify a statement are not sufficient to show its validity, an observation on a drawing is not sufficient to prove that a statement is true • to allow the teacher to get a much better knowledge of the procedures of his pupils. 2. Correction and assessment Criteria for a good research narrative The action of narration is not an easy activity, but one can retain some elements that are to be emphasized and encouraged by the corrector of the copies. • Writing style • The accuracy of the narrative: all ideas, all trials are described thoroughly • The sincerity of the narrative Criteria for good research To help students to better understand what is expected, it could be useful to refer to intermediate assessment means. • An assessment of the analysis of a problem by the formulation and explanation of conjectures • An assessment of the research phase: identifying and comparing strategies • Another assessment of the “research” phase: using hints • An assessment of the overall attitude • An assessment of an oral presentation.

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Chapter 5

Prospective Secondary School Mathematics Teachers’ Use of Digital Technologies to Represent, Explore and Solve Problems Alexánder Hernández, Josefa Perdomo-Díaz, and Matías Camacho-Machín

Abstract This research aims to analyze the work done by a group of prospective secondary school mathematics teachers while solving mathematics problems using GeoGebra. Our focus is on analyzing the Mathematical Activity as they progress solving one problem, through the different problem-solving episodes (SantosTrigo & Camacho-Machín, 2013): understanding, exploration and the search for multiple approaches. The results show evidence of mathematical processes and activities that include extending and posing new problems and finding novel paths to reason and solve the tasks with technology. We could observe them to generate and pursue new routes to represent mathematical objects, transforming representations, formulating conjectures and observing and justifying relationships and conjectures. Keywords Mathematical problem-solving · Digital technologies · Prospective mathematics teachers

1 Introduction In the last decade, the emergence of digital technologies and social media has called traditional educational models into question. How and to what extent should available digital technologies be employed? Which of these digital tools are useful in school? What can they offer to students to engage them in developing problemsolving competencies? These issues are addressed and discussed in various studies. A. Hernández · J. Perdomo-Díaz · M. Camacho-Machín (B) Universidad de La Laguna, San Cristóbal de La Laguna, Spain e-mail: [email protected] A. Hernández e-mail: [email protected] J. Perdomo-Díaz e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_5

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In answering these questions, Santos-Trigo and Reyes-Martínez (2018) uphold that teaching must rely on the coordinated use of technologies. Mobile devices (tablets or phones), the use of websites and digital applications that function as communication platforms (Google Classroom, Padlet, Instant Messaging, etc.), information platforms (Khan Academy, YouTube, Wikipedia) and, more specifically, the use of technologies designed for mathematical activity, such as Wolfram Alpha and GeoGebra, must be incorporated into classrooms. Working on mathematics tasks through the use of GeoGebra offers students the opportunity to ask questions, make conjectures, represent objects and explore their mathematical properties. Thus, the teacher is faced with new challenges, for example: The reflective use of technologies, find a reconciled way to teach traditional and digital content or explore opportunities technologies for teach other contents. Eventually, the design and implementation of tasks that allow secondary education mathematics students to use digital technologies. It is important for prospective Secondary School Mathematics teachers to be aware of the importance of different modes of technology use. Thurm and Barzel (2022) point out that there are several common modes of technology uses (to support working with multiple representations, discovery learning, individual learning, to support practicing and reflection) that can facilitate the combination of technical skills and mathematics understanding. They also emphasize the importance of beliefs about their use in teaching problem solving and state that […] self-efficacy is a central construct for higher level aspects of teaching mathematics with technology and that beliefs about potential benefits of technology use and beliefs about the time requirements of implementing technology may outweigh the importance of teachers’ beliefs about negative effects of technology use. (p. 59)

In research studies where students are encouraged to use DGS to carry out tasks and activities (Contreras, 2014), researchers have noticed that working in an environment that provides an opportunity for experimentation makes it possible to formulate conjectures, which are expressed with varying degrees of rigor, depending on the mathematical understanding of the users. In the activities that were implemented, the use of technology is entirely necessary to follow a process that transitions from experimental to formal through the discovery of properties. The elements required for this are dragging or orderly moving objects, the quantification of object attributes and the possibility of finding the solution using a different route from that taken with pencil and paper. The research also considers the difficulties in the use of technology, which can result from previous cognitive obstacles or from a lack of theoretical knowledge. This thus requires teachers to intervene in order to guide the students when formulating a conjecture and when deciding how much detail to consider during their mathematical justification process. Santos-Trigo et al. (2015) note that the use of technology will yield changes in the teaching–learning of mathematics, and point out that: The use of digital technology offers learners an opportunity to extend ways of reasoning about the problems; however, representing and exploring mathematical tasks through the use of digital technologies bring in new challenges for teachers that include the development of an expertise in the tools use in order to identify and analyze what changes to contents and teaching practices need to be considered. (p. 256)

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A preliminary study with future teachers (Hernández et al. 2020) highlighted the different skills that a future teacher needs, given that mathematical concepts emerge in Secondary Education that not only include new ideas and extensions of elementary concepts, but that there is also a change in the nature of the mathematical thinking involved (Kilpatrick, 2015). For example, there is a greater emphasis on the need to follow the initial premises, to then drawing deductive inferences of math properties. Teachers thus need an understanding of mathematics that allows them to prepare and shape more complex activities such as: looking for examples and counterexamples to justify the validity or falsity of a conjecture; extending properties to mathematical systems that are broader than the initial ones (Conner et al., 2011); or presenting interesting problems that arise from the problem-solving process itself (Contreras, 2014; Jacinto & Carreiras, 2017). In short, Secondary Education teachers must have an understanding of mathematics that allows them to help their students acquire a sufficiently rigorous and formal conception of the discipline, in keeping with their educational level. In this paper, researchers analyzed the work of prospective secondary teachers as they solved a problem in a virtual learning environment. We observed at different moments while they interacted with GeoGebra and other digital resources. At first, they took actions to understand and explore the problem. They then made GeoGebra constructions and investigated different way to solve it. The analysis of those interactions and the teachers’ abilities are the objective of this research.

2 Conceptual Framework The components analyzed in this work correspond to three of the components that the MUST (Mathematical Understanding for Secondary Teaching) model specifies for the Mathematical Activity (Heid et al., 2015). When we talk about Mathematical Activity, we consider the specific mathematical actions that a teacher uses when teaching in the classroom. It can also be understood as a series of activities that comprise mathematical “know-how” and that the teacher, being aware of them, uses to involve students in the study of mathematics. The actions they identify are grouped into three components: Mathematical Noticing, Mathematical Reasoning and Mathematical Creation. Zbiek and Heid (2018) using these components from several Algebra examples. […] the mathematical Activity perspective reveals mathematical actions that cut across School Algebra and Abstract Algebra. They are activities in which professors and Abstract Algebra students, as well as School Algebra teachers and students engage frequently. (p. 193)

In general, the actions considered for each component are: Mathematical Noticing: Groups the actions of recognizing and identifying the mathematical characteristics specific to the different structures, the different notations or symbolic forms, as well as the ability to notice when a mathematical argument, whether expressed simply or rigorously, is valid, and the ability to connect

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mathematical ideas with each other (representing ideas in different structures and connecting different concepts) and with the real world (explaining problems in real contexts through mathematics). Hernández (2021) characterizes a series of actions carried out by a group of prospective secondary mathematics teachers when they solved an optimization problem (finding the maximum or minimum area of a family of figures) using digital technology that make it possible to establish descriptors to analyze mathematical noticing. These actions were as follows: (i) distinguish the important elements of a dynamic exercise that can be quantified to analyze the problem, (ii) identify the different mathematical characteristics of the graphical, algebraic and spreadsheet views in GeoGebra, and (iii) relate the mathematical ideas to one another, specifically that of the geometric locus of a point when moving the figure with that of an area function with domain restrictions. These actions are related to GeoGebra utilities that are present in a dynamic construction. In this case, they involve the possibility of quantifying parameters, the use of different representation systems and the generation of geometric loci as part of the functional approach. Mathematical Reasoning: Groups the observation, conjecture and justify or prove activities by using deductive logic, mathematical properties, regularities and patterns, generalizations of specific cases, restricting properties and extending to other structures. In the analysis of the solution to the same problem, Hernández (2021) also describes a group of actions that characterize mathematical reasoning using digital technologies, noting that these actions are: (i) formulating valid conjectures about the solution based on the observations made about the dynamic construction, (ii) restricting a mathematical argument, such as the domain of a function, to make its graph match a geometric locus, (iii) justifying conjectures made by observing the controlled movement in the graphical view through the formal use of logic, (iv) justifying conjectures made by observing the quantification of attributes through the formal use of logic. In analogous fashion to mathematical noticing, these actions are related to the GeoGebra utilities that are present in a dynamic construction and with the aspects necessary to appropriate the use of GeoGebra. In this case, the actions are the quantification of attributes, the movement of elements (drag) and the controlled movement that influence the statement of mathematical arguments. Hernández et al. (2020) note that when using GeoGebra in problem solving, the action of observing movement in a dynamic construction to conjecture mathematical properties is essential, and they indicate that the actions related to mathematical reasoning are more frequent than those of the other components. Mathematical Creation: Implies the ability to find new paths to express mathematical objects, generate new ones and transform their representation. This is related to choosing representations of objects that highlight their structure, restrictions or properties, when new objects are defined and when they are manipulated by changing their form, but not their representation. The actions described for mathematical creation by Hernández (2021) when prospective teacher solve problems with digital technologies are: (i) dynamically construct a family of figures with fixed properties such that when a point is moved,

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the figures of the family are displayed; (ii) choose the dimensions of a family of figures to create an automatic record table where each of its members can be identified; (iii) represent the area of a family of figures as an analytical function. In conclusion, constructing and manipulating a dynamic construction generates useful mathematical objects for the teacher, who must be able to choose one representation or another to highlight aspects of said object, or design problems in which students can interpret and analyze different approaches to solve them. Jacinto and Carreira (2017) observed the necessary skills when doing a dynamic construct, too. The authors also identified techno-mathematical fluency as a way of understanding the effectiveness of combining mathematical knowledge and the affordances of digital tools in a problem-solving context. The other element that comprises the conceptual framework used in our work is that described by Santos-Trigo et al. (2022). This framework integrates the use of digital technologies to organize what the authors call the Digital Wall, with three intertwined components (a problem-solving approach, resources and support) that students can use to reflect, reinforce and control their own learning. In particular, we will focus on the first component in terms of what Santos-Trigo et al. (2022) point out using a problem-solving approach: […] based on conceptualizing the discipline as a set of dilemmas that students need to elucidate and solve in terms of using mathematics concepts, resources and strategies. (p. 32)

and using tasks as the main element, since Tasks are the vehicle for students to engage in problem-solving activities and looking for multiple ways to solve the tasks becomes an important activity for them to develop their mathematical thinking. (p. 33)

The second component of the framework is also important in this work. The coordinated use of online platforms (Wikipedia, Wolfram Alpha, etc.) greatly facilitated the solution process of the tasks proposed (Santos-Trigo et al., 2022). Santos-Trigo and Camacho-Machín (2013) note that solving problems with GeoGebra provides an ideal scenario for teachers to stimulate mathematical reflection in their students. This requires starting with dynamic constructions and extending a specific problem, to subsequently arrive at the study of a family of cases. This is why this type of work must be carried out during their training period as teachers. The authors consider five stages (understanding, exploration, search for multiple approaches, connections and extensions, and retrospective view and reflections) to describe the process of solving problems with technology. In previous research, we focused on analyzing future teachers’ problem solving when they had finished. On that occasion, our research instruments were written reports, oral presentations, and a short interview. This time we focus the analysis on the problem-solving process in person. In this research, we focus on analyzing mathematical activity over the course of the first three episodes, which are described below: Understanding: The use of technology focuses on building a dynamic model as a means to raise and explore questions that lead students to understand and make sense of the problem.

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Exploration: The possibilities offered by GeoGebra (controlled movement, quantification of attributes, functional representation, geometric loci, etc.) make it possible to observe regularities, invariant attributes that result in properties and in the formulation of conjectures. Search for multiple approaches: The conjectures and empirical solutions deduced when exploring the problem yield a series of properties that lead students to devise and carry out different plans to find the solution. It is necessary to promote among students the importance of thinking about and solving a problem in different ways. Each solution method must use different mathematical concepts, representations or resources. The goal is for the students to connect the different mathematical ideas that they know and have used. We are interested in analyzing and documenting in detail how the use of tasks involving problem solving with GeoGebra makes the Mathematical Activity (Heid et al. 2015) in terms of the “actions one takes with mathematical objects” (p. 18) that emerge in future teachers, and we ponder the following research question: What actions related to the components of Mathematical Activity arise when solving problems with GeoGebra? Which of these actions appear in the understanding stage? And in the exploration stage? And during the search for multiple approaches? The goal of this research is to analyze the most relevant aspects of the components of Mathematical Activity—mathematical noticing, reasoning and creation—that arise when prospective secondary teachers use GeoGebra to solve mathematics problems.

3 Methodology Data were collected from a Problem-Solving Workshop implemented during nine two-hour sessions in the Mathematics for Teaching course, where students solve the tasks in groups of two using GeoGebra. The workshop was taught in the computer laboratory to seventh-semester mathematics majors who, in a short time (two years), will be qualified to be secondary school teachers. There were eighteen students enrolled in this course, whose main goal was to present the knowledge for teaching secondary school mathematics to university mathematics students. Only 12 students were selected for the analysis presented in this paper (the rest were discarded due to insufficient information), labeled as Gk (with k = 1, …, 6). In the workshop, the students solved a total of four problems that had previously been analyzed by the research team. Three types of data were collected for the analysis: the GeoGebra files, the handwritten notes and the recordings of the student groups’ work sessions, where the resolution of the problem was recorded live simultaneously with the conversations of the pairs. In this paper, we present an analysis of the work done by the students while solving the Connecting Islands problem. Connecting islands: We want to connect three islands (G, P and H) in a fiber optic network in a way that uses the least amount of cable. The distance between islands is GP 79,322 m, GH 64,514 m and PH 95,932 m. Where should the connection point be located to minimize the amount of cable needed?

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4 Data Analysis and Results This is a triangle (GPH) whose side lengths is known, where each vertex represents an island. Then, to find the solution to the problem, a free point can be added to the plane, called C, which represents the connection point between the islands. The sum of the distances from the three vertices to the connection point (CableLength = C G + C P + C H ) can then be calculated. GeoGebra can be used to find a solution in a simple way by moving the point along the plane and gradually approaching the place where the value of the Cable Length is minimized. The six groups made a GeoGebra’s construction for the analysis of the problem. Figure 1 is a synthesis of the proposals of future teachers. Once the position of the point is determined, other questions may be asked. Why is that the solution? Can this point be found using a robust method? Does said point have other properties? Answering these new questions turns the problem into a task to develop the mathematical reasoning of the participants (Fig. 2).

4.1 Episode 1: Understanding the Problem We analyzed recordings and observed the math discussion of each group when they made dynamic constructions. When using GeoGebra to solve the Connecting Islands problem, the groups initially focused on understanding the meaning of “connection point.” Once they decided that they had to find a point whose sum of distances to the islands was as small as possible, they proceeded to look for how to represent that relationship in the dynamic construction. To represent the islands, they decided to use three points that comprised the vertices of a triangle. Some groups used images

Auxiliary elements

Objects of the construction G, P and H points on the plane such that

C free point on the plane Cable Length value of the sum of the lengths of the CG, CP and CH segments B barycenter of triangle GPH O orthocenter of triangle GPH I incenter of triangle GPH X circumcenter of triangle GPH F Fermat point of triangle GPH P' vertex of the equilateral triangle with base HP H' vertex of the equilateral triangle with base GH G' vertex of the equilateral triangle with base PG C' 60° rotation of point C about H

Fig. 1 The problem, objects and auxiliary elements (researcher’s proposal)

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G free point in the plane. P point on the circumference of center G and radius 79,322. H intersection point between the circles with centers. G and P and radii 64,514 and 95,932, respectively.

Fig. 2 Elementary construction to explore the problem

of the islands to illustrate the non-mathematical context of the problem. They did this by placing images of the three islands, or of the archipelago, in the background of the construction, and tracing each vertex over the images of each island. The professor did not intervene in this problem to give instructions on how to make the constructions. Two key differences were identified in the steps carried out by the groups when doing their constructions: the way they represented the islands, and how they constructed the connection point (Table 2). The way they represented the islands reflects the need to transfer the distance data given in the problem statement to the construction in GeoGebra. For the most part, the groups chose to use three points separated by an arbitrary distance, instead of considering the distances specified (Table 2). The second difference involved how they used GeoGebra to find and build the connection point. Half of the groups (shown with an * in Table 2) used a free point of the plane, C, from which they drew line segments to the vertices of the triangle, defining the CableLength variable as the sum of their lengths (Table 1_Step 2_Option 1). As a result, they were able to move C along the plane and see the sum of the distances at each point. The other half calculated and recorded this sum in a different variable for each point they wanted to verify (Table 1_Step 2_Option 2). In short, it is evident that the participants made different dynamic constructions (Table 1). Some took into account the distances given in the statement (Table 1_Step 1_Option 2), while others relied on arbitrary points (Table 1_Step 1_Option 1). In addition, some constructions included as an auxiliary element a point that could be used to find the desired connection point directly (marked with *), while others did not. Since the professor did not intervene to give instructions on how to make the constructions, there was only one change by Group 1 (G1 ), who at one point in the process deemed it necessary to represent the distances and decided to change their construction. Group 4 (G4 ) also considered including the distances, but eventually opted to continue with their initial idea and work with arbitrary points. Using basic tools such as Circumference (center, radius) and Point, it is possible to include the distances in the statement in a dynamic construction. So, the lack of

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Table 1 Basic steps for building a triangle of arbitrary dimensions that can provide a general answer, which would then have to be applied to the specific case presented in the problem statement

This table is research team work. It is a rewrite of the steps of the problem that the groups followed

geometric knowledge to build a triangle given the length of its sides, or a lack of practice, could be reasons why the groups did not use that option. Instead, they used the Segment with Given Length tool. The inputs to this tool are a point on the plane and a length that is entered on the keyboard; GeoGebra’s response is a segment of said length and a point that marks the end of the segment. By using the tool several times, three segments can be drawn with the desired distances; the difficulty lies in lining up the ends (Table 3). In conclusion, on the one hand, we determined that the techno-mathematical fluency (Jacinto & Carreira, 2017), knowing how to

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Table 2 Types of dynamic constructions used by the groups Characteristic differentiating the construction types (ways to represent the islands)

G1

G2

G3

G4

G5

G6

Arbitrary points to represent the islands

◻*

◻*

–

◻

◻

◻

Take distances into account

◼*

–

◻*

▲

–

–

◻ Dynamic construction made at the beginning ◼ Figure reconstructed as the solution progresses ▲ Inconclusive reconstruction of the figure that is not used to explore the problem *Free point included to calculate Cable Length

use the tools of GeoGebra and existing cognitive difficulties (Iranzo & Fortuny, 2009), drawing a triangle knowing its sides, influenced the approach to the problem. Whether the participants worked toward the solution with one type of construction, or another determined their reasoning to a certain extent. The construction that relaxed the initial conditions (Santos-Trigo & Reyes-Martínez, 2018) allowed questions to be asked about whether the shape of the triangle was related to the location of the point. Conversely, when working with the construction that included distances, the formulation of the conjectures was directed to the characteristics of the connection point. In the Conceptual Framework, we described this episode as the interval where the use of technology focuses on building a dynamic model, although, there are some Table 3 Discussion by G4 on using the Segment with Given Length tool

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characteristic actions of the understanding stage that are present in other episodes; for example, the use of GeoGebra tools for representing mathematical objects. Sometimes, future teachers add new objects to a dynamic construct after exploring a problem. This is the reason why the end of the understanding episode for this problem is difficult to pinpoint, since the construction of the connection point is intertwined with the exploration of the problem. In this report, we place it at the moment when the groups calculated or represented with a parameter the sum of the distances from a point to the vertices. The analysis of this problem provides certain results related to the participants’ Mathematical Activity. The realization of different dynamic constructions, some of which satisfied every problem condition and others that did not include the distances, to which in some cases a free point was added as an auxiliary element to find the solution, provided different ways to highlight the mathematical creation of the participants. On the one hand, the students who wanted to include distances in their construction encountered problems involving the mathematics (building the triangle given its sides) and the appropriation of the DGS (using tools such as “segment” given its length). The participants who managed to overcome the difficulties were able to represent on the graphic view a figure that allowed them to find the optimal connection point, and then explore its properties. In contrast, when the groups worked without taking distances into account, they used the graphic view to represent a generic case that allowed them to find a general solution to the problem.

4.2 Episode 2: Exploration The participants’ exploration of this problem, with the help of dynamic construction, relied on adding auxiliary elements to the construction (Table 4). These elements were selected mainly after viewing information online on building the centers of the triangle. They also searched for the names of the centers based on how they were built. Table 4 Elements used (shown with an ◼) by each group at some point in the problem exploration episode Auxiliary elements

G1

G2

G3

G4

G5

G6

Cable length

◼

◼

◼

–

–

–

C free point on the plane

◼

◼

◼

–

–

–

F Fermat point

◼

◼

◼

◼

◼

◼

Circumcenter of triangle GPH

◼

◼

–

◼

–

◼

Barycenter of triangle GPH

–

◼

–

◼

–

◼

Incenter of triangle GPH

–

–

–

◼

◼

◼

Orthocenter of triangle GPH

–

–

–

◼

◼

–

Sums of distances from the vertices to the centers

◼

–

–

◼

◼

◼

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The way they checked the sum of the distances from the centers to the vertices made a difference in the reasoning employed during this episode. The groups that used the free point C on the plane were able to find the solution directly. Following this realization, they continued two ways. On the one hand, they determined if that free point coincided with any known center; on the other, they studied the properties the point once it was in the optimal position. The other groups, not knowing where the connection point was, explored by comparison: they built different centers and confirmed which of them had the lowest sum of distances to the vertices. These groups also took advantage of the ability to move in GeoGebra to drag the vertices (Islands) and see if the connection point matched for different triangles. The conjectures formulated by the groups during this episode (Table 5) were independent of the type of construction made, since they stemmed from a common idea: the point sought is a center in triangle GPH. The type of construction made a difference in another way: groups 1, 2 and 3, who used the free point, were able to verify if their conjectures were true or not, while groups 4, 5 and 6, having formulated the same conjectures, could not check them, they could only see which center was the best approximation to the solution. Upon reviewing the recordings and discussions of the groups, we concluded that, for this problem, the conjectures were formulated before working with the DGS. They used the tools in GeoGebra to trace the mid-perpendiculars, bisectors, medians and heights of triangle GPH to then analyze, respectively, the circumcenter, incenter, barycenter or orthocenter, in any of the ways described above. Specifically, there were two conjectures based on the observation of the construction (the last two in Table 5) related to the size of the angles with the vertex at the connection point and with a certain order between the centers in reference to the sum of distances to the vertices. Eventually, the groups didn’t find a point of concurrency in the triangle GPH that solved the question. Then, they made use of other resources (Santos-Trigo et al., 2022). They resorted to looking online to find a solution. The result they used was the definition of Fermat Point in Wikipedia. In this entry, they were able to see that Table 5 Conjectures formulated by groups during the exploration episode Conjectures formulated

G1

G2

G3

G4

G5

G6

×

The circumcenter of triangle GPH is the solution

◼

◼

–

◼

–

◼

×

The barycenter of triangle GPH is the solution

–

◼

–

◼

–

◼

×

The incenter of triangle GPH is the solution

–

–

–

◼

◼

◼

×

The orthocenter of triangle GPH is the solution

–

–

–

◼

◼

–

×

Using the sum of the distances to the vertices, the centers can be arranged as follows: Fermat point-Incenter-Barycenter-Orthocenter-Circumcenter

–

–

–

◼

–

–

√ √

The Fermat point of triangle GPH is the solution

◼

◼

◼

◼

◼

◼

The size of each angle GFP, PFH and HFG is 120°

–

–

◼

–

–

–

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the Fermat Point satisfied the condition of being a point such that “the total distance from the three vertices of the triangle to the point was the shortest possible”. They also found instructions for building it with a straightedge and compass. Discovering the Fermat point and recognizing it as a general solution to the problem marked the end of the exploration episode. They then went on to look for a formal justification that the Fermat point is the desired connection point. Group 3 (G3 ) stands out among the rest in this problem. After finding the connection point using free point C, G3 wondered if point C satisfied any condition that can identify it with any known center. They confirmed that C is not the center of the circle that passes through G, P, and H, nor the center of the circle that is tangent to the sides of the triangle. They also confirmed that the lines perpendicular to the sides that pass-through C do not coincide with the heights or the mid-perpendiculars. This group was the first to discover that the solution was related to Fermat’s point by doing an online search (Table 6). During this episode, differences were observed when exploring, determined by how the groups used GeoGebra. If they used the free point on the plane and the value CableLength, they noticed that one center built did not coincide with the connection point, and thus discarded it directly. By contrast, if they only built the centers and compared them, they knew which one was better, but not if any of them was the Table 6 Discussion by G3 to build the Fermat point based on the Wikipedia entry

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desired point. In other words, the action of formulating conjectures was influenced by integrating the advantages provided by GeoGebra. The participants’ mathematical reasoning was manifested differently depending on how they interacted with GeoGebra; thus, the use of DGS to solve the problems influenced their Mathematical Activity. In Hernández (2021), this and other differences were used to classify the groups according to GeoGebra problem-solving ability.

4.3 Episode 3: Search for Multiple Approaches Since only one session of the Workshop was devoted to the Connecting Islands problem, the groups spent little time formalizing the plans or paths to the solution they had outlined. During that time, the groups pointed out three properties, two related to the Fermat point and another that considered the sum of distances function (Table 7). Taking these properties into account, the groups tried to find a formal approach to the solution, which they thought required them to justify that the Fermat Point was the connection point sought. Discovering that the angles with a vertex at F had a constant size of 120°, and that the circles that inscribed the equilateral triangles intersected at F, led them to relate the two observations to the concepts of central angle and inscribed angle of a circle. They used this idea to justify why the angles measured 120°, but not to justify why the Fermat point was the best connection point. By the end of the session, the groups had made progress in two approaches to the problem. These could be called the geometric and analytical approaches to Connecting Islands problem. Table 8 provides an outline of the steps in both. The geometric approach is a well-known proof of this problem, and can be found online. Group 3 (G3 ) found an outline for this solution during the session, which they partly implemented before finishing. Then, at home, they shared in the virtual classroom a proof in which they justified each step. At the professor’s request, the groups spent the final minutes of the session giving a numerical answer to the problem. To give this answer under the terms requested in the problem statement, the participants who had not taken into account the distances between the islands in their construction found that they needed to transfer that information to their construction. In general, they solved it by dragging points G, P, and H to position them such that the distances between them were approximately Table 7 List of properties observed by the participants that gave rise to different approaches Properties observed

Geometric approach

Analytic approach

The size of each angle GFP, PFH and HFG is 120° ●

◌

The circles that inscribe the equilateral triangles intersect at F

◌

◌

The cable length can be expressed as a function

◌

●

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Table 8 Steps for the geometric and analytical approaches that arose in the workshop session

This table is a research team work. It is a rewriting of the proofs of the problem that the groups raised

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those given in the statement. In this way, the distances from the vertices to the Fermat point coincided with the answer to the problem. They were able to go from the general to the specific with the help of technology (Contreras, 2014). In the search for an approach to this problem, connections were found with mathematical reasoning, because the justification was preceded by the observation and formulation of conjectures that relied on constructing the triangle’s various centers, until they eventually constructed the Fermat Point and formulated new properties related to it (Table 7). The mathematical noticing of the participants, another component of the Mathematical Activity, was observed when the participants, guided by Group 3 (G3 ), were able to understand and transcribe the geometric approach by connecting the construction steps in GeoGebra with those of the formal proof. This was evidenced when they dynamically constructed each of the steps in the proof and then formally wrote it in one of the documents delivered as part of the TRPG task. An analysis of this problem reveals certain results related to the Mathematical Activity of the participants. The realization of different dynamic constructions, some of which satisfied every problem condition and others that did not include the distances, to which in some cases a free point was added as an auxiliary element to find the solution, provided different ways to highlight the mathematical creation of the participants. On the one hand, the students who wanted to include distances in their construction encountered problems involving the mathematics—building the triangle given its sides—and the appropriation of the DGS—using tools such as “segment”, given its breadth. The participants who managed to overcome the difficulties were able to represent on the graphic view a figure that allowed them to find the optimal connection point, and then explore its properties. In contrast, when the groups worked without taking distances into account, they used the graphic view to represent a general case that allowed them to find a general solution to the problem. Recognizing Fermat’s point as a solution led them to focus on finding a justification of why the characteristics of that point yielded the property of minimizing the sum of the distances. From the perspective of Mathematical Activity, this ability to realize the validity of a series of arguments, whether expressed simply or rigorously, is related to mathematical noticing. As with other components of the MUST, the use of technology expands the actions of mathematical noticing; in this case the ability to connect mathematical ideas with one another is extended thanks to the simultaneous representation of the different views provided by GeoGebra.

5 Conclusion When using technology to solve problems, steps involving mathematical reasoning stand out over others. As Zbiek and Heid (2018) noted, the very nature of an activity can cause certain components to stand out over others. Accordingly, the use of a DGS to solve problems leads to actions such as observing, formulating conjectures and justifying. This is because by using the ability to move objects in GeoGebra, specific cases can be visualized. During the workshop, seeing the elements of the

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construction move allowed the participants to discover mathematical relationships that led them to formulate conjectures. Having GeoGebra and other digital resources favor the emergence of a large number of conjectures is a result that has already been mentioned in many studies, as noted in the Background and Conceptual Framework sections, something that was observed too in our analysis “in live” of the process of resolution. It was also possible to identify how the action of formulating conjectures was not always subsequent to that of visualizing in GeoGebra; rather, the participants resorted to the construction to verify or refute a previous conjecture. These courses of action taken by the participants show, on the one hand, that the interaction with the use of digital technologies facilitates the process of refining conjectures (Contreras, 2014). On the other, that GeoGebra acts as an interactive partner (Granberg & Olsson, 2015), giving the user the opportunity to represent an idea, to analyze the answer of the DGS. From the perspective of Mathematical Activity, the actions of observing and formulating conjectures are directly related to mathematical reasoning. Visualizing the motion extends the action of observing from a specific case to an action that approaches generalization or to a process that can be used to refine conjectures. These actions are intertwined with mathematical creation, since anytime there is a change in the meaning of the actions, new elements have to be represented that allow it. That is, when the conjecture precedes the observation, it is necessary to add new elements to the construction and then visualize their attributes when they move. In line with the approaches made by Leung (2017), the prospective Mathematics Secondary Teachers participating in this research evidenced elements of Mathematical Activity by drawing from their own experience as problem solvers with digital tools. All this facilitated a reflection on how their future students learn using different digital work environments. In this sense, Santos-Trigo (2019) emphasizes the importance for teachers to […] work on problems and discuss ways in which technology help them restructure their teaching practices that pay attention to the type of reasoning that emerges throughout the problem-solving process. (p. 65)

We also consider that our research results have contributed to highlight the importance of incorporating in their training the technology affordances that allowed them to have opportunities for learning specific strategies and heuristics derived from the use of digital technologies, for example (Santos-Trigo, 2019) “moving orderly objects within the model, quantification and exploration of objects’ attributes, finding and analyzing objects’ loci; using sliders, and arranging data in tables”. (p. 66). Acknowledgements This work has been supported by the National Plan Grant EDU2017-84276R, PID2022-139007NB.I00 of the Spanish State Research Agency (MICINN) and FORMULATIC ProID2021010018. The authors would like to thank the referees who provided useful and detailed comments on a previous version of the manuscript.

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References Conner, A., Wilson, P., & Kim, H. (2011). Building on mathematical events in the classroom. ZDM Mathematics Education, 43, 979–992. Contreras, J. (2014). Solving optimization problems with dynamic geometry software: The airport problem. Journal of Mathematics Education at Teacher College, 5(2), 17–27. Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. The Journal of Mathematical Behavior, 37, 48–62. https://doi.org/10.1016/j.jmathb.2014.11.001 Heid, M., Wilson, P. S., & Blume, G. W. (Eds.). (2015). Mathematical understanding for secondary teaching: A framework and classroom-based situations. NCTM and IAP. Hernández, A. (2021). Resolución de problemas con GeoGebra en la formación inicial de profesores de matemáticas: Un análisis desde la actividad matemática. Unpublished Doctoral Thesis. Universidad de La Laguna, Spain. https://riull.ull.es/xmlui/handle/915/26865 Hernández, A., Perdomo-Díaz, J., & Camacho-Machín, M. (2020). Mathematical understanding in problem solving with GeoGebra: A case study in initial teacher education. International Journal of Mathematical Education in Science and Technology, 51(2), 208–223. https://doi.org/10.1080/ 0020739X.2019.1587022 Iranzo, N., & Fortuny, J. M. (2009). La influencia conjunta del uso de GeoGebra y Lápiz y Papel en la adquisición de competencias del alumno. Enseñanca de las ciencias, 27(3), 433–446. https:// doi.org/10.5565/rev/ensciencias.3653 Jacinto, H., & Carreira, S. (2017). Mathematical problem solving with technology: Technomathematical fluency of a student-with-GeoGebra. International Journal of Science and Mathematics Education, 1115–1136. https://doi.org/10.1007/s10763-016-9728-8 Kilpatrick, J. (2015). Background for the mathematical understanding framework. In M. K. Heid, P. Wilson, & G. W. Blume (Eds.), Mathematical understanding for secondary teaching: A framework and classroom-based situations (pp. 1–8). NCTMand IAP. Leung, A. (2017). Exploring techno-pedagogic task design in the mathematics classroom. In A. Leung & A. Baccaglini-Frank (Eds.), Digital technologies in designing mathematics education tasks (pp. 3–16). Springer. https://doi.org/10.1007/978-3-319-43423-0_1 Santos-Trigo, M. (2019) Mathematical problem solving and the use of digital technologies. In P. Liljedahl & M. Santos-Trigo (Eds.), Mathematical problem solving, current themes, trends and research ICME-13 monographs, (pp. 62–89). Springer. https://doi.org/10.1007/978-3-030-104 72-6_4 Santos-Trigo, M., & Camacho-Machín, M. (2013). Framing the use of computational technology in problem solving approaches. The Mathematical Enthusiast, 10(1&2), 279–302. Santos-Trigo, M., & Reyes-Martínez, I. (2018). High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies. International Journal of Mathematical Education, 50(1), 1–20. Santos-Trigo, M., Reyes-Martínez, I., & Gómez-Arciga, A. (2022). A conceptual framework to structure remote learning scenarios: A digital wall as a reflective tool for students to develop mathematics problem-solving competencies. International Journal of Learning Technology, 17(1), 27–52. Santos-Trigo, M., Reyes-Martínez, I., & Ortega-Moreno, F. (2015). Fostering and supporting the coordinated use of digital technologies in mathematics learning. International Journal Learning Technology, 10(3), 251–270. Thurm, D., & Barzel, B. (2022). (2022) Teaching mathematics with technology: A multidimensional analysis of teacher beliefs. Educational Studies in Mathematics, 109, 41–63. https://doi.org/10. 1007/s10649-021-10072-x Zbiek, R. M., & Heid, M. (2018). Making connections from the secondary classroom to the abstract algebra course: A mathematical activity approach. In N. H. Wasserman (Ed.), Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers, (pp. 189–210). Springer. https://doi.org/10.1007/978-3-319-99214-3_10

Chapter 6

Primary School Teachers’ Behaviors, Beliefs, and Their Interplay in Teaching for Problem-Solving Benjamin Rott

Abstract How do teachers in German primary schools organize teaching for problem solving? Lessons on this subject by 13 teachers were videotaped. The behaviors of the teachers were coded using a grid combining Pólya’s phases of problemsolving processes with the way in which teachers emphasize their pupils’ strategic diversity in problem solving. After their lessons, the teachers were interviewed; the interviews were coded to obtain the teachers’ beliefs regarding mathematics and problem solving. Comparisons of both the lesson and the interview codings reveal significant interrelations and, thus, the importance of beliefs in the context of teaching for problem solving. Keywords Mathematical problem solving · Teachers’ beliefs · Epistemological beliefs

1 Background of the Study Internationally, (non-routine) problem solving [PS] is considered to be an important aspect of education (e.g., OECD, 2013). In this context, teaching for, about, and via PS are distinguished (Schroeder & Lester, 1989) to address becoming a better problem solver, learning more about the subject of problem solving, and learning more about certain mathematical topics by thinking about problems, respectively. The study at hand focuses on “learning for problem solving” (i.e., to foster students’ problem-solving competencies) but the other ways of addressing PS are also included as long as they serve the development of students’ PS competencies. In Germany, teaching for PS is obligatory, that is, all teachers are required to include this in their classes’ individual curricula, but they are often not told how to do this. Generally, teachers in Germany have a lot of freedom to design the way they teach in their classes; but regarding mathematical problem solving, they are regularly B. Rott (B) Institute of Mathematics Education, University of Cologne, Cologne, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_6

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not educated about more or less preferable ways to teach this topic (Kuzle & Rott, 2017). Thus, it is of interest to see how they cope with this requirement, especially since one can assume that their beliefs play a central role in this (cf. Ernest, 1989; Rott, 2019). As education regarding PS should start early (Lesh et al., 2013), it is of interest to investigate how pupils are taught for PS at an early age; therefore, this study focuses on primary schools, which addresses in most German federal states the grades 1 (5–6 year-old pupils) to 4 (8–9 year-old pupils). Research questions are: (1) What is primary school teachers’ behavior in PS lessons? (2) What are the teachers’ beliefs regarding PS and the nature of mathematics? (3) What is the interplay of the teachers’ beliefs and their behavior in PS lessons?

2 Theoretical Framework Rott (2019) introduced a grid to sort and classify teachers’ behavior, combining the way in which they use direct instruction (closely managing their pupils’ approaches), open teaching (emphasizing different approaches and strategies), or try not to interfere at all (stay neutral) with Pólya’s (1945) four phases of PS processes (Table 1). Even though teachers do not necessarily plan lessons with these phases in mind, their behavior can still be described this way as all lessons who feature problem solving prominently have phases in which the pupils (with or without the help of their teachers) try to understand the problem, try to solve it (planning and carrying out the plan), and will compare and discuss their results. Teachers’ beliefs—which are “psychologically held understandings, premises, or propositions about the world that are thought to be true” according to Philipp (2007, p. 259)—about the nature of mathematics are often depicted in a three-partite way (e.g., Dionné, 1984; Ernest, 1989; Grigutsch et al., 1998). Following the terminology by Ernest, in this study, three beliefs or views about mathematics are distinguished: the instrumentalist view (mathematics as an accumulation of facts, rules, and skills); the Platonist view (mathematics as a static but unified body of knowledge); and the problem-solving view (mathematics as a dynamic, continually expanding field of human creation and invention). A possible interplay of teachers’ behavior and beliefs was already predicted by Ernest (1989, p. 26): […] the instrumental view of mathematics is likely to be associated with a transmission model of teaching, and with the strict following of a text or scheme. […] Mathematics as a Platonist unified body of knowledge corresponds to a view of the teacher as explainer, and learning as the reception of knowledge, although an emphasis on the child constructing a meaningful body of knowledge, is also consistent with this view; mathematics as problemsolving corresponds to a view of the teacher as facilitator, and learning as autonomous problem posing and solving, perhaps also as the active construction of understanding.

Several researchers have addressed this interplay of behavior and beliefs (e.g., Beswick, 2012; Stipek et al., 2001; see also Philipp, 2007). Even though most studies

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Table 1 Categories to describe different options for teachers to organize phases in PS lessons Phase

Closely managed

Neutral

Emphasizing strategies

Understand the problem

Teacher explains the problem (formulation)

Teacher does not comment on the problems and does not answer students’ questions

Teacher gives hints but does not explain the problem or highlights importance of strategies

Devise a plan

Teacher tells students which (“correct”) approach to use

Teacher does not give any guidelines and strategic support

Teacher hints at different approaches and encourages students to follow their ideas

Carry out the plan

Teacher gives students concrete content-related support (often early in the process)

Teacher gives (almost) no (strategic) help and does not answer students’ questions related to the problem

Teacher gives staggered aids (motivational/ feedback/general strategic/task-specific strategic/ content-related)

Look back

Teacher fixates on results and neglects approaches

Different approaches are Teacher highlights presented, but strategic approaches and ideas are not highlighted strategies over results explicitly

show consistencies between teachers’ behavior and their beliefs in the sense of Ernest, there is still need for additional research for at least two reasons (cf. Rott, 2019): (a) Some researchers report inconsistencies between teachers’ beliefs and their classroom behavior (e.g., Cross Francis, 2015; Skott, 2001). (b) Almost no studies deal with teaching for PS competencies; instead, observed lessons cover other topics like adding fractions.

3 Methodology To be able to analyze teachers’ behaviors in PS lessons, it is not enough to just interview them about their teaching, they have to be observed. The participants of this study are German teachers who voluntarily agreed to being filmed in their lessons; this is a convenience sample of teachers that had contact to the author or students of his. As of May 2022, 30 teachers of primary and secondary schools participated in this study, with codenames being capital letters starting with “A”. In this article, data from all 13 teachers of this sample, who work at primary schools (teachers N, O, S, T, U, V, W, X, Y, AA, AB, AC, and AD) are analyzed. All participants of this study are female, which is not a coincidence as almost 90% of German primary teachers are female (Statista, 2022). The teaching experience of those 13 teachers ranges from 0 years (just finished their second phase of teacher education) to 35 years (shortly before their retirement) with a mean of 15, and a median of 10 years teaching

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experience. For space reasons, only teachers N (about 35 years old with 10 years of experience), S (about 60 years old with 35 years of experience), and T (about 35 years old with 9 years of experience), who all work in North Rhine-Westphalia, Central Germany, are reported in detail (the video of teacher N was recorded by Fröhlich, 2018; S and T were both recorded by Reusch, 2018). The methodological decisions of this study are as follows (for details, see Rott, 2019—there for lower secondary teachers). Each teacher was asked to plan a lesson (researchers did not interfere) with the topic PS, that is building pupils’ PS competencies. The lessons were observed and videotaped (camera pointing towards the blackboard, students are mostly seen from behind; still, their parents gave consent to being videotaped); thereafter, each teacher was interviewed to gain additional information on the lesson and especially on her beliefs. Lesson videos were coded using the grid (Table 1); for each lesson, one code for each of Pólya’s phases was given. The teachers’ beliefs were coded from the interview statements (independent of the lessons), using the three views by Ernest (1989): instrumentalist, problem-solving, or Platonist view. Teachers’ behavior in their lessons was coded by two independent raters (the author and students writing their master theses in the project) with excellent interrater agreement (Cohen’s kappa > 0.9; see Rott, 2019, for details); divergent codes were re-coded consensually. The interviews have been discussed and interpreted in a small team of researchers (using consensual validation).

4 Results 4.1 Teachers’ Behavior Teacher N: She starts the lesson in her 3rd grade class with “I have brought something new for you, an empty house and numbers” and shows her pupils the situation in Fig. 1. “A number should be placed in each room and the roof. How can you calculate in this house so that the numbers fit?” Without defining what “fit” could mean, she tells her pupils to work in pairs to “explore” how this house works. Teacher N has prepared (1) an empty house with easy/difficult numbers (to choose for weak/strong pupils), (2) a second house with different numbers, and (3) an empty house to fill in with numbers thought up by the pupils. She makes her pupils repeat her instructions. (This phase lasted almost 8 min.) The pupils work in pairs with their houses. Teacher N walks around and asks pupils what they have done so far, but she does not give any tips. (Almost 22 min). In the final phase of the lesson, she makes all children come to the blackboard. Alternately, girls and boys present their ideas. Teacher N loudly repeats all correct ideas and ignores wrong ones. The correct way is to put small numbers in the lower row, multiply them for the second row, and add those numbers for the roof (“always the largest number”). These houses are called “times-plus-houses.” (10 min).

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Fig. 1 Initial situation by Teacher N

She announces that next lesson, they would continue with these houses. (1 min). Interpretation: The situation stems from the German project “PIKAS” (https:// pikas.dzlm.de/) that publishes commented materials for primary teachers. The teaching unit on these houses (https://pikas.dzlm.de/node/784) suggests teachers to exchange numbers in the rows or to add 1 to one of those numbers to have children explore the commutative and distributive laws. The first proposed lesson focuses on finding the rules; teacher N adapted this for her lesson. In the first phases of her lesson, she acts very neutral and does not give away how to put numbers into the houses. In the last phase, comparing ideas and looking back, she closely manages her pupils (see Fig. 2 for her profile). Teacher S: She starts the lesson in her 1st grade class with the following question: “On Wednesday, four boys and two girls will visit me. How many Choco Santa Claus do I need for them?” The pupils give answers like “nine”, “seven”, or “six”, and other numbers. Teacher S asks four boys and two girls from the class to step up and she counts them loudly. She then writes “4 + 2 = ” onto the blackboard and says that the correct answer is “six.” (This phase lasted for about 6 min.) Afterwards, teacher S draws two big and three small ducks on the blackboard and asks for a corresponding equation. She does not accept “three plus two”, but only “two plus three” and writes “2 + 3 = 5” under her drawing. Finally, she draws six apples and asks for an equation. The pupils guess several additions, but the teacher only accepts “6 + 0 = 6”. She then announces that in their book, the pupils would find a page with lots of animals and objects (a “Wimmelbild”); they are supposed to Fig. 2 Profiles of the lessons by Teachers S and T

Phase

c. m.

n.

Understand

S

NT

Plan

S

NT

Carry out

S

NT

Look back

S N

e. s.

T

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find an equation for each object. She then exemplarily solves the first tasks in front of the class. (10 min). In the working phase, the pupils are asked to work alone. Teacher S walks around and gives explicit hints to pupils that struggle. (17 min). In the final phase of the lesson, she initiates a comparison of the results. Pupils are asked to read one of their equations aloud. If they are correct, teacher S says so; if they are incorrect, she corrects them immediately. She finally announces the homework: a worksheet with another Wimmelbild. (6 min). Interpretation: In all phases of this lesson and, therefore, in all of Pólya’s phases of the PS process, teacher S closely manages her pupils’ approaches (see Fig. 2 for the profile of her lesson). Teacher T: In her 3rd grade class, she starts with a short recapitulation of the last lesson. “Who remembers what we did last week?” The children recall that they had built meter squares (1 m2 of cardboard); with those, they had measured areas in the classroom, for example the windows or the blackboard. “Today,” teacher T announces, “we try to measure very large areas. With our meter squares.” There would be two groups each to measure the soccer field and the schoolyard. Teacher T forms four groups (to prevent friends from working together all the time) and gives three of the meter squares to each group. (8 min). The pupils go outside and try to cover the areas with their meter squares. All groups soon start to place their squares on one edge of their areas; all three squares in a row, then the last one is lifted up and placed at the other end of the line. This way, they count the number of meter squares in a line on the edge. One group then places a second and third line next to their first, whereas another group just measures the second edge that is perpendicular to the first one. Teacher T observers her pupils, but does not comment on their approaches. She just asks them to take care of the meter squares. (20 min). Back in the classroom, the pupils present different numbers for their area sizes. Teacher T does not comment whether the results are correct or not; instead, she asks how the pupils had got to them. Some pupils report that they had measured the edges and identified the number of squares for the whole area by multiplication. Others agree and are asked by teacher T to explain this idea. The class then compares the numbers of squares in a line for each edge and computes the area sizes together. (17 min). Interpretation: In the first phase, teacher T does not give the pupils any hints or strategies how to work. For the Understanding phase, her actions are therefore coded as neutral. The same is true for the Planning and Implementation phases while the pupils are outside. In the Looking Back phase, teacher T does not focus on the results but emphasizes strategies (see Fig. 2 for her lesson profile). Overall, the three lesson profiles show that the cells from the grid (Table 1) can be helpful in giving a very concise overview of teachers’ behavior in a lesson with the topic “problem solving,” even though these lessons took place in different grades with different problems. Interestingly, for the most parts of their lessons, the behaviors of teachers N and T look pretty similar—both act very neutral—except for comparing

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results at the end of their lessons. In the final phase, teacher N focuses on one correct approach, whereas teacher T highlights pupils’ ideas.

4.2 Teachers’ Beliefs Teacher N: Being asked about the term “problem,” teacher N says that those are tasks that are too difficult for pupils; however, often the children would not read carefully enough and could solve such tasks if they did. Apart from that, a problem would be a puzzling task or a brain teaser. Teacher N then adds that for her, it is important for children to learn calculation rules; but it is also important for the pupils to understand those rules and not apply them blindly. Discovery learning like in today’s lesson would be a good way to discover and understand rules. Her beliefs were coded in between the instrumentalist and problem-solving views by Ernest (1989), with the first view prevailing. Teacher S: In the interview, teacher S states that PS is very important for her and her teaching and that she would often use them. The way she talks about “problems,” however, makes it clear that for her, every task is a problem. She then highlights the importance of learning algorithms and schemata—in the case of her lessons, the rules of addition and subtraction—before being able to solve problems. For pupils, to solve a problem is to find the correct rule to apply. Overall, teacher S was coded as holding the instrumentalist view by Ernest (1989). Teacher T: For teacher T, a problem is a task for which no algorithm has previously been learned. Characteristics of working on problems are trying, making errors, and coming up with one’s own strategies. She uses PS in her lessons whenever possible; she states that the videotaped lesson was part of her regular lesson plan and not intended for the study. However, she also states that she knows about the importance of basic skills like arithmetic rules and knowing the multiplication table. Overall, teacher T was identified as mainly holding the problem-solving view by Ernest.

4.3 The Interplay of Teachers’ Behavior and Beliefs Teacher N: Teacher N lets her pupils discover the rules how to fill in the “houses,” which fits to the problem-solving view. However, in the end, she highlights the correct approach without recapitulating different ideas or errors by her pupils, which better fits to the instrumentalist view. Teacher S: Teacher S acts as if there were only one correct way to approach a task, a single solution, and a defined way to write it down. She only accepts solutions in the way that she expects them to be and tells her pupils in a detailed way how to work on tasks. This fits to her instrumentalist view. Teacher T: In the observed lesson, teacher T highlights procedures and strategies over results. While planning to and working on the problem of the lesson, she does

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Fig. 3 Profiles of the lessons by Teachers N, O, S, T, U, V, W, X, Y, AA, AB, AC, and AD sorted by their beliefs (Abbreviations: c. m. = closely managed; n. = neutral; e. s. = emphasizing strategies)

not interfere with her pupils’ thinking processes and ideas. In the last part of the lesson, she lets different groups of pupils present different solutions and especially different strategies and initiates a discussion about those strategies. This behavior perfectly fits to the problem-solving view (cf. Ernest, 1989; Rott, 2019). These behavior patterns by teachers N, S, and T and the fundamental differences between them can also be seen in the lesson profiles in Fig. 2: The behavior by teacher S is entirely coded as closely managed, whereas teachers N and T acted neutral through most parts of the lessons. However, whereas N closely managed a comparison of results, T emphasized strategies in the last part of her lesson. This procedure of coding teachers’ behavior in the lessons and their beliefs from the interviews was applied to all 13 teachers in this study. Overall, in this sample, there was no teacher mainly holding the Platonist view; eight teachers mainly holding the instrumentalist view; and five teachers mainly holding the problem-solving view. The results are summarized in Fig. 3, the teachers’ profiles are sorted by their beliefs and show a clear pattern: Teachers holding the instrumentalist view tend to closely manage their pupils’ approaches and solutions, especially in the Look back phase of the lessons. They emphasize procedures that can be trained (one “correct” way to solve a problem), neglecting their pupils’ strategies and errors. In contrast, the teachers holding the problem-solving view seem to avoid closely managing their pupils’ approaches; instead, they act neutral or emphasizing strategies, also especially in the Look back phase. They do not care as much as the instrumentalist teachers about correct solutions but about ideas and strategies.

5 Discussion and Conclusion The research literature on teaching and beliefs has previously shown significant interplays of teachers’ behavior and beliefs, i.e. showing that the ways in which teachers act in their classrooms are consistent with their beliefs. However, most of these studies (e.g., Siswono et al., 2019; Wilkins, 2008) use only closed self-report instruments instead of observations and interviews to gain insight into teachers’ behaviors and beliefs, respectively. Also, studies examining the interplay of teachers’ actions and beliefs often do not focus on PS lessons, but on routine procedures like adding fractions (cf. Rott, 2019). Therefore, studies like the one presented in this

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paper are needed. The results of this study are consistent with the literature, extending them to PS lessons in primary schools.

5.1 Limitations of the Study Looking back at this study, some limitations need to be addressed. So far, the teachers’ behavior (lesson profiles, especially in the Look back phase) and their beliefs (instrumentalist or problem-solving view) always match. However, the results presented here rely on only 13 teachers; thus, no generalization is intended. Also, using the grid (Table 1) to code teachers’ behavior might be seen as very simplistic. Actually, it is the intention of the grid to be simple; using the grid to code lessons was never intended to replace more thorough, qualitative analyses of PS lessons. Instead, it was designed to allow for comparisons of very different lessons; this requires some kind of abstraction and simplicity. This also allowed to identify patterns in the behavior codes as presented in Fig. 3. In university seminars and school internships, describing lessons via the grid proved to be easily usable even by pre-service teachers and to be a powerful tool for analyses and discussions of observed lessons. Another limitation is that only beliefs about the nature of mathematics (in the sense of Ernest, 1989) were considered, neglecting beliefs about teaching and learning of mathematics (that often but not always match with beliefs about mathematics, cf. Beswick, 2012; Safrudiannur & Rott, 2020). Other factors that influence teachers’ behavior, like knowledge and goals (cf. Schoenfeld, 2010) or especially their specialized knowledge about problem solving (cf. Chapman, 2015), have also not been included. In this case, however, all teachers had the same goal (teaching for PS) and uniformly little knowledge about mathematical PS and teaching for PS (confirmed in their interviews).

5.2 Outlook So far, the analyses were descriptive with no intention to evaluate “good or bad teaching.” However, empirical studies and theoretical frameworks suggest the student orientation and highlighting independent ideas and strategies might result in better overall learning outcomes than close guidance (e.g., Hattie & Yates, 2014). If teachers’ behavior that aims at strategic diversity (emphasizes strategies) instead of closely managing students’ approaches (“one correct way of solving a problem”) is favored, then this study could have direct implications for teacher education and professional development. Such courses and programs might only be successful if the participating teachers’ beliefs are taken into account (cf. Stipek et al., 2001)—said the other way around: without addressing their beliefs, knowledge about teaching for PS might not be enough to change teachers’ behavior. However, changing beliefs

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about mathematics and problem solving has shown to be a difficult, complex, and time-consuming endeavor (cf. Philipp, 2007; Stipek et al., 2001). It seems advisable to enable extensive PS experiences for pre-service teachers already in their education at universities. For example, Bernack-Schüler et al. (2015) were able to show that experiences that last at least one semester might have a permanent effect on pre-service teachers’ beliefs about mathematics. However, at least in Germany, such experiences are (still) no integral and mandatory element of teacher education at most universities—even though there are notable exceptions (e.g., Grieser, 2013).

References Bernack-Schüler, C., Leuders, T., & Holzäpfel, L. (2015). Understanding pre-service teachers’ belief change during a problem solving course. In C. Bernack-Schüler, R. Erens, A. Eichler, & T. Leuders (Eds.), Views and Beliefs in Mathematics Education: Proceedings of the MAVI 2013 Conference (pp. 81–94). Springer. Beswick, K. (2012). Teachers’ beliefs about school mathematics and mathematicians’ mathematics and their relationship to practice. Educational Studies in Mathematics, 79, 127–147. Chapman, O. (2015). Mathematics teachers’ knowledge for teaching problem solving. LUMAT, 3(1), 19–36. Cross Francis, D. I. (2015). Dispelling the notion of inconsistencies in teachers’ mathematics beliefs and practices: A 3-year case study. Journal of Mathematics Teacher Education, 18(2), 173–201. Dionné, J. (1984). The perception of mathematics among elementary school teachers. In J. Moser (Ed.), Proceedings of the Sixth Annual Meeting of the PME-NA (pp. 223–228). University of Wisconsin. Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching: International Research and Pedagogy, 15(1), 13–33. Fröhlich, M. C. (2018). Problemlösen im Klassenraum—eine empirische Analyse mit dem Schwerpunkt Primarstufe [Problem solving in the classroom: An empirical analysis with a focus on primary school]. Unpublished Master Thesis, University of Cologne. Grieser, D. (2013). Mathematisches problemlösen und beweisen—Eine entdeckungsreise in die mathematik [Mathematical problem solving and proving: A journey of discovery into mathematics]. Springer. Grigutsch, S., Raatz, U., & Törner, G. (1998). Einstellungen gegenüber Mathematik bei Mathematiklehrern [Mathematics teachers’ attitudes towards mathematics]. Journal Für MathematikDidaktik, 19(1), 3–45. Hattie, J., & Yates, G. (2014). Visible learning and the science of how we learn. Routledge. Kuzle, A., & Rott, B. (2017). Arbeitskreis Problemlösen [Working group “problem solving”]. In U. Kortenkamp & A. Kuzle (Eds.), Beiträge zum Mathematikunterricht 2017 (S. 1441–1443). WTM. Lesh, R., English, L., Riggs, C., & Sevis, S. (2013). Problem solving in the primary school (K-2). The Mathematics Enthusiast, 10(1), Article 4. Organization for Economic Co-operation and Development (OECD). (2013). PISA 2012 assessment and analytical framework: Mathematics, reading, science, problem solving and financial literacy. https://doi.org/10.1787/9789264190511-en Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect (Ch. 7). In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–315). Information Age. Pólya, G. (1945). How to solve it. University Press.

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Reusch, J. (2018). Problemlösen im Klassenraum—ein Vergleich von zwei Lehrerinnen an einer integrativen Grundschule [Problem solving in the classroom: A comparison of two teachers at a primary school]. Unpublished Master Thesis, University of Cologne. Rott, B. (2019). Teachers’ behaviors, epistemological beliefs, and their interplay in lessons on the topic of problem solving. International Journal of Science and Mathematics Education. https:// doi.org/10.1007/s10763-019-09993-0 Safrudiannur, S., & Rott, B. (2020). Measuring teachers’ beliefs: A comparison of three different approaches. EURASIA Journal of Mathematics, Science and Technology Education, 16(1). https://doi.org/10.29333/ejmste/110058 Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. Schroeder, T. L., & Lester, F. K. (1989). Understanding mathematics via problem solving. In P. Trafton (Ed.), New directions for elementary school mathematics (pp. 31–42). National Council of Teachers of Mathematics. Siswono, T. Y. E., Kohar, A. W., Hartono, S., Rosyidi, A. H., Kurniasari, I., & Karim K. (2019). Examining teacher mathematics-related beliefs and problem-solving knowledge for teaching: Evidence from Indonesian primary and secondary teachers. International Electronic Journal of Elementary Education, 11(5), 493–506. https://doi.org/10.26822/iejee.2019553346 Skott, J. (2001). The emerging practices of a novice teacher: The roles of his school mathematics images. Journal of Mathematics Teacher Education, 4(1), 3–28. Statista. (2022). Anteil der weiblichen Lehrkräfte an allgemeinbildenden Schulen in Deutschland im Schuljahr 2020/2021 nach Schulart [Percentage of female teachers at general schools in Germany in the school year 2020/2021 by type of school]. https://de.statista.com/statistik/daten/ studie/1129852/umfrage/frauenanteil-unter-den-lehrkraeften-in-deutschland-nach-schulart/ Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17(2), 213–226. Wilkins, J. L. (2008). The relationship among elementary teachers’ content knowledge, attitudes, beliefs, and practices. Journal of Mathematics Teacher Education, 11(2), 139–164. https://doi. org/10.1007/s10857-007-9068-2

Chapter 7

Movie Clips in the Enactment of Problem Solving in the Mathematics Classroom Within the Framework of Communication Model Tin Lam Toh and Eng Guan Tay

Abstract In this chapter, we propose that the teaching of mathematical problem solving can be understood through a classical model of communication. The use of movie clips for the teaching of mathematical problem solving can be seen as a communication process. The role of the movie clips, serving in addition to being a narrative hook, presents the mathematical problem in a way understandable and relatable to students through its appropriate contextualization of the mathematical problem. The chapter further discusses the characteristics of two movie clips that can be used for teaching mathematical problem solving. Keywords Mathematical problem solving · Communication model · Movie clip

1 Introduction Mathematical problem solving has been the heart of the K-12 mathematics curriculum in many countries in the world, such as the United Kingdom (Cockroft Report, 1982), the United States (NCTM, 1989), Australia (Australian Education Council, 1990) and Singapore (Ministry of Education (MOE), 2006a, 2006b). Despite a world-wide push for problem solving since the 1980s and much educational research conducted on problem solving, the enactment of problem solving in authentic classrooms and success stories in the acquisition of problem solving by students, including continue to be relatively limited. In addition, the reasons why problem solving is often not predominant in the mathematics classroom, could be both cognitive and affective. Firstly, problem solving is T. L. Toh (B) · E. G. Tay National Institute of Education, Nanyang Technological University, Singapore, Singapore e-mail: [email protected] E. G. Tay e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_7

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generally perceived to be difficult for most students. Even the first two Polya stages (Polya, 1954) of Understanding the Problem and Devising the Plan, already pose much difficulty to the average student (Gunbas, 2015; Rahman & Amar, 2016). On an affective level, anecdotal evidence from authentic mathematics classrooms shows that students may not be interested in problem solving (or mathematics in general), and problem solving lessons are usually treated as esoteric lessons which are disjoint from the usual mathematics lessons. A survey conducted by Toh and Lui (2014) among the Singapore mathematics teachers showed that low attaining students did not like mathematics as they did not find the subject relevant to daily life. The students also were of the opinion that most mathematical tasks in the national examinations require high cognitive effort. On the other hand, the survey pleasingly reported that many Singapore teachers had ackowledged the importance of considering the affect of their students in addition to the mathematical content in designing their mathematics lessons. They had also focused their attention to address their students’ affective needs in mathematical activities during the lessons. This attention to affect is aligned to the Singapore syllabus document, which explicitly states that students’ affect (attitude towards learning mathematics) is one of the five dimensions that need to be addressed in mathematical problem solving (see Fig. 1 for the Singapore mathematics curriculum framework). The many innovative approaches to motivate their students in mathematics used by the mathematics teachers in Toh and Lui’s survey included the adaptation of movie clips to contextualize mathematical tasks, in order to motivate them to learn mathematics.

Fig. 1 The Singapore mathematics curriculum framework

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Contextualizing mathematics is one important aspect that has attracted the attention of educators and researchers. The use of contexts which are meaningful, either relevant or fun, to learners will likely have positive impact on developing their problem solving processes (e.g., Goldstone & Son, 2005), and, more importantly, to persuade them to be willing to persevere with a problem-solving task, which is normally perceived as difficult. In fact, researchers such as Gunbas (2015) asserted that contextualization will likely increase a learner’s ability to transfer their skills to novel situations. Gunbas’s (2015) study advocated the use of appropriate stories to help learners understand the proposed problems (understanding the problem is the first stage of problem solving according to Polya), because stories can make the tasks more meaningful (either relevant or interesting) to learners. This in turn facilitates the activation of real-world knowledge, which may help learners devise the correct solution (Inoue, 2008). We believe that the use of exciting movie clips in mathematics classrooms can engage students both cognitively and affectively in problem solving, through presenting the mathematical content using contexts that students find more easily comprehensible and appealing. In fact, we are not alone in proposing the use of movie clips for classroom instruction; many other researchers have already started this approach for instruction (e.g., Butterworth & Coe, 2004; Greenwald & Nestler, 2004; Han & Toh, 2019; Reiser, 2015). After the covid-19 pandemic, instructional practice has permanently changed to include modes of instruction to modes other than face-to-face instruction, such as flipped learning and asynchronous instruction. We believe that with this new landscape of educational instruction, movie clips might have a greater role to play for classroom instruction.

2 Movie Clips In this chapter, we adopt an operational definition of the term movie as a series of animated scenes which convey a storyline. Thus, movie clips can include the typical anime which is the fancy of many schoolgoing children, parts of a film, or even a self-developed animated video clip specifically designed for instructional purpose. The use of movie clips for classroom instruction is not too far fetched as the global trend of mathematics instruction is moving towards the application of mathematics in the real world, e.g., the emphasis by the Singapore Ministry of Education (MOE) (MOE, 2000, 2006a, 2006b, 2012). The use of movie clips for classroom instruction is not a recent innovation. For example, Park and Lamb (1992) reported the use in a Physics lesson of a scene from Superman II, where Superman saved a boy from falling into the depths of the Niagara Falls, to contextualize concepts involving acceleration due to gravity. Use of movies for instruction has started to receive increasing attention of researchers worldwide. For example, Salleh and Tan (2006) held that movies and television commercials can promote critical thinking in the science and mathematics classroom. Compared to

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static representation such as photographs and diagrams, movies are able to adequately capture the dynamic constructs of the concepts involved (Bell et al., 2012; Pierce et al., 2005). Russo et al. (2021) urged educators and researchers to consider extending the use of pictures and story narratives to encompass movie clips and short films for classroom instruction. In addition to solving the mathematics problem presented through a movie clip, the context of the movie clips could further serve as a motivation for students to stretch their own learning in extending the problem by considering other scenarios. The underlying principle of this is to tap on the students ‘ interest in the context so that they will persevere in solving the problem and further creating other challenging problems within the same context. The aim of this chapter is to discuss how movie clips can be integrated into a mathematics problem solving lesson, and the principles of selecting/designing such a movie clip. The details of of enacting problem solving in a mathematics classroom using Polya’s four-phase problem solving model and the strategies required for students’ sense making in working on problem solving tasks has been discussed at great length (e.g., Toh et al., 2008a, 2008b; 2011; 2013), and hence will not be elaborated here.

3 Some Elements of Communication Theory Communication can be simply stated as the transmission of a message from a sender to a receiver (Baran, 2019). Lasswell (1948) put it succinctly that communication serves to answer the questions: (1) who says? (2) says what? (3) through which channel? (4) to whom? (5) with what effect? As an application of a simple communication process to the classroom setting: Suppose a teacher attempting to teach problem solving insists on speaking at the high level technical language, which is not accessible by students. Even though there is sender and message, in this case it will likely not get the desired response of the students (to engage with the message). To further define the term communication more precisely, the process of communication includes establishing a commonness or sameness of thought between two groups: the sender and receiver (Schram, 1955). Researchers have started to connect education with entertainment, and in this direction, various theoretical views and empirical research on entertainmenteducation (or edutainment) have been carried out extensively with a focus on the impact of the communication on the receiver (e.g., Aksakal, 2015; Okan, 2003; Singhal & Rogers, 2002). This could further suggest that perhaps education can be understood using the lens of the theory of (mass) communication. In this section, the key features of a communication model are discussed. The theoretical model of a communication process consists primarily of a sender and receiver, who, through the process of communication, will be able to achieve some form of commonality with the sender. The establishment of this commonality is not an easy task (Belch & Belch, 2004), as its success depends on many factors.

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Source/ Sender Feedback

Encoding

Channel message

NOISE

Decoding

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Receiver

Response

Fig. 2 A communication model [The model is re-drawn from Belch & Belch (2004, p. 139)]

In addition to the two fundamental components of sender and receiver, a typical communication model further consists of the communication tools (message and channel), and four communication functions and processes (encoding, decoding, response, and feedback). Lastly,“noise” in the communication model refers to any interference that hinders effective communication. It ranges from the source/sender to the receiver. The entire communication model can be summarised in Fig. 2. The sender, in attempting to ensure that the message can be understood by the receiver, encodes it using a suitable form for the message to be transmitted, usually in a symbolic form. The sender next decides the channel of communicating the message to the sender. The channel of communication can either be personal channels or nonpersonal channels (Smith & Vogt, 1995). The receiver next decodes the message by transforming the message into the original ideas or information that was intended by the sender. Throughout the entire process of communication described above, the message could be subject to interference or distortion, which is unplanned. This unplanned distortion or interference is referred to as noise. It should be noted that noise could occur in any stage of the communication process described above. The receiver’s reaction after encountering the message is typically termed as a response in communication theory. The response goes far beyond merely understanding the message conveyed by the sender. In particular, in the field of advertising, the response could be manifested as the consumer’s immediate desire of the advertised product, which manifests as an immediate order of the product, and possibly followed by other responses (e.g., persuading friends to purchase the same product). The response also includes an addition part that is communicated back to the sender. This is termed as feedback of the communication process. Communication study posits various models for the response part of the communication process, and has been depicted as a hierarchy of stages that a receiver went through, starting from being unaware of the message to having received the message. We present one such response hierarchy model commonly used in communication theory: the AIDA model, which was originally developed to depict the stages that a customer undergoes in the selling process by a salesperson (Strong, 1925). The receiver (or buyer) of a “succesful” communication goes through the stages of attention, interest, desire and action, with the end result being action (or buying the product). These stages can be classified categorically as the cognitive stage, affective

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stage and behaviorial stage. In fact, other response hierarchy models such as hierarchy effects model (Lavidge & Steiner, 1962), innovative adoption model (Rogers, 1962), and information processing model (McGuire, 1978) posited similar stages in response to communication, although with different terminology. The commonality across the various hierarchy response models is that communication does not end with the receiver being cognizant of the message; the impact of the communication on the receiver’s affective and behavioral domains is an equally important part of the process. The above discussion clearly resonates with an educator, as there are many points analogous to classroom instruction.

4 Mathematical Problem Solving with a Little Help from Communication Theory Some educators have recognized that teaching and learning in the classroom is a communication process (e.g., Puro & Bloome, 1987). Hence, it is a natural progression to study the classroom instruction of mathematics (or any other discipline) through the lens of communication. In this chapter, we specifically focus on mathematical problem solving, a process skill even an above-average student will find challenging. We draw a parallel of the entirety of the teaching and learning problem solving process with the communication process which is summarized in Fig. 3. In conjunction with our use of communication theory in teaching problem solving in the mathematics classroom, we use the Polya’s four-step model (Fig. 4) which was also used by Toh et al. (2008a, 2008b, 2011) in their task of enacting problem solving in the mathematics classrooms. Note that the problem solving model in Fig. 4 highlights the non-sequential nature of the four steps; rather there are multiple loops across any two of the stages. The planning stage of the teacher begins with the message, which is the problem solving task that the teachers want to engage their students. In the process of encoding, the teacher considers the contextualization of the task in order to facilitate students’ understanding of the problem and kindle their interest to solve the problem. The teacher then selects the channel of communication, which in our discussion, is the use of movie clip. This includes the selection/development of appropriate movie clips to embed the mathematical task, in order to sustain the students’ interest to engage and to persevere in the task. With the choice of mathematical task embeded in the movie clip (with the context of the movie clip deemed appropriae for the students), the teacher plans how best to provide the scaffold the students in decoding the message, that is, to fully understand the embedded mathematical task and to solve it. This corresponds to the scaffolding provided by Toh et al. (2008a, 2008b, 2011) for their enactment of problem solving lesson. The scaffold in Toh et al. facilitated the students in going through the four stages of Polya (Fig. 4). The scaffolding related to the movie-based problem solving

7 Movie Clips in the Enactment of Problem Solving in the Mathematics …

Problem solving task

Contextualization (encoding)

Presentation to students (channel message)

Personal channel e.g., classroom instruction Nonpersonal channel e.g., blended learning, selflearning, etc.

Scaffolding (decoding)

Teacher-guided scaffolding Students’ independent use of scaffolding worksheet

Positive emotion with the task

Solution of the task (receiver)

Posing related tasks and solving them (Behavior)

Solving the problem and extending the problem based on positive emotion with the assigned task Fig. 3 A proposed communication in problem solving model

Understand the Problem

Devise a Plan Carry out the Plan Check and Extend Fig. 4 Polya’s problem solving model

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lesson could include generic scaffold similar to those used by Toh et al., or taskspecific scaffold. The planning of the appropriate scaffold to facilitate students to decode the message thus forms a crucial part of the lesson planning process of a problem solving lesson. The next planning phase consists of considering the message channel of the communication, that is, how the movie clip can be used for instructional purposes. These movie clips could be used in the classroom, facilitated by the teacher (personal communication), or used for flipped learning (nonpersonal communication). The eventual planning phase should further consider the message reaching the receiver who has successfully solved the problem. Ideally, the problem solving experience should provide the students with positive emotion and experience with the process, so that there is a behavioral change in the student. This behavioral change is translated to the students being willing to engage in further extension of the task, that is, posing a new problem modified from the given problem solving task. This corresponds to Polya’s stage 4, that of checking their solution of the correctness of the original problem, and extending or generalizing to further problems. The above description thus shows that indeed the communication model (Fig. 3) provides a holistic perspective in designing a problem solving task that not only addresses problem solving from the perspective of Polya (Fig. 4), but also addresses the affective needs of the students. Unlike the situation of mass communication, the entire process of communication (Fig. 3) interpreted in the education setting depicts that the teacher plays a dual role of both the sender of the message (i.e., the problem-solving task), and also a facilitator of the students in decoding and further engagement with the message. We begin by considering the use of the problem-solving tasks in Toh et al. (2011) as an illustration of the enactment of a problem-solving lesson. In the proposed lessons in Toh et al. (2011, 2013), a set of 17 problems was proposed to introduce the various aspects of problem solving in the specialized problem solving lessons for teaching about problem solving. The set of 17 problems in Toh et al. (2011) are reproduced using the same numbering in the website http://math.nie.edu.sg/mprose.

4.1 Use of Movie Clip in Problem Solving Consider the case of Problem No. 1 in http://math.nie.edu.sg/mprose (only the first part of the full problem is shown here): You are given two jugs, one holds 5 L of water when full and the other holds 3 L of water when full. There are no markings on either jug and the cross-section of each jug is not uniform. Show how to measure out exactly 4 L of water from a fountain. As discussed in detail in Toh et al. (2011), this is a mathematically rich problem to engage students in various problem-solving heuristics, and especially enabling students to appreciate the importance of understanding a problem (First Stage of Polya’s problem solving model, see Fig. 3). The context of the problem is that the jugs are not marked or the cross-section is not uniform, so that one could not measure

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a fraction of the capacity of each jug. Each jug can either measure full or zero capacity of the jug. The possible scaffoldings and the levels of scaffoldings (problem specific and generic related to Polya’s stages) that teachers could provide, and the possible students’ responses, have been discussed in Toh et al. (2011). The mathematical content knowledge of the problem is one in which students ranging from lower secondary to undergraduate level can easily understand, and the content of the problem could involve elementary arithmetic at the lower secondary level to advanced Number Theory at the undergraduate level. In this sense, this problem can thus be used as a form of differentiated instruction in most classrooms. Note that the cognitive resource required for this problem is minimal (only the four basic operations with integers are required). Through Acting It Out or Drawing a Diagram, having already understood the problem, students would be led to devise an algorithm to find a possible solution of the problem. The students could also model the situation using the equation 5x + 3y = 4, interpreting a positive integer n as filling the container n times and a negative integer -m as discarding the water in the container m times. Students could be led to explore the uniqueness or otherwise of the solution of this equation. The higher ability students (or undergraduate students) could be led on to find the general solution of the diophantine equation. In a typical school mathematics classroom, such a problem would likely appear as a mind-game for most school students, because it is difficult to contextualize the mathematical content under real-world mathematics. Students might regard such questions as IQ questions, that is, the category of problems that discomfit the students or perceived of having the intention to pull them down psychologically. Bearing this pitfall in mind, this problem, if contextualized using a context which is appealing to students, might be able to offer a different twist to the students and to arouse their eagerness to solve the problem, especially if it is part of an action movie. A movie clip corresponding to Problem 1 contextualizing this problem can be found from the Hollywood Movie Die Hard Part 3 and is available in the YouTube website https://www.youtube.com/watch?v=BVtQNK_ZUJg. The movie clip contextualizes by problematize the problem into an important mission to prevent a detonation of a bomb in a park. In this part of the movie, the movie characters, in order to deactivate a bomb which was to be detonated within a short while, were required to measure exactly 4 gallons of water from two unmarked containers, one 5-gallon and the other 3-gallon. The measurement needed to be exact to deactivate the bomb. This movie clip contextualizes the given problem, thereby providing the students the motivation in the real-world context to solve this problem in order to save human life. Understanding the problem involves being cognizant of the given situation (e.g., the crucial fact that the containers are not marked, so that it is impossible to measure any volume of water that does not fill the container completely). The movie clip includes such a description by the characters in order to facilitate the viewers’ understanding of the problem. The channel of communication of the encoded message (in the form of a video clip) could be either personal communication through the classroom instruction during synchronous classroom instruction, or nonpersonal communication, through asynchronous mode as the prelude to the problem-solving lesson, such as in the case of

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flipped learning (Bhagat et al., 2016). The advantage of using it in asynchronous instruction is that students are able to stop and rewind the movie clip in order to achieve a better understanding of the problem. Moreover, this clip makes it suitable for flipped learning as the solution of the original problem was clearly elucidated in the movie so that students will be able to acquire the steps in solving the problem. In this aspect, various pedagogical approaches for the synchronous and asynchronous instruction can be infused to achieve students’ understanding of the problem and devising the plan to solve the problem. The decoding of the message forms the main part of the problem-solving lesson. Much discussion of the problem-solving approach to tackle this problem has been detailed in Toh et al. (2008a, 2008b, 2011), Toh and Lui (2014). The role of the teacher is to deliberate on the scaffolding process for the students (e.g., the use of the practical worksheet proposed by Toh et al. (2008a; 2008b; 2011)). Note that the characters in the movie clip articulate their thinking processes and the solution of solving Problem No. 1. The context could lead the students to develop greater curiosity in this genre of problems, following which the remaining part of Problem No. 1 could be shown fully to the students. Note that teachers could ride on the affordance of this problem for extending the problem, by changing the capacity of the two jugs and the volume of water required to be measured. However, a change of the capacity of the two jugs to measure a required volume of water might not lead to a feasible solution, as illustrated below. Entire Problem No. 1. You are given two jugs, one holds 5 L of water when full and the other holds 3 L of water when full. There are no markings on either jug and the cross-section of each jug is not uniform. Show how to measure out exactly 4 L of water from a fountain. Show also how you could obtain the following (if possible): (i) Get 2 L from 3 to 7 L jugs. (ii) Get 6 L from 12 to 16 L jugs. (iii) Get 12 L from 18 to 24 L jugs. The main intention of using the context of the movie clip to help students understand the mathematics problem, excite them to engage and further to persevere in problem solving (cognitive stage), and beyond that, to get emotionally (affective stage) attracted to the problem presented in the movie clip. Eventually, the students should go beyond solving the original problem to extend the original problem (as suggested by parts i, ii and iii in the entire problem) (behavioral stage). Using the perspective of communication theory from the advertising industry, the receivers (consumers) should ideally not stop at being cognizant of the product; analogously in our case, having solved the problem solving. The students should be emotionally attracted to the problem and are willing to extend the problem and eventually pose new problems within the given context and solve these newly created problems. Ideally, students could recognize the underlying mathematical structure of the generic problem.

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4.2 A Self-developed Movie Clip to Teach Problem Solving for Low Attaining Students In an attempt to engage low attaining students to the processes of mathematical problem solving, Han and Toh (2019) reported the development of an animated movie strip to engage students in thinking about percentages. The content of the clip shows a boy intending to buy roses from two potential florists: one florist offering a 10% discount and the other florist offering a 5% discount followed by another 5% discount. The boy has to decide which is a better deal. The content of the movie clip was about more advanced manipulation of a combination of two percentages. This belongs to the lower secondary mathematics topic on percentage, and the subtopic on combination of percentages. The combination of percentages may be counterintuitive to students’ understanding (e.g., an increase of 10% followed by 5% is not equivalent to an increase of 15%). The movie clip problematizes the task of determining the equivalence or otherwise of a discount of 10% and a combination of a discount of 5% followed by another 5% in a real world scenario. This problem is of the same genre of Problem 7B found in http://math.nie.edu.sg/mprose/elearning. Unlike the video on the jugs problem, the context of the storyline in this movie clip is very much related to the students’ daily life and is sufficiently interesting to engage them due to its relevance. This video clip contextualizes the problem using a real-world situation of a young boy intending to purchase roses for his girlfriend on Valentine’s Day. The boy has to decide which of two florists to purchase from: both florists offer the same selling price of $10. One florist offered a flat 10% discount, and the other florist offered two successive discounts of 5%. The video attempts to engage students to check for the difference in price offered by the two florists. The audience is asked to assist the boy to make decide the better deal offered by the two florists. The movie clip next provides a detailed mathematical explanation of the two offers. The later part of the video consists of other extended problems related to the composition of two percentages using exact numbers. The movie clip continues with more new scenarios which are modifications of the animated video clip. As with the previous movie clip, this clip can be used both for nonpersonal communication or personal communication in the mathematics classroom. Within the mathematics classroom, the movie clip can be used to make explicit the mathematical principles underlying this task, or if the part of explaining the solution of the task is not given, it could be used as a problem-solving lesson with the teacher providing appropriate scaffolding.

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5 Common Features of the Two Movie Clips In selecting, editing or developing video clips for mathematical problem solving instructions (or indeed for any classroom instruction), we argue that teachers need to adhere to some fundamental principles to best enable student learning of problem solving (or at least to make problem solving appealing to the students and to make them willing to persevere in the challenging task), in addition to their usual pedagogical considerations. In this section we attempt to examine the two movie clips described above, which are of very different nature and quality, and identify some characteristics for teachers to consider.

5.1 Storyline Relatable to Students The storylines of both movie clips are easily relatable to most students of school going age, hence the context is relevant and interesting to them. In the first clip, the context was one involving survival in spite of challenging obstacles to achieve success. Survival forms the most exciting storyline in recent movies (Beck, 2017). The second clip, involving a boy purchasing roses for his girlfriend, is one that is real and relatable to the students. Both movie clips show an ordinary person or even an underdog performing amazing feats (in their own ways), which is much favored by movie goers (Beck, 2017).

5.2 Contextualized as a Decision-Making Task for Students Decision-making is a part of an individual’s cognitive development and is an indicator of development of one’s logical reasoning (Celik, 2017). The problem-solving tasks in both movie clips are typical tasks which are problematized using the storyline. The tasks are transformed into decision-making tasks for the viewers. This would not only serve to arouse their curiosity, but also empower them to make decisions within a real-world context.

5.3 Inclusion of Humour Elements of humour were infused into the storyline in both video clips. Humour in movie clips is a useful pedagogical tool to increase students’ information retention and appreciation of information (Kaplan & Pascoe, 1977).

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5.4 Reduction of “Noise” in the Movie Clip In the classical model of communication, “noise” is any interference that might disrupt the communication process. The first movie clip was edited to remove noise which might otherwise adversely affect the objective of the video clip. The noise in this movie clip includes the vulgarity and rascist jokes which are not conducive for the problem solving process. This was completely removed from the edited movie clip. In the second movie clip, which was developed by the authors, the graphics in the movie clip, which includes the characters and the background scene, was not overtly elaborate but kept to the minimal, while the colours were sufficient but not excessive, but still sufficiently appealing to the students. The designers saw the noise in this movie clip as an excessive use of colours and comic characters. Such a design was made with the aim for students to focus on the essentials shown in the clip, thereby reduce the noise of communication through the movie clip.

5.5 Use of Simplified Language in the Movie Clips As the movie clips were intended for laymen, as with the nature of most popular movies, the language used in both were not the precise mathematical language that is found in typical mathematics discourse, but a form that most layman could understand. In the first movie clip, the explanation of the mathematical steps in solving the jug problem was discussed using a simple language that even a laymen can easily understand. In fact, in the second self-developed movie clip, a form of colloquial language (known as Singlish and commonly used in the Singapore community) was used in the narration and conversation portions in the clip. This was part of the deliberate design by Han and Toh (2019), based on the consideration that an overly sophisticated language could distract or discourage the students, especially the low attaining students, from the mathematical content which is being conveyed by the movie clip.

5.6 Task Generalizable to New Problems The tasks which were presented in the two movie clips, through the humorous presentation of the tasks, have the ability to encourage the students to generalize and extend the problem with “what if” considerations as they watch the movies. Note that viewers of movies are not passively watching the movies, but also are the active creators of various alternative scenarios as well.

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5.7 Suitable for Both Personal and Nonpersonal Communication Both movie clips serve more than just a “narrative hook’; the movie clips are, as discussed above, “complete” so that they could be used for both personal communication (or synchronous instruction, such as face-to-face classroom instruction) or nonpersonal communication (or asynchronous instruction, such as flipped learning). The solution steps of each of the problems were clearly elucidated in the clips, so that they could be used for the individual’s own viewing in the nonpersonal communication. For personal communication, the teacher can use various methods to use the clip: either show it all, or dispense with the part of the explanation of the solution of the problem so as to engage the students in producing the solution.

6 Use of Movie Clips for Mathematics Instruction Some researchers have gone so far as to comment that the context of mathematics problems that are presented in most school textbooks have not been generally successful in arousing the interest of learners (e.g., Walkington et al., 2012). Perhaps an appropriate use of movie clips could fill this gap by contextualizing the mathematical tasks in a meaningful and interesting way for them to acquire the related mathematical concepts. Gunbas (2015) claimed that contextualization could motivate students to struggle through the difficult problem solving processes, and asserted that such an approach could increase the chance of the learners to transfer their learning to another new context. Not many studies have appeared on the use of movies for mathematics instruction, although education literature for using movies and films as part of classroom instruction in other disciplines exist. The use of films in acquisition of vocabulary in linguistic lessons (e.g., Ashcroft et al., 2018), and education at the tertiary level (e.g., Lee, 2019) have started to appear in recent years. In enacting a mathematics lesson, in particular a problem-solving lesson, adapting or developing movie clips, the teacher needs to identify his or her role during the lesson: as a facilitator or as an instructor during the lesson. This identification of role will then affect how the movie clip is used in the lesson. If the teacher perceives his or her role as a facilitator rather than an instructor, we would refer the reader to the different levels of scaffolding proposed by Toh et al. (2008a, 2008b, 2011), Toh and Lui (2014), i.e. emphasis on Polya stages and control, specific heuristics, and problem specific hints. For whatever role the teacher perceives, the context of the movie clips should be “easy” enough to provide the opportunity for students to extend and generalize the problems using “what if” consideration, thereby stretching students to the highest possible level of problem solving.

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Table 1 Sub-questions for teachers in designing their lessons based on ADDIE model Processes

Sub-questions

Analysis

• What content/concept in problem solving we want to teach students? • What prior knowledge on problem solving do students know? • Which aspects of problem solving fit well with the movie clip?

Design

• What are our lesson objectives? • How will these objectives be covered with the movie clip?

Development

• Which part of the movie clip will achieve which lesson objectives? • What additional learning experience will enhance students’ learning of problem solving using the movie clip?

Implementation • How should students be engaged during the instruction? Evaluation

• How successful students find the lesson? • What modifications need to be made to make the lesson more successful?

While much has been suggested on the benefits of movie clips for the teaching of mathematical problem solving, researchers (e.g., Salleh & Tan, 2006) have advised the use of sound principles of instructional design to create a meaningful learning experience among the students. One such design principle is Gagne’s et al. (2005) ADDIE model. The model consists of five processes: Analysis, Design, Development, Implementation and Evaluation. We adapt the ADDIE model with selected sub-questions for teachers to consider in designing their instruction using movie clips (Table 1).

7 Final Remarks The use of movies or videos for mathematics learning for students is a growing area of interest for educators and researchers. In fact, some researchers (e.g., Butterworth & Coe, 2004; Greenwald & Nestler, 2004; Niess & Walker, 2010; Russo et al., 2021) believe that movies can serve more than a narrative hook: it can be instrumental in supporting mathematics instruction from the introduction of new concepts, exploration of mathematics in the real world, and to build up students’ image-bank that would be useful for expressing their mathematical ideas. At the current stage, literature on the use of movie clips for mathematics instruction is relatively scarce. There is little suggestion in the education literature on the difficulties likely to be encountered by students and teachers with this new mode of instruction. However, it is a truism that the development of movie clips for the delivering of one lesson is highly demanding of time and effort and, as most may feel, inefficient. Researchers and educators have begun identifying existing movies which could be used for mathematics instruction. For example, Reiser (2015) compiled a set of mathematics activities for secondary school classroom by building on existing movies and television programmes. Other educators have also developed resource by building on well-known movies (e.g., Butterworth & Coe, 2004; Greenwald &

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Nestler, 2004). It appears that the collection of resource of such activities building on existing movies and television programmes is expanding. We hope not only on the expansion of such collection, but also an increase in research of re-looking at mathematics classroom instruction using the lens of a mass communication model. Hopefully, this chapter could spur further interest in this relatively new way of examining classroom instructions.

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Lavidge, R. J., & Steiner, G. A. (1962). A model for predictive measurements of advertising effectiveness. Journal of Marketing, 24, 59–62. Lee, S. C. (2019). Integrating entertainment and critical pedagogy for multicultural pre-service teachers: Film watching during the lecture hours of higher education. Intercultural Education, 30(2), 107–125. McGuire, W. J. (1978). An information processing model of advertising effectiveness. In H. J. Davis & A. J. Silk (Eds.), Behavioral and management science in marketing (pp. 156–180). Ronald Press. MOE. (2000). Mathematics syllabus—Primary (2001). Author. MOE. (2006a). Mathematics syllabus—Primary (2007). Author. MOE. (2006b). A guide to teaching and learning of O-level mathematics 2007. Ministry of Education, Curriculum Planning & Development Division. MOE. (2012). Primary mathematics teaching and learning syllabus (2013). Author. NCTM (National Council of Teachers of Mathematics). (1989). Principles and standards for school mathematics. NCTM. Niess, M. L., & Walker, J. M. (2010). Digital videos as tools for learning mathematics. Contemporary Issues in Technology and Teacher Education, 10(1), 100–105. Okan, Z. (2003). Edutainment: Is learning at risk? British Journal of Educational Technology, 34(3), 255–264. Park, J. C., & Lamb, H. L. (1992). Video vignettes: A look at physics in the movies. School Science and Mathematics, 92(5), 257–262. Pierce, R., Stacey, K., & Ball, L. (2005). Mathematics from still and moving images. Australian Mathematics Teachers, 61(3), 26–31. Polya, G. (1954). How to solve it. Princeton University Press. Puro, P., & Bloome, D. (1987). Understanding classroom communication. Theory into Practice, 26(1), 26–31. Rahman, A., & Ahmar, A. (2016). Exploration of mathematics problem solving process based on the thinking level of students in Junior High School. International Journal of Environmental and Science Education, 11(14), 7278–7285. Reiser, E. (2015). Teaching mathematics using popular culture: Strategies for common core instruction fro, film and television. McFarland. Rogers, E. M. (1962). Diffusion of innovations. New York Free Press. Russo, J., Russo, T., & Roche, A. (2021). Using rich narratives to engage students in worthwhile mathematics: Children’s literature, movies and short films. Education Science, 11, 1–19. Salleh, H., & Tan, C. (2006). Using television commercials and movies to promote critical thinking in primary science and mathematics education. NEOS, 83–98. Schram, W. (1955). The process and effects of mass communications. University of Illinois Press. Singhal, A., & Rogers, E. M. (2002). A theoretical agenda for entertainment—Education. Communication Theory, 12(2), 117–135. Smith, R. E., & Vogt, C. A. (1995). The effects of integrating advertising and negative word-ofmouth communications on message processing and response. Journal of Consumer Psychology, 4(2), 133–151. Strong, E. K. (1925). The psychology of selling. McGraw-Hill. Toh, T. L., Leong, Y. H., Tay, E. G., Quek, K. S., Toh, P. C., Dindyal, J., Ho, F. H., & Yap, R. A. S. (2013). Mathematics mathematics more practical: Implementation in the schools. World Scientific Publisher. Toh, T. L., & Lui, H. W. E. (2014). Helping normal technical students with learning mathematics—A preliminary survey. Learning Science and Mathematics Online Journal, 2014(1), 1–10. Toh, T. L., Quek, K. S., Leong, Y. H., Dindyal, J., & Tay, E. G. (2011). Making mathematics practical: An approach to problem solving. World Scientific. Toh, T. L., Quek, K. S., & Tay, E. G. (2008a). Mathematical problem solving—A new paradigm. In J. Vincent, R. Pierce, & J. Dowsey. (Eds.), Connected maths: MAV yearbook 2008 (pp. 356–365). The Mathematical Association of Victoria.

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Toh, T. L., Quek, K. S. & Tay, E. G. (2008b). Problem solving in the mathematics classroom (Junior College). National Institute of Education & Association of Mathematics Educators. Walkington, C., Sherman, M., & Petrosino, A. (2012). “Playing the game” of story problems: Coordinating situation-based reasoning with algebra representation. Journal of Mathematical Behavior, 31, 174–195.

Chapter 8

On Teaching of Word Problems in the Context of Early Algebra Nicolina A. Malara and Agnese I. Telloni

Abstract This study reports a project on teaching word problems in early algebra for primary and lower secondary students. The word problems approach consisted of a set of connected activities in which problem solving, argumentation and problem posing were infused. The project aims to promote in students, within the usual classroom activities, inquiry attitude in problem solving, and awareness of the emerging mathematical objects. It engages them in argumentation activities to justify mathematical properties and in modelling processes. Results of one of the experiments of the project in a class of 5th grade students are reported. Within this study, we aim to: (a) orient the students to make argumentations of a solving process in general terms, (b) facilitate them to explore the corresponding numerical expressions for different series of numerical data, and (c) engage them in solving new questions that arise during their activities. The viability of the project is discussed in light of the results of the experimentation. The chapter is concluded with suggestions for the research and reflections on the teacher’s role in such a problem solving approach. Keywords Problem solving · Problem posing · Metacognition · Argumentation · Early algebra · Teachers education

1 Introduction Students, especially those in early grades, are usually not comfortable with mathematics learning experiences. It is well known that traditional teaching, based on rote learning, induces stereotyped behaviors and emotional detachment in the students. In this respect since the 1980s, many authors (e.g., Garofalo & Lester, 1985; Lester, 1983; Mason et al., 1985; Schoenfeld, 1985, 1987; Silver, 1985, 1987) suggested N. A. Malara Department of Physics, Mathematics and Computer Science, University of Modena & Reggio E., Modena, Modena, Italy A. I. Telloni (B) Department of Education, Cultural Heritage and Tourism, University of Macerata, Macerata, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_8

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to teach problem solving focusing on inquiry and metacognition, with the goal of inducing students to make their thinking processes explicit. Other scholars (e.g., English, 1997, 1998; Kilpatrick, 1987; Silver, 1994, 1997) promoted problem posing activities because such activities are conducive for solving problems and developing in students a good attitude toward mathematics. The mathematics syllabuses around the world generally propose a richer view of problem solving, including an emphasis on open-ended problems. These problems are viewed today as an opportunity for educating students to work collectively and facilitating flexibility in thinking. Teachers should devolve to students problem situations that may bring out the emergence of mathematical concepts and properties. The underlying belief of this approach is that the students are creators of their own knowledge that develops through whole-class mathematical discussions (Bartolini Bussi, 1998). This teaching approach allows the students to experience mathematics, developing their “natural power” (Mason, 2016) to solve problems and understand the origin of the mathematical objects. Argumentation (Krummheuer, 1995) plays a key role in this approach: the students learn to express ideas, discuss different perspectives and monitor their own or other solving paths. Written argumentations allow students to organize and communicate their thoughts formally. Often, many teachers neither teach problem solving nor use problems to promote mathematical thinking, but simply assess the students’ ability to apply procedures or to do computations. Generally, standard problems continue to be conventionally used in teaching problem solving, while problem posing is ignored almost everywhere. Our study aims at investigating the viability of infusing problem posing within problem solving in early algebra, within an engaging environment that stimulates an inquiry attitude in the students, looking also at the teacher’s behaviours. It is aligned with the recent research, which focuses on teaching problem posing in the class, the role of the teacher, and the effectiveness of the teacher—researcher partnership (Caj & Hwang, 2020; Liljedahl & Cai, 2021; Liljedahl & Santos-Trigo, 2019; Papadopulos et al., 2022; Singer et al., 2015; Stoyanowa, 2000;). Specifically, the project reported here deals with the teaching of word problems for primary and lower secondary school, within early algebra, focusing on arithmetic teaching from an algebraic perspective (Cusi et al., 2011; Malara, 2003; Malara & Navarra, 2003, 2018). The project puts the word problems as the core of a set of connected activities that infuses problem solving (S), argumentation (A) and problem posing (P), hence is named SAP. It promotes generalization and the representation of relationships in verbal and formal terms. The early algebra aspects of SAP deal with generational activities (Kieran, 1996) for students, to make sense of objects as mathematical expressions, identities, equations, and functional relationships, to approach concepts such as generic number, unknown, variable, equation, and proof.

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2 Theoretical Frameworks 2.1 Problem Solving and Problem Posing Frameworks Polya’s books “How to solve it” (1945) and “The mathematical discovery” (1962), are still seen today as cornerstones of the problem-solving literature (Felmer et al., 2016, p. v). Polya articulated problem solving process as encapsulated in the following four steps: (1) understanding the problem; (2) devising a plan; (3) carrying out the plan; (4) looking back. The organizational structure of the SAP project is aligned with these steps. Moreover, following Polya, we use problem-solving to introduce students to typical ways of mathematical reasoning (analogy, generalization, formulation of conjectures, and variation of conditions). Mason (2022) claimed that the main feature of the book Thinking Mathematically (Mason et al., 1985), is “to concentrate on the lived experience of mathematical thinking”, including the engagement with problems and reflection. In agreement with Kilpatrick (2016), he stressed that Polya’s looking back step refers to the metacognitive dimension, leading to the search for different strategies as well as to the generalization of problems and the problem posing. Other studies focused on metacognition in the teaching of problem solving (e.g., Schoenfeld, 1985, 1987, 1992). According to these studies, students should make their thoughts explicit in planning and solving problems. In particular, Schoenfeld highlighted that a good solver is capable of monitoring the solving process and exploring strategies, deciding when to pursue a strategy or abandon it. Metacognitive activities constitute a basic aspect of the SAP project. Silver (1987) indicated metalevel processes (planning, monitoring, and evaluation) as important components of mathematical problem-solving behaviour. In particular, he emphasized that students should be asked to design an explicit plan for the solution of a problem before calculation. Aligned to this aspect, we adopted a qualitative approach to problem solving in SAP. Some researchers believed that problem posing is crucial for problem solving, hence it needs to be included in mathematics curriculum (e.g., Brown & Walter, 1983; Ellerton, 2013; English, 1997, 1998; Kilpatrick, 1987; Leung & Silver, 1997; Silver & Cai, 1996; Silver et al., 1996). For SAP, we take reference of design of problem posing activities from Silver (1994, 2013). According to Silver problem posing is aligned to inquiry-oriented instruction. It improves students’ problem solving. Moreover, he stressed that problem posing activities humanize mathematics, i.e., provide students the opportunity to develop a personal relationship with mathematics.

2.2 Early Algebra Frameworks Kieran (1989, 1992) highlighted some students’ predominant misconceptions and stereotyped behaviours when approaching algebra. These are attributed to teaching

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arithmetic instrumentally. Further, she broadened the vision of elementary algebra, fostered by three types of activities, namely, generational activities, transformational activities, and global meta-level activities, at increasing levels of complexity. The first level, the generational activities, aimed at the construction of the objects of algebra by linking meanings to the experience (Kieran, 1996). Bell (1985, 1996) introduced the construct of the essential algebraic cycle, which constituted three interrelated actions: representing, transforming, and interpreting, to reshape the teaching of algebra. He also supported the introduction of pre-algebra, as a teaching area for the development of “pre-concepts” to bridge the gap between arithmetic and algebra (Linchevski, 1995). Some scholars (e.g., Da Rocha Falcão, 1995; Meira, 1996; Radford, 2000) proposed linguistic and socio-constructive approaches for teaching algebra; others focused on the development of ‘symbol sense’ (Arcavi, 1994), promoting multiple interpretations and metacognition (Arzarello et al., 1993; Gray & Tall, 1993; Kaput, 1991; Mason, 1996). We take reference to these autors to define our approach to early algebra.

2.3 Theoretical Aspects of Our Approach to Early Algebra We hypothesised that there is a strong analogy between modalities of learning natural language and algebraic language. When a child learns the native language, he acquires its meanings and the rules step by step; only later, at school, he learns to read and reflect on the structural aspects of language. Hence, at early grades, we promote the development of an environment that stimulates the representation of verbal sentences into the arithmetical or algebraic language, carried out by collective classroom discussions. Through such activities, the students construct and elaborate their first arithmetic/algebraic expressions; they also experience the use of letters and interpret the meaning of formal sentences within specific situations. Therefore, the appropriation of the new language occurs experimentally, and its rules gradually evolve from initial syntactical inaccuracies. We call algebraic babbling this process of construction/ interpretation/refinement of ‘raw’ writings. In this process the students’ collective discussions are crucial to explore and solve questions and to reflect on what has been done and on what yet could be done. The teacher coordinates the students’ interactions to foster the exploration and the mathematical construction, paying attention to the students’ oral and written argumentations. The translation between languages (i.e., the representation in formal terms of verbal expressions and, vice versa, the interpretation of formal sentences) is the key point of this approach. To facilitate it, we revised the arithmetical teaching in an algebraic perspective and devised some theoretical constructs which are expressed by the four dualities: (1) canonical and non-canonical representations of a number; (2) transparent and opaque representations; (3) representing and solving; (4) process

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and product. The first duality allows the students to conceive of the canonical representation of a number, i.e., the symbolic representation of the number. Any arithmetical expression having the number as a result is referred to as non-canonical representation of the number. As an illustration, the sentences 3 + 4, 2 + 5, 2 × 3 + 1, 10 − 3 are non-canonical representations of the number 7. Looking at different representations of the same number, the students learn to recognize equivalences among them and express justification for their equality. The second duality allows the students to understand that, depending on the goal of the mathematical activity, a non-canonical representation of a number can be more ‘transparent’ than the canonical one, giving the opportunity to see hidden relationships. This type of experience allows the students to overcome the narrow operative view of arithmetic and attain a symmetric view of the equal sign. The third duality focuses the students’ attention in solving a problem on ‘representing’ as a key action for postponing calculations and reifying (in the sense of Sfard, 1994) arithmetical procedures. This shifts students’ attention from computation to the numerical sentence representing the solving process. The fourth duality, closely linked with the third, brings the students to see an arithmetical expression as both a calculation process and its product. As we have shown in our quoted studies, these theoretical constructs allow the students to interiorize pre-algebraic ways of looking at arithmetic. This study relies on these theoretical constructs and pursues the same goal within the frame of arithmetic problem solving.

2.4 The Teacher’s Role In the frame of our earlier studies, we developed a theoretical construct to study the roles played by the teachers to promote students’ attitude toward inquiry and awareness, and their view of the algebraic language as a tool for thinking (Cusi & Malara, 2015). This construct is a refinement of Lester’s analysis and focuses on metacognition in teaching problem solving (Lester, 2013). It includes a set of teacher’s actions aimed at effectively guiding the students in the processes of reasoning through algebraic language. More precisely, the teacher: (a) plays the role of an investigating subject, stimulating the students’ attitude to the research, and acts as an integral part of the class in facing a question, tacitly orienting its development; (b) acts as a practical/strategic guide making explicit and sharing with the students’ behaviors, thoughts, reasons. (c) acts as a reflective guide in identifying practical/strategic models; (d) stimulates and provokes the enactment of fundamental skills, such as the appropriation of specific systems of representation and their coordination, (e) plays the role of activator of interpretative processes and of anticipating thoughts, and (f) fosters meta-level attitudes, being an activator of metacognitive attitudes oriented toward the control of processes and strategies and in making them explicit. The teachers involved in our research, before starting the experiments in their classes, take part in specific educational programs. They collaborate in all the phases of the research (design, didactical implementations, assessment) and are engaged in

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critical analysis of the classroom discussions and their own behaviorus toward the ability to notice (Mason, 2002).

3 The SAP Project The SAP project proposes broadening the usual teaching of verbal problems combining both problem solving and problem posing in an arithmetical setting. It is aligned with our studies (e.g., Malara, 1993, 1999, 2012; Malara & Gherpelli, 1994, 1997; Malara & Navarra, 2000, 2003). The considerations in planning and designing the project are: (a) the texts of the verbal problems; (b) the identification and the naming of the variables, (c) the construction of the solving process and its written argumentation; (d) the transition from a problem to a class of problems; (e) the intertwining between problem posing e problem solving that can be actualized, starting from a verbal problem.

3.1 The Texts of the Verbal Problems Zan (2016) defined narrative a problem that is formulated as a little story, that develops over time with connections of causality, and where the end question is not ‘on the story’ but ‘within the story’, i.e. it has ‘an aim’ functional to the story. In agreement with her, we believe that narrative problems not only motivate the students in solving problems but also engage them in a progressive reduction of the text of the problems. Starting from a narrative problem and focusing on the mathematics of the story, the parts of the text that do not offer any quantitative information will be erased. The abridged text facilitates the students’ connections and the individuation of the solution process. This approach aspires also to ‘humanize’ the problems that students typically read in textbooks, bringing them to understand the sense of their concision.

3.2 Identification and Naming of the Variables In order to facilitate students to monitor the meaning of their problem solving, they should be led to make explicit what the numbers represent. This shifts the attention to the (concrete or abstract) entities of which the numbers represent the quantity. When a numerical datum is related to an action that has taken place, often the students have to do a linguistic transformation to identify a term that expresses the result of the action. This might be demanding for students with poor knowledge of the language. In analyzing the text of a problem, the teacher needs to organize a classroom discussion to lead the students to name a variable abstracting from the action that generated it.

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Referring to this interpretative and linguistic process, we speak of objectivation of the variables, since it can be framed into the theory of objectivation (Radford, 2010). We believe that the objectivation of the variables is the first fundamental step to promote the deferring of the calculations. This favours an approach to problems focused on highlighting the relationships between its variables. This allows metacognitive control on the steps of reasoning for solving a problem and promotes the formulation of written argumentations in general terms.

3.3 Construction and Argumentation of the Solving Process To promote the students’ formulations of linguistic and formal sentences for solving a problem—which are forms of algebraic babbling—we argue that it is useful to bring the students not only to name the intermediate variables but to express them by referring to the ones which generate them. This shifts the students’ attention from the operative to the relational plane. In this way, both the linguistic expression of the connection between intermediate variables and their arithmetical representation through non-canonical representations are clarified. So, students will progressively understand how an intermediate variable generated by two or more steps of connection can be represented through a nested arithmetical expression. This also allows the students to produce more easily their argumentations of their solving processes in general terms, and to construct the solving arithmetical expression before doing the calculations. We define relational these argumentations, which focus on the relationships between variables and produce forms of early algebraic thinking (Radford, 2011; Schifter, 2018).

3.4 The Intertwining Between Problem Posing and Problem Solving The shift of attention from the numerical values to the variables of a problem could lead students to appreciate: (a) a problem as a class of problems; and (b) a solving numerical expression as a representation of the mathematical structure of this class of problems. To promote an algebraic view of a numerical expression, students must be guided to the joint analysis of many arithmetical expressions. This is done by asking them to write the solving arithmetical expressions of the problem for different numerical values, out of which many arithmetic questions with a pre-algebraic character may arise. Working on them, the students may face problems posing questions involving: transformation of arithmetical expressions; recognition of numerical equivalences between them; detection of structural analogies; noticing of numerical regularities, their generalization and justification. All these problem posing activities facilitate the development of algebraic thinking.

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3.5 The Structure of the SAP Project In the design of the SAP project, activities involving problem solving and problem posing are linked by: (a) the arithmetical expressions arising from the solving procesess of a problem, and (b) introducing what if questions about other possible numerical data for the problem. In more detail, from the side of problem solving, the SAP activites are: S1. Interpretation of the text of a (narrative) problem and its new formulations S2. Identification and naming of the involved variables (both data and intermediate variables) S3. Written argumentations of solving processes and writings of the related arithmetical expressions S4. Comparison of argumentations and strategies through their numerical expressions S5. Calculations. From the side of problem posing, the SAP activities are: P1. Variation of the numerical data of a problem P2: Construction of the solving arithmetical sentences of a problem for different series of numerical data P3. Study of sets of solving arithmetical expressions for particular series of data: discovery of regularities and their justification P4. Formulation of the texts of problems analogous to a given one in some suggested contexts. Here the word problems are the core of an interconnected system of activities that merges linguistic and argumentative aspects, problem solving and problem posing in an early algebra perspective. Taking into account Polya’s four steps of problem solving with respect to the SAP project, we can say that S1 is related to Polya step 1 (understanding the problem) and step 2 (devising a plan); S2 is related to Polya step 1 for the linguistic processes of naming of the data quantities, while for the identification of the intermediate quantities—which contribute to the construction of the solving process—is related to Polya step 2 (devising a plan); S3 may be linked with Polya step 3 (carrying out the plan) as well as S5. Activity S4, which deals with metacognitive activities, may be framed in Polya step 4 (looking back). The activities P1 to P4, which are related to variations and new questions arising from a given problem, may be framed in the Polya steps 2 and 4.

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4 Methodology The teacher involved in the project had participated in an educational program devoted to primary and middle school teachers and concerning early algebra and problem solving and posing. The experiment was implemented in 2019 in a class of 22 5th-grade students at a public school in the neighborhood of Turin (Italy). Three students in the class had special learning needs. The students were accustomed to participating in classroom discussions and producing written argumentations, but they had no prior experience with early algebra. Before the intervention, the teacher discussed with the students the concepts of canonical and non-canonical representations, and of transparent or opaque representations. The classroom work was performed in bi-weekly sessions of 2.5 h each, for a total of 35 h, in the presence of one of the two researchers. We wished to verify whether the students could be brought to: 1. consider the study of the verbal problems in a more articulate way, focused on the variables and independence of the numerical values of the data, and to conceive the problem-solving process of a problem as the construction of a thinking process; 2. start producing relational argumentations of the solving process problem, possibly giving reasons for their strategical choices; and using the generative noncanonical representations of the intermediate variables to represent this process and its result; 3. compare solving expressions of a problem related to different solving strategies, searching for the arithmetical reasons for their equivalence; 4. assign different ordered sets of the numerical data of a problem, becoming able to notice some regularity and to justify it; moreover, to represent the related processes in verbal terms and by the letters through opportune algebraic babbling processes; 5. formulate analogous verbal problems in a different context. More generally, we wished to examine the students’ development of inquiry attitude, metacognitive abilities, and appreciation for problem solving and posing. The implementation of the project has been developed in cycles of: (i) collective discussion about the simplification and interpretation of the text; (ii) individual activity of problem solving with argumentation about the construction of the solving strategy and the related arithmetical expression; (iii) metacognitive discussion about their productions; (iv) individual activity on the variation of the numerical data of the problem and related argumentation; (v) metacognitive discussion on the students’ productions and (vi) problem posing on the questions which emerged. The classroom discussions were recorded. Field notes and comments were jotted down by the researcher and the teacher. Additional data included individual students’ productions and students’ comments on their experience in the experiment, and the teacher’s reflections. The data analysis concerned the kind of the students’ participation; the feature of the individual students’ mathematical contributions; the quality of the attained

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results; and the emotional and motivational impact on the students. On the solution of a verbal problem, we examined: (1) the students’ approach to problem solving; (2) the strategies used; (3) their argumentations; (4) their approach to constructing the arithmetical expression and the difficulties encountered. For the tasks on the variation of the numerical data of a problem, we studied the quality of the choices made by the students, the development of their reasonings, and the type of argumentation they produced. In the students’ individual reports on the classroom discussions, we examined the clarity, the completeness and the kind of the argumentation; the reasoning developed on the numerical examples, the presence of algebraic formulations.

5 The Didactical Path The didactical path has been structured on the three word problems in Table 1. The first problem, inspired by Ferrari (2006), was chosen because we wanted to verify whether: (a) the students would organize their reasoning in relational terms (elaborating the bits of information) or in procedural terms (following the order of text); (b) the students would consciously choose the new numerical data. At the same time, we wanted to facilitate them to make explicit the constraints posed by the structure of the problem with the values of the variables. The second problem was chosen to induce the students to change it into a typical textbook example. This problem presents difficulties related to the identification and naming of the variables and to argumentation. We wanted to explore whether the comparison of two solving processes could facilitate the arithmetical justification of the equivalence of the corresponding solving expressions. The third problem is a narrative problem adopted from a textbook. It has been chosen for its particular formulation. It anticipates the conclusion of the story adding later further information, thus hampering the setting up of suitable mental images in the students. We wanted to verify the incidence of the work made on the objectification of the given and intermediate variables and the effects of the re-formulation of the sentences for the understanding of the situation. Moreover, when the data are variated maintaining the equality that there is among three values of them, the correspondent solving expressions present an interesting regularity, therefore we wanted to verify whether the students could collectively notice it and justify it.

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Table 1 Narrative word problems studied in the experiment 1. The library problem. Since some years ago the teacher Jane organized a small library of high-quality books. With Christmas approaching she thinks she will enrich it with new books to lend to her students for the holidays. She decides to make an inventory for getting an idea about how many books she needs to buy. She discovers that there are 58 books available for loan. To be sure that there will be enough books for the loan, she buys 26 new books. Some students, that are passionate readers see the new books in the library and ask to read them. Jane gives them 8 books on loan. Unfortunately, during the night some thieves enter the school. The next day, Jane finds her classroom in disarray, she discovers that 19 books have been stolen. She is muddle-headed, almost unable to reason. Please, help the teacher Jane, write a numerical sentence representing the number of books remaining in the library so she can soon start again the loan 2. The Pinocchio problem. Pinocchio goes back home after many mischiefs. The Blue Fairy who supervises him, punishes him saying: «You are just unruly! From now, for each lie you will say I shall lengthen your nose by 3 cm». Pinocchio is sad and he cannot imagine his nose, already 5 cm long, can lengthen. In the evening, the Fairy wishes him goodnight, saying: « Dear Pinocchio it is a pity you are growing up as a liar, nobody will trust you. Please, try to be sincere. I wish to help you, hence from for each true answer you give I shall shorten your nose by 2 cm, but for each lie your nose will lengthen by 3 cm». The next morning he goes out and spends all day wandering, and as usual he tells several lies. He returns home at sunset, but when he opens the door he hears the Talking Cricket who scolds him: «Pinocchio, look at your nose, it is more than 5 cm!». The Talking Cricket goes down with a jump by the wardrobe where he was comfortably seated, takes out of his breast pocket a rolled meter and he measures Pinocchio’s nose. Then he shouts «Your nose is 20 cm long! You must have told lies the whole day. Have you forgotten the promise you made? What will you answer her when she will ask how many true answers you have given today?» Pinocchio turns pale, and he says to the Talking Cricket: « Today I have counted my lies, they were 7, but I do not remember whether I have given true answers». The Cricket replies: «Get moving! You may still find how many true answers you have given». Imagine you are there, next to Pinocchio and try to help him 3. The canaries problem. Mister Peter has many canaries in an enormous aviary. He has tried many times to count them without any success. So he decides to distribute the canaries within smaller aviaries. Initially he thinks that seven middle-size aviaries may be sufficient, so he takes them and allocates in each of them the same number of canaries, but 2 canaries remain out. Then he solves the allocation problem by equally distributing the canaries in 9 aviaries. To achieve this, he simply takes away 2 canaries from each of the previous aviaries, adding them to the 2 remaining canaries, and distributed in equal parts all the canaries into the added aviaries. Help Mr. Peter to discover how many canaries there are in each aviary and how many canaries he owns

6 Some Key Episodes of the Experiment 6.1 The Library Problem As expected, the objectification of the variables was not critical for the students. The only element which needed negotiation was the characterization of the variable corresponding to the number 58. The sentence that emerged from the discussion was “58 is the number of books which are initially in the library”, simplified into “58

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is the initial number of books”. Next, the students developed the problem-solving strategy following the story, and simultaneously made calculations. Most students made numerical-descriptive argumentations and wrote the arithmetical expressions without using brackets. We forced the students to express their argumentations without reference to the numbers. The successive classroom discussion was focused on the relational strategy of the problem. The students were brought to reflect on the fact that the arithmetical expression makes explicit a condition on the variables. This became a guide for them to choose new values.

6.2 The Pinocchio Problem Most of the students were initially stuck. They progressively perceived the numerical data as values of variables and became aware of (a) the generative formulation of the intermediate variables to write the solving expression of the problem; and (b) the possibility of choosing new numerical values based on the arithmetical expression itself. We present some episodes of the classroom work, showing the interplay between problem solving and problem posing. On the objectivation of the variables of the problem. The naming of the variables is a demanding interpretative activity that must be facilitated by the teacher. The Pinocchio problem was reduced to the following synthetic version: The nose of Pinocchio is 5 cm long. When Pinocchio tells a lie, the Blue Fairy makes his nose become 3 cm longer, but when Pinocchio gives a true answer the Fairy makes his nose become 2 cm shorter. At the end of the day, Pinocchio has told 7 lies and his nose is now 20 cm long. We have to represent the number of true answers given by Pinocchio in the day.

The interpretative work on this problem was crucial for the objectification of the variables implicitly introduced by actions. The discussion started with the question, posed by the teacher, to make explicit the meanings of the various numbers in the text. The key part of the discussion was the search for a name for the variable expressed by the following sentence “When Pinocchio tells a lie, the Blue Fairy makes his nose become 3 cm longer”. The teacher engaged the students to search for a noun expressing the result of the action ‘to lengthen’. Through the analysis of other opportune verbal predicates and related nouns, the students discovered the linguistic generative role of the term ‘lengthening’. Then, she introduced the term ‘unitary’, to complete the denomination of the variable at stake. This variable was defined as ‘unitary lengthening of the nose’. Similarly, the variable ‘unitary shortening of the nose’ was defined. At the end, the variables listed on the interactive whiteboard (IW) were: the initial length of the nose, the unitary lengthening of the nose, the unitary shortening of the nose, the number of the lies said, the final length of the nose. The numerical value of each variable was reported next to it. The teacher engaged the students to solve the problem by asking them to write their argumentation of the solving process and its arithmetical expression before

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calculations. This aimed to assess if they were able to define the intermediate and final variables. Most of the argumentations were operational; only some students made explicit their steps of reasoning and named some intermediate variables. Two solving paths emerged, one based on ‘the length of the nose for all the lies’, and the other based on ‘the lengthening of the nose for all the lies’. Difficulties of representing a number through the arithmetical operation that generated it, and the wrong or missed use of the brackets were observed. The naming of the intermediate variables. In the session devoted to the review of the students’ argumentations, the teacher reverted to the identification and naming of the intermediate and final variables and to the writing of the solving expressions. She led the students to name the intermediate variables during the collective discussion. The variables that were defined were ‘length of the nose for the lies’, ‘lengthening of the nose for the lies’, ‘lengthening of the nose for lies and true answers’, and ‘shortening of the nose for all the true answers’. The teacher asked the students to express by words such variables by means of the variables of the problem, or by intermediate variables defined earlier. This task was performed by the students through an interpretative passage from the numerical to the linguistic register, using operational expressions. For example, to the request to explain how ‘lengthening of the nose for the lies’ is obtainable without any reference to the numbers, several students answered ‘I multiply 3, that is the lengthening for one lie, by the number of the lies, that is 7’. The teacher tacitly reformulated the sentence in an impersonal form, assuming the result of the action as a subject. This is the sentence she wrote on the IW: ‘The lengthening of the nose for the lies is obtainable as the product between the unitary lengthening and the number of the lies’. She later further refined the sentence changing the predicate ‘it is obtainable’ into the more abstract predicate ‘it is equal’, and then into a simple ‘it is’. These linguistic refinements were done to allow the shift of students’ attention from operational to a more abstract level, in order to facilitate their understanding of the ways of building and interpreting a solving expression, and fostering a relational view. The intermediate variables were synthesized at the IW as we reported in Table 2. This teaching step highlights the importance of the teacher’s awareness in counsel (Mason, 2008). The synthesis in Table 2 made evident that the intermediate variable ‘shortening of the nose for the true answer’ had different numerical representations according to the strategy used. This allowed the teacher to pose a new pre-algebraic question to be discussed below. From problem solving to problem posing: a first early algebra question. The teacher asked the students to find a justification for the equivalence of the two expressions representing the shortening of the nose for the true answers (Table 2, row 5, columns 2 and 4). The teacher led them to arithmetically examine the two expressions to search for a relationship. The outcomes of the discussion highlighted the students’ recognition of the structural similarities of the solving expressions and their control of the intermediate variable at stake.

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Table 2 Verbal and Arithmetical Representations of the intermediate and final variables The Pinocchio problem First solving strategy

Second solving strategy

Intermediate variables Arithmetical and result expression

Intermediate variables and result

Arithmetical expression

• The lengthening of 7 × 3 the nose for the lies is the product of the number of the lies said in the day (7) and the unitary lengthening of the nose (3)

• The lengthening of the nose for the lies is the product of the number of the lies said in the day (7) and the unitary lengthening of the nose (3)

7×3

• The length of the 5 + (7 × 3) nose for the lies is the sum of the real length of the nose (5) and the lengthening of the nose for the lies (7 × 3)

20 − 5 • The lengthening of the nose for the lies and the true answers is the difference between the final length of the nose (20) and the real length of the nose (5)

[5 + (7 × 3)] − 20 • The shortening of the nose for the true answers is the difference between the length of the nose for the lies [5 + (7 × 3)] and the length of the nose at the end of the day (20)

(7 × 3) − (20 − 5) • The shortening of the nose for the true answers is the difference between the lengthening of the nose for lies (7 × 3) and the lengthening of the nose for lies and true answers (20 − 5)

• The number of the true answers given is the quotient between the shortening of the nose for the true answers [5 + (7 × 3) − 20] and the unitary shortening (2)

[5 + (7 × 3) − 20]/2 • The number of the true answers given is the quotient between the shortening of the nose for the true answers (7 × 3) − (20 − 5) and the unitary shortening (2)

[(7 × 3) − (20 − 5)]/2

This aspect is aligned to the “algebraic activities of metacognitive type” suggested by Kieran (1996), where it is the meaning that suggests the algebraic transformations. Another remarkable emerging aspect concerns a purely arithmetical side of the question, based on the subtraction property. The students’ recognition of this property in action allowed the teacher to explain that it is the reason for the equivalence of the two arithmetical expressions. Another early algebra question: the variation of the numerical data. The students were asked to vary the numerical values in Pinocchio’s problem. About half the class choose correct ordered sets of new numerical data. The others followed the previous operative process, changing the data by trial and error, but without success. A sample

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of a student’s new approach and her processes of thought in choosing the new data is presented below. First option. Initial Length [of the nose] = 10 cm; Lengthening of the nose for one lie = 5 cm; Shortening of the nose for a true answer = 3 cm; Quantity of lies = 9; Final day length nose = 22 cm. She writes: {[(9×5)+10]-22}÷ 3 = 11, then writes the second solving expression [(9×5) - (22-10)]÷ 3 = 11. Choice of the data. I have chosen these data thinking to add 2 to each [original] data, except the shortening of the nose for each true answer and the initial length [of the nose]: I decided to double the initial length while I have chosen [3 for] the unitary shortening because the result of the brackets is divisible by 3. Second option. Initial Length [of the nose] = 6 cm; Lengthening of the nose for one lie = 4 cm; Shortening of the nose for a true answer = 2 cm; Quantity of lies = 5; Final day length nose = 14 cm. She writes: {[(5×4)+6]-14}÷ 2 = 6 then write the second expression [(5×4) - (14-6)]÷ 2 = 6. Choice of the data. Initially I thought that the shortening of the nose for a true answer could remain 2, so within the brackets, it was enough that the result was even. Then I thought that the result could be 12. Then I thought that the result within the round brackets should be 20, formed by 5×4 and then plus 6 [I subtract 14] in a way that the result is 12, that divided by 2 gives 6.

The student provided two explicit series of numerical data. She showed precise reasoning about the variables. The explanation of her reasoning in the second option is not direct even if the thinking path is understandable. The observation of the expression related to the second strategy was probably made because it allowed her to choose the missing data in an easier way. Through this sample, the contextualized interpretation of the expressions has evidently supported the arithmetical transformations. Moreover, one notices that the student felt the need to close the expression with its result, showing her operative vision of the expression. From the research point of view, this and other similar protocols indicate a students’ good attitude and learning: they were able to reason using the variables, identifying exploring strategies effectively referring to the numerical expression.

6.3 The Problem of the Canaries The students were very engaged with this problem, but they needed to be guided on the grammatical and lexical plane before understanding the text, and were very active in the objectivation of the variables and the intermediate variables. The students’ argumentations on the solution of the canaries problem. All the students adopted the same strategy to find the number of canaries in each aviary; they added the number of all the canaries subtracted by the aviaries to the number of the ones that remained out from the first distribution and then divided this sum by two. Differently from the previous problems, the quality of students’ argumentations improved. The 12 out of 22 written argumentations were procedural. There were some justificative argumentations of procedural type, which we defined as “transitional argumentations”, where the subject explained the choices made and the meaning of the intermediate results (‘to have this I did …). Among these, some argumentations

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stood out for the meta-level feature, shown by the use of words such as ‘I have thought that … I had to do …’, and the impersonal form in some cases. An example of such an argumentation is shown next: I thought that Peter takes away two canaries from each aviary, then I made 2×7, that is the two canaries that he takes away from each of the 7 aviaries, to understand how many canaries he shifts in total, he shifts 14 canaries, then I have added the two canaries and the result is 16=14+2. The 16 canaries to be repositioned have to be equally divided in the two aviaries bought, for finding the number of canaries in each of the 7 aviaries, because the number of canaries in only one aviary is equal to the number of canaries in each of the 9 aviaries. Then 16÷2 gives 8, therefore there are 8 canaries in each of the 9 aviaries. After I have multiplied the 8 canaries of each aviary by 9 aviaries, that is the number of the canaries in each aviary by the number of the aviaries, and I have found 27, the total number of the canaries [She inverts the digits and writes 27 instead of 72]. I have also understood that before, in the 7 aviaries there are 10 canaries in each of the aviaries with the remainder of two canaries.

This argumentation shows a student’s coherent mental image of the situation and full control of the problem solving process. The student was on the metacognitive dimension at the beginning (I have thought that …) and in conclusion (I have understood that …). She reflected simultaneously on the variables and on their numerical values. She flexibly moved between the verbal plane and the numerical. There are signs of impersonal formulation of sentences (‘the 16 canaries have to be divided …’). She also used operative predicates (‘16/2 gives 8, I have found …’) with some linguistic inaccuracies (I multiplied the 8 canaries by the 9 aviaries). This argumentation shows the state of transition of the student shifting from an operational way of argumenting to a meta-level one with a relational feature. Concerning the solving expression, some students correctly constructed it before calculating. They expressed the represented intermediate variable for each part, as illustrated by Fig. 1a. The student did not close each part of the expression with its result; moreover, he used the verbal predicate ‘to represent’ to indicate the intermediate variable, showing his understanding of the role of the non-canonical representations for a number. We observed the tendency to perform the calculations among the students who made procedural argumentations. Some of them, after having written the solving expression, reproduced it as many times as the operative steps, substituting each time the result to the operation, as exemplified by Fig. 1b. This shows the student’s control of the meaning of each number in the expression and her need to coordinate the arithmetical sentence with the steps of the operative process, making the involved intermediate variables explicit. The teacher worked on the students’ argumentations and encouraged them to convert sentences from colloquial to impersonal form, which is typical of mathematical communication. The aim was to bring the students to formulate relational sentences using the predicate “to be”, becoming aware that it legitimizes the interchange of different representations of the same quantity. The teacher functioned as a reflective guide and promoter of linguistic-interpretative acts of meta type, discussing with the students how to carry out suitable linguistic transformations, and also as a model of behavior in noticing equivalences in various expressive forms.

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Fig. 1 Students’ interpretations of the solving numerical expression

The variation of the numerical data and related argumentations. The teacher asked the students to change the numerical data in the problem, solve it justifying the choices made, write the new solving expressions and interpret them. This was to test whether the students were able to take into account the constraints posed by the situation. Moreover, the problem has the peculiarity of having three equal numerical data (the number of the canaries not allocated in the first distribution; the number of the added aviaries; the number of the canaries subtracted by each of the initial aviaries). We wanted to test whether the students noted this equality in changing the numerical data, and whether they were able to refer to the variables, to express their reasoning in a general way, and to pose themselves at meta-level in their argumentations. 11 students varied the data in an acceptable way, although they used a limited range of values. Those who assigned different values to the three variables had shown a clear formulation of the argumentations, avoiding reasoning on the numbers and moving toward a generality. The students who had chosen adequate series of numerical data, maintaining the equality among the three variables, made transitional argumentations on the variables, and wrote correctly the arithmetical expressions. Some of them started to express themselves in an impersonal way but shifted toward personal linguistic forms (which were more usual for them). Most part of the students who had chosen incompatible data did not take into account the second condition. Their argumentations were mainly of procedural type and sometimes incomplete, even if the effort to refer to the variables prevailed. Several of them wrote the arithmetical expression after the calculations. Interpretative difficulties with these expressions were noticed. The type of numerical data chosen, the related solving expressions, and the types of questions that emerged are listed in Table 3. Problem posing in early algebra. We discuss the work made around two numerical questions arising from an exploration of the solving expressions written by the

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Table 3 Types of numerical data chosen by the pupils, related solving expressions and arising questions Types of numerical data chosen Solving sentences generated by the students by the students

Types of questions arising

1. Compatible data that [[(11 × 3) + 3]/3; [(9 × 4) maintain the equality among + 4]/4 the three variables [(9 × 3) + 3]/3; [(8 × 3) + 3]/3

Q1. Observe the sentences. Is there a regularity between its values and its result? If yes, try to identify it

2. Compatible data that do not [(6 × 2) + 4]/2; [(8 × 2) + maintain the equality among 4]/2 the three variables [(9 × 4) + 4]/5; [(13 × 2) + 4]/5

Q2. Some of these sentences are transformable in simpler forms. Try to identify them and explain how/why

3. Incompatible data (1st condition is not observed)

[(6 × 8) + 6]/2; [(4 × 1) + 4]/1

Q3. Are these data possible? Think about their meanings and justify your answer

4. Incompatible data (2nd condition is not observed)

[(9 × 2) + 4]/4; [(6 × 3) + 4]/5

Q4 Explain why these data are not acceptable. Try to change a value in each so that the expression becomes solvable in the natural numbers

5. Lacking data

[(12 × _) + 4]/2; [(10 × _) + 3]/3

Q5 Which type of number might you put for the missing datum? Explain why

students concerning the numerical values which maintain the equality among the three variables. Formulation of a conjecture and its proof by collective discussion. The teacher drew Table 4 and invited the students to recognize possible regularities. After some interventions on specific numbers, a student articulated “the result was one more than the initial number”, and another student added “for the number 9 it happened also when the other three numbers changed”. Other tests presented analogous expressions, such as: [(10 × 4) + 4]/4; [(12 × 3) + 3]/3; [(20 × 5) + 5]/5 and also analogous but not plausible values for the problem, such as: [(4 × 7) + 7]/7 or [(3 × 8) + 8]/8, and the regularity had been confirmed. To shift the question to the numerical plane, the teacher posed the problem to try to understand the reasons for the regularity analyzing the exemplary case: [(4 × 7) + 7]/7. The discussion developed around the points: 1. The analysis of the expression 4 × 7 + 7 and its transformation in 5 × 7 2. The substitution of 4 × 7 + 7 with 5 × 7 in the expression [(4 × 7) + 7]/7 3. The analysis of the meaning of the expression (5 × 7)/7, its interpretation with reference to the operators “ ×7” and “/7”, the recognition that it represents 5 and the writing of the equality (5 × 7)/7 = 5 4. The representation of 5 as 4 + 1 and the recognition of the equality (4 + 1) × (7/7) = (4 + 1)

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Table 4 A problem for students to recognize the regularity through pattern recognition Initial number Number of the Number of the Number of the of the aviaries canaries canaries added aviaries remained out subtracted by each aviary

Generative expression of the number of canaries in each aviary

7

2

2

2

[(7 × 2) + 2]/ 8 2

11

3

3

3

[(11 × 3) + 3]/3

9

4

4

4

[(9 × 4) + 4]/ 10 4

9

3

3

3

[(9 × 3) + 3]/ 10 3

8

3

3

3

[(8 × 3) + 3]/ 9 3

Canonical representation of the number of the canaries in each aviary

12

5. The re-examination of the initial expression [(4 × 7) + 7]/7, its transformation in (4 + 1) × 7/7 and it successive transformation in 4 + 1. The most problematic part of the discussion regarded the meaning of the expression 4 × 7 + 7: L.1 Teacher: What may we say of the number (4×7) + 7? L.2 Student A.: It is 35, 5×7, one more than four L.3 Teacher: 35 = 5×7, to what do you refer when you say ‘one more than four’? L.4 Student A.: To 5×7 L.5 Teacher: Ok, but what do you want to express when you say ‘one more than four’? L.6 Many: One plus four is five L.7 Teacher: I would like to know what he wanted to say. Please A. try to say better what intended to say, it seems to me interesting L.8 Student A: I have 4×7, I add 7 it does 35 that is 5×7, and 5 is 1 more than 4 L.9 Teacher: Ok. Now we try to understand the reasons for this equality. On one side we have 4×7+7, on the other side we have 5×7. What does 4×7 mean? L.10 Class: silence L.11 Teacher: We go a little bit back, what does 2×7 represent? L.12 Stud. B: 2+2+2+ up to seven 2 L.13 Teacher: And if we want to write the double of seven? L.14 Many: 7+7 L.15 Teacher: May we write it with multiplication? L.16 Stud. C: 7×2 L.17 Teacher: Ok but tell me, are 2×7 and 7×2 different numbers? L.18 Chorus: no L.19 Teacher: If the two expressions denote the same number, 2×7 can be interpreted also as twice 7?

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L.20 Many: Ah, yes. L.21 Stud D: But the commands are different L.22 Teacher: You are right, the commands are different, but we say better, the expressions are different but the number that they represent is the same. This is an important distinction. If we look at them as expressions we say that they are equivalent, if we look at them as numbers, we say that they are equal. I wish to know whether we have yet any doubt L.23 Chorus: No. No, it is well L.24 Teacher: Ok. Then we come back to 4×7. We read it as 4 times 7. What do you say to me? L.25 Stud. A: Adding yet 7 it is 5 times 7. L.26 Teacher: Let us say it in a complete way. If to 4 times 7 we add yet one 7, it is the same if we say 4 times 7 plus 1 time 7 . Do you agree? (many faces and eyes express yes). Ok. Then we can write (the teacher writes on the IW) 4×7+1× 7 = 5×7 (and at the same time she says), in this way we understand from where appears the number 5, it is the sum of the numbers of the times. From this excerpt, we notice that the teacher, taking advantage of an intervention of a student (line 2), stimulated him to express his insight (lines 5, 7). The student expressed more clearly his thought, shifting from an operative-transformational level. The teacher, as a participant and activator of interpretative acts, engaged the students in the analysis and clarification of the meaning of the expression 4 × 7 (line 9). This revealed the difficulty of the students in monitoring the mathematical meanings. To overcome the impasse, the teacher stimulated them at the interpretive level in simpler cases. As an outcome, the model of multiplication as a repeated sum emerged for the different roles played by the factors (lines 12–16). In order to get the students to conceive the first factor as an operator on the second factor, the teacher focused the students’ attention on the commutative property of multiplication, which justifies the interpretation of the first factor as the multiplier. Some criticality emerged due to rigid interpretation of multiplication emerged by the interventions of two students (lines 16, 21). This concerns the classical dilemma process–product (Davis, 1975; cited in Radford, 2018). The teacher recognized the double interpretation of the writing, as an expression corresponding to a meaning and as a quantity, but induced the students to distinguish these two meanings (line 22). Regarding the interpretation of the expression 4 × 7 + 7, the teacher posed herself as a reflective and interpretive guide. She brought the students to interpret (5 × 7)/7 as the action of the composition of the operators × 7 and /7 on factor 5. In the last part of the discussion, the number 5 was represented as 4 + 1 again and the syntactic transformations allowing to pass from [(4 × 7) + 7]/7 to 4 + 1 were considered. At this point, the teacher introduced the terms ‘successor’ and ‘antecedent’ of a number. The transformation of similar expressions allowed the students to understand the reasons for the regularity. To prove the regularity the generic number n was introduced and the expression [(4 × n) + n]/n was transformed. The emerging discussion allowed facing further explorations.

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Returning to the expressions from another perspective: the game of the picked number. The teacher proposed that the students analyze the following equalities: • (8 × 3) + 3]/3 = (8 + 1) × 3/3 = 8 + 1 = 9 • [(9 × 3) + 3]/3 = (9 + 1) × 3/3 = 9 + 1 = 10 • [(11 × 3) + 3]/3 = (11 + 1) × 3/3 = 11 + 1 = 12 Here, she highlighted the underlying operative scheme, i.e. the composition [(… × 3) + 3]/3 and asked the students: “If the result of the expression is 15, what is the number that should be filled in the blanks? And if the result is 47? And if it is 174?” The students soon gave the correct answer. The most engaging aspects of this collective discovery have been its translations into a riddle “pick a number”, formulated through a procedural reading of the expression. In detail: (1) Pick a non-zero number; (2) Pick another non-zero number; (3) Multiply the first number by the second; (4) Add the second number to the result; (5) Divide the result by the second number; (6) Tell me the result and I’ll tell you the number you picked. The exciting and happy atmosphere in the class increased the students’ motivation. The teacher asked the students “Consider the work done and explain why it is not necessary to calculate the solving expression of the problem to know its result”. The students’ productions showed that they had reached a good understanding, despite various types of ‘dinginess’, such as improper linguistic expressions. The students gave a justification of the regularity through the analysis of numerical cases, seen in general terms, referring to the meanings of (a) the sum of a number and one of its multiples; (b) the effects of the composition of two numerical operators one inverse of the other. About half of the students roughly used the letters and felt the need to show a particular case of the rule. Two students’ responses are shown below (with inaccuracies underlined). Student (SN): If a number is multiplied by another number, then it is added to the same number by which we have multiplied it, and later that we divide it by the number for which we have multiplied and added it, the result will be the successor of the initial number. For example, if I multiply 6 by 2 and then I add 2 it is as if I multiply seven times 2. Therefore 6×2+2 = is equal to 7×2. After having made 7×2 one divides by 2, but seeing that the multiplication is the inverse calculation of the division, ×2 ÷2 has not any value, then the result is 7. a = number times n = numerical operator. (a×n+n) ÷ n = a+1. Student RN: Take a number, if you multiply it by a number then you add again that number and divide the result by the same number, the result will be the successor of the initial number. Initial numbers = a, number = n number succ. / prev. = N + / N - ; [( a × n )+ n ] = a +1

a n

n

n a+1

[(7×2) + 2] ÷2 = 8 . [The number] n is always the same, a+1 the successor of the initial number a.

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The two argumentations are examples of general type, generated by the observation of some analogous numerical sentences where the initial and final terms appear correlated. The second is notable because the student was dyslexic and had serious difficulties in keeping attention.

7 Analysis of the Results 7.1 Students’ Behaviours Within the Collective Discussions The type of participation deeply changed along the development of the path. As to the first problem (the classroom library), at the beginning the students had a natural shyness to intervene. Later the support of the teacher nurtured the interaction among peers. For Pinocchio’ problem, there were three classroom discussions, two devoted to problem solving and one to problem posing (on the equivalence of the solving expressions). In the first discussion, devoted to the interpretation of the text and the objectivation of the variables, the students gave appropriate answers only if asked. In the second discussion, devoted to the generational representations of the intermediate variables, the students began to participate and talked about the representations of the intermediate variable “shortening of the nose for the true answers”. The third discussion developed from these interventions. There was spontaneous participation of several students within an atmosphere of collaboration. In the discussions on the problem of the canaries the students’ participation increased and had to be handled while the atmosphere became always more animated. The students’ contributions were pertinent and productive. Some solicited intervention received clipped or imprecise answers. The more meaningful results on the side of problem solving were: the naming of the variables; the formulation of more adequate argumentations; and the construction of arithmetical representation of the intermediate variables and of the result by the variables. The more interesting results on early algebra were: the justification of the equivalence of the two solving expressions (the Pinocchio problem); the discovery of the regularities about analogous solving expressions and the study of the arithmetical problems that emerged (the canaries problem). The teacher played an important role in leading the students to understand the relationship between the sum of a number and of a multiple, and to see the generality through the analysis of a numerical case. The class appeared motivated and attracted by the experience throughout the lessons and proud of their achievements.

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7.2 Analysis of the Students’ Productions About the Individual Activities In analysing the students’ productions, we distinguish between those related to problem solving (looking at the types of argumentation, ways of construction of arithmetic expressions) and those related to problem posing in early algebra (looking at strategies of variation of the numerical data, ways of justification of equivalences or regularities). Productions related to problem solving: the types of argumentation. In the students’ productions about the first problem, the solving process focused on standard calculations. The argumentations were purely operational of a descriptive type and formulated in first person, according to the scheme: “I read the text, above all I did this and I found …, I did this other and I found, then I did this …; … and I found the result”. In Pinocchio problem, a first evolution of the students’ argumentations was observed. The students adopted the scheme: “Above all I want to find this, then I do ….To know this other I do, … To obtain the result I do …”. There were planning operational argumentations referring to the variables and focused on the subject’s decisions. In some of them the intermediate variables were not expressed in generative terms. In the canaries’ problem, several students shifted to the relational plane. The argumentations they made had a hybrid formulation, sometimes with a reflective feature. Some sentences written in the first person were connected with sentences in an impersonal form (for instance, I thought that to have this it necessary to do…), there were references to the variables, and the generative definitions of the intermediate variables appeared. Moreover, some sentences were of a general type; but the operative feature dominated, the use of the verb “obtain” prevails over the use of the verb ‘to be’. There were transitional argumentations towards the relational form. These results showed that it is possible that students progressively succeed in doing argumentations with a feature of generality. Productions related to problem solving: the construction of arithmetic expressions. The students had no difficulties with the first problem. They constructed the solving sentence without the use of the brackets, simply making the arithmetic transposition of the operational steps suggested by the problem. The students did not monitor the solving process in the second problem. They did not construct the solving expression. In the responses related to the assignment of new numerical data, about half of the students showed a significant improvement in the construction of the solving numerical expression. The students were able to construct the arithmetic expression with the correct use of the brackets for the third problem. This partially happened in the next task, concerning the change of the numerical data: more than half the class produced correct numerical expressions. Some difficulties remained for students whose approach was purely operational. They were able to represent the result of the operation done in columns by indicating the generating expression, but they did

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not seem to understand the reciprocal connections. They were not able to change all the data and generate new numerical expressions. This shows the fragility of these students in monitoring the process, and their rigidity in accepting an expression as a representation of its result or in substituting a number with the arithmetic sentence generating it. Productions related to pre-algebraic problem posing. The variation of the numerical data in the canaries’ problem posed different questions to the class. The analysis of several numerical expressions generated by compatible numerical data brought the class to discover a regularity (between the number of the initial aviaries and the final number of the canaries in each aviary). The final numbers were represented in a noncanonical form, through their generative expressions. The students’ responses on the reasons for such regularity are classified in three types. (1) Most of the students have faithfully retraced what was done in the class, precisely: the path of the discovery of the regularity; the justification process, developing the reasoning by an exemplar case but verbally expressing it in general terms; the justification in general terms. (2) Other students, after the analysis of some numerical sentences (point 1), reasoned on the general plane, using the letters, and then verified the equality by giving the letters some numerical values. (3) Some students, among those who faced the problem in a sharply operational way, did not finish the report, avoiding justifying the regularity. This was a consequence of the difficulty to understand the arithmetic transformations at stake. Some students were not able to recognize that (4 × 7) + 7 is 5 × 7. The outcomes of the study highlight the potentialities of developing pre-algebraic activities through solving arithmetic expressions of some given problems. When the students refer to different solving strategies, it is crucial to discuss with them the arithmetic justification of the equivalence of the solving sentences. Moreover, the variation of the numerical data of a problem, and the collection and classification of the students’ productions could be crafted into activities with different levels of difficulties. Regularities that arise within the arithmetical sentences (e.g., the third problem) and their analysis provide affordances for justification and generalization. Such activities could turn out to be very engaging for the students, offering them the pleasure of discovery.

8 Conclusion A key and pervasive aim of the project is orienting the education of the students toward relational argumentations, with planning and justification features. This kind of argumentation is rarely found in our schools. However, it is strongly recommended in our National Indications (MIUR, 2012) from kindergarten to upper secondary school. It is considered fundamental for the development of a cultural view of mathematics and, more broadly, for promoting the students’ dialectic and rhetorical competencies needed for the citizens of democratic societies, as our society, which has its roots

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within the Greco-Roman culture (Russo, 1998). This study points out the possibility of pursuing this goal. For most of the protocols examined, there are important signals of the students’ transition toward generality. An important result is that more than half of the students transited from argumentation as a description of a calculation process towards a new more evolved concept of argumentation, explicitly planning a solving strategy, aligned with the “devising a plan” step by Polya, and in a qualitative way, in the sense by Silver (1994). However, we believe that long-term work is needed in order to make this approach a mental habit for all students. Moreover, for the most part of the students’ argumentations, there is an explicit reference to the individual actions or thoughts. The use of the first person in expressing a problem solving process prevents the acquisition of a suitable distance to formulate relational argumentations, that should be impersonal. The balance of these two aspects suggests further studies. It could be interesting to distinguish between informal argumentation and argumentation for social communication. We believe that the students’ shift from the colloquial register to that of social communication has to be driven through specific didactical interventions. This topic will be the object of our future research. Another emerging aspect deals with the difficulties in the construction of the solving expression met by some students anchored to a view of a problem as a sequence of calculations. Focused interventions had to be planned to foster the students’ overcoming of this difficulty. From the research perspective, the intertwining between problem solving in a realistic context and problem posing in the realm of natural numbers was didactically fruitful, since the contextual interpretation of the expressions favours the justifications of arithmetic properties. An example of this aspect arose when the students justified the equivalence of the two solving expressions for Pinocchio’s problem. Moreover, it allowed highlighting arithmetic questions about the concept of the multiple of a number and its related properties. An example of these questions has been discussed in the context of the canaries’ problem. Concerning the dissemination of the experimental path, some conditions are needed: first of all, the teacher should be aware of the educational aims of the project. Aligned with our vision of the teacher who poses him/herself as a model of aware and effective attitudes and behaviours the teacher should adopt the following key roles along the didactical path: strategic guide, to address the students towards the control of their actions, to foster the connection of the knowledge and the anticipatory thought; interpretive and linguistic guide, to support the students in formulating relational argumentations of procedural processes and in coordinating verbal and formal languages, activator of metacognitive thoughts to bring the students to become aware of their knowledge and capability. However, these conditions are necessary but not sufficient. The teacher needs also the sensitiveness to grasp the opportunities arising from the classroom activities, whether they be unexpected insights of the students or potential sources of deepening emerging from their productions (Mason, 2008).

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Chapter 9

Problem Posing by Mathematics Teachers: The Problems They Pose and the Challenges They Face in the Classroom Alina Galvão Spinillo, Síntria Labres Lautert, Neila Tonin Agranionih, Rute Elizabete de Souza Rosa Borba, Ernani Martins dos Santos, and Juliana Ferreira Gomes da Silva

Abstract Problem posing is an important aspect of mathematics education. It has received increasing attention in the literature on curricular and pedagogical innovation. Some investigations on problem posing have looked at how students formulate mathematical problems, while others looked at how teachers pose them. In this chapter, this issue was discussed from two different perspectives both referring to how mathematics teachers deal with problem posing. Study 1 analyzed the characteristics of problems they posed, by asking them to formulate eight mathematical word problems whose resolution would involve multiplication and/or division. Study 2 analyzed the practice of teachers who use problem posing to teach mathematics to their students. In this study, participants answered in writing the following questions: What do your students learn by formulating math problems? What do they need to know to be able to formulate math problems? What are the main difficulties they face in formulating math problems? What are the main difficulties you encounter when A. G. Spinillo (B) · S. L. Lautert · R. E. de Souza Rosa Borba · J. F. G. da Silva Federal University of Pernambuco (UFPE), Recife, Brazil e-mail: [email protected] S. L. Lautert e-mail: [email protected] R. E. de Souza Rosa Borba e-mail: [email protected] J. F. G. da Silva e-mail: [email protected] N. T. Agranionih Federal University of Paraná (UFPR), Curitiba, Brazil e-mail: [email protected] E. M. dos Santos Pernambuco University (UPE), Recife, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_9

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working with the formulation of math problems in the classroom? The data obtained in the first study allowed to know the characteristics of mathematical word problems posed by teachers, especially in relation to the types of problems they formulate. On the other hand, data in the second study allow to know their opinions about their students and their teaching practice regarding problem posing in the classroom. The results of both studies were discussed in an articulated way, emphasizing the formulation of mathematical problems as part of the didacticknowledge that needs to be included in teacher education programs. Keywords Problem posing · Characteristics of mathematical problems · Teachers’ opinion · Teacher knowledge

1 Introduction Problem posing is an important aspect of mathematics education (Brown & Walter, 2013; Silver, 1994). It has received increasing attention in the literature on curricular and pedagogical innovation (Baumanns & Rott, 2021, 2022; Singer et al., 2013). Several authors state that problem posing goes hand-in-hand with problem solving (Abu Elwan, 2016; Arıkan & Ünal, 2015; Cai & Hwang, 2002; English, 1997; Xie & Masingila, 2017). The activity of problem posing is recognized in public policies in several countries, such as the United States, China, Italy and Korea. In Brazil, the National Common Curricular Base—BNCC (Brasil, 2017, 2018) recommends that in Elementary School mathematics teaching involves solving and formulating problems in different contexts. In High School, the document recommends that problem posing be considered a tool for learning mathematics to investigate causes, develop and test hypotheses and create solutions for situations in different areas of knowledge. Studies have investigated this topic in students (English, 1997; Lowrie, 2002; Silver, 1994) and in teachers. Some investigations seek to develop problem posing skills in prospective and in-service teachers, as well as instruct them in the creation of activities to be applied in the classroom (Cai et al., 2019; Crespo, 2003; Lavy & Shriki, 2007; Pelczer et al., 2014). Other investigations, in turn, examine prospective and in service teachers’ ability to pose problems and the characteristics of the problems they pose (Agranionih et al., 2021; Lee et al., 2018; Leung & Silver, 1997; Ramirez, 2006; Xie & Masingila, 2017). Intervention studies have unquestionable value both in terms of the training they provide to participants and in terms of their applicability in teaching situations. However, given the focus of this chapter, the following discussions are about the characteristics of the problems that mathematics teachers pose.

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2 The Characteristics of Mathematical Problems Posed by Teachers In general, studies in this perspective examine the impact of certain factors on the ability to pose problems such as the teacher’s mathematical knowledge and teaching experience. An example of this was the study conducted by Leung and Silver (1997) with prospective teachers who were asked to formulate arithmetic word problems. The problems they posed were analyzed in terms of quality and their complexity. The quality referred to whether the problem was mathematical or not, whether it had plausible steps or not, and whether or not it presented the necessary and sufficient information to solve it. The complexity referred to the number of steps required to solve it. Although almost all of the problems contained enough information, more than half of the problems were simple, requiring a single step for their resolution. Data also revealed that participants with greater mathematical knowledge tended to formulate more complex and better quality problems than those with less mathematical knowledge. The conclusion was that future teachers need more experience to improve the quality and complexity of problems they pose, and that the development of the ability to formulate problems should be considered in training coursesfor these professionals. A similar conclusion was pointed out by Ramirez (2006),who also investigated the formulation of problems in in-service teachers and prospective teachers. The author observed that difficulties experienced by participants had several causes, including the absence of mathematics teaching based on problematization during the education of future teachers. Sengul and Katranci (2014) asked the prospective teachers in their study to formulate problems from a problem presented as a model, to describe the difficulties they experienced, and and suggest how to overcome the difficulties identified. The main difficulties identified were: formulating a word problem different from the model presented, choosing the appropriate numerical expressions, formulating problems related to everyday life situations, and formulating problems compatible with the level of knowledge of students for whom the problems were intended. According to the teachers, one way to overcome these difficulties would be to analyze the characteristics of the problems that served as a model and solve the problem they posed before drafting its latest version. The need to have more experience with problem posing was also mentioned by the teachers. The existence of a close relationship between problem posing and problem solving is defended by Xie and Masingila (2017), when they state that problem posing contributes to problem solving which, in turn, supports problem formulation. In a study carried out with prospective teachers, participants were asked to solve mathematical problems they had formulated. The main result was that good problem solvers were usually able to formulate more complex mathematical problems. Apparently, problem formulation and problem solving influence each other and, together, can serve to develop the understanding of mathematical concepts.

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According to Lee et al. (2018), the variability in problem formulation depends on teachers’ knowledge of the subject and familiarity with this type of activity. This conclusion was derived from the results obtained in an interview conducted with Middle and High School teachers. Respondents reported that they face difficulties when trying to incorporate problem formulation in the classroom, such as the limited time they have to carry out the activities and the fact that there is little variety of problems in textbooks. However, they also mentioned the benefits obtained from learning mathematical concepts and success in solving everyday problems by the students. The authors comment that teacher training courses need to address problem formulation, since the lack of instruction on the subject can make it difficult to put activities of this nature into practice when they become teachers. Agranionih et al. (2021) examined whether the characteristics of additive and multiplicative problems formulated by Elementary School teachers varied according to the school grade in which they worked. Participants were divided into two groups: those who taught in the 1st and 2nd grades, and those who taught in the 3rd, 4th and 5th grades. In both groups, the vast majority of the problems formulated required a single step for their resolution. This result indicates that, regardless of the school grade in which they teach, teachers tend to think in terms of a single operation for each problem, so that problems formulated were very simple, not bringing challenges to the solver. Data also revealed that there was little variability regarding the types of problems formulated, which was observed in both groups of participants. According to the authors, such limitations seem to result from the lack of familiarity teachers have with the activity of elaborating problems and the conception they have about mathematical problems in general. To overcome such limitations, it is necessary to develop a broad understanding of what a mathematical problem is, to know the properties of the concept involved in problems to be formulated and the resolution procedures that can be adopted. The results of the aforementioned studies indicate that: (i) mathematical knowledge, teaching experience and instruction on the formulation of mathematical problems are important aspects for teachers to be able to formulate complex, challenging and varied problems; (ii) problem posing is an important didactic tool in the teaching and learning of mathematics, and (iii) this topic, still unfamiliar to teachers, is related to teacher knowledge that needs to be considered in training courses.

3 Teacher Knowledge The above studies illustrate the essential knowledge teachers must have for effective teaching, as pointed out by Shulman (1986) and Ball et al. (2008). Shulman (1986) defends that teaching involves conceptual and operational mastery of the content to be taught, but these are not sufficient. An understanding of what is to be learned and how it can be taught are also required. The knowledge of characteristics of the group of learners it is intended for, the complexities of the content and conditions of the

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school environment are also necessary. In other words, basic knowledge for teaching involves the interaction of content, curriculum and pedagogical knowledge. Indicating Mathematical Knowledge for Teaching, Ball et al. (2008) presented six sub domains of the knowledge proposed by Shulman (1986): (1) common content knowledge: this is not exclusive to teachers but rather used by all that are able to solve varied problems involving certain contents; (2) specialized content knowledge: this is a specific domain for teachers and requires understanding of characteristics of the content to be taught; (3) horizon content knowledge: this allows teachers to recognize relations between different contents and also knowing ways of deepening their understanding; (4) knowledge of content and students: this allows teachers to predict possibilities, interests, motivations and difficulties students may have in learning mathematical contents; (5) knowledge of content and teaching: this allows teachers to evaluate the activities to be developed in classrooms, in terms of their advantages and disadvantages; and (6) knowledge of content and curriculum: this provides better planning by knowing what is prescribed in official educational documents and what is presented in didactic materials. Santos et al. (2021) stress that not only do teachers need to master knowledge concerning the content of the problems they pose, but they also need specific knowledge– such as knowing different problem types and the adequacy of these types to different school grades. Chapman (2012) agrees with this statement, but goes beyond, adding that previous experience with problem solving and creativity are required, as well as the knowledge of different meanings for mathematical contents. The knowledge of different meanings for mathematical contents refers to what is proposed by Vergnaud (1983) about the understanding of mathematical concepts by individuals. According to his theory of conceptual fields, a single situation may involve different mathematical concepts, and the same mathematical concept may be intertwined in different situations. In order to develop a broad and adequate understanding of a given concept, it is necessary to present that concept in different situations. In this theoretical approach, one may say that teacher knowledge involves mastering a wide variety of mathematical situations that must be proposed in the classroom. Thus, in addition to formulating challenging problems, teachers need to be able to formulate different problems, in order to propose to their students a high diversity of situations that allow them to cover the different aspects of the same mathematical concept. From this perspective, it is important to investigate if a problem posed by teachers presents a variety of situations that involve the multiplicity and complexity of relations that characterize mathematical concepts. It is thus clear that there is an association between teacher knowledge and the quality of teaching, and that what teachers know and think has a strong relationship with what they are able to propose and accomplish in their classrooms. Therefore, problem posing is knowledge that needs to be acquired by those adopting problem formulation as a didactic tool to teach mathematics. Building on the above investigations, two studies are presented below. Study 1 aimed to investigate the characteristics of problems posed by Elementary and Middle School teachers whose resolution involves multiplication and division. Study 2 explores teachers’ views on their own experience teaching mathematics through

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problem formulation. The aim of these two studies is to explore two facets of this topic. One of them deals with the teacher as a problem poser and the other facet deals with the teacher as a mediator of teaching and learning situations. As a problem poser, the teacher expresses his/her views about the relations that characterize mathematical concepts. As a mediator of teaching and learning situations, the teacher reflects on his/ her practice, expressing opinion about it, in this case, opinion about his/her practice of teaching mathematics through the formulation of problems. These studies adopt different methodological paradigms in an attempt to achieve different goals; however, they converge towards investigating the same phenomenon: by teachers of mathematics. Articulating these two studies in the same chapter allows looking at the formulation of mathematical problems by teachers from different perspectives, expanding the scope of discussions on this topic. Thus, the purpose of this chapter is to look at the teacher as a problem poser and as a mediator in teaching and learning situations.

4 The Report of the Two Studies 4.1 Study 1: Teachers Posing Word Problems of Multiplicative Structure The formulation of problems by teachers has, for a long time, been of interest to scholars of mathematics education in general, and theorists who are dedicated to listing the teaching knowledge necessary for those who teach mathematics (Ball et al, 2008; Shulman, 2005). Due to this interest, as previously mentioned, research in the area has been characterized by intervention studies carried out with teachers, and (ii) studies that analyze the characteristics of the problems they pose. This study is part of this second group of studies, aiming to investigate the types of problems that Elementary School teachers formulate, focusing specifically on problems of multiplicative structure (see Vergnaud, 1983, 1994). Thirty-seven teachers who teach students attending 1st to 9th grades of Elementary School in state schools of the northeast of Brazil, voluntareed to take part in the study. They were divided into three groups: Group 1: thirteen 1st and 2nd grade teachers; Group 2: thirteen 3rd and 4th teachers, and Group 3: eleven 5th to 9th grade teachers. The majority of the 1st to 5th grade teachers had a degree in pedagogy and the majority of 6th to 9th grade teachers had a degree in Mathematics. The teachers took part in a course on multiplicative structures and initially were individually asked to pose, in writing, eight different word mathematical problems that could be solved through the operation of multiplication and/or of division. The order in which the problems were produced was free, as was the time allocated to complete the task that was applied in one session. In addition, these teachers did not have specific instruction on problem posing, neither was this emphasized in previous official Brazilian documents (Brasil, 1997).

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These documents emphasize problem solving, history of mathematics, computational technology and games. The current document (Brasil, 2017, 2018) emphasizes both problem solving and problem posing. The data were analyzed in two ways. Initially, characteristics of the problems formulated by the teachers were analyzed and, then, the profile of interviewees was examined for the variability regarding the types of problems they formulated. The categories were developed by means of thorough data analysis. Disagreements were discussed by judges to consensus. The types of problems posed by teachers, A total of 296 productions were elaborated by the participants who were classified as follows: (1) math exercises (3.7%): “Solve the following operations: 257/2 = ; 384/3 = ; 674/4 = ” (2) word problems of addition and subtraction instead of division and multiplication as requested (8.1%): “Maria went to the bakery with R$30.00. Her mother asked her to buy R$5 of bread, R$4 of ham and R$4 of cheese. How much change did Maria return home?” (3) word problems that were impossible to solve because they were characterized by inadequate statements in which relevant information was omitted, which presented linguistic inaccuracies that hindered their interpretation (5.1%): “If I have an apple box with 8 apples, how do I distribute to the boys at recess. How many left? How much was left for each?” From the corpus obtained, 246 productions (83.1%) were considered for analysis, which were effectively word problems that involved the operations of multiplication and division for their solution and that could effectively be solved. These were classified according to the typology proposed by Magina et al. (2016) based on Vergnaud (1983, 1994),1 according to the problems briefly described below and exemplified in Table 1. Simple proportion problems: they involve a direct proportional relationship characterized by a constant relation between two variables. Multiplicative comparison problems: they involve two magnitudes of the same nature that arecompared in a multiplicative way. The agent that transforms a magnitude indicates how many times a magnitude will be greater (or less) than the other. Product of measures problems: they envolve a relationshipconfigured as the combination of elements from two or more different sets that results in a third measure that is the product of that combination, generating a third set of elements. The classification of each problem was made through discussion between judges.2 Disagreements were discussed to consensus. The distribution of these types of problems in each group is illustrated in Fig. 1. 1

Although other types of multiplicative structure problems are documented in the literature, only these three types were identified in the corpus analyzed in this study. 2 The judges were the authors of this chapter.

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Table 1 Examples of the types of problems posed by teachers Types of problems

Examples

Simple proportion

A restaurant supplies 170 lunches to a factory per day. How many lunches should the restaurant provide in 7 week days? Dad bought a bike and as he couldn’t pay in cash, he decided to pay it in 12 installments of R$36.00. How much was the bike Dad bought? Dona Mariabought a stereo that costs R$ 450.00. She paid the purchase amount in 5 equal installments. What is the value of each installment?

Multiplicative comparison

Last year, 245 oranges were harvested on a farm. This year three times as many oranges were produced. How many oranges were produced this year? Paulo and João are brothers. Paulo is 3 times the age of João. If Paulo is 24 years old, how old is João? Pedro is 18 years old and his younger brother is 1/3 (one third) of his age. How old is Pedro’s younger brother?

Product of measures

Joana has 5 blouses and 3 skirts. How many ways can Joana dress herself without repeating the skirt/blouse combination? Simone has 2 blouses (1 green and 1 blue), 1 short and 1 pants to be used as a school uniform. How many ways can Simone match the pieces without repeating them? The cafeteria has 3 types of juice on a menu: orange, lemon and papaya. And 2 types of snacks: chicken croquette and pie. How many different possibilities of mealcan we eat in this cafeteria?

100 90

88.2 82.5

80.2

80 70 60 50 40 30 19.7

20 10

8.2 0

7.5

10

3.5

0 Group 1: 1st and 2nd grade teachers

Simple proportion

Group 2: 3rd and 4th grade teachers

Multiplicative comparison

Group 3: 5th to 9th grade teachers

Product of measures

Fig. 1 Percentage of types of problems formulated by teachers in each group

The results show big differences between the types of problems in each group. As shown in Fig. 1, in all three groups the percentage of simple proportion problem was high (Group 1: 80.2%, Group 2: 88.2% and Group 3: 82.5%) and smaller, and not so different, percentages of multiplicative comparison and product of measures problems.

9 Problem Posing by Mathematics Teachers: The Problems They Pose … 45 40

41

40

159

42

40

36 33

35 30 25 20 15 10

5

6

4

5 0 Group 1: 1st and 2nd grade teachers Mutiplication only

Group 2: 3rd and 4th grade teachers Division only

Group 3: 5th and 9th grade teachers

Division and multiplication

Fig. 2 Number of problems involving division and multiplication operations in each group

The groups of participants were also compared according to the operation involved in problem solving, that is, multiplication, division or both in a combined way (Fig. 2). As shown in Fig. 2, in Group 1 and Group 2, there was no predominance of problems requiring one or the other operation. In Group 3, however, problems requiring the multiplication operation were more frequent than the division ones. It was possible to observe that problems combining the division and multiplication operations were infrequent in the three groups. Teachers’ profile. The teachers’ profile was determined from the diversity of types of problems formulated by the same participant. For this analysis, the three types of problems produced were considered: simple proportion, multiplicative comparison and product of measures. Three profiles were identified: Profile 1 (no variability): teachers whose all problems posed were of only one of these types; Profile 2 (little variability): teachers whose posed problems were of two of these three types; and Profile 3 (high variability): teachers whose posed problems were of these three types. Figure 3 illustrates the distribution of these profiles in each group of participants. As can be seen, Group 3 teachers were concentrated on Profile 2 (little variability), whereas Group 1 and Group 2 teachers were concentrated on Profile 1 (no variability) and Profile 2 (little variability). Profile 3 (high variability) was rarely observed, and was absent among Group 1 teachers. Teachers posing word problems of multiplicative structure: some conclusions. Some conclusions can be derived from the data in this Study 1. In general, teachers basically tend to formulate simple problems involving only one step to solve them. This finding corroborates those documented by Leung and Silver (1997) about problems posed by teachers and preservice teachers who did not formulate complex and challenging problems. There was a little variability in the formulation of problems that were limited to a restricted number of mathematical situations, since they focused on simple proportion

160 13 12 11 10 9 8 7 6 5 4 3 2 1 0

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7

7

7

6 5

3 1

1

0 Group 1: 1st and 2nd grade teachers

Profile 1 (no variablility)

Group 2: 3rd and 4th grade teachers

Profile 2 (litle variability)

Group 3: 5th to 9th grade teachers

Profile 3 (high variability)

Fig. 3 Number of teachers in each profile in each group

problems. Apparently, simple proportion represents a prototypical problem that the teacher has in mind when referring to situations of a multiplicative nature. Perhaps, this prototypical representation stems from, among other factors, the fact that there is a high frequency of simple proportion problems in textbooks adopted in the early years of schooling in Brazil, as documented by Lautert et al. (2019) and Souza and Magina (2017). This little variability expresses a limited conception that moves away from the theoretical perspective that mathematical concepts are involved in different situations, as proposed by Vergnaud (1983, 1994). By prioritizing one type of problem over others, the teacher restricts the possibility of considering the different facets of mathematical concepts that are impossible to emerge in a single situation. For example, simple proportion problems require, for their resolution, the establishment of relations between two variables, while solving product of measures problems requires combining elements from two (or more) different sets that results in a third set of elements. When focusing on the first type of problem, as observed in this study, the teacher not only expresses a limited view of mathematical problems, but also limits the learning opportunities of their students. One may say that teachers have difficulties in posing complex and challenging mathematical word problems that make use of various aspects characterizing the concepts of multiplication and division. In particular, it seems that the participants lacked specialized content knowledge (Ball et al, 2008) concerning these operations, that is, lacked understanding of multiplicative properties that characterized them. It seems that this limited conception stems from possible gaps in teacher training courses. Thus, teachers require additional knowledge concerning problem posing in order to create mathematical problems that are cognitively challenging and that involve a large variety of situations that may help their students to understand multiplicative concepts. The results also revealed that the school grade in which these teachers taught did not appear as a factor that influenced the characteristics of the problems they formulated. It is important to mention that by dividing the participants

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into groups according to the school grade they taught, it was thought that, possibly, teachers who worked in more advanced school grades would possibly formulate more elaborate and more varied problems. This, however, did not occur. A similar result was obtained in previous studies conducted in Brazil (Agranionih et al, 2021; Souza & Magina, 2017). The results of this first study provide information about the teachers as a problem poser, revealing important aspects about the characteristics of the problems they formulate which, in turn, reveal the notions they have about their knowledge about problems of multiplicative structure. The other facet investigated in this research, which is the teacher as a mediator in teaching and learning situations, is addressed in Study 2 described below.

4.2 Study 2: Teachers’ Opinion About the Teaching of Mathematics Through Problem Posing Several investigations have looked at teachers’ and prospective teachers’ opinion about the teaching of mathematics (Gellert, 2000; Hannula et al, 2007). The idea justifying this interest on the part of researchers is that the way of conceiving mathematics education influences the way of teaching this subject. In these studies, general aspects were investigated, such as the opinion of respondents about the nature of mathematics they intend to teach and their opinion about the goals of teaching mathematics. This second study in this chapter starts from that same idea, however, it focuses on a specific aspect, which is the opinion of teachers about the challenge of teaching mathematics through activities involvingproblem formulation. The opinion of interviewees was obtained by a questionnaire with four open questions that dealt with two perspectives: one related to students and another related to teachers. Participants were sixty-one teachers who taught students attending 1st, 2nd, 3rd, 4th and 5th graders of Elementary School in Brazil. They taught in state schools in the south of Brazil and were participating voluntarily in a continuing education course involving problem solving. According to information provided, they used problem formulation in the classroom to teach mathematics as recommended by the current official documents. During the course, each participant individually answered, in writing, four questions, printed on sheets of paper, with enough space for an answer, namely: What do your students learn by formulating math problems? (Question 1), What do they need to know to be able to formulate math problems? (Question 2), What are the main difficulties they face in formulating math problems? (Question 3), and What are the main difficulties you encounter when working with the formulation of math problems in the classroom? (Question 4). The first three questions were related to students and the fourth was related to the teacher. The order in which the questions were answered was free, as was the time allocated to complete the task that was applied in one session.

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The answers, in a total of 244, were analyzed through discussion between three judges. The types of answers were created based on the analysis of the corpus of responses given by the participants to each question. Disagreements were discussed by judges to consensus. The answers provided to the three student-related questions (Questions 1, 2 and 3) were classified into five types, and those related to the teacher (Question 4) were classified into two types as described and exemplified in the sections that follow. Types of answers in Questions 1, 2 and 3. From the analysis of the judges, it was possible to identify five types of answers expressing the skills and knowledge that can be acquired by students through problem posing activities proposed in the classroom. These types are described below and exemplified in Table 2. Type 1 (intellectual abilities): responses in which teachers mention that through problem posing, students acquire intellectual skills such as attention, creativity, ability to formulate hypothesis and development of logical reasoning. Type 2 (linguistic abilities): responses in which teachers mention that through problem posing, students acquire skills related to reading and writing, specifically regarding the ability to understand and produce texts. Type 3 (mathematical knowledge): responses in which teachers mention that through problem posing, students acquire specific mathematical knowledge (arithmetic operations, quantification, counting, decimal number system) and knowledge related to problem solving (use of strategies, identify what is asked in the problem, select relevant numerical information). Type 4 (socio-emotional abilities): responses in which teachers mention that through problem posing, students form positive feelings about themselves (selfesteem, self-efficacy), develop motivation, autonomy and maturity. Type 5 (no response, vague, confused): the respondent does not respond or makes a general, vague and unrelated comment to the question. The distribution of these types of answers in Question 1, Question 2, and Question 3 is shown in Fig. 4. In Question 1 (What do your students learn by formulating math problems?) large proportions were detected in Type 1 (77%), followed by Type 3 (46%) and Type 2 (36%) responses. This result indicates that, according to the teachers, through the formulation of problems, students learn mathematical knowledge and develop linguistic skills; but the main gain is intellectual. In Question 2 (What do they need to know to be able to formulate math problems?) large proportions were detected in Type 3 (74%) responses, followed by Type 2 responses (39%) and Type 1 (25%).Thus, in the teachers’ opinion, in order to formulate mathematical problems, students need to master linguistic and mathematical knowledge, the latter being considered the most determining aspect of the ability to formulate problems. In Question 3 (What are the main difficulties they face in formulating math problems?), similar frequencies were detected in Type 1 (38%) Type 2 (36%) and Type 3 (36%) responses. Apparently, according to the teachers interviewed, the students’ difficulties in formulating problems reside in intellectual and linguistic limitations

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Table 2 Examples of types of answers in Question 1, Question 2 and Question 3 Types of responses

Question 1 What do your students learn by formulating math problems?

Question 2 What do they need to know to be able to formulate math problems?

Question 3 What are the main difficulties they face in formulating math problems?

Type 1 (intellectual abilities)

They learn to think, acquire knowledge about the subject, formulate hypotheses

First, it is necessary to develop the ability to reason and think

Organization of ideas and thoughts

They learn logical reasoning, creativity

Logical reasoning, idea Lack of concentration proposition

Type 2 (linguistic abilities)

They learn to interpret Read and write the text of word correctly and logically problems understand what is written

Understand the problem text

They learn to write and formulate a question

Read and understand

Restricted vocabulary, reading and writing of the word problem

Notion of numbers and the four operations

Relate the problem information to the resolution

Depending on the concepts involved in the problems, they can learn about various mathematical concepts

Numerical knowledge, having ideas involving additive and multiplicative thinking

The calculation in simple resolutions (division operation)

They feel challenged to try to resolve situations through reflection. They feel victorious when they succeed in solving them

Family stimulation is lacking, which leaves children without motivation to participate in the class

Lack of autonomy

They need to have something that generates motivation or provokes an inquiry about the reality in which they live

Initiative is lacking. Most await a response from colleagues or the teacher

Students want an immediate and prompt response

Type 3 (mathematical They learn knowledge) mathematics in practice and not only in theory

Type 4 (socio-emotional abilities)

Type 5 (no response, vague, confuse)

We can work with Solve and hit problem solving with the student in any Show more difficulty school grade, adapting to the needs of each student

They are not prepared for this The teacher needs to work on the subject and then ask the students to do something

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100 90 80

77

74

70 60 46

50 40

39

36

30

38

36

25

20 10

36

13 8 3

2

11

5

0 Question 1: What do your students learn by Question 2: What do they need to know to be Question 3: What are the main difficulties they formulating math problems? able to formulate math problems? face in formulating math problems?

Type 1

Type 2

Type 3

Type 4

Type 5

Fig. 4 Percentage of types of response in Questions 1, 2 and 3 (out of 61 responses). Note Type 1 (intellectual abilities); Type 2 (linguistic abilities); Type 3 (mathematical knowledge); Type 4 (socio-emotional abilities), and Type 5 (no response, vague, confused)

and lack of mathematical knowledge. These percentages were very close, and there did not seem to be a concentration on one of the types of answers, as occurred in relation to the other two questions. It is worth mentioning the fact that responses related to socio-emotional abilities (Type 4) were not frequent. Thus, although they are part of the teachers’ repertoire, these aspects do not appear as determinants in their responses. As illustrated in Fig. 4, intellectual skills and mathematical knowledge appear as determining instances of teachers’ conceptions about the gains and needs of students when teaching mathematics through problem formulation. Types of answers in Question 4. The answers given to Question 4 (What are the main difficulties you encounter when working with the formulation of math problems in the classroom?) were classified into two types, which are described below and exemplified in Table 3.3 Didactic-pedagogical difficulties: responses in which the interviewees mentioned difficulties related to the dynamics of the classroom, limited teaching resources and difficulties related to limitations they have in terms of their ability to carry out problem formulation activities with students. Difficulties associated with students: responses in which teachers mention difficulties related to students, arising from limitations in certain skills and mastery of mathematical knowledge. Of the 59 responses, 66.1% mentioned difficulties of a didactic-pedagogical nature, while 33.9% mentioned difficulties associated with the students. Thus, in the teachers’ opinion, most difficulties they face have a didactic-pedagogical nature 3

Two teachers stated that they did not have any difficulty in working with the formulation of problems in the classroom. Thus, 59 teachers reported difficulties, this number being taken as a basis for the percentages reported below.

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Table 3 Examples of types of answers in Question 4 Types of difficulties

Question 4 What are the main difficulties you encounter when working with the formulation of math problems in the classroom?

Didactic-pedagogical difficulties

Too many students in the classroom hinders a more individualized monitoring Little time for activities Lack of concrete material to exemplify better Difficulty in elaborating situations that lead to reflection and that leave the obvious or traditional A great challenge is knowing how to promote meaningful and interesting situations to stimulate unmotivated and inattentive children

Difficulties associated with students Little creativity and simplicity in the statement produced by most children The biggest difficulty is that many students are not yet literate, which makes it difficult to interpret the problems in order to solve them autonomously The student, many times, does not have conditions to work the formulation, does not have attention, concentration. Often lacks maturity and interest Difficulty students have in organizing ideas and lack of prior knowledge Students don’t like to think

resulting from limitations imposed by the classroom functioning (excessive number of students, little time for activities), the lack of didactic resources to facilitate the accomplishment of the activities and limitations related to the teacher’s ability to know how to promote such activities. Hybrid responses. Data analysis allowed to identify hybrid responses that consisted of the combination of types in the same response. Examples of these responses are presented in Table 4 for each question. Of the 244 responses provided to the four questions, only 79 were hybrid (32.4%). This result indicates that teachers tended to consider only a single aspect in their answers. The hybrid responses were distributed as follows: Question 1 (37.9%); Question 2 (31.7%); Question 3 (25.3%) and Question 4 (5.1%). As can be seen, these answers were rare in the question concerning teachers’ difficulties with the formulation of math problems in the classroom. Teachers’ opinion about the teaching of mathematics through problem posing: some conclusions. As mentioned at the beginning of the description of this second study, the interview with teachers was about the opinion they had about their students and about themselves when carrying out, in the classroom, activities involving the formulation of mathematical problems. Considering the teachers’ opinion about their students, it is generally observed that: (i) the main gain with the formulation of problems in the classroom lies in the

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Table 4 Examples of hybrid responses in each question and the nature of hybridisms identified Nature of hybridisms

Question 1:What do your students learn by formulating math problems?

Intellectual, linguistic and socio-emotional abilities

To sharpen logical reasoning, to interpret, to dynamize knowledge. Self confidence

Intellectual and socio-emotional abilities

Concentration, motivation, knowledge, development of thought

Intellectual and linguistic abilities

They learn to organize ideas, to think, to analyze possibilities, to write in agreement

Nature of hybridisms

Question 2: What do they need to know to be able to formulate math problems?

Linguistic abilities and mathematical knowledge

Interpretation, understanding the four operations, understanding the terms

Intellectual and linguistic abilities

Reading, interpretation and reasoning

Linguistic abilities and mathematical knowledge

Interpret texts, produce small texts, master the techniques for solving operations

Nature of hybridisms

Question 3: What are the main difficulties they face in formulating math problems?

Linguistic abilities and mathematical knowledge

In writing and reading comprehension, in understanding the decimal number system, also in problem statements

Linguistic and intellectual abilities

Produce the statement with clarity and argumentation, with sequence. Have creativity

Socio-emotional and linguistic abilities

Realize that they are capable, master reading to improve their knowlede

Nature of hybridisms

Question 4: What are the main difficulties you encounter when working with the formulation of math problems in the classroom?

Didactic-pedagogical difficulties and difficulty associated with the student

Their attention and time to explore their hypotheses with each of them

Didactic-pedagogical difficulties and difficulty associated with the student

Time to carry out activities. Reading skills of the students

Didactic-pedagogical difficulties and difficulty associated with the student

My difficulty is to adapt the amount of content to the children’s time and immaturity

acquisition of intellectual abilities; (ii) what they most need to know in order to be able to formulate math problems is to master mathematical content related to the school grade they attend; and (iii) the students’ difficulties in formulating problems are varied, due to intellectual and linguistic limitations and related to the lack of mathematical knowledge. It is worth mentioning the fact that socio-emotional abilities were not mentioned as a relevant aspect in the teachers’ responses. Interestingly in the literature in the area there are studies that associate teaching practices through the formulation of problems with the development of motivation and autonomy on the part of students

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(Bonotto, 2013; Brown & Walter, 2005; Sengul & Katranci, 2014). However, in the perception of the teachers interviewed, this association was not identified since their opinion was strongly marked by the cognitive focus (intellectual abilities and knowledge about mathematical concepts). Regarding the teachers’ opinion about the challenges they face in the classroom, intellectual limitations and lack of mathematical knowledge on the part of students and, above all, difficulties of a didactic-pedagogical nature arising from operational limitations and relative to the teacher’s ability to promote problem formulation activities, were pointed as difficulties. These were also documented by Lee et al. (2018), such as difficultiesin introducing problem posing into their teaching practices due to the short time available in the classroom, and lack of familiarity with this activity. The lack of familiarity and expertise to carry out these activities concerns the pedagogical knowledge mentioned by Ball et al. (2008), and Shulman (1986). In particular, it seems that the participants lacked specialized content knowledge concerning As will be discussed in the concluding remarks, this knowledge needs to be addressed in initial and continuing training courses for teachers.

5 Concluding Remarks In order to articulate the theoretical discussion made at the beginning of this chapter and the discussion generated from the data obtained in the studies described above, four questions are answered in this final part.

5.1 How Important is Mathematical Problem Posing for Teaching? Posing mathematical problems is as important as solving them, and is even considered a facet of problem solving (Abu Elwan, 2016). In addition, its didactic value for learning mathematical concepts and for the development of logical reasoning in students is undeniable (Singer et al., 2013). Also, posing mathematical problems seems to stimulate the students’ creativity, motivation, and autonomy when learning (Bonotto, 2013; Brown & Walter, 2005; Sengul & Katranci, 2014). Silver (1994) emphasizes the importance of sharing the responsibility for formulating problems with students instead of concentrating this responsibility exclusively in the hands of teachers and textbook authors. Thus, as Kilpatrick (1987, p. 123) recommends, “problem formulating should be viewed not only as a goal of instruction but also as a means of instruction.” In Study 2, the relevance of problem formulation for students’ cognitive development and for learning mathematical concepts was recognized by respondents who use problem formulation as a means of instruction.

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5.2 Why is It Relevant to Investigate the Characteristics of Mathematical Problems Posed by Teachers? In order to effectively become an instructional activity, problem formulation must be considered teacher knowledge, which leads to this second question that ultimately justifies this study. The characteristics of problems formulated by teachers reveal, in some way, their conceptions about what constitutes a mathematical problem and about the knowledge they present about the concepts involved in problems they formulate (Agranionih et al., 2021; Leung & Silver, 1997). As revealed in Study 1, the low variability in problem formulation characterized the profile of most teachers interviewed, indicating a limited view of relationships between the different facets constituting mathematical concepts and the mathematical situations in which these facets emerge. This limited perspective can, as a result, limit the learning opportunities proposed in the classroom. Knowing the characteristics of mathematical problems posed by teachers can also be important in the elaboration of intervention studies that seek to expand teachers’ mathematical knowledge, their ability to formulate and solve problems and their pedagogical knowledge (Lavy & Shriki, 2007; Leung, 1994, 1996; Pelczer et al., 2014). This discussion highlights the importance of knowing more and more the teacher as a problem poser in relation to a specific field of mathematical knowledge and knowing the gains that are acquired by the teacher who becomes a good formulator of mathematical problems, as discussed below.

5.3 What Difficulties Do Teachers Face When Teaching Mathematics Through Problem Formulation? In Study 2, teachers mention as difficulties, students’ limitations of a didacticpedagogical nature that involved operational factors concerning the classroom functioning and the teacher’s limited ability to promote problem formulation activities. These two types of difficulties were also reported by Lee et. al (2018). It is important to mention that the teacher’s limited ability to promote such activities is related to pedagogical knowledge that seems to be absent in teacher training courses, as recent studies mention (Leavy & Hourigan, 2021; Penget al., 2022). This statement generates the fourth question.

5.4 Why Should Teachers Be Problem Posers? There are several reasons to make teachers mathematical problem posers, as mentioned in answering the second question presented in this final part of the chapter. Formulating problems would be another ability to be developed by the teacher that

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would enable them to critically analyze textbooks, identifying their limits, formulating problems that can add and deepen the mathematical ideas they want to develop in the classroom and create situations deemed appropriate to the interests of their students. It is not intended to say that problems formulated by teachers should replace those present in textbooks, but to create a partnership between teacher and textbook in favor of increasingly efficient instructional practices. As Divrik et al. (2020) emphasize, the textbook is an educational tool with a prominent role in education. Nevertheless, we emphasize that it is not the only source of problems to be presented to students, including in this scenario the challenging problems teachers might pose if they are prepared for such and the problems that students may pose if taught to do so.

5.5 Teacher as Problem Poser and a Mediator of Teaching and Learning Situations Through the Formulation of Problems It seems relevant to emphasize the importance of articulating the different roles that the mathematics teacher plays, as discussed in this chapter when looking at the teacher as a problem poser and as a mediator in teaching and learning situations. This perspective is shared by Cai and Leikin (2020) who comment that studies on problem posing should consider both content knowledge (Study 1) and aspects related to affect, beliefs, attitudes and values (Study 2). Liljedahl and Cai (2021) comment that the way a mathematical problem is posed has an impact on the person who will solve it, either in relation to the motivation to solve it, or in terms of the possibility of understanding the mathematical concepts and relationships involved in the problem. In this sense, the teacher’s specialized knowledge is fundamental, such as, for example, their knowledge about what is involved in multiplicative structure problems, as investigated in Study 1. In the words of Li et al. (2020, p. 326), “…the problems that a teacher poses shape the mathematical learning in their classes”. Thus, this chapter sought to articulate, through two studies, different but complementary facets of problem formulation, focusing on those who teach, in this case, the mathematics teacher. According to Cai and Hwang (2019) very little is known on teachers’ opinions about teaching through problem posing, which makes Study 2, described in this chapter, an attempt to fill the gap in the literature on this subject. Furthermore, these authors point out the need to examine problem posing from both students’ and teachers’ perspectives. Future research could turn to the learner, investigating students as problem posers and their opinion about the learning of mathematics through problem formulation. Acknowledgements The authors are grateful to the following institutions: (i) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) of the Brazilian Ministry of Science and Technology for the grants awarded to the first and to the third authors; and (ii) Coordenação de

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Aperfeiçoamento de Pessoal de Ensino Superior (CAPES) of the Brazilian Ministry of Education for the financial support given through the Education Observatory Program (Observatório da Educação—OBEDUC).

References Abu Elwan, R. (2016). Mathematics problem posing skills in supporting problem solving skills of prospective teachers. In C. Csíkos, Rausch, & J. Szitányi (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 3–10). PME. Agranionih, N. T., Spinillo, A. G., & Lautert, S. L. (2021). Characteristics of mathematical problems posed by teachers. Acta Scientiae, 23(1), 233–264. https://doi.org/10.17648/acta.scientiae.6183 Arıkan, E. E., & Ünal, H. (2015). Investigation of problem-solving and problem-posing abilities of seventh-grade students. Educational Sciences: Theory and Practice, 15(5), 1403–1416.https:// doi.org/10.12738/estp.2015.5.2678 Ball, D., Thames, M., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554 Baumanns, L., & Rott, B. (2021). Rethinking problem-posing situations: A review. Investigations in Mathematics Learning, 13(2), 59–76. https://doi.org/10.1080/19477503.2020.1841501 Baumanns, L., & Rott, B. (2022). The process of problem posing: Development of a descriptive phase model of problem posing. Educational Studies in Mathematics, 110, 251–269. https://doi. org/10.1007/s10649-021-10136-y Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37–55. https://www.jstor.org/stable/23434195 Brasil. (1997). Parâmetros curriculares nacionais: matemática [National Curriculum Parameters: Mathematics]. Ministério da Educação. Secretaria de Educação Fundamental. MEC/SEF. Brasil. (2017). Base Nacional Comum Curricular. Educação Infantil e Ensino Fundamental [National Common Curricular Base. Early Childhood Education and Elementary School]. Ministério da Educação. Secretaria da Educação Básica, SEB. http://basenacional comum.mec.gov.br/images/BNCC_EI_EF_110518_versaofinal_site.pdf Brasil. (2018). Base Nacional Comum Curricular. Ensino Médio [National Common Curricular Base. High School]. Ministério da Educação. Secretaria da Educação Básica, SEB. http://basena cionalcomum.mec.gov.br/images/historico/BNCC_EnsinoMedio_embaixa_site_110518.pdf Brown, S. I., & Walter, M. I. (2005). The art of problem posing. Psychology Press. https://doi.org/ 10.4324/9781410611833 Brown, S. I., & Walter, M. I. (2013). Problem posing; reflections and applications. Psychology Press. https://doi.org/10.4324/9781315785394 Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. Journal of Mathematical Behavior, 21(4), 401–421. https://doi.org/10.1016/S0732-3123(02)00142-6 Cai, J., & Hwang, S. (2019). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 101391. https://doi.org/10.1016/j.ijer.2019.01.001 Cai, J., Chen, T., Li, X., Xu, R., Zhang, S., Hu, Y., Zhang, L., & Song, N. (2019). Exploring the impact of a problem-posing workshop on elementary school mathematics teachers’ conceptions on problem posing and lesson design. International Journal of Educational Research, 1–13.https:// doi.org/10.1016/j.ijer.2019.02.004 Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics, 105(3), 287–301. https:// doi.org/10.1007/s10649-020-10008-x

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Lowrie, T. (2002). Young children posing problems: The influence of teacher intervention on the type of problems children pose. Mathematics Education Research Journal, 14(2), 87–98. https:// doi.org/10.1007/BF03217355 Magina, S., Merlini, V. L., & Santos, A. (2016). A Estrutura Multiplicativa à luz da Teoria dos Campos Conceituais [The multiplicative structure in light of the theory of conceptual fields]. In J. A. Castro Filho (Ed.), Matemática, cultura e tecnologia: perspectivas internacionais [Mathematics, culture and technology: international perspectives] (pp. 65–82). CRV. Pelczer, I, Singer, F. M. E., & Voica, C. (2014). Improving problem posing capacities through inservice teacher training programs: challenges and limits. In Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 1, pp.401–408). Peng, A., Li, M., Lin, L., Cao, L., & Cai, J. (2022). Problem posing and its relationship with teaching experience of elementary school mathematics teachers from ethnic minority area in Southwest China. Journal of Mathematics, Science and Technology Education, 18(2). https://doi.org/10. 29333/ejmste/1153 Ramirez, M. C. (2006). A mathematical problem-formulating strategy. International Journal for Mathematics Teaching and Learning, 7, 79–90. https://www.cimt.org.uk/journal/ramirez.pdf Santos, E. M., Araújo Gomes, C. R., & Gomes, A. S. (2021). The posing of mathematical problems by University Students of Mathematics. In A. G. Spinillo, S. L. Lautert & R. E. S. R. Borba (Eds.), Mathematical reasoning of children and adults (pp. 267–292). Springer. https://doi.org/ 10.1007/978-3-030-69657-3_12 Sengul, S., & Katranci, Y. (2014). Structured problem posing cases of prospective mathematics teachers: experiences and suggestions. International Journal on New Trends in Education and Their Implications, 5(4), 190–240. http://www.ijonte.org/FileUpload/ks63207/File/17..sengul. pdf Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.2307/1175860 Shulman, L. (2005). Conocimiento y enseñanza: fundamentos de la nueva reforma. Profesorado. Revista de currículum y formación del profesorado, 9(2) 1–30. https://www.ugr.es/~recfpro/rev 92ART1.pdf Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. http://www.jstor.org/stable/40248099 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. https://doi.org/ 10.1007/s10649-013-9478-2 Souza, E. I. R. S., & Magina, S. M. P. (2017). Concepção do Professor do Ensino Fundamental sobre Estruturas Multiplicativas. [The conception of elementary school teachers concerning multiplicative structures]. Perspectivas da Educação Matemática, 10(24), 797–815. https://per iodicos.ufms.br/index.php/pedmat/article/view/2930/4163 Vergnaud, G. (1983). Multiplicative structures. In R. Lesch & M. Landau (Eds.), Acquisition of mathematics concepts and procedures (pp. 127–174). Academic Press. Vergnaud, G. (1994). Multiplicative conceptual field: What and why? In H. Gershon & J. Confrey. (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 41–59). University of New York Press. 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. https://doi.org/10.1007/s10649-017-9760-9

Chapter 10

Problem Posing Among Preservice and Inservice Mathematics Teachers Ma. Nympha Beltran Joaquin

Abstract This study explored whether significant differences can be found in word problems composed by preservice teachers and inservice teachers in the field. Forty word problems constructed by preservice and inservice teachers in two semistructured problem-posing tasks were examined and analyzed in terms of type, content, context, and cognitive load. The findings of the study show that while both groups of teachers used varied content topics that dealt with numbers, algebra, geometry, measurement and a little on data, the problems produced were of the routine reformulated type which were mostly on applications of mathematical concepts. Only one of the problems which was constructed by a preservice teacher was of the process type. Personal context dominated both sets of problems but it was that noted that the inservice teachers tend to produce problems that are more theoretical in nature. Moreover, it was noted that both groups did not explore other contexts such as occupational, societal and scientific, and constructed word problems were bordering on the low cognitive load. The study underscores the potential of problem posing for assessing both mathematical content knowledge and pedagogical skills of teachers. Recommendations for focus training on crafting good, effective and varied word problems that are realistic and accurate are put forward. Keywords Problem posing · Preservice and inservice teachers · Characteristics of word problems · Content · Context · Cognitive load · Nonroutine · Reformulated problems · Misconception

1 Introduction Problem solving and problem posing are two related skills that lead to better understanding of mathematical concepts and their applications. Mathematics teachers are expected to be adept at these skills. Inservice teachers may have honed their skills in the art of constructing and solving word problems through years of teaching while Ma. N. B. Joaquin (B) University of the Philippines—Diliman, Quezon City, The Philippines e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_10

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preservice students who have not been exposed to problem posing tasks may find creating word problems a challenge. However, certain studies such as the one by Kilik (2017), showed that only a little more than half of elementary teachers can pose a word problem satisfying a specified strategy. Moreover, Kozaklı Ulger and Yazgan (2021) noted that prospective mathematics teachers have the potential to pose problems that are nonroutine in nature and recommended further studies on this. On the other hand, some studies have attempted to characterize word problem solving using different frameworks and categories. It is in this light that this study was conducted. Its main interest was to know how the preservice and inservice teachers differ in terms of constructing word problems. It aimed to investigate and characterize the word problems created by both groups of teachers using a combination of categories gleaned from previous studies.

2 Problem Solving Problem solving has been acknowledged as both a goal for teaching mathematics and as a conduit to achieve this goal. This dual role of problem solving in the field of mathematics education makes it ubiquitous in the classroom sessions. The term problem solving refers to “mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development” (NCTM, 2010, p. 1). It is one of the twenty-first century skills that we need to develop among our students (UNESCO IBU, 2020). Knowing how to solve problems is essential for success in this volatile, uncertain, complex and ambiguous world. Future leaders as well as the members of the workforce need to be equipped with this skill in order to be able to navigate their way through the challenges that come across their paths. Problem solving, on the other hand, can be viewed as a means to teach mathematical concepts. As a pedagogical approach, problem solving has been proven to be beneficial to learners in terms of higher mathematics achievement, more positive and broad attitudes toward mathematics, and better creativity skills (Albay, 2019; Kiymaz et al., 2011; Stein et al., 2003). In this approach, students are normally presented with a problem, a solution for which would require knowledge and application of the lesson to be introduced. The students immediately see the relevance of the lesson due to the problem posed by the teacher. Ernest (1988) acknowledged that teaching students through problem solving encourages and fosters reason ability, meaningful learning, and flexible thought, among others.

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3 Problem Posing Crucial to the problem-solving process is the problem-posing stage. Silver (1994) noted that problem posing, which is often referred to as generation or reformulation of problems, occurs during the process of problem solving. The development of problem solving skills among students is highly dependent on the quality of word problems to which the students are exposed. Moreover, the understanding and appreciation of the mathematics lessons are reliant on the quality of word problems used to elicit them. The art and skill of problem posing has gained attention among mathematics education researchers worldwide. Several studies have focused on how teachers pose word problems in their mathematics classes. (Baumanns & Rott, 2022; Cai & Hwang, 2021; Kozaklı Ulger & Yazgan, 2021; Yao et al., 2021). Problem posing has been found to be correlated to general mathematics performance and has been shown to improve mathematics achievement as well as creativity (Guvercin & Verbovskiy, 2014; Silver, 1994). Aguilar (2017) studied the characteristics of word problems constructed by high achieving students with either verbal or visual learning styles in terms of complexity, clarity, sufficiency, originality, and integration. Recently, Baumanns and Rott (2022) proposed a framework for characterizing problem-posing activities. The framework had three dimensions. Dimension 1 distinguishes a generated problem from a reformulated problem, with the latter requiring less effort. Dimension 2 differentiates routine from nonroutine problems; while Dimension 3 classifies them according to high or low metacognitive behavior.

4 Characterizing Word Problems in Mathematics Teacher-constructed word problems may be characterized in a number of ways. Stoyanova (1997) classified problem-posing situations as: free, when students are asked to construct a problem from a naturalistic situation; semi-structured, when they are given a situation in which they are asked to formulate a problem; and structured, when students are asked to formulate a new problem based on a given problem or solution to a problem. In an earlier study, Ernest (1988) employed a different scheme in categorizing mathematics problems into 4 types. The problem types that emerged were translation problems that translate real world problems into mathematical expressions; process problems that involve different cases for the solver to use; applied problems that allow students to use mathematical concepts and skills to solve realistic problems; and puzzle problems that give them opportunity to experience recreational mathematics. On the other hand, Bauman and Roth (2022) acknowledged that word problems may be classified as generated or reformulated word problems. They could be “based on an open situation for which mathematical knowledge, skills, and concepts from previous experiences have to be applied” (p. 32) or they could be obtained by varying conditions of a given problem (Schoenfeld, as

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cited in Bauman & Roth, 2022). Aside from categorizing word problems according to type, word problems may be classified according to the content area in mathematics on which they are anchored. The Trends in International Mathematics and Science Study (TIMSS), which assesses mathematical literacy among participating countries every four years uses the 3 content domains number, measurement and geometry, and data for Grade 4. For Grade 8, it uses the 4 content domains number, algebra, geometry, and data and probability (Lindquist et al., 2017). On the other hand, the Programme for International Student Assessment (PISA), which is given to 15year olds among its participating countries uses the 4 content domains quantity, uncertainty and data, change and relationships, and space and shape. It is from these that students are expected “to draw on to reason, to formulate the problem, to solve the mathematical problem, and to interpret and evaluate the solution determined” (OECD, 2018, p. 10). Another possible area for classifying mathematics word problems is through the contexts used in formulating them. Problem contexts depict the extent to which teachers are able to integrate with other fields of study. Teacher-constructed word problems allow educators to evaluate whether the contexts of the word problems they provide in classes are aligned with the different exposures of students as they go about their daily activities. Gogus (2012) noted that the theory of constructivism states that learning takes place when students construct meaning by interpreting information in the context of their own experiences. Moreover, the study conducted by Bottge (1999) supports the use of contextualized problems to enhance the problem-solving skills of students in both general and remedial classes. Meanwhile, PISA uses four context categories for classifying their items. These are personal, occupational, societal and scientific. “Problems classified in the personal context category… include those involving food preparation, shopping, games, personal health, personal transportation, sports, travel, personal scheduling and personal finance… Problems classified in the occupational context category … involve such things as measuring, costing and ordering materials for building, payroll/accounting, quality control, scheduling/inventory, design/architecture and job-related decision making… Problems classified in the societal context category… involve such things as voting systems, public transport, government, public policies, demographics, advertising, national statistics and economics… Problems classified in the scientific category … include such areas as weather or climate, ecology, medicine, space science, genetics, measurement and the world of mathematics itself.” (OECD, 2018, pp. 29–30).

Additionally, word problems may be categorized according to cognitive load. Gupta and Zheng (2020) identified 3 types of cognitive load, namely intrinsic, extraneous and germane. The intrinsic cognitive load in mathematics problems refer to the difficulty of content, which is related to students’ prior knowledge. In contrast, the extraneous cognitive load refers to the mental load which is brought about by improper instructional design. Germane cognitive load on the other hand refers to mental load induced by the efforts to construct new knowledge.

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5 Conceptual Framework It is of interest to analyze the teacher-constructed word problems that are being used during lessons. It gives us insights on how mathematics learning takes place in the classroom. Delivery of the teacher-constructed word problems reflect how the dual role of problem solving is portrayed in an ordinary class session. It speaks of how the goal of developing the skill of problem solving is being attained. It likewise gives us an idea on the extent to which mathematics lessons are taught through problem solving. Based on the review of literature, word problems created by both preservice and inservice teachers could be analyzed using a framework that uses four dimensions, namely type, content, context, and cognitive load. Figure 1 presents the Word Problem Classification Framework used in this study. Problem type is based on three classifications. The first is Ernest’s (1988) word problems classification according to translational, applied, process and puzzle types of problem. The second type refers to Bauman and Roth’s (2022) dimension on whether the problem is generated or reformulated, while the third deals with routine versus non-routine word problems. The framework acknowledges that there could be a combination of these types of problems. For example, here could be reformulated (or generated) routine (or nonroutine) word problems that are either of the translational, applied, process or puzzle type, as long as they are straightforward and based on problems typically found among textbooks. Nonroutine problems are complex problems that require solvers a certain level of creativity and uniqueness in solving them (Pramayudi et al., 2019). The content dimension in the Word Problem

Fig. 1 Word problem classification framework

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Classification Framework has categories number, measurement, geometry, algebra and data which are gleaned from TIMSS Grade 4 and Grade 8 levels. On the other hand, the problem contexts in the framework is based on the PISA categories, namely personal, occupational, societal and scientific. Lastly, the cognitive load dimension is restricted to Gupta and Zeng’s (2020) intrinsic cognitive load which can be classified as high and low in terms of difficulty.

6 Method This is a qualitative descriptive study that aimed to analyze and compare mathematical word problems created by preservice and inservice teachers. A total of 40 original word problems created by both groups of teachers based on semi-structured tasks were analyzed in this study. The data were gathered from the two lone classes on mathematics pedagogy in a state university, one at the undergraduate and the other at the graduate level. The university typically offers only one course on mathematics pedagogy per semester, per level. The first class was a high school mathematics methods course for pre-service teachers who have not had lessons on construction of word problems in mathematics. The other class was composed of in-service high school teachers of mathematics with 2–9 years of classroom experience and who were enrolled in a graduate course on instructional planning and procedures for mathematics. Ten volunteers from each class participated in the study. They were included in the study due to their willingness to take part in the research and availability during data collection. Both groups of pre-service and in-service teachers were instructed to construct a word problem based on each of two specific semi-structured problem posing tasks. Task 1 required them to construct a word problem with “18” as the final answer, while Task 2 required them to use the numbers “2, 5, 1/2 and 20.7” in constructing a word problem. The tasks were adapted from Joaquin’s (2007) Mathematical Thinking Skills Test that measures the basic, critical and creative thinking of students in mathematics. The semi-structured tasks were chosen to ensure that the teachers, particularly those who were already teaching would not simply submit a word problem that they had previously given to their classes, as these were most likely to have been revised already. Both the 10 preservice and 10 inservice teachers were given an hour to accomplish the two tasks individually during one session of their respective methods/ instructional planning courses. They were not allowed to open their books and notebooks, nor were they allowed to access the Internet. A total of 40 word problems (2 word problems, one for each task, from each of the 10 preservice and 10 inservice teachers) were gathered for this study. All the 40 generated word problems were analyzed using the framework described earlier. Coding was done to systematically identify the type, content, context and cognitive load of each problem starting with answers to Task 1 from the preservice teachers followed by Task 1 answers of inservice teachers before proceeding to Task 2 word problems, following the same order. The analysis called for interviews in

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order for each of the teacher participants to explain how they constructed their word problems. Nonstructured interviews were held the week after the teachers were asked to construct the word problems.

7 Results and Discussions Results show that with regard to problem type, most of the word problems posed by both the preservice and inservice teachers in the two assigned tasks could be classified as reformulated routine problems since they were straightforward questions that are usually asked among traditional mathematics books and do not pose much challenge to the problem solver. The use of mathematical word problems that require nonroutine thinking and real-world knowledge instead of mere routine problems should be emphasized in the classrooms (Pongsakdi et al., 2020). Unfortunately, all the problems generated by the two groups of teachers did not exhibit this. Moreover, they can mostly be classified as Ernest’s (1988) applied problems since they required direct use of mathematical concepts or formulas to be solved. None of the problems produced by both groups could be classified as translational problems nor the puzzletype problems but there was one from the preservice teachers who was able to produce a process problem that required considering cases when solved. Figure 2 presents this problem. The problem created by Preservice Teacher G for Task 2 did not strictly follow the stipulated procedure as it included another number (4) in the problem. Moreover, the first part could lead the problem solver to assume that the triangle is to be cut from the longest side going to the opposite vertex. However, analyzing the solution provided would lead us to appreciate the intention of the problem poser. The problem is about the Triangle Inequality Theorem, the solution for which called for considering different cases. Such is an example of Ernest’s (1988) process problem. In terms of content topics, it is evident that for Task 1, both groups produced problems with Measurement (perimeter and area) as the most dominant content, but the ones posed by the inservice teachers mostly required knowledge on algebra in order to be solved. Hence, it is noted that certain problems integrate the domains Measurement and Algebra. Interestingly, for Task 2, where the semi-structured task required them to use only the specified values in the construction of the problem, a slightly different scenario was obtained. While the preservice group still maintained area and perimeter as the most frequently occurring content topic for their problems, the rest were about Numbers (operations on fractions and decimals). In contrast, the inservice group produced more problems about operations on fractions and decimals, with only two items on area and perimeter. Generally, both groups of teachers were able to produce word problems that cover all five categories in the content dimension of the Word Problem Classification Framework. The other word problems generated by preservice teachers were on specific topics such as partitive proportion, distance, length, conversion of units, and computing change from a purchase, while some of

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Fig. 2 Preservice Teacher G—Task 2

the problems produced by the inservice teachers dealt with speed, simple length, average, Pythagorean theorem, 30-60-90 triangles, and quadratic equations. Problem context refers to the situation or circumstances surrounding the problem. It was noted that most of the preservice teachers’ output used places or objects from daily life situations such as car, stick, handkerchief, ribbon, lace, garden plot, and fence. All these problems fall under the PISA’s personal context category and were noted to be aligned to the daily experiences of the teachers. The use of realia, on which learners could easily relate to, has been proven to be effective in promoting understanding of concepts. However, the few other problems posed by this group were hypothetical situations that either simply asked for a direct use of a formula or presented a figure on which computations would be based. On the other hand, it was highly notable that the contexts of the problems constructed by inservice teachers were too general, such as those about a travelling object, a rectangular field with certain dimensions, or about a simple geometric figure like a rectangle, trapezoid or triangle without a compelling situation. The only and most concrete objects used by this group of teachers were the ladder, bookshelf, and fence. Gravemeijer and Doorman (1999) describes context problems as those in which the situation is experientially real to the learner. Realistic Mathematics Education (RME) states that problem context offers learners opportunities for progressive mathematizing (Gravemeijer & Doorman, 1999). Hence, it is essential that word problems be situated in

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varied contexts but within the problem solver’s field of experience. Interestingly, none of the problems from both groups of teachers could be classified under PISA’s occupational, societal and scientific contexts since they lacked details that would situate them within the real world of high school students. Interviews were conducted to better understand how the content and context of the problems were conceptualized. Most preservice teachers claimed to have used the strategy of working backwards particularly for Task 1, which already specified “18” as the answer to the problem. Some recalled going over the factors of 18 as 2 and 9, and then composing a problem about the area of a rectangle with 2 and 9 as dimensions. Another simply thought of an object such as the ribbon which could be easily added and formulated a problem on finding the total length of ribbons. The inservice teachers, on the other hand, admitted that they initially thought of word problems that they recently used in their classes that would satisfy the criteria for the given tasks. This explains why there were problems involving Algebra, the subject most of the inservice teachers were teaching. There were likewise problems on theorems on special triangles composed by those teaching Geometry. Apparently, even if there were restrictions on the semi-structured tasks, they did not prevent some teachers from reusing word problems. However, none admitted that the problems they submitted for this study were the exact word problems they have previously used in their classes. They acknowledged that to a certain extent, they were still confronted by the challenge to tweak their classroom problems to conform to the stipulations in the given tasks. Such word problems produced fall under Baumann and Rott’s (2022) reformulated problems since they were mostly based on existing problems that the teachers already presented to class. The rest of the constructed word problems, including those original outputs from the preservice teachers may be classified as generated problems. Unexpectedly, the preservice teachers who were mostly enrolled in calculus courses, did not construct word problems about calculus nor analytic geometry, which is a co-requisite of calculus. Some teachers from this group expressed their intention to produce only simple word problems in compliance with the given tasks since they were informed that their outputs were not graded. As regards cognitive load, this study focuses on the intrinsic type, with high and low difficulty as its categories. The analysis of the output of the two groups of teachers showed that while all the problems posed by both groups were application problems on the different fields of mathematics, the cognitive load of these problems was still at the low level. Most of the preservice group’s problems can be solved by simple operations on rational numbers, while the inservice teachers’ problems required use of basic algebraic concepts such as solving linear equations. Both sets of problems generated involved only the basic applications of lessons in the respective fields of mathematics. None were found to go beyond the fundamental concepts that could prove to be challenging to students. Figure 3 presents a sample output from the preservice group. The problem created by Preservice Teacher A for Task 1 may be termed as a multi-step problem because it involves many operations. However, it can easily be solved using arithmetic concepts only. It does not necessitate deeper knowledge in

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Fig. 3 Preservice Teacher A—Task 1

high school mathematics and therefore can be classified under problems with low cognitive load. Meanwhile, a closer look at the generated word problems showed that some participants from both groups had either assumptions, misconceptions, and difficulties in constructing clear, accurate and realistic word problems. The succeeding discussion elaborates more on this using sample outputs from the teachers. Figure 4 presents the problem created by one of the preservice teachers for Task 1. The problem composed by Preservice Teacher D involves two concepts, namely perimeter and area of a square. Its solution assumed that the squares are of the same dimension but Teacher D did not explicitly state this crucial information in the problem. Otherwise, two squares of dimensions 2 × 2 and 4 × 4 would be another solution which gives a bigger total area of 20 m2 . Further, he did not specify that he was asking for the maximum area his fencing materials could enclose. This implies that another acceptable solution would have been enclosing two 1 × 1 squares, giving

Fig. 4 Preservice Teacher D—Task 1

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Fig. 5 Preservice Teacher G—Task 1

only a total area of 2 m2 , which would still answer the question. Students’ performance on word problem solving is related to their text comprehension (Pongsakdi, Kajamies, Veermans, et al., 2020) and hence, problems should be clearly stated and without biases or assumptions. Figure 5 gives us another sample problem from the preservice group. Preservice Teacher G gave a problem involving a 36-cm2 handkerchief. A 6-cm by 6-cm handkerchief does not sound realistic, much more asking for its perimeter when folded in half. Perhaps if the context were more elaborated, for example, stating that a company is packing folded tissue paper with the given dimensions, then the students could easily relate to the problem. They would find it more practical rather than theoretical or absurd. Freudenthal (as cited in Laurens et al., 2018) emphasized that students should be given an opportunity to re-discover mathematics by processing real-world situations. By using realistic contexts, students would be able to comprehend easily and see the applications of their lessons. The preceding examples highlight the lack of mathematical pedagogical content knowledge on the part of the preservice teachers, as the problems they composed were wanting in clarity and accuracy. However, it should be noted that at this point they have not had sessions on word problem formulation. Interestingly, an analysis of the problems posed by the inservice teachers who already had at least two years of teaching experience yielded somewhat similar results. Figure 6 shows the work of Inservice Teacher E. Note that although computing heights could be considered part of daily life situations, this problem created by Inservice Teacher E talked about the total height of two girls reaching 20.7 feet, which is absurd. It should be noted that Teacher E is a senior high school teacher with two years of experience. The problem he produced is similar to the preservice teachers’ problems which were found to be using personal

Fig. 6 Inservice Teacher E—Task 2

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Fig. 7 Inservice Teacher C—Task 1

Fig. 8 Inservice Teacher E—Task1

contexts but with unrealistic situations. Moreover, the first and last statements of this particular problem seemed to contradict each other and could confuse the problem solver. This implies that even actual teachers in the field need training on how to construct good mathematics word problems. Figure 6 presents another sample from the inservice teachers’ group (Fig. 7). The problem posed by Inservice Teacher C involves both perimeter and area, just like the one created by Preservice Teacher G. Although the cognitive load is higher in nature, in that it involves solving two equations in two unknowns, or substitution of variables, it seems inaccurate because he is equating the area to the perimeter. Technically, these two terms have different units of measure and hence, cannot directly be compared. Inservice Teacher C had taught high school geometry for 3 years and was currently teaching Algebra. Clearly, problems posed by teachers speak of their depth of knowledge on the subject matter. Figure 8 presents another interesting problem by another inservice teacher. It is obvious that the Inservice Teacher E has a misconception because he talked about the hypotenuse of an equilateral triangle, which this type of triangle clearly does not have. Inservice Teacher E has been teaching senior high school for 5 years. The nature of the word problems created by teachers for classroom use greatly affect the role of problem solving in the mathematics curriculum. Mathematics lessons are supposedly learned through the problem solving activities in the class sessions (problem solving as a means) while answering these teacher-constructed word problems are assumed to hone their problem solving skills (problem solving as a goal).

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8 Conclusions and Recommendations The findings of this study showed that both preservice and inservice teachers tend to produce application type reformulated routine word problems incorporating various mathematical content but are of low cognitive load. The context used by both groups lean toward the personal side, with some problems from the inservice teachers that are theoretical in nature. Based on the foregoing discussions, it may be concluded that preservice and inservice mathematics teachers do not differ much in terms of the type, content, context, and cognitive load of the problems they create. Although the problems both groups generated vary at certain aspects, in general, it can be said that the problem posing skills of preservice and inservice teachers are on a par with each other. The years of teaching experience did not prove to be an advantage for the inservice teachers. It is noted that both groups did not have explicit training on composing mathematical word problems prior to data collection. Findings suggest that exposure to problem posing through retooling workshops and trainings could be explored in order to develop this highly important skill among inservice teachers. Inclusion of the development of such skill in the curriculum for preservice teachers is likewise recommended. Cai and Hwang (2021) mentioned that teaching how to construct word problems has not been consistently and substantively included in the mathematics curriculum. Teachers need to be trained to create nonroutine word problems that promote critical and creative thinking in their solutions. Concentration should not just be on producing application type of word problems, but rather on the process and puzzle types, as these have a higher cognitive load and develop higher-order thinking. Teachers should likewise be trained to employ a variety of real life contexts with accurate information in the occupational, societal and scientific fields aside from the personal contexts commonly used. The results of the study further highlighted how the problems posed by the preservice and inservice teachers reflect their orientation, knowledge of content and as well as their pedagogical skills. Word problems constructed by teachers play a significant role in attaining the dual goals of problem solving in the mathematics curriculum. A good mathematics problem should be challenging, clear, accurate and free from ambiguities and misconceptions. Inaccuracies in formulating these word problems could lead students to confusion and teacher misconceptions that are reflected in them could be passed on to the problem solvers. The potential for assessing teacher mathematics content and pedagogical knowledge through analysis of the word problems they create should be further explored. Teachers who are steep in content should be able to produce word problems that are free from mathematical errors and misconceptions while teachers with high mathematical pedagogical knowledge should be able to reach out to students through posing mathematically sound nonroutine problems in various contexts that are high in cognitive load but are within the field of experience of the students. Moreover, studies on how the problems posed by teachers explicitly hone the problem solving skills of students should be further investigated.

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Similar studies may be conducted on a wider scale using different ranges of years of teaching experience for the inservice group.

References Aguilar, A. (2017). Mathematics problem construction of high achieving verbal and visual students: Characteristics and thought processes. Unpublished doctoral dissertation. University of the Philippines. Albay, E. (2019). Analyzing the effects of the problem solving approach to the performance and attitude of first year university students. Social Sciences & Humanities Open, 1(1). Baumanns, L., & Rott, B. (2022). Developing a framework for characterising problem-posing activities: A review. Research in Mathematics Education, 24(1), 28–50. https://doi.org/10.1080/ 14794802.2021.1897036 Bottge, B. A. (1999). Effects of contextualized math instruction on problem solving of average and below-average achieving students. Sage Journals, 33(2). https://doi.org/10.1177/002246699903 3002 Cai, J., & Hwang, S. (2021). Teachers as redesigners of curriculum to teach mathematics through problem posing: Conceptualization and initial findings of a problem-posing project. ZDM Mathematics Education, 53, 1403–1416. https://doi.org/10.1007/s11858-021-01252-3 Ernest, P. (1988). The problem-solving approach to mathematics teaching. Teaching Mathematics and its Applications: An International Journal of the IMA, 7(2), 82–92. Gogus, A. (2012). Constructivist learning. In N. M. Seel (Eds.), Encyclopedia of the sciences of learning. Springer. https://doi.org/10.1007/978-1-4419-1428-6_142 Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1/3), 111–129. http:// www.jstor.org/stable/3483163 Gupta, U., & Zheng, R. Z. (2020). Cognitive load in solving mathematics problems: Validating the role of motivation and the interaction among prior knowledge, worked examples, and task difficulty. European Journal of STEM Education, 5(1), 05. https://doi.org/10.20897/ejsteme/ 9252 Guvercin, S., & Verbovskiy, V. (2014). The effect of problem-posing tasks used in mathematical instruction to academic achievement and attitude toward mathematics. International Online Journal of Primary Education, 3(2). https://www.iojpe.org/index.php/iojpe/article/view/122 Joaquin, M. N. B. (2007). Achievement goal modification: Effects on student mathematical thinking skills and self-efficacy. Unpublished dissertation. Kilik, C. (2017). A new problem-posing approach based on problem solving strategy: Analyzing pre-service teachers’ performance. Educational sciences theory and practice, 17, 771–789. Kiymaz, Y., Sriraman, B., & Lee, K. H. (2011). Prospective secondary mathematics teachers’ mathematical creativity in problem solving. In B. Sriraman & K. H. Lee (Eds.), The elements of creativity and giftedness in mathematics. Advances in creativity and giftedness (Vol. 1). SensePublishers. https://doi.org/10.1007/978-94-6091-439-3_12 Kozaklı Ulger, T., & Yazgan, Y. (2021). Non-routine problem-posing skills of prospective mathematics teachers. Eurasian Journal of Educational Research. https://ejer.com.tr/wp-content/upl oads/2021/08/ejer.2021.94.7.pdf Laurens, T., Batlolona, F., Batlolona, J., & Leasa, M. (2018). How does realistic mathematics education (RME) improve students’ mathematics cognitive achievement? EURASIA Journal of Mathematics, Science and Technology Education. https://doi.org/10.12973/ejmste/76959 Lindquist, M., Philpot, R., Mullis, I. V. S., & Cotter, K. E. (2017). TIMSS 2019 Mathematics Framework. In I. V. S. Mullis & M. O. Martin (Eds.), TIMSS 2019 assessment frameworks.

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Retrieved from Boston College, TIMSS & PIRLS International Study Center. http://timssandp irls.bc.edu/timss2019/frameworks/ National Council of Teachers (2010). Why is teaching with problem solving important to student learning? NCTM Research Brief. Organization for Economic Co-operation and Development (OECD). (2018). PISA Mathematics Framework (Draft). https://pisa2022-maths.oecd.org/files/PISA%202022%20Mathema tics%20Framework%20Draft.pdf Pongsakdi, N., Kajamies, A., Veermans, K., et al. (2020). What makes mathematical word problem solving challenging? Exploring the roles of word problem characteristics, text comprehension, and arithmetic skills. ZDM Mathematics Education, 52, 33–44. https://doi.org/10.1007/s11858019-01118-9 Pramayudi, A. A. A. S., Sudiarta, I. G. P., & Astawa, I. W. P. (2019). Classification of students’ nonroutine problem-solving skill. Journal of Physics: Conference Series, Volume 1503. International Conference on Physics and Natural Sciences, August 2019, 30–31, Bali, Indonesia. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–48. Stoyanova, E. N. (1997). Extending and exploring students’ problem solving via problem posing. Edith Cowan University Research Online Institutional Repository. https://ro.ecu.edu.au/theses/ 885/ Stein, M. K., Boaler, J., & Silver, E. A. (2003). Teaching mathematics through problem solving: Research perspectives. In H. L. Schoen & R. I. Charles (Eds.), Teaching mathematics through problem solving: Grades 6-12 (pp. 245–256). National Council of Teachers of Mathematics. UNESCO International Bureau of Education. (2020). Twenty-first century skills. http://www.ibe. unesco.org/en/glossary-curriculum-terminology/t/twenty-first-century-skills Yao, Y., Hwang, S., & Cai, J. (2021). Preservice teachers’ mathematical understanding exhibited in problem posing and problem solving. ZDM Mathematics Education, 53, 937–949. https://doi. org/10.1007/s11858-021-01277-8

Chapter 11

An Approach to Developing the Problem-Posing Skills of Prospective Mathematics Teachers: Focus on the “What if not” Heuristics Zoltán Kovács

Abstract The ability to create problems is included in the Hungarian National Core Curriculum as a mathematical skill to be developed in students, which is one of the reasons why problem-posing is a fundamental pedagogical skill of mathematics teachers. Therefore, it is necessary to include the development of this competence in the training program for prospective mathematics teachers. The study results were obtained during the “Mathematics Competitions” course for prospective mathematics teachers, where one of the objectives is to develop problem-posing skills. The students involved have already completed problem-solving courses, so they are already familiar with the essential aspects of problem-posing. In particular, they have used successive reformulations of problems during the problem-solving process. However, they have not systematically addressed the practice of problem-posing based on an initial problem. The author compares the problem-posing products of two groups: novices and experts. Novices had only problem-solving practice, whereas the expert group received training in the “what if not” method of problem-posing. By adapting the existing evaluation categories in the literature, the author created a system of categories suitable for describing the corpus and providing adequate information for evaluating problem-posing products. The conclusion is that competent problem-posing is a learnable activity. After dedicated training, prospective teachers with similar mathematical backgrounds and problem-solving experience posed problems more competently. The difference can be explained by the intervention resulting in better problem perception by the expert group than by the novices. The comparative analysis’s pedagogical implication is the need for focused training for problem posing in teacher education. Additionally, the author says that the skills needed to pose a problem go beyond those needed to solve a problem. Keywords Problem-posing · Mathematics competitions · Modern elementary mathematics · “what if not” strategy · Mathematics teacher’s training Z. Kovács (B) Eszterházy Károly Catholic University, Eger, Hungary e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_11

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1 Introduction One of Hungary’s traditions in mathematics education is to enable talented students to participate in mathematics competitions (Gy˝ori et al., 2020). Consequently, the mathematics teacher has a role in preparing students for competitions, and this task is also included in mathematics teacher training programs (Ministry of Human Resources, 2013). This paper is based on the author’s experiences teaching the course “Mathematics Competitions” to prospective mathematics teachers at the University of Debrecen (Hungary). However, since it is hardly possible to gain significant experience in solving problems at the competition level in a one-semester course, the author’s didactical consideration in this course was instead to teach prospective teachers how to approach a problem from the perspective of ‘modern elementary mathematics’ (Koichu & Andž¯ans, 2009). Modern elementary mathematics can be defined as the union of two types of entities: elementary methods and problems that can be solved by means of these methods. Elementary methods are methods recognized by a mathematical community as not depending on any specific branch of modern mathematics, but broadly utilized in many branches. Examples of elementary methods are induction, the invariant method, considerations of symmetry and of equivalence. Elementary methods are often used in obtaining estimations, analyzing singular cases, etc. This field of mathematics continuously presents new problems and results. (p. 11)

Another goal of this course was to move away from the “exercise paradigm” (Skovsmose, 2001) and into a “landscape of investigation,” where students could freely explore problem situations. Based on the extensive research available (Cai et al., 2015; English, 2020; Kilpatrick, 1987; Koichu, 2020; Kontorovich & Koichu, 2016; Pehkonen, 1997; Silver, 1994; Singer et al., 2013), the author hypothesizes that the problem-posing method can aid in achieving these objectives. Specifically, an apriory epistemological analysis of a problem utilizing the problem-posing approach might assist the prospective teacher in responding appropriately to the inevitable unexpected twists and turns that occur while working with talented pupils (Mason, 2015). The desired level of empowered problem posing is described by Crespo (2015), who argues that this level has the qualities of openness, mathematical interest, challenge, and social relevance. However, previous studies have shown that prospective teachers often lack advanced problem-posing skills. For example, Crespo describes the phenomenon of disempowered problem-posing as a closed reformulation of an initial problem, an overly narrow interpretation of the mathematical background, or a problem setting that underestimates the problem’s difficulty and does not acknowledge the mathematical background. The author’s rationale behind this research is how he can help improve prospective teachers’ problem-posing practice. However, problem-posing integrates several subsystems (Kontorovich et al., 2012), one of which is the knowledge of heuristic strategies for problem-posing. Therefore, the research described in this article is guided by whether short-term focused training in problem-posing based on the “what if not” strategy (Brown & Walter, 1990) can help prospective mathematics teachers become competent problem-posers.

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The author uses a comparative analysis of two groups’ problem-posing products to answer this question. Earlier, both groups practiced in Polya’s fourth look-back problem-solving phase (Polya, 1945). Therefore, prospective teachers were expected to be able to think about, e.g., how a problem could be used in other situations and how it could be made more general. However, one group had received targeted training in the use of the “what if not” strategy in three sessions (the group of experts in what follows), while the other had not (the group of novices); they could only draw on their experience of problem-solving. The starting point was a competition task for junior high school students, which was discussed in groups with several solutions. Then, working independently, the prospective teachers had to create problem variations. The research was conducted in the context of methods of competition preparation with prospective mathematics teachers of average mathematical ability, and the author believes that this aspect also fills a gap in the literature. In the present study, the expert group was more successful at problem-posing than the novice group. The result confirms that problem-posing is a learnable activity but also shows that problem-posing is an activity that needs to be learned. The author highlighted two perspectives regarding the learning process. The first is the significance of problemposing heuristics, and the second is that otherwise simple competition tasks with a rich mathematical background are effective learning materials.

2 Theoretical Background 2.1 Problem-Posing It is generally accepted that problem-posing is a component of mathematical literacy (Niss & Jablonka, 2014). Although problem-posing has been studied for decades, suffice it to point to Koichu’s (2020) overview and English’s (2020) commentary, there is no uniformity in researchers’ understanding of problem-posing. Several definitions have appeared in the literature; see (Baumanns & Rott, 2021) for a detailed analysis. Silver’s (1994) approach has been widely cited. It involves inventing new problems based on specific situations and reformulating existing problems. Stoyanova and Ellerton (1996) defined problem-posing as the elaboration of personal interpretations of concrete situations and the formulation of concrete situations as meaningful mathematical problems. Papadopoulos et al. (2021) classify the possible ways of problem-posing into five categories: generating new problems only; reformulating existing or given problems only; generating and reformulating problems simultaneously; posing questions; and modeling. In this study, the author shares the view of Cai and Hwang (2020, p. 2): By problem posing in mathematics education, we refer to several related types of activity that entail or support teachers and students formulating (or reformulating) and expressing a problem or task based on a particular context (which we refer to as the problem context or problem situation).

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Kontorovich et al. (2012) suggest a framework for interpreting the problem-posing process from a problem-solving perspective. The framework has a modular structure that may be utilized in different contexts. The five modules are: (1) task organization; (2) knowledge base; (3) problem-posing heuristics; (4) group dynamics; and (5) aptitude. When a teacher creates a problem-posing activity, the task-organization component accounts for all didactical considerations. The knowledge base includes mathematical schemes and related mathematical communication skills; see also (Schoenfeld, 2014). Problem-posing heuristics are organizational units of the activity. An example is the well-known “what if not” strategy. Finally, aptitude includes fitting the task, the possible evaluators, and the possible solvers. Brown and Walter (1990) systematize the “what if not” problem-posing heuristic strategy. The function of the strategy is to pose new problems from already solved ones by varying the conditions or goals of the given one. Here, the problem-posing heuristic is guided by a specific question. The process leading to the new problem consists of four steps, namely: (1) listing of attributes of the original problem; (2) posing “what if not?” questions for attributes; (3) posing mathematical questions about the altered problem; and (4) problem analysis: discovery of the self-posed situation.

2.2 Problem-Posing in Teacher Education There is a long history of research on the role of problem-posing in higher education, both in general and in teacher education in particular. According to Osana and Pelczer’s (2015) meta-analysis of the literature from 1990 to 2012, the role of problem-posing in mathematics teaching is recognized, and the ability to pose problems is considered a part of teacher competence. The classroom-based study by da Ponte and Henriques (2013) involved investigation activities and problemposing in a university course. These activities were intended to stimulate students’ curiosity in asking questions and promote their ability to investigate. The results show that the investigations provided opportunities for students to experience mathematical processes, including posing questions and formulating and testing conjectures. Ellerton (2013) demonstrates how problem-posing can be incorporated into a mathematics teacher education curriculum. The results also show the significance of problem-posing activities in teacher education programs. The research demonstrated the efficacy of these activities in developing prospective teachers’ concepts of the significance of problem-posing and investigation activities in classroom mathematics. Based on their research, Tichá and Hošpesová (2013) argue that students enrolled in teacher education typically have naïve beliefs about mathematics and the nature of mathematics education. The research done by the authors shows that this belief could be changed by posing problems, especially when this is done with shared reflection. Crespo (2015) considers the teacher and the students as creators of mathematics problems themselves and suggests a framework for the problem-posing activity of teachers in the classroom and sets four possible screenplays: (1) posing

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problems to students, (2) posing problems with students, (3) posing a mathematical problem of personal interest, and (4) posing socially relevant mathematical problems. Problem-posing can be a tool for the teacher to use when planning lessons; it can also be a means of mathematical instruction, i.e., the teacher creates a situation in which they use student problem-posing to implement the lesson’s task order (Cai & Hwang, 2020). In all classroom implementations, the teacher must have the ability to pose problems and must be able to anticipate the problems that the students will raise (Mason, 2015). Although research has demonstrated that students and teachers are capable of posing interesting and significant mathematical problems, researchers have also found that some students and teachers pose problems that are not mathematical, problems that cannot be solved, and irrelevant problems (Cai et al., 2015). When Voica and Pelczer (2010) compared the problem-posing activities of in-service teachers (experts) and prospective teachers (novices), they found that teachers are guided by pedagogical objectives and consider their class when posing a problem. On the other hand, prospective teachers regard problem-posing as a self-referential activity focused on problems with no specified audience. The authors found that teachers pay much more attention to how the problem is worded than prospective teachers do. For example, many of the problems that prospective teachers come up with have vague statements or wrong suggestions for solutions, which does not happen very often with teachers. Chapman (2012) reports that students’ sense-making of problem-posing depends on their mathematical knowledge, imagination or creativity, and experience with problem-solving. Students were conflicted with their prior experience, which exposed them mainly to closed problems. Tichá and Hošpesová (2013) found that the conceptual shortcomings of teachers are reflected in problem-posing products.

2.3 Evaluation of the Problem-Posing Products There are several methods for evaluating problem-posing activities, each of which addresses a different dimension of problem-posing. Crespo (2015) introduces two main categories for assessing students’ problem-posing products: empowered and disempowered. A disempowered approach means that the problem-poser gives a closed reformulation of the original problem, narrows the mathematical background of the original problem, or ‘blindly’ poses the problem. The mathematical difficulty of the posed problem is underestimated in the latter situation; problems are posed without any prior solution or complete comprehension of the mathematics behind them. The empowered approach includes posing open problems, mathematically challenging, interesting, or socially relevant mathematics problems. In this category, problems require solvers to explain their work and communicate their ideas. Questions and problem-adaptations scaffold rather than lead the solver’s thinking. They use mathematical aesthetic criteria such as surprise, novelty, simplicity, and fruitfulness. Posing socially relevant problems means engaging in understanding and addressing social issues with mathematics.

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Tabach and Friedlander (2015) assessed the problem-posing activities of sixth and seventh-grade students by examining the creativity traits of fluency, flexibility, and originality. This method provides information on the performance of a group. Group fluency is measured by the average number of problems created in the group. Only new problems are considered when assessing the group’s flexibility. The new problem, as interpreted by the authors, means that the mathematical structure of the problem is different from the original problem. Furthermore, originality is measured by the average number of unconventional modifications. Leikin and Elgrably (2020) embedded problem-posing in an investigation activity and approached the performance of prospective teachers in terms of creativity. Their model for evaluating problem-posing is based on the model for evaluating creativity described in Leikin (2009). They evaluate students’ problem-posing processes in terms of fluency, defined as the number of explored properties; flexibility, defined as the number of different problems posed; and originality, defined as the novelty and rarity of the properties explored. The novelty was assessed by the participants’ learning history and rarity in the collective space of problems raised.

2.4 Hungarian Traditions of Mathematical Competitions and Competition Preparation The author briefly reviews Hungary’s traditions of mathematical competitions and competition preparation to place the research in a broader context. One of the pillars of the Hungarian mathematics teaching tradition is talent management and, in this context, the system of mathematical competitions (Gy˝ori et al., 2020). These traditions can be traced back to 1893, when a journal for secondary school students and their teachers, the Középiskolai Mathematikai Lapok [Mathematical Journal for Secondary Schools], was launched. A national mathematics competition, the Eötvös Competition (1894), named after the famous physicist Loránd Eötvös (1848–1919), was started up a year later. This competition was so successful that the competition material was published in Hungarian (Kürschák, 1929) and later in English (Kürschák et al., 1963). Notes in these books, which often accompany the solutions, shed light on the problems’ higher mathematics background and provide generalizations and new solutions. Since the 1950s, school mathematics circles and booklets have helped students prepare for competitions. This practice of mathematics circles was very similar in its aims and means to the out-of-school activities described by Koichu and Andž¯ans (2009) but organized within a school (but extra-curricular) framework, under the guidance of enthusiastic mathematics teachers. The domain of the math circle movement can be associated with modern elementary mathematics (Koichu & Andž¯ans, 2009). The themes in this area also appear in secondary school mathematical lessons, although their solutions are more complex than those mandated by regular curricula (Kontorovich, 2020). However, it should be noted that this practice of mathematics

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circles has become less common in Hungarian practice, and out-of-school activities have appeared, see, e.g., (Juhász, 2019). The mathematics competition preparation course is Hungary’s tradition of mathematics teacher training. The textbooks for the courses, e.g., Molnár (1983), Radnainé Szendrei (1988), build on the Hungarian problem-solving tradition of Polya and deal with problem-posing in a targeted way. Molnár (1983) continues the tradition of Kürschák (1929). The book also embraces the Polya heritage (Kovács, 2022), intensively involving the look-back phase (Polya, 1945) and applying the problemposing method. “New problems emerge as a result of generalization or specialization, analogies, reformulations, and variations in data and conditions,” the author writes (p. 36). The author also emphasizes that the habit of problem recognition and problem generation enhances problem-solving skills. Radnainé Szendrei (1988) also uses the methods of searching for generalizations, multiple solutions, and problem-posing. The author writes in the introduction: Even after solving the problem, we play with it by examining it from several angles by exploring generalization possibilities. The aim is to incorporate the newly acquired knowledge more deeply into the solver’s previous knowledge and to give the reader a complete picture of how mathematics is done. (p. 3)

In this short quotation, the author formulates two crucial insights: on the one hand, through problem variation, the learner’s schema for each mathematical knowledge or concept is extended, and on the other hand, the learner “cultivates” mathematics, i.e., behaves as a mathematician. The current Hungarian legislation governing the training of prospective mathematics teachers (Ministry of Human Resources, 2013) also requires that the training include the development of problem-solving and problem-posing skills. Furthermore, in line with Hungarian tradition, the legislation also addresses talent management, specialized activities (math circles), and mathematical competitions in teaching mathematics.

3 Method 3.1 Research Question The current study is guided by the question of what role heuristic strategies of problem-posing, dedicatedly the “what if not” strategy, play in competent problemposing. Namely, can short-term focused training in problem-posing based on the “what if not” strategy help prospective mathematics teachers become competent problem-posers? The research corpus consists of the problem-posing products of prospective teachers. The problem-posing task was completed in individual work in a closed classroom setting. The research method employed was a qualitative analysis of the

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products, during which the author evaluated the problem-posing products. In addition, the author conducted a comparative analysis of the performance of the two groups when analyzing the results. By competent problem-posing, in the context of this research, the author means that the problem-poser can form a related mathematical problem that can be included in the schema of the original problem. The “schema of the problem” is approached from the perspective of modern elementary mathematics and problem-posing. It includes the mathematical background of the problem, the heuristic strategies that can be used to solve it, and additional problems that can be related to the problem by questions of mathematical nature, such as “what if not.” Relatedness means that there is a semantic similarity between the original problem and the new problem; some of the decisive attributes of the new problem and the original problem are the same.

3.2 Participants The research involves two groups of Hungarian prospective teachers from the University of Debrecen. The author taught them the compulsory course “Mathematics Competitions” in the penultimate semester of the theoretical training before school teaching practice, as shown in Table 1. The course aims to develop prospective teachers’ problem-solving skills by applying heuristic strategies and incorporating high-level mathematical concepts into problem-solving practice in algebra, number theory, combinatorics, geometry, and precalculus. At the same time, a dedicated goal is to prepare prospective teachers for talent management, including preparing gifted students for competitions. Both groups have experience in problem-solving, having completed a “Problem-Solving Seminar” course, where they were introduced to Polya’s well-known four-phase problem-solving model (Polya, 1945), including practice in the look-back phase. In addition, the group of experts received three sessions of targeted training in the “what if not” problem-posing strategy, while the novices did not. The ethical consideration of the experiment is that the novice group received the same training in the “what if not” strategy after the data collection event. Table 1 The groups participating in the experiment Group

Date

Number of prospective teachers

Serial numbers

Novices

2019

22

N1-N22

Experts

2018

13

E1-E13

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3.3 Research Design Intervention in the expert group. The three lessons on problem-posing were designed using the principles of Brown and Walter (1990). The author acted as the group’s instructor, leading the lessons once a week during regular class hours. The activities in each lesson were undertaken in the same order. The author used the cooperative think-pair-share method while addressing problems. First, the author gave the prospective teachers a problem to solve, and they were instructed to do it autonomously. Then, after approximately twelve minutes of independent work, they discussed their concepts in pairs, and during this activity, the author observed the pairs’ progress. Based on the observation of the pairs, the author chose a pair to present their ideas, or perhaps the entire solution, to the group. The group could also contribute to the solution or give fresh ideas. Then, the focus turned to problem-posing in the following phase, and the lessons continued with the same flow of activities. They identified the attributes of the original problem, developed “what if not, but” questions, and then, in pairs, selected one of the questions they wanted to address in the remaining time of the lesson. The author guided in selecting problem variations because of the possible difficulties and recommended that the more challenging problems be solved as homework. At the end of the lessons, some selected questions were discussed. The author lists the three problems selected for problem-posing training, each with a possible variation, in the Appendix of this paper. Task organization. The data collection event took place in the first session of the semester for the novice group and in the fourth session for the expert group. It was possible to use a calculator, the internet, and a smartphone so that the student could even use digital tools. The organization of the data collection event is as follows: 1. Setting the initial problem and dealing with it independently, possible with several solutions. 2. Discuss solutions to the initial problem. First, the prospective teachers presented their solutions, and then the author complemented them with some solutions that had not yet been presented. Of the six solutions presented in the next section of the study, the first five occurred in the author’s classroom practice (not necessarily at the time of the experiment). The sixth solution is the approach to the problem discussed in the booklet, which is the source of the problem. 3a. The assignment in the novice group. The author asks the group to work independently on Polya’s look-back phase. After a summary of the aim of the lock-back phase, the author quoted Polya, “Can you use the result, or the method, for some other problem?” and asked the prospective teachers to set and solve as many related problems as possible. 3b. The assignment in the expert group. The author asked the prospective teachers to set and solve as many problem variations as possible. Two different task organizations correspond to the prior knowledge of the teacher candidates. For the novice group, the author could only rely on Polya’s look-back

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phase, whereas for the expert group, it could be expected that the term “problem variation” would be familiar to them. The time available for both groups for problem variations and solutions was 45 min. The starting problem and solutions. The starting point for the problem variation is taken from the high school booklet for 5–8 graders (Fried, 1970). As the author summarized earlier, these booklets were used to support the work of extra-curricular mathematics circles organized within a school setting. Problem Is there any multiple of 7 that consists of only 9s? One of the features of this problem is that it can be solved based on various mathematical backgrounds and using different strategies. 1. Try and error, check 9, 99, 999, . . . , 999999 (Fig. 1). 2. Thinking backward, and applying the standard multiplication algorithm in an “incomplete multiplication” situation, see Fig. 2. (Incomplete multiplication is a common task in Hungarian textbooks at the primary level: the pupil has to find the missing digits of a multiplication.) 3. Pattern finding, applying modular arithmetic. 7|106 − 1, (see 2nd row of Table 1) or 7|9 · 111111, (3rd row), see Table 2. 4. Simple mathematical knowledge is the divisibility rule for 7.  999  999 , 7|999 − −

+

999 = 0. Note that this divisibility rule, which can easily be derived from the previous solution, is not part of the Hungarian curriculum.

Fig. 1 Try and error with the CASIO CLASSWIZ pocket calculator

Fig. 2 Solution of the problem based on the standard multiplication algorithm

Table 2 Solution of the problem based on modular arithmetic 100

101

102

103

104

105

106

Remainder divided by 7

1

3

2

6

4

5

1

Sum of reminders

1

4

6

12

16

21

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199

5. Advanced mathematical knowledge is Euler’s totient theorem. While 7 is a prime number, ϕ(7) = 6, where ϕ is the Euler’s totient function. Moreover, 10 and 7 are relatively prime numbers, thus 106 ≡ 1 mod (7). 6. Pigeonhole principle. Consider seven consecutive positive integer powers of 10: 10n , 10n+1 , . . . , 10n+6 . None of these numbers is divisible by 7, so there are two that give the same remainder when divided be 10k and  byk−l7. Letthese two powers l k l l k−l 10 , where k > l. Then 7|10 −10 = 10 10 − 1 , i.e., 7|10 −1 = 99 . . . 9. Discussion of the opening problem. Naturally, the question of whether the opening task is a problem is determined by the problem solver’s prior knowledge and problemsolving experience. The author’s assertion that this task aligns with modern elementary mathematics, as discussed by Koichu and Andž¯ans (2009), may be questionable given the trial and error procedure’s prompt yield of solution. The network of solutions, on the other hand, adds to the picture of the problem. Furthermore, looking at the problem through the ‘colored glasses1 ’ of problem-posing, the problem has the potential to unfold into a rich series of problems, thereby entering the world of modern elementary mathematics. Through problem variations, the problem can even become a competition challenge. Approaching the task from this angle already breaks free from the exercise paradigm (Skovsmose, 2001), allowing for inquiries and inviting prospective teachers to participate in exploratory processes. From this point of view, this problem has a rich scheme. At the same time, the problem provides an opportunity for those new to problem-posing and who only have experience in problem-solving to explore since several questions can be asked in Polya’s look-back phase. Another issue is that the first two solutions do not explain why they ‘work.’ The two types of mathematical proofs—those that prove a proposition’s correctness and those that explain why it is correct—are distinguished in the literature (Hanna, 1989). The author argues that “proofs that explain should be favored in mathematics education over those that merely prove” (p. 45). With these considerations in mind, what did the author expect from this task? These expectations came from the author’s analysis of the task from an epistemological point of view and his work with other groups, which gives ample problem space. The first expectation was that the prospective teacher would ask about all possible solutions (the “how many?” question). All but the first two solution methods show that the conditions of the problem are satisfied by infinitely many numbers, which can consist of 6, 12, 18, … digits. The second expectation was that the prospective teacher would generalize the task, i.e., solve the problem not only for concrete numbers but generally (the “is it always true?” question). For example, Euler’s totient theorem shows that seven can be replaced by any number that is a relative prime to 10; moreover, multiples of even numbers are even, and multiples of numbers divisible by five end in 0 or 5 (and not in 9). Third, the author expected prospective teachers to use methods that explain why the solution works, i.e., modular arithmetic, the pigeonhole principle, or Euler’s totient theorem. Finally, it was not the author’s expectation that prospective teachers would set problems at a higher level, 1

Late Marion Walter’s words in Baxter (2005).

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which falls into Crespo’s (2015) “posing a mathematical problem of personal interest” category. The task design was not suitable for posing this type of problem; it would have required more time and in-depth work. Such a possible problem might be, for example, giving a “better” solution, i.e., a solution with fewer digits than the one that follows from Euler’s totient theorem.

3.4 Evaluation of Problem-Posing Products The author examined three dimensions of prospective teachers’ problem-posing products. The first dimension qualifies the task based on the cognitive demand on a nominal scale ranging from a blind task to a flexible problem. The second dimension describes which attributes of the original task the prospective teacher has changed. Finally, the third dimension examines the prospective teacher’s heuristic strategy and mathematical background to solve their problem-posing products. The cognitive demand of the problem-posing product. Figure 3 shows the process of coding the outputs, where the rectangles contain the final codes. The features of codes are as follows: The result of the problem-posing process is blind when a task is posed without a solution or faulty solution. Inadequate task means that the prospective teacher reformulates the original problem in closed form or provides context only. Example 1 (Inadequate task). Prove that 999999 is a multiple of seven! (N12). As a decision rule, we classify a problem variation as inadequate if its relationship to the original problem is vague. An example of the latter type is the following task. Example 2 (Inadequate task). Which digit in the 2017th place is in the decimal form of 2017/7? (N6). If the generated question leads directly to an arithmetic solution, or the problem can be handled with a simple numeric example up to 100, the question is qualified as an exercise. Example 3 (Exercise). Is there any multiple of 11 that consists of only 7s? (E1).

Fig. 3 The process of coding

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77 is a simple answer. Problem-posing leads to a simplified problem if the mathematical scope narrows. Most of the simplified problems posed by prospective teachers require the application of divisibility rules from the Hungarian school curriculum. Example 4 (Simplified problem). Is there any multiple of 6 that consists of only 8s? (N2). The prospective teacher’s answer is in Fig. 4. (Note that the prospective teacher does not take sufficient care when writing down the number constructed by three, six, or nine 8s.) If the prospective teacher answers their question using the same methods and strategies as in solving the original problem, the new task is an analogous problem. Example 5 (Analogous problem). Is there any multiple of 17 that consists of only 9s? (N17, Fig. 5)? Flexible problem alteration means that the modified problem’s mathematical structure differs from the initial problem’s. The seeds of a flexible problem can be found in several works, but unfortunately, most of them were communicated by prospective teachers without a solution or with a faulty solution. Example 6 shows

Fig. 4 An example of a simplified problem

Fig. 5 Analogous problem by N17, using Euler’s totient theorem

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a product submitted with a correct solution, where the prospective teacher invokes the divisibility rule by eleven. The prospective teacher establishes that numbers with this property consist of even 9s. Example 6 (Flexible problem). Determine all positive integers that are multiples of 11 and consist of only 9s. (E12). It is essential to clarify how the term problem is used in the context of this study. The measurement instrument of the research does not allow the author to investigate how the prospective teachers imagine the role of the problem they are constructing, i.e., what is the instructional purpose and the reason behind the problem. Therefore, the author considers both simplified, analog, and flexible problem-posing products as the result of a competent problem-posing process, with a different meaning than Crespo’s (2015) term empowered. The product attributes that have changed. Looking at the starting problem through the lens of the “what if not” strategy, one may conclude that the task is rich in attributes. Possible attributes are 1. 2. 3. 4. 5.

The goal of the task is to answer an “existence” question. The divisor in the task is seven. All the digits of the multiple are 9. The multiple is made up of identical digits. Divisor-multiplier context.

Heuristic strategies and the mathematical background used by the prospective teachers. Six categories in this dimension are the same as the solution strategies in the initial problem presentation. Later, during the corpus processing, new categories had to be added to the system, which the author will report on in the next section.

4 Results 4.1 The Cognitive Demand for the Product Table 3 contains the result of the coding. Table 3 Prospective teachers’ problem-posing products Blind task

Inadequate task

Exercise

Simplified problem

Analogous problem

Flexible problem

Sum

Novices

12

17

8

7

14

0

58

Experts

10

4

4

7

16

1

42

Sum

22

21

12

14

30

1

100

11 An Approach to Developing the Problem-Posing Skills of Prospective … Table 4 Problem- and non-problem-type tasks in the group of novices and experts

Table 5 Number of problems per prospective teacher in the novice and expert groups

203

Non-problem

Problem

Novices

37

21

Sum 58

Experts

18

24

42

Sum

55

45

100

Problems per prospective teacher Mode

Max

Average

Novices

1

3

0.95

Experts

2

4

1.85

One possibility to compare the performance of the two groups is to compare the problem type (simplified, analogous, flexible) and the non-problem type (blind, inadequate, exercise) tasks, see Table 4. 38% of all problem-posing products in the novice group were problems, while in the expert group, the result was 57%. Fisher’s exact test was used to determine if the difference was statistically significant, i.e., whether there was a significant association between the novices/experts variable and the non-problem/problem type products. The result p = 0.04 indicates a significant relationship between the two variables. The author then focused on individual performance and looked at how many problems (simplified, analog, flexible) each prospective teacher had created (Table 5). The average number of problems in the expert group was 1.85, and only one prospective teacher did not set a problem-type product. The value for the novice group was 0.95, while eight prospective teachers in the novice group did not set a task on the problem level.

4.2 The Product’s Attributes that Have Changed The author also investigated which attributes of the original task the prospective teacher altered. Table 6 shows the number of problems in which the attribute in question has changed. Tables 7 and 8 contain the same information as Table 6 but give the data per person.

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Table 6 Attributes of the original task the prospective teachers altered Attribute

Number of products

Products per prospective teacher

Novices

Novices

Experts

Experts

0

No change

11

4

0.50

0.31

1

The goal of the task is to answer an “existence” question

2

5

0.09

0.38

2

The divisor in the task is seven

5

7

0.23

0.54

3

All the digits of the multiple 12 are 9

8

0.55

0.62

4

The multiple is made up of identical digits

1

4

0.05

0.31

5

Divisor-multiplier context

6

1

0.27

0.08

2&3

The divisor in the task is seven AND all the digits of the multiple are 9

20

12

0.91

0.92

2&4

The divisor in the task is seven AND the multiple is made up of identical digits

1

1

0.05

0.08

58

42

2.64

3.23

Table 7 Attributes of the original task the novice prospective teachers altered (number of problems for every prospective teacher, novice group) Attr Novice prospective teacher serial number

Sum

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0

1

1 1 2

1

1

2

1

3

1 1

4

1

1

2

1 1

1

1

5

1

3

3

2

12 1

1

2&3 1 1

11 2

1 1

1

5 2&4

2

2 2 1

2

3

1

2

1 3

1 2

6

3

20

1

Sum 2 4 1 1 2 2 3 3 3 2

1 4

4

1

4

1

4

3

3

3

3

1

4

58

4.3 Heuristic Strategies and the Mathematical Background The strategies used by prospective teachers to solve their problems are summarized in Table 9. The last category (tracing back to the original problem) means that the prospective teacher implied that the problem should be solved in the same way as

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Table 8 Attributes of the original task the expert prospective teachers altered (number of problems for every prospective teacher, expert group) Attr

Expert prospective teacher serial number 1

2

3

5

1

1

6

7

1

0 1 2

1

3

Sum 8

9

2

1

10

12

13

2

1

1

1

1

1

1

1

1

5

1

1

5

1

1

7

2

1

8

1

4

1 2

2

1

1

2

1

2

1

1

1

2&4 Sum

11

4

1

2

4 2&3

4

2

5

2

4

3

2

2

4

4

3

12 1

3

5

3

42

the original problem without explaining the details. “0” means that the prospective teacher provides a context only or does not provide a solution. Finally, “simple arithmetic” means the assignment of a simple arithmetic operation, e.g., performing a division. Table 9 Heuristic strategies and the mathematical background used by the prospective teachers

Novices Experts Sum 0

16

6

22

Simple arithmetic

11

5

16

Try and error

5

3

8

Thinking backward

12

0

12

Modular arithmetic

5

5

10

Divisibility rule

8

17

25

Euler’s totient theorem

1

0

1

Pigeonhole principle

0

0

0

Tracing back to the original problem 0

6

6

42

100

58

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5 Discussion 5.1 Aptness One of the most remarkable differences between the performance of the two groups is the low occurrence of inadequate tasks in the expert group, while 29% of the tasks in the novice group are inadequate; see Table 3. A plausible reason is that the expert group was familiar with the problem-posing procedure and understood what to expect from the product. In addition, the expert group had a better understanding of the purpose of the problem-posing, while the novice group tended to reformulate the problem in a closed way, looking for context and setting a problem that did not fit the scheme of the original one. The author also explains the higher number of computational tasks in the novice group for this reason, but the difference is not as pronounced. This observation confirms the role of the aptness facet (Kontorovich et al., 2012) in the problem-posing process. The pedagogical implication is that problem-posing ability does not emerge naturally from problem-solving practice but must be intentionally taught to prospective teachers.

5.2 Fluency, Flexibility, and Novelty The quantitative analysis immediately shows a difference between the two groups. The expert group has higher fluency, whether we consider all problem-posing products or only the tasks classified as problems. The number of total products per prospective teacher is 3.23 in the expert group and 2.64 in the novice group. The number of tasks classified as problems is 1.85 per prospective teacher in the expert group and 0.95 in the novice group. This difference is also reflected in the mode descriptor, with experts most often giving two problem variations and novices giving one. There is also a notable difference in the attribute change between the two groups. In the expert group, two prospective teachers (15%) made one type of change, namely E01 and E06 (Table 8). In contrast, in the novice group, eight prospective teachers (36%) made only one type of modification (Table 7). In both groups, the most common alteration was to change attributes 2 and 3 simultaneously. However, there is a difference in the pattern of the data. It was more common for novices to create only a product in which the two attributes were changed together (eight prospective teachers did this, N1, N7, N8, N11, N12, N14, N16, N18), and only two prospective teachers changed the second or third attribute separately in addition to the simultaneous change (N2, N17). In contrast, among the eight expert prospective teachers who changed attributes 2 and 3 simultaneously, five also changed the attributes separately (E3, E9, E10, E12, and E13). The author explains these differences by the fact that the experts had a more analytical view of the original problem; they were more conscious of separating the

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Fig. 6 Listing the attributes in E06’s work

attributes, as this step was part of the “what if not” strategy. This fact is evident in six experts’ work. An example is shown in Fig. 6. The prospective teacher lists four out of the five attributes. (The first, the task goal, is missing, but there is a vague reference to the residue classes, whose appearance was probably influenced by modular arithmetic as a way to solve the problem.) Flexibility and novelty were not significant factors in this experiment, and there was no difference between the results of the two groups in this respect. The most original problem attempts were those in which the prospective teacher changed the divisor-multiplier relation (see Example 10 later), or the problem required a multistep proof in which the statement did not follow directly from Euler’s totient theorem (see Example 8 later). However, these attempts ended in blind tasks. A possible explanation for this phenomenon is the task design in the experiment, i.e., the prospective teacher had limited time and had to tackle the task individually. The lack of a robust mathematical background may also have played a role. Because problemposing and problem-solving are closely related processes, failure to solve problems has contributed to the creation of blind problems. The success of problem-solving depends on the mathematical resources available (Schoenfeld, 2014). For example, in dealing with their problems, the prospective teacher used Euler’s totient theorem in only one case (Table 9). In most cases, the more straightforward solution methods were chosen (divisibility rules, thinking backward), although modular arithmetic was used eight times.

5.3 The Look-Back Phase Five products from the expert group and two from the novice group addressed the cardinality of the solution set. The example in Fig. 7 illustrates how the careful

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Fig. 7 Scaffolding the activity by E08

application of the “what if not” strategy leads to this question. The scaffolding in the picture made the prospective teacher want to know how many solutions there were. This experience has shown that mastering the “what if not” strategy also helps uncover the problems’ structure and supports the look-back phase. Several prospective teachers addressed the possibility of generalization, the most general question being as follows. Example 7 (Blind task) Is there a multiple of n that consists of only the digit k? (E12). Before putting this question, the prospective teacher changes the second and the third attributes of the problem separately and then simultaneously; see Table 8. However, the prospective teacher could not answer the question in this general context. The question of generalization has most often led to the statement that any digit can take the place of nine in the case of the divisor being seven. Only E09 realized that any prime greater than five could replace seven, but that was as far as they got.

5.4 The Role of the Control and Mathematics Resources The proportion of blind tasks was 19% in the novice group and 24% in the expert group (Table 3). Within the blind tasks, the number of incorrectly solved tasks was four in the novice group and three in the expert group. The difference between the two groups in this research is not significant, and the author argues that the main reason for faulty solutions is the lack of adequate control in the problem-solving process. The following example demonstrates this. Example 8 (Blind task). Is there a multiple of 17 in which only the seven occur? (N16, Fig. 8).

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Fig. 8 CASIO Classwiz calculator result

The task is formally a proper problem-variation, and Euler’s totient theorem guarantees the positive answer, namely, 17 is a prime number, thus 17|1016 − 1 ⇒ 17| 99 . . . 9 ⇒ 17| 11  . . . 1 ⇒ 17| 77  . . . 7 .   16 digits

16 digits

16 digits

However, the prospective teacher used the trial-and-error strategy with the help of the calculator and obtained an incorrect argument (Fig. 8). The multiplication by the last digits (9 · 7 = 63) shows that the result of the division in question cannot be ended by 9, i.e., the calculator gave a rounded result. In fact, the integer part of the division ends with 8, and the next decimal place is 6. This example shows that if one does not approach the problem using the paradigm of modern elementary mathematics or if one does not try to approach it by explaining the solution (see Hanna’s (1989) description of the role of proofs), one can go astray. Another phenomenon is when the prospective teacher is not aware of the mathematical background of the problem. In the following example, the prospective teacher has altered attribute 5 (divisor-multiple context). As a result, the mathematical background was changed to the fundamental theorem of number theory, but the prospective teacher did not realize this and could not solve the problem. Example 9 Is there any power of seven that consists of only 9s? (E04). In the exercise and simplified-type products, it was observed that even simple questions were often solved by the prospective teachers using advanced methods previously used to solve the original problem. In Example 10, the answer to the question is obvious and can be quickly decided from the final digit of the number. However, the prospective teacher solved the problem (correctly, by the way) by applying the modular arithmetic method used not long before. Example 10 Is there any multiple of 8 that consists of only 9? (E05, Fig. 9). The reason for this phenomenon lies in the complex relationship between problemsolving and problem-posing. The knowledge we have just used in the problemsolving process can impact the tools we use for a new task that is close in time. The pedagogical implication is that if the lens of problem-posing means asking ourselves specific questions (e.g., “what if not”), we should remember Polya’s advice: ask yourself if you could do it more simply. Then, after a “what if not” variation, the mathematical background of the problem can be simplified, which the poser needs to recognize.

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Fig. 9 Using modular arithmetic instead of a simple reason in E05’s work

Overall, the research confirms a holistic approach to problem-posing and problem-solving (Kontorovich et al., 2012). Schoenfeld’s (2014) well-known factors (resources, heuristics, control, and beliefs) also strongly influence the problemposing activity.

6 Conclusion The research question that guided this study was, “Can short-term focused training in problem-posing based on the “what if not” strategy help prospective mathematics teachers become competent problem-posers?” By competent problem-posing, in the context of this research, the author means that the problem-poser can form a related mathematical problem that can be included in the schema of the original problem. The notion of “schema of the problem” is approached from the perspective of modern elementary mathematics and problem-posing. It includes the mathematical background of the problem, the heuristic strategies that can be used to solve it, and additional problems that can be related to the problem by questions of mathematical nature. The author compared the problem-posing products of two groups of prospective mathematics teachers, one group receiving a brief, targeted training in the “what if not” strategy (experts) and the other not (novices). The initial problem is a number theory problem with a rich schema. The results of the research are summarized in the following highlights:

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• The understanding of the “what if not” strategy increased the problem perception. The expert group had a better understanding of the purpose of the problem formulation, while the novice group tended to reformulate the problem in a closed way, looking for context and setting a problem that did not fit the scheme of the original one. • Knowledge of the “what if not” strategy increased the fluency of the expert group; more competent problems per person were posed, and the vast majority of experts could pose at least one competent problem—the “what if not” strategy positively impacted the analytic approach to the task. However, many members of the novice group spontaneously arrived at possible problem variations (not using “what if not” questions), albeit to a lesser extent. • Knowledge of the “what if not” strategy had no detectable effect on flexibility and novelty. • The success of the problem-posing process in both groups was influenced by gaps in problem-solving ability, including the mathematical background. Based on these results, we conclude that the “what if not” strategy can enhance the competence of prospective mathematics teachers in problem-posing, particularly in problem perception and fluency.

7 Limitations and Further Research The differences between the results of the two groups may have explanations other than specific training on the “what if not” heuristic. Among these, the possession of mathematical resources may be decisive. The author attempted to eliminate this effect by investigating an elementary number theory problem that all prospective teachers confidently solved. The author will also investigate the mid-term impact of the “Mathematics Competitions” course on a larger sample, namely how competent problem-solving characteristics emerge in prospective teachers’ products after completing the course. Future research should also explore what types of tasks could be appropriate for the learning trajectory. Acknowledgements This study was funded by the Research Program for Public Education Development of the Hungarian Academy of Sciences (KOZOKT2021-16). Competing Interest Statement The author declares that he has no competing financial interests or personal relationships that could have influenced the work reported in this paper.

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Appendix: Problems Applied During the Expert Group Intervention Which of the numbers 2011, 2012, 2013, and 2014 can be written as the sum of two or more consecutive positive odd numbers? (Hungarian competition problem for 9–10 graders). The essence of the problem is to write a positive number as a sum of an arithmetical sequence. Here, the difference can be varied, e.g., what if the difference is not 2 but 1? A possible problem variation is the following. Is it possible for 2012 to be written as the sum of consecutive positive integers? Prove that if the area of a triangle is 1/2 unit, then its perimeter is greater than 3 units. (Hungarian competition problem for 11–12 graders.) This problem was transformed into the following more general problem, which was the starting point of the problem-posing. Prove that the perimeter of a regular triangle is the smallest of triangles of the same area. What if not a regular triangle is given? A possible problem variation is the following. Prove that the perimeter of the square is the smallest of quadrilaterals of the same area. Connect the midpoints of the two opposite sides of the parallelogram with a point on each of the other two sides. Show that the area of the quadrilateral defined by the four connecting segments is half of the area of the parallelogram. [A problem for seventh graders (Kovács et al., 1988).] What if we start from a trapezoid, and if the fixed points are the midpoints of the bases? Prove that 21 bm ≤ t ≤ 21 am, where t is the area of the inscribed polygon, b is the shortest base of the trapezoid, and a is the longest base of the trapezoid.

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Kontorovich, I., & Koichu, B. (2016). A case study of an expert problem poser for mathematics competitions. International Journal of Science and Mathematics Education, 14(1), 81–99. https://doi.org/10.1007/s10763-013-9467-z Kontorovich, I., Koichu, B., Leikin, R., & Berman, A. (2012). An exploratory framework for handling the complexity of mathematical problem posing in small groups. Journal of Mathematical Behavior, 31(1), 149–161. https://doi.org/10.1016/j.jmathb.2011.11.002 Kovács, C., Szeredi, É., & Sztrókayné Földvári, V. (1988). Matematika. Általános iskola 7. [Mathematics for seventh graders]. Tankönyvkiadó. Kovács, Z. (2022). The tradition of problem-posing in Hungarian mathematics teaching. Teaching Mathematics and Computer Science, 20(2), 233–254. https://doi.org/10.5485/TMCS.2022.0546 Kürschák, J. (1929). Matematikai versenytételek [Mathematical competitions]. Privat edition. Kürschák, J., Hajós, G., Surányi, J., Neukomm, G., & Rapaport, E. (1963). Hungarian problem book: 1894–1905. Random House. Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Sense Publishers. https://doi.org/10.1163/9789087909352_010 Leikin, R., & Elgrably, H. (2020). Problem posing through investigations for the development and evaluation of proof-related skills and creativity skills of prospective high school mathematics teachers. International Journal of Educational Research, 102, 101424. https://doi.org/10.1016/ j.ijer.2019.04.002 Mason, J. (2015). Responding in-the-moment: Learning to prepare for the unexpected. Research in Mathematics Education, 17(2), 110–127. https://doi.org/10.1080/14794802.2015.1031272 Ministry of Human Resources. (2013). 8/2013. (I.~30.) EMMI rendelet a tanári felkészítés közös követelményeir˝ol és az egyes tanárszakok képzési és kimeneti követelményeir˝ol [Decree No 8/ 2013 (I.~30.) of the Ministry of Human Resources on the common requirements for teacher preparation and on the training and outcome requirements for certain teacher specialisations]. Molnár, E. (1983). Matematikai versenyfeladatok gy˝ujteménye (1947–1970) [Collection of mathematical competition problems (1947–1970)]. Tankönyvkiadó. Niss, M., & Jablonka, E. (2014). Mathematical literacy. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 391–396). Springer. https://doi.org/10.1007/978-94-007-4978-8_ 100 Osana, H. P., & Pelczer, I. (2015). A review on problem posing in teacher education. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing (pp. 469–492). Springer. Papadopoulos, I., Patsiala, N., Baumanns, L., & Rott, B. (2021). Multiple approaches to problem posing: Theoretical considerations regarding its definition, conceptualisation, and implementation. Center for Educational Policy Studies Journal, 12, 13–34. https://doi.org/10.26529/cep sj.878 Pehkonen, E. (1997). Introduction to the concept “open-ended problem.” In E. Pehkonen (Ed.), Use of open-ended problems in mathematics classroom (pp. 7–11). University of Helsinki. Polya, G. (1945). How to solve it. Princeton University Press. Radnainé Szendrei, J. (1988). Szakközépiskolai versenyek matematikafeladatai mindenkinek [Mathematics problems for all in vocational secondary school competitions]. Tankönyvkiadó. Schoenfeld, A. H. (2014). Mathematical problem solving. Elsevier Science. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. 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. https://doi.org/ 10.1007/s10649-013-9478-2 Skovsmose, O. (2001). Landscapes of investigation. ZDM—International Journal on Mathematics Education, 33(4), 123–132. https://doi.org/10.1007/BF02652747 Stoyanova, E., & Ellerton, N. (1996). A framework for research into students’ problem posing in school mathematics. Technology in Mathematics Education.

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Tabach, M., & Friedlander, A. (2015). Problem posing as a learnable activity. Didactica Mathematicae, 37, 93–110. https://doi.org/10.14708/dm.v37i0.844 Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143. https://doi.org/10.1007/ s10649-012-9455-1 Voica, C., & Pelczer, I. (2010). Problem posing by novice and experts: Comparison between students and teachers. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 121–130). Institut National de Recherche Pédagogique. www.inrp.fr/editions/cerme6

Chapter 12

Regulation of Cognition During Problem Posing: A Case Study Puay Huat Chua

Abstract The regulatory phases of cognition during problem posing are being presented through a case study of a grade 9 student, Tan, engaged in a geometric problem-posing task. The present study follows Schoenfeld’s notion of regulation of cognition as it is used in his episode-based framework for analysis of problem-solving protocols. The study paints the different phases in the regulation of cognition during problem posing, namely, property noticing, problem construction, checking solution, and looking back. The looking back phase is not strongly exhibited. Discussion of these phase descriptors in classroom problem-posing instructions are also made. Keyword Problem posing · Problem solving

1 Problem Posing and Problem Solving Although the field of problem posing in mathematics education research has received active research inquiry only in the “last two decades or so” (Silver & Yankson, 2017), several authors have pointed to the importance of students’ mathematical problem posing. Much of the work is linked to students’ exploration in mathematics (Cai et al., 2015a; Cifarelli & Sevim, 2015; Cai, 2003) and to the teaching and learning of mathematics (Crespo, 2003; English, 1997; NCTM, 2000). Bransford et al. (1996) opined that to develop mathematical thinking needed to solve complex real world problems, it is important for students to be able to generate and to formulate their own problems. Researchers like Brown and Walter (1993) highlighted that problem-posing activities within mathematics lessons can help in reducing mathematics anxiety, in surfacing misconceptions and in promoting group learning and that “we learn mathematics when we were actively engaged in creating not only the solution strategies but the problem that demand them” (p. 187). There are also links between doing problem-posing tasks and students’ being able to make connections and make sense of mathematics (Carrillo & Cruz, 2015). P. H. Chua (B) National Institute of Education, Nanyang Technological University, Singapore, Singapore e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_12

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Hansen and Hana (2015) also argued that problem posing can give students the much needed ownership of their learning environment, since it is a natural component of inquiry-orientation and is grounded in the “belief of giving priority to the question over the answer.” Elsewhere, Cai et al. (2015b) found that problem-posing activities can promote students’ conceptual understanding, foster their mathematical communication and capture their interest and curiosity. In a classroom milieu, many of the students’ mathematics experiences are solutiondriven. The problem-solving process often ceases when a solution is reached although much can be gained, for example, by making the arrived solution as a starting point for further mathematics exploration. Problem posing as a classroom activity can therefore offer a platform for students to move away from the fixation on problem solving where thinking is chained by prior knowledge and by set ways of seeing things. Besides, the posed problems can also show how much students understand a mathematical topic because problem posing cannot be done without a context. If learning mathematics is taken to involve creating meaning, then the ability to pose problems is an essential skill for creating that meaning to the learner. Such a problem-posing classroom approach would have its place in supporting the classroom enactment process within The Singapore Teaching Practice framework, which is a model that “makes explicit how effective teaching and learning is achieved in Singapore schools.”

2 Research Background In Singapore, schools are expected to implement education policies set by the Ministry of Education (MOE) with some degree of customization and innovation at the local level (Ho, et al., 2022). In the MOE document, Mathematics Syllabus (2012), students are encouraged to “connect ideas within mathematics (p. 8).” The Singapore Mathematics Curriculum Framework (Mathematics Syllabus, 2012, p. 14) also places importance on the need for thinking skills and metacognition in mathematical problem solving. Such skills can be engendered through engaging students in problem-posing activities since posing a problem requires them not just to have the necessary concepts but also the ability to link these coherently together to form a problem. Students’ responses to problem-posing tasks could provide a window through which to view students’ ability to make connections within mathematics and a mirror that reflects the content and the character of their school mathematics experience. The importance of problem posing as a mathematical activity that could promote engaged learning provides the main impetus to the present study. But there are few local studies in Singapore on mathematical problem posing, and even fewer studies that look at promoting student problem posing in the classroom (Chua, 2011). For the present study, Silver’s (1994) notion of problem posing as “the creation of a new problem from a mathematical situation or experience” will be used. The status of problem posing in the literature of mathematics education research have been increasingly recognized and acknowledged, specifically for its potential influence on

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problem-solving work. Many researchers (Christou et al., 2005a, 2005b; Weber & Leikin, 2016) pointed to a complementary relationship between problem posing and problem solving. Although much work has been done in investigating the metacognitive phases during problem solving (Schoenfeld, 1992; Silver & Marshall, 1990), less work has been done on such processes during problem posing. The emerging significance of problem posing as a mathematical activity that could promote engaged learning provides the main impetus to the present study.

3 Research Framework The interpretative framework of mathematics problem posing shown in Fig. 1 synthesizes most of the ideas espoused in previous studies and includes a tentative classification of the underlying processes (Chua, 2011). The context of the framework is on students posing their own problems to a given stimulus. Consistent with Kontorovich and Koichu’s (2009) four-facet framework to describe problem posing, the interpretative framework draws on Schoenfeld’s (1985a) model comprising categories for understanding problem solving. Various research also suggests the close relationship between the processes of problem solving and problem posing (Mamona-Downs & Downs, 2005; Polya, 1971; Silver, 1995). The framework also draws on the notion of recursion in the Pirie-Kieren model of how students develop mathematical understanding (Pirie & Kieren, 1994). The interpretative framework characterizes much of the actions and behaviours during problem posing. However, it is not intended to depict any sequencing of problem-posing actions. Strategizing the problem formulation involves drawing topics (from the poser’s resources) to be used for posing the problem. Christou et al. (2005a) suggested that the posing may involve processes like association, analogy (Kilpatrick, 1987), editing, selecting, comprehending and translating of quantitative information. The poser starts by sieving the key components of a given mathematical stimulus, and explores how the inter-related components can be linked to the objective of formulating the problem. For example, given the stem “2x 2 + x − 1” to pose a problem, the poser may draw on the related quadratic concepts for inclusion into the problem formulation. In problem posing, students have to decide which of the related concepts should be included in the building of the initial state and the goal state for the emerging problem. This strategic action affects the formulation of the problem. Setting of initial and goal states involves creating the context for the emerging problem. Students’ knowledge of problem-solving heuristics may influence the setting of these states because they may think about the solutions as they pose their problems (Cai, 2003; Lowrie, 2002). Students could use problem-posing strategies such as “what-if-not” (Brown & Walter, 1993), or formulate problems that involve maximum or minimum conditions. The problem-posing actions could be seen in the example:

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Schoenfeld ProblemSolving Framework (1985a) 1. Resources – knowledge that solver was capable of using “deeply intertwined”

2. Heuristics – the know how, rules of thumb 3. Control – of cognition, resource management and allocation 4. Beliefs influence managerial actions

Problem-Posing Actions

Strategic Formulation– draw resources through analogy, association (Kilpatrick, 1987), use of edit, select, comprehend, translate quantitative data (Christou, et al., 2005), ‘fold back’ (Pirie & Kieren, 1994). Setting Initial and Goal States - context building, link topics, solve emerging problem (Cai, 2003), check solution paths, use strategies, e.g., max/min, extend, modify, generalize, what-if-not (Brown & Walter, 1993).

Recursion between States (Pirie, 2002) – work back, check solution, make changes, consider sense-making. Problem Presentation - scaffold, use of a diagram, direct / indirect formulation.

Regulation of Cognition – episode-based framework (Schoenfeld, 1985a) property noticing, problem construction, checking solution, looking back. (heuristics influences posing as there are problem-solving acts within posing)

Fig. 1 Towards a conceptual framework of problem posing (Chua, 2011)

Mary has 20 coins that she had to put into two boxes with one of them having to contain at least six coins. [initial state] What is the maximum number of coins that the other box can contain?” [goal state arrived at by using maximum/ minimum strategy]. (Chua, 2011)

Recursion between states is about the switching between the initial state and the goal state to validate the emerging problem and to check whether the states are consistent with the solution path. Through planning and analysis, successive refinements are made to the emerging problem, for example, by checking if the posed problem makes sense. Proulx and Maheux (2017) argued that the posing of a problem by itself is not static because the “posing triggers a solving process that in turn transforms the initial posing” (p. 163). Chua (2011) noted that recursion in problem posing involves “a complex process of folding back between prior knowledge in the poser’s resources and the checking on the emerging problem.” For example, with the initial state as “given that 2x 2 + x – 1 = 0” and a tentative goal state as “find x”, the poser may fold back on his or her prior knowledge about the properties of quadratic graphs and the quadratic discriminant to modify the goal state which could appear as “by sketching an appropriate graph, find the number of real roots.” Problem presentation is about creating the final form of the posed problem. The final form of the posed problem can be a direct or an indirect problem, or a problem containing various degrees of scaffolding. This can be done, for example, by the inclusion of a diagram in the original problem to provide contextual support. The problem

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find the number of real roots in 2x 2 + x − 1 = 0

is a direct problem since the discriminant is positive. Although the problem if the quadratic px2 + x − 1 = 0 has real roots, what is the minimum value of p?

could be considered indirect since p is an unknown to be computed and is constrained by having to be a minimum (Chua, 2011). Studying problem posing purely as a cognitive process would not be complete given the complexities involved in its processes. One needs to study how the cognitive process is being regulated, as argued by Garofalo and Lester (1985): “successful cognitive performance depended on having not only adequate knowledge but also sufficient awareness and control of that knowledge” (p. 163).

3.1 Regulation of Cognition In problem posing, the initial and goal states have to be created and the accompanying solution paths have to be considered. Far from being a straight-forward linear process, it involves recursion between the two states, as the poser checks for coherence as the solution path is being worked on. The knowledge of the mathematical ability of the intended solver is also another consideration in the process of problem posing. Because the problem poser needs to keep track of what he or she is doing and thinking, the poser requires a control of the cognitive processes. Described simply, metacognition is “thinking about thinking” (Livingston, 1997). In mathematics education research, the use of the term metacognition does not have a common definition. In the context of acquisition and of application of learning skills, researchers like Sperling et al. (2004) and Schraw and Moshman (1995) pointed to two of the metacognitive components that are consistent with Flavell’s (1981) original notion of metacognition. The component on knowledge about cognition points to one’s level of understanding of one’s own memories, of the cognitive system and of how one learns. The other component on the regulation of cognition is about how one could regulate one’s own approach to learning, including goal setting, choosing and applying strategies and monitoring of one’s actions. Georghiades (2004) however noted that different researchers provided different definitions, portraying different focuses on the processes and the mechanisms associated with metacognition. This led to the development of different instruments to study the construct. Some instruments that had been developed served a more general purpose, like studying metacognition in reading comprehension which are not suitable for use in this study. Schoenfeld (1985a) specifically pointed out that “the techniques of the psychological community for exploring metacognition, while useful, would prove far too limited for the purposes of mathematics education” (p. 379). He suggested the need for a variety of techniques for the analysis of problem-solving protocols such as clinical interviews.

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Problem-posing studies can draw from Schoenfeld’s (1985b) notion of “control” which is about resource allocation during problem-solving performance and which he pointed as being a “major determinant of the problem-solving outcome” (p. 143). Just like in problem solving, where “it is not only what you know but how you use it that matters” (Schoenfeld, 1987, p. 192), decision making in problem posing comes about from drawing from the poser’s prior knowledge and understanding the context of the proper use of the knowledge to formulate the problem. Schoenfeld’s (1985a) “control” of cognition in problem solving refers to “global decisions” regarding the selection and implementation of resources and strategies. Processes include planning, monitoring and assessment. The present study follows Schoenfeld’s notion of regulation of cognition as it is used in his episode-based framework for analysis of problem-solving protocols. His framework which focuses on decision-making behaviour during problem solving involves the parsing of verbal protocols into episodes which he described as “periods of time during which the problem solver was engaged in either one large task or in closely related body of tasks in the service of the same goal” (p. 316). Schoenfeld (1985a) pointed to five episodes during problem solving (read, analyze, explore, plan / implement and verify). The solver’s decisions at the transition points between episodes could have implications on the outcomes of the solution attempts. According to Schoenfeld (1985a), reading the problem could be overt or silent. In analysis, the solver attempts to understand the problem by identifying task-specific knowledge, including familiar problems and being cognizant about the initial conditions and goal state. In exploration, as the need dictates, the solver makes a decision on the progress of the solution path, including working on less structured ways of solving. Planning/implementation involves making selection of solution steps and strategies and checking of the follow through in the planned course of action. Verification involves checking the solution for sense making and meeting the goal state. Given the complexity of problem solving, the five phases may not occur in a linear manner. The framework specifies the characteristics of each episode which could be compared with students’ observed problem-solving behaviour.

3.2 Phases in Problem Posing A summary of the descriptions about the phases in problem solving and in problem posing is shown in Table 1. Yimer and Ellerton (2006) noted that numerous metacognitive frameworks are “minor variations” of Polya’s (1971) four-stage model. The proposed problem-posing phase framework also takes reference from earlier work from Polya’s (1971) framework on problem solving and also pulls descriptions from Schoenfeld’s episode-based model to describe the problem-posing phases. The resulting framework is useful in putting the problem-posing phases into the contexts in which they occurred. In the Property Noticing phase, a problem poser has to decide on what to use from his or her knowledge to set the initial and goal states. Under the Problem

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Table 1 Regulation of cognition in problem posing and in problem solving (Chua, 2011) Problem-posing Brief description (with phase framework Schoenfeld’s terms)

Polya’s Brief description problem-solving framework (1971)

Property noticing Decide on what to use from resources to set initial and goal states. [Schoenfeld’s reading, exploration episodes]

Understand the problem

Include understanding initial and goal states, representing the problem. [Lester’s (1985) ‘orientation’ category]

Problem construction

Create context and use posing Devise a plan strategies to link states. [Schoenfeld’s planning-implementation episode]

Use problem-solving strategies and processes. [Lester’s (1985) ‘organization’ category]

Checking solution

Check on solution path of emerging problem, modifying states if necessary. [Schoenfeld’s episodes of exploration, implementation, analysis]

Work on the plan—check if it works, otherwise go back to first step. [Lester’s (1985) ‘execution’ phase]

Looking back

Reflect on how posed problem can Look back be done differently, reflect on quality of posed problem and confidence

Carry out the plan

Reflect on solution, any alternatives. [Lester’s (1985) ‘verification’ category]

Construction phase, the problem poser has to build context and use posing strategies to link the two states. During Checking Solution, a problem poser after finding that the solution path is not compatible with the initial and goal states, may have to revert back to the Property Noticing phase. The problem poser then retrieves other suitable knowledge or creates new connections to continue with the posing. The problem poser may have to make recursions between the states as he or she checks and makes changes to the emerging problem. The problem poser may reflect on the validity of the problem and again may recursively make changes during the phase on Looking Back (Chua, 2011). A series of related problem-posing actions in the conceptual framework for problem posing may be linked to the problem-posing phases. For example, during the Property Noticing and Problem Construction phases, the problem poser may have to

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draw upon his or her prior knowledge through analogy and association to start the strategic formulation of the problem and to set the initial and goal states. During problem posing, students have to consider the mathematics involved in relating the initial and the goal states of a problem and a possible solution path linking the states. Fluency in problem representation involves posed problems that are, for example, free from ambiguity, contradiction, over-conditioning (extraneous information), and implicit assumptions. That a student when posing a problem must consider the ability of the solver means problem posing requires reflection and planning. Such reflection involves the use of cognitive skills and the regulation of cognition. Drawing from the extant work on mathematical problem solving by Polya (1971), Kilpatrick (1987), Schoenfeld (1985a, 1985b), and Pirie and Kieren’s (1994) model for the growth of mathematical understanding, Chua’s (2011) model on the regulation of cognition during problem posing, posited the phases of property noticing, problem construction, checking solution, and looking back. In posing a problem, property noticing is a distinct phase when students check on possible related topics to set the context to build the problem. In the problem construction phase, students engage resources to build up the initial and the goal states. They create the context and use posing strategies to link the states by checking on possible solution paths, recursively checking and making changes if necessary to the states as they evaluate the emerging problem. Adaptive planning is then made if necessary to the initial plan. Students present the problem and then decide if scaffolding is needed to facilitate its solving. In checking solution, students check the solution of the emerging problem and may modify the initial and the goal states. The looking back involves students reflecting on how the posed problem can be done differently. In his investigation of the phases of the regulation of cognition during problem posing, Chua (2011) conducted four case studies of grade 9 students from two secondary schools in Singapore when they were working individually on a geometric problem-posing task. Students were novice problem posers. This paper reports one of the case studies about a grade 9 female student, called Tan.

4 Research Design A case study approach was used to study the problem-posing processes of Tan, as she worked on the task. Tan was first asked to practice the think-aloud protocol while working on a multiplication task. The researcher then administered the problemposing task. The act of these on-going commentaries by Tan enabled the researcher to see at first-hand, the process of posing through to the forming of the final product. A post-task interview was conducted immediately after she had completed the task to have a better understanding on the problem-posing moves.

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4.1 Case Study Tan was a vivacious female student and whom was being described by her teacher as an average performer in mathematics. She believed that some people were naturally good in mathematics, and that she was not. In her view, mathematics “should be about working on a set of rules to get to the correct answer in solving a mathematics problem …” and that mathematics problems (presented in classroom by teachers) should all be solvable because from her experience, there had always been an algorithm to solve them. This very instrumental view about mathematics may have greatly influenced the way she posed some of the problems. Tan was tasked to write a problem for her friends based on this problem stem, and to provide a solution to her posed problem: In triangle ABC, angle CAB is 60°, AB is 4 cm and AC is 3 cm.

The task was chosen to engage posers on their awareness of their knowledge of trigonometry and it was within the capacity of a grade 9 student to complete the task. The geometric nature of the task was targeted to elicit a variety of possible problems and associated posing strategies.

5 Analysis Tan started by changing the given length of AC to 12 cm as shown in Fig. 2. This over-conditioning of triangle ABC immediately caused an inconsistency which she was totally unaware. There was also an error in writing “CF” instead of “EF” in her posed problem. Her overly focus on the procedural solution was reflected the use of the property of similar triangles as shown in Fig. 3.

5.1 Property Noticing Tan began with a long preamble about the possibilities from the given conditions. She started by noticing the unequal lengths of the given triangle and proceeded to Fig. 2 Tan’s modified triangle and additional diagram

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Fig. 3 Tan’s posed problems and her proposed solutions

explore various topics that could be used to formulate the problem, for example, using the notion of a triangle enclosed within a circle, or about the area of triangle in a semi-circle, and about congruency and similarity. Her association of topics extended to the use of simultaneous equations and algebraic representation and manipulation: may be I could add length of BC like x + 4, and then x as the unknown.

In building a context to the problem, she was mindful of the audience to which the problem was intended, as shown in her verbalization: “my friends study this [topics] quite well, may well practice it”, and she “could add in other unknown chapters [topics]” making the problem “more confusing.” It was clear that she was situating the problem in the context of her understanding of her friends’ ability to solve the problem. Although her foray into possibilities were extensive, she did not monitor nor evaluate these probable topics for use in the problem formulation. She finally settled for the use of two similar triangles to start the posing. She explained that the choice was made because the topic was just recently taught. This recency effect on topic selection perhaps influenced the way she developed the problem.

5.2 Problem Construction She started by setting the goal states and worked on solution path, often moving retrospectively: “start with a diagram and then add on the details” approach. Tan was moving recursively between posing and checking on a solution path at the same time: I can start writing BC [reference to EF] as 1.5 cm. DE and AB similar, so change length to 2 cm, so now the 2 triangles are similar in a way that their shapes are the same but their lengths are totally different. I can ask them to find the ratio of BC to EF [first goal state] and then I will start solving … this is supposed to be an easy question…so BC is 3 cm, EF is 1.5.

Her second goal state was to find the unknown DF, labelled, x: “so I treat DF as the unknown.” This was followed by her specification of the intended solution path: “and I will state that they should not use the Pythagoras theorem.” She was conscious of the potential solver, and considered the difficulty the problem may present. In the retrospective interview, she alluded that her specification of not using the Pythagoras theorem was a convenient means to make the problem appears “easier” because she needed.

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to ask them to find all those stuff about similarities, so I think I will just make their life easier.

This was another example of her awareness of the problem-solving ability of her friends. But the triangle DEF constructed was inconsistent, and that made the problem implausible. There was also evidence of the use of a planning-implementation sequence. She explicitly verbalized the plan to use similar triangle to develop the goal state to find the ratio of BC to EF: So if I solve it, 3 divided by 1.5…so 2 will be equal to 12 over x...so I bring it, cross multiply so 2x equal 12… so I will bring the 2 over and then I will get 6, so this is supposed to be the solution.

5.3 Checking Solution Evidence of a partial review of Tan’s posed problem could be found in her reflection that the angles ACB and DFE should be the same. Her confidence in the posed problem was also reflected later in the retrospective interview: no need to check [the solution] they [friends] will say that it is easy. I know that there is no problem with the question.

It is a case of a misplaced confidence, i.e., confidence through her making surfacelevel checks, specifically her checking on the rationalization of the solution involving use of similar triangle, but being unaware of the inconsistencies that she created in the two triangles.

5.4 Looking Back Reflecting from the solver’s perspective, Tan articulated a possible reaction from her friends: they could be too confident and so they may write the sign wrongly like, for congruency…you have to write the equal sign and the wavelike line on the top…

Her focus on the surface level of the anticipated solution reflected her concern about the need for proper written notation. Little was reflected on the nature of the posed problem in looking back, specifically on the validity of the posed problems. This appeared to be a weak phase in her problem posing. Her very targeted approach to working towards a solution without checking on the validity of the posed problem, and her specific concerns about the surface presentation of the solution could perhaps reflect her instrumental view about mathematics and learning mathematics.

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6 Concluding Remarks The present findings offer a glimpse of the regulation of cognitive process during problem posing. The explication of the phases provides useful descriptors for researchers who are examining the stages of the students’ problem formulation. Such descriptions may be helpful for teachers who are engaging students in classroom problem-posing exercises. In planning, it would be instrumental to study the extent which students’ pre-existing domain knowledge (Property Noticing) shapes the posing of problems. Such findings could shed light on how students are able to draw from the specific domain knowledge in posing problems. Given a problemposing task, the teacher could demonstrate to students how to set the initial and the goal states to start the problem-posing process, for example using Brown and Walter’s (1983) “what-if-not” technique. Sensitizing students on what could occur during the property noticing phase and the problem construction phase could also help students appreciate how the emerging problem structures could be linked to the solution path. Through the posed problems, the teacher could check students’ ability to construct meaningful context that involves, for example, quadratics and their competency with the use of the quadratic equation formula. In this case, problem posing could be used as a diagnostic tool to gauge students’ learning. Examining the type of problems individuals posed for themselves and for others to solve may therefore shed important insights into their mathematics development and help guide future instructions. But because teachers are at the “heart of implementing any educational innovation or improvement,” therefore further work has to look into how they use classroom problem posing to teach mathematics (Cai & Hwang, 2020). Since teachers who do not feel comfortable about problem posing would unlikely involve students in problem posing, the learning from this study can also inform teacher professional development in problem posing. The study suggests that students’ problem-posing actions like setting initial and goal states, and strategic formulation may be at play during the problem-posing phases. Further studies may unravel the close interplay of these actions during these phases in problem posing. Because problem-posing tasks have to be situated within a context, future studies may also investigate to what extent are these problem-posing actions task-specific, and whether they may differ across different mathematical domains. Given the close relation between problem solving and problem posing, findings from these studies may shed light on the types of students’ problem-posing strategies beyond the what-if-not strategy, and in that process, support instructions on problem-posing within mathematics lessons. Just like being proficient in solving problems, posing good problems requires effective metacognition (A. H. Schoenfeld, personal communication, February 10, 2006). Explicit discussion with students on problem posing may bring about metacognitive experiences, just as learning problem solving would. For example, with the teacher role-modeling the problem-posing metacognitive processes, students can pick ideas about the components of a problem, then use that knowledge in formulating problems. In particular, the explicit classroom demonstrations of regulatory factors

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of checking and of looking back during problem posing could engender such good habits among the students. This may have a good ‘translation’ effect in improving students’ problem solving. Mathematical problem posing can be learned just like problem solving. There are various problem-posing strategies and pedagogies advocated by researchers to improve students’ learning in the mathematics classroom. For example, Manouchehri (2001) developed the “Four Point Instructional Model” to promote “genuine mathematical inquiry” in a grade 6 classroom through problem posing. Such interventions can create a classroom environment amenable to problem-posing activities, and in that process also encourage students’ explicit reflection on their own thinking processes. With the push for more subject discipline classroom engagement in The Singapore Teaching Practice, there is increased emphasis in getting students to go beyond just problem solving and having the sense of inquiry in the real-world context which can be facilitated by problem posing. Given the importance of problem posing as an emerging field of study in mathematics education research, the findings of this exploratory study have the potential to add to the body of local knowledge about how problem-posing instructions can be engendered in the classroom to bring about deeper classroom engagement in mathematics. Specifically, the study underlines the importance of planned approaches for the use of problem-posing activities in the classroom. The exploratory study can contribute to the local metacognitive knowledge of how students do problem-posing. This is timely as much of our mathematics learning in the classroom is solution driven. It also points to the need to further explore the very close relationships between problem solving and problem posing, specifically in the regulation of cognition in both processes. Beyond this study, cross national comparison studies involving students of different grade levels, would be useful to deepen our understanding of the regulation of cognition during problem posing.

References Bransford, J. D., Zech, L., Schwartz, D., Barron, B., & Vye, N. (1996). Fostering mathematical thinking in middle school students: Lessons from research. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 285–302). Lawrence Erlbaum. Brown, S. I., & Walter, M. I. (1983). The art of problem posing. The Franklin Institute Press. Brown, S. I., & Walter, M. I. (Eds.). (1993). Problem posing: Reflections and applications. Lawrence Erlbaum. Cai, J. F. (2003). Singaporean students’ mathematical thinking in problem solving and problem posing: An exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719–737. Cai, J. F., & Hwang, S. (2020). Learning to teach through mathematical problem posing: theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.01.001

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Cai, J. F., Hwang, S., Jiang, C., & Silber, S. (2015b). Problem-posing research in mathematics education: Some answered and unanswered questions. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing (pp. 3–34). Springer. Cai, J. F., Jiang, C., Hwang, S., Nie, B., & Hu, D. (2015a). How do textbooks incorporate mathematical problem posing? An international comparative study. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems (pp. 4–22). Springer. Carrillo, J., & Cruz, J. (2015). Problem-posing and questioning: Two tools to help solve problems. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems (pp. 23–36). Springer. Christou, C., Mousoulides, N., Pittalis, M., & Pantazi, D. P. (2005a). Problem solving and problem posing in a dynamic geometry environment. The Montana Mathematics Enthusiast, 2(2), 125– 143. Christou, C., Mousoulides, N., Pittalis, M., Pantazi, D. P., & Sriraman, B. (2005a). An empirical taxonomy of problem posing processes. Zentralblatt fur Didaktik der Mathematik, 37(3). Retrieved from http://www.umt.edu/math/reports/sriraman/Int_Reviews_Preprint_Cyprus_Sri raman.pdf Chua, P. H. (2011). Characteristics of problem posing of grade 9 students on geometric tasks (Doctoral dissertation, National Institute of Education, Nanyang Technological University, Singapore). Retrieved from https://repository.nie.edu.sg/handle/10497/4500 Cifarelli, V. V., & Sevim, V. (2015). Problem posing as reformulation and sensemaking within problem solving. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing (pp. 177–194). Springer. Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52, 243–270. English, L. D. (1997). The development of fifth-grade children’s problem posing abilities. Educational Studies in Mathematics, 34, 183–217. Flavell, J. H. (1981). Cognitive monitoring. In W. P. Dickson (Ed.), Children’s oral communication skills (pp. 35–60). Academic Press. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176. Georghiades, P. (2004). From the general to the situated: Three decades of metacognition. International Journal of Science Education, 26(3), 365–383. Hansen, R., & Hana, G. M. (2015). Problem posing from a modelling perspective. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing (pp. 35–46). Springer. Ho, J., Hung, D., Chua, P. H., & Munir N. B. (2022). Integrating distributed with ecological leadership: Through the lens of activity theory. Educational Management Administration and Leadership. February 2022. https://doi.org/10.1177/17411432221077156 Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Lawrence Erlbaum. Kontorovich, I., & Koichu, B. (2009). Towards a comprehensive framework of mathematical problem posing. In M. Tzekaki, M. Kaldrimidou & C. Sakonidis (Eds.), Proceedings of the 33rd PME, Vol 3 (pp. 401–408). Psychology of Mathematics Education. Lester, F. K. (1985). Methodical consideration in research on mathematical problem solving instructions. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving—Multiple research perspectives (pp. 55–69). Lawrence Erlbuam. Livingston, J. A. (1997). Metacognition: An overview. Retrieved from Livingston—Metacognition—An Overview | PDF | Metacognition | Cognition (scribd.com). Lowrie, T. (2002). Young children posing problems: The influence of teacher intervention on the types of problems children pose. Mathematics Education Research Journal, 14(2), 87–98. Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. Journal of Mathematical Behavior, 24, 385–401.

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Manouchehri, A. (2001). A four-point instructional mode. Teaching Children Mathematics, 8, 180– 186. Mathematics Syllabus (2012). Retrieved from mathematics_syllabus_primary_1_to_6.pdf (moe.gov.sg). National Council of Teachers of Mathematics. (2000). Professional standards for teaching mathematics. NCTM. Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 165–190. Polya, G. (1971). How to solve it (2nd ed.). Princeton University Press. Proulx, J., & Maheux, J. F. (2017). From problem solving to problem posing, and from strategies to laying down a path in solving: Taking Varela’s ideas to mathematics education research. Constructivist Foundations, 13(1), 160–167. Schoenfeld, A. H. (1985a). Mathematical problem solving. Academic Press. Schoenfeld, A. H. (1985b). Metacognitive and epistemological issues in mathematical understanding. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving—Multiple research perspectives (pp. 361–379). Lawrence Erlbaum. Schoenfeld, A. H. (1987). What’s all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–216). Lawrence Erlbaum. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). Macmillan. Schraw, G., & Moshman, D. (1995). Metacognitive theories. Educational Psychology Review, 7(4), 351–371. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. Silver, E. A. (1995). The nature and use of open problems in mathematics education: Mathematical and methodological perspectives. International Review on Mathematics Education, 27, 67–72. Silver, E. A., & Marshall, S. P. (1990). Mathematical and scientific problem solving: Findings, issues and instructional implications. In B. F. Jones & L. Idol (Eds.), Dimensions of thinking and cognitive instruction (pp. 265–290). Lawrence Erlbaum. Silver, E. A., & Yankson, K. (2017). Roots and sprouts: Cultivating research on mathematical problem posing. Journal for Research in Mathematics Education, 48(1), 111–115. Sperling, R., Howard, B., & Staley, R. (2004). Metacognition and self-regulated learning constructs. Educational Research and Evaluation, 10(2), 117–139. The Singapore Teaching Practice. Retrieve from AST | The Singapore Teaching Practice (moe.edu.sg) Weber, K., & Leikin, R. (2016). Recent advances in research on problem solving and problem posing. In Á. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 353–382). Sense Publishers. Yimer, A., & Ellerton, N. F. (2006). Cognitive and metacognitive aspects of mathematical problem solving: An emerging model. In P. Grootenboer, R. Zevenbergen & M. Chinnappan (Eds.), Identities, cultures, and learning spaces, Proceedings of the 29th Annual Conference of the Mathematics Education Research Group of Australasia, Canberra (pp. 575–582). MERGA.

Chapter 13

Problem Posing in Pósa Problem Threads Lajos Pósa, Péter Juhász, Ryota Matsuura, and Réka Szász

Abstract Problem posing is considered as a crucial element of learning and doing mathematics, and there is a constant quest for how this can be taught effectively. The Pósa method is a Hungarian instructional method of mathematical guided discovery through specifically designed problem threads, where problem posing plays an important role. The chapter presents how this method encourages students to pose problems, and hence create their own problem threads. To illustrate this pedagogical approach, to delineate various problem-posing heuristics, and to describe how a teacher guides his middle school students to engage in problem posing, we describe a sample problem thread involving geometric transformations. Keywords Problem posing · Pósa method · Guided discovery · Problem thread

1 Introduction There has been a growing interest in teaching students to pose problems in mathematics classrooms. Problem posing enhances students’ problem solving (Cai et al., 2013; Silver & Cai, 1996), creativity (Silver et al., 1990; Van Harpen & Sriraman, 2013; Voica & Singer, 2013), self-confidence (Mason, 2010), and attitude towards mathematics (Headrick et al., 2020; Winograd, 1991). Moreover, the experience of posing problems fosters the development of students’ conceptual understanding and their mathematical reasoning skills (Silver, 1994). The value of problem posing for L. Pósa · P. Juhász (B) Alfréd Rényi Institute of Mathematics, 1053 Reáltanoda u. 15-13, Budapest, Hungary e-mail: [email protected] R. Matsuura St. Olaf College and Budapest Semesters in Mathematics Education, 1520 St. Olaf Ave, Northfield, MN 55057, USA e-mail: [email protected] R. Szász Budapest Semesters in Mathematics Education, 1520 St. Olaf Ave, Northfield, MN 55057, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_13

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students is also reflected in standards. The National Council of Teachers of Mathematics (2000) recommends that “[p]osing problems, that is, generating new questions in a problem context, is a mathematical disposition that teachers should nurture and develop” in students (p. 117). Problem posing is a central element in the tradition of Hungarian guided discovery (Connelly, 2010), which has earned international acclaim through the work of Pólya (1945). The core philosophy of the Hungarian tradition is that students, regardless of their age, should do mathematics as real mathematicians (Gosztonyi, 2016). Problem posing is inseparable from problem solving (Kilpatrick, 1987), hence it is an integral part of the work of mathematicians. In the classroom, problem posing activities can provide an authentic learning context for students (Singer et al., 2013). The Pósa method, named after the first author, is a recent branch of the Hungarian pedagogical tradition (Juhász, 2019), with unique ways of involving students in problem posing. Despite its prominent role in the work of doing mathematics, problem posing and its role in learning and teaching has been an understudied topic by the mathematics education community. Singer et al. (2013) advocate for further research to understand and support advantages provided by problem-posing activities in the classroom. Researchers (e.g., Silver et al., 1996) have studied the strategies and cognitive processes used by teachers when posing problems. This chapter builds on such work, but places the focus on students and their thinking. Through classroom episodes, we will illustrate how a Hungarian teacher engages his middle school students in a problem-posing activity. Following in the tradition of Pólya, we describe various problem-posing heuristics, as well as the ways in which the teacher can guide students to develop and utilize those heuristics.

2 Background: Problem Threads and Problem Posing 2.1 Problem Threads The Pósa method stems from the tradition of Tamás Varga (Gosztonyi, 2016) in the Hungarian style guided discovery approach of teaching mathematics. In many Hungarian classrooms, there is an explicit emphasis on problem solving and the excitement of discovery. The teacher prepares a set of problems that guide the students to the lesson’s goal (Stockton, 2010). The Pósa method, in particular, is structured around problem threads (Juhász, 2019), which refer to a series of tasks with a specific structure and purpose, that provide students with opportunities to struggle productively towards understanding. In structure, the tasks are carefully sequenced; they build on each other like building blocks, gradually guiding students to learn a particular mathematical idea. In order to make the next step less apparent for students, sequential tasks in a problem thread are not assigned in succession, but with some time apart. In a lesson, students will work on tasks from different problem threads, so that the themes in multiple problem threads are developed simultaneously. In purpose,

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the tasks serve as vehicles for fostering students’ problem solving, problem posing, reasoning, and communication skills. Tasks can take on many different forms: a mathematical problem to be solved, the process of developing a definition, an experiment, or a game, just to name a few. The overall aim is to help students learn to think like mathematicians (Juhász, 2019). Such problem threads were originally designed and used in weekend camps for gifted students (Gordon Gy˝ori & Juhász, 2017), but were later adapted to the mainstream curriculum.

2.2 Question Posing Heuristics Barabé and Proulx (2015) identify ‘generating problems from specific constraints’ as a major category of problem posing in the mathematics classroom, where a subcategory is ‘generate from a previously solved problem’: after having solved a specific problem, learners are asked to create other problems based on this solved problem. A typical example of this is Pólya’s (1945) ‘Looking back’ heuristic (making analogies, modifying, generalizing, specifying and studying variations). Similarly, in the Pósa method, problem posing usually emerges as a follow-up activity to a previously solved problem. Proponents of inquiry-based methods argue that fostering students’ problem solving skills is more essential for student learning than teaching them procedures (Maaβ & Artigue, 2013). According to the Pósa philosophy, the same holds for problem posing. When posing problems, students do not receive explicit guidance about which problem posing strategies to use. They are simply asked, usually after solving a problem: “What would you ask next?”, and thus the problem posing task is a completely open situation. Strategies are discussed explicitly as a follow-up to problem posing, when the teacher facilitates a reflective discussion on the problems that students posed. The teacher guides students to formulate heuristics underlying the problems posed, which will help students to use these heuristics in the future. The most common problem posing heuristics identified are: H1. Change the parameters of an existing question. H2. Find a common property among the solutions to an existing problem or among examples (e.g., none of the shapes are bounded) and ask, “Is this property necessary?” or “How can we modify this property?” H3. Introduce analogous or related concepts. H4. Attempt to use more precise language. This often leads to development of new ideas and questions. H5. Ask “out of the blue” questions, which are unexpected, mathematically interesting questions. Such questions are rare and coincidental, and we do not have an explicit method for posing them, but we find that reflecting on how the question arose can be beneficial. When first tasked to pose their own questions, students are usually hesitant. But with time and experience, they gain confidence and competence in problem posing.

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3 Problem Thread on Symmetry We present a problem thread on symmetry in the two-dimensional plane that arose through questions posed by students. After each question we indicate which of the above problem posing heuristics were used. By posing (and answering) these questions, students learned about various geometric ideas such as infinity, boundedness, and transformations. A class of 30 middle school aged students worked on the thread over two weekend mathematics camps, while also working on other problem threads simultaneously (as explained in the previous section). These high performing students were selected based on their results in national mathematics competitions and teachers’ recommendations. Some of them attend schools with special mathematics classes, but the majority are from regular schools. Since elementary plane geometry, basic constructions, and geometric transformations are taught from 4th grade in Hungary, these students have a strong background in these topics. Students learned through a mix of problem-solving and problem-posing phases. They worked in small groups of 2–4 students, with the teacher circulating between groups, listening to student dialogues and providing hints when needed. Problemsolving phases also included whole class discussions where students shared answers, and students and the teacher gave hints for those who had not yet solved the assigned problems. Problem posing usually happened as a whole class discussion, and sometimes as homework. Students also decided which problems they posed should then be assigned to the whole class to solve. This way, the questions posed by individual students were collectively used to compose a problem thread for the whole class. In the description below, we highlight one or two main themes for each problemposing phase, indicated in parentheses after the phase heading. In Phase 2, for instance, students engage in free exploration while posing their own problems; they also learn a new concept as a consequence of the problem-posing process. While the problem-posing heuristics above describe the underlying mathematical reasoning behind problem-posing, these themes reflect on the process from a pedagogical point of view.

3.1 Phase 1: Problem Solving The teacher began the lesson by posing the following question. Q0. Is there a shape in the plane with more than one point of symmetry? (Note: A “point of symmetry” (POS) of a shape refers to a point at which if the shape is reflected about the point—or rotated by 180° around the point—then the image and the original shape are identical.) To clarify the concept of POS, the teacher asked: “Who can construct the POS of an equilateral triangle with a compass and straightedge?” This was a loaded question: although an equilateral triangle has no POS, the question suggested that there is one

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and we want to know if they can construct it. Most students raised their hands thinking of the centroid (which in this case is also the orthocenter and the center of the incircle and the circumcircle). After some discussion, students realized that this is not a POS, and that an equilateral triangle has no POS, which helped deepen their understanding about POS. Then students moved onto thinking about Q0. When students tackled the problem of finding a shape with more than one POS, many thought of “shapes” as polygons and concluded that there is no such shape. When reminded that a “shape” need not be a polygon, some students came up with a line as their answer. In the follow-up discussion, the teacher asked: What are the POS of a line? Does anyone have a different solution? Students came up with other solutions such as: two parallel lines, three parallel lines, the whole plane, etc.

3.2 Phase 2: Problem Posing (Free Exploration & Learning a New Concept) Next, the teacher asked: What would you ask now? There are properties of question Q0 that foster problem posing by students. First, it is low threshold and accessible, so that students can engage with the task without much prior knowledge. It is also concrete and open-ended—in a sense that there may be multiple solutions and they do not necessarily rely on a prescribed approach— which invites student experimentation and heightens their curiosity. Students did not have specific guidance about what type of questions to ask or what heuristics to use. The aim of this type of free exploration is to foster creativity in students. It was only after the students formulated the questions that the teacher guided them to reflect on the heuristics employed. Some student questions include the following. Each question is accompanied in parentheses by the problem posing heuristic that was used. Q1. Is there a shape with exactly two POS? (H1) (Note: “Shape” refers to planar shapes.) Q2. Is there a shape with more than one, but a finite number of POS? (H2) Q3. Is there a shape with more than one POS that does not contain a line? (H2) With the original problem Q0, another common property that students observed is that all the shapes with more than one POS were unbounded. But since they had not previously learned about the notion of boundedness, students struggled to articulate their question. They posed questions such as: Is there a finite shape with more than one POS? If a shape has more than one POS, must it go out to infinity? The teacher probed the students for more mathematical precision, asking them what they meant by a “finite” shape or that a shape “goes out to infinity.” After much discussion, students suggested that a “finite” shape is something that fits inside a square. This depicts how the process of problem posing led to the development of

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a new concept. The teacher agreed and informed the students that they just derived the notion of boundedness, except that mathematicians like to use a disc (i.e., the interior of a circle) rather than a square. After discussing the formal definition of boundedness, students posed the question again: Q4. Is there a bounded shape with more than one POS? (H2, H4) The above episode about question Q4 illustrates how posing a question brought out the need for more precise mathematical language. This, in turn, led to the development of a new definition of boundedness. The teacher supported the students through this process by encouraging them to articulate their ideas with more clarity and not settle for a vague description such as a “finite” shape. Students attempted to solve question Q4 and were certain that there was no such shape. With a bit of playfulness, the teacher asked them to close their eyes while he drew the solution on the board. When they opened their eyes, students saw that the board was erased clean. A student exclaimed, “It’s the empty set!” From this experience, students realized that they needed further precision to pose the question that they really wanted to ask. Here is the final version: Q5. Is there a non-empty, bounded shape with more than one POS? (H4)

3.3 Phase 3: Problem Solving After posing questions 1 through 5, students started thinking about how to solve them. The same underlying idea is behind Q1, Q2 and Q5, since these questions ask about the existence of a (non-empty, bounded) shape with more than one POS. Students had the intuition that these are not possible, but it was difficult for them to make the proof precise. After a while they realized the key to the proof: if we reflect a POS about another POS then we obtain a new POS (this involves proof by contradiction, which was new to most students). Thus, if there are at least two POS, then there will be infinitely many of them. An additional idea for Q5 (that the teacher also used as a hint for students who were stuck) was using the fact that the shape has a point (since it’s non-empty), which yields another proof by contradiction. Q3 posed a challenge to students even though they understood the concepts so far, as it is difficult for them to think beyond the usual point sets, such as polygons and the circle. Initially they thought about bounded shapes, and, in particular, the usual ones. Then came the straight line and the plane. For this problem they needed to find less common shapes such as the integers on the number line, or the set of grid points on the plane.

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3.4 Phase 4: Problem Posing (Using an Analogous Concept & Question Aesthetics) Next, the teacher introduced an analogous concept: axis of symmetry. Students practiced the concept by finding axes of symmetry of familiar shapes such as a square, a rectangle, and an equilateral triangle. The teacher then asked students to pose questions involving the axis of symmetry. This task illustrates our belief that students do not need to be experts in using a concept (such as the axis of symmetry) for posing problems about it. On the contrary, problem posing provides an opportunity to practice and deepen their understanding of the concept. Students posed Q6 through Q12, shown below. (Note: “Axes” refers to “axes of symmetry.”) Q6. If a shape has infinitely many axes, do these always intersect in one point? (H3) Q7. Is the intersection of two axes always a POS? (H2) Q7 is a very natural question. However, the solution is difficult, and it requires advanced techniques that these middle school students had not yet experienced. So the teacher did not assign it for students to solve, but used the opportunity to discuss the notion of question aesthetics: what mathematicians mean when they call a problem “beautiful.” This includes (but is not limited to) whether one enjoys thinking about the problem, and whether the problem and its solution give them a sense of appreciation. Q7 highlights a connection between POS and axes of symmetry, and mathematicians tend to find such connections to be aesthetically pleasing. The teacher added that this sense of mathematical beauty is different for every person, so that mathematical taste is another aspect of diversity that people must respect in each other. Q8. A shape has infinitely many axes. Can it be bounded? (H2) Students realized soon that they needed to add that the shape is not empty. Q9. A non-empty shape has infinitely many axes. Can it be bounded? (H4) Q10. A shape has at least two, but a finite number of axes. Do these intersect in one point? (H2) Q11. A shape has exactly two axes. If these intersect, will the intersection be a POS? (H2) Our remarks above regarding aesthetics in Q7 apply to Q11, too. Q12. A shape has both a point and an axis of symmetry. Does the point lie on the axis? (H2)

3.5 Phase 5: Problem Solving The teacher assigned Q6, Q9, Q10, and Q12 for the whole class. As students began to work on question Q12, many asked, “Which point and which axis of symmetry

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are we talking about?” The students realized that the question was not clearly stated. Once again, the need for precision arose, which led to the next problem-posing phase.

3.6 Phase 6: Problem Posing (Arising from the Need for Precision) After students thought about problems assigned in the previous phase, the teacher called for a class discussion where they acknowledged that Q12 is unclear (i.e., it is not clear whether the shape can have more than one POS and axis, and if so, which POS and which axis the question refers to). This was a good opportunity for the teacher to lead a discussion about the need for precise language when posing questions. The discussion led students to formulate the following questions: Q13. Can a shape have exactly one point and one axis of symmetry? (H4) Q14. Is there a shape with infinitely many points and axes of symmetry such that none of the points lie on an axis? (H4)

3.7 Phase 7: Problem Solving Students continued to work on problems Q6, Q8, and Q10, with Q14 added. Q6 reinforced previous ideas. As before, students needed to think beyond the usual shapes such as polygons and the circle. Previous examples such as the straight line and the entire plane worked as counterexamples for this question. Q8 brought a surprise to students, as they started to look for complicated shapes, but then they realized that a circle is a simple answer. When thinking about Q10, many students realized that reflection of an axis about another axis yields a new axis. For other students, the teacher suggested thinking back to analogous problems with POS. Students enjoyed thinking about Q14, and they found various answers. Here are some examples (Fig. 1).

3.8 Phase 8: Problem Posing (Restricting the Topic) At this point, the teacher asked students to pose further questions that involved the notion of infinity. Here, the teacher was intentionally restricting the topic of the problems posed by students to a particular concept. A natural complement to free exploration from Phase 2, the aim of this pedagogical approach is for the teacher to guide students towards focusing on a specific topic, which in this case is the connection between infinity and boundedness.

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Fig. 1 Examples for Q14

Students posed the following questions: Q15. Is there a shape with infinitely many POS and no axis? (H1) Swapping the roles of POS and axis, they also asked: Q16. Is there a shape with infinitely many axes and no POS? (H1)

3.9 Phase 9: Problem Solving Q15 is an interesting and moderately difficult question. For students who needed help, the teacher suggested that they look for a shape with one POS but no axis. For this they found a parallelogram that is not a rhombus. The next step was to alter this shape in a way that it has infinitely many POS, but still no axis. Students found the parallelogram grid. Here, the teacher is modeling the problem-solving heuristic of first posing and working on a simpler problem (which also highlights the intimate connection between problem posing and problem solving), and then revising/extending its solution to address a more complex scenario. For Q16, students were looking for more complex constructions than needed, but after a while they realized that a half plane works.

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3.10 Phase 10: Problem Posing (Looking for Complexity) After students realized that the half plane is a solution for Q16, the teacher asked if they could modify the question in a way that would require more complex solutions. Even though Q16 is not easy to answer, the fact that a simple shape satisfies it leads to the intellectual curiosity about how to modify the question so that the answer is more complex. Both posing and solving problems with complex solutions encourage students to think about different solution approaches, which broadens their understanding. This is also a good opportunity for the teacher to remind students to use problem posing heuristics, and then use problem solving strategies to solve the modified question. As a guiding question, the teacher asked: “Do you find something common in the solutions you found so far?”, thus modeling the use of the heuristic H2. Students observed that in all of the solutions the axes were parallel, so they restricted Q16 by requiring that the shape would have intersecting axes: Q17. Is there a shape with infinitely many axes, no POS, with at least two intersecting axes? (H2)

3.11 Phase 11: Problem Solving As students worked on Q17, they recalled the question posed when defining POS, whether it is possible to construct the POS of an equilateral triangle. They learned that the equilateral triangle has three axes which intersect at one point, but no POS. This idea helped them construct the following answer, shown in Fig. 2 (the dotted lines are the axes). Some groups included the interior of the triangles, some only the sides. After students found the construction in Fig. 2, the teacher encouraged them to prove that the shape does not have a POS.

3.12 Phase 12: Problem Posing (as a Means of Assessment) At this point the teacher asked students to pose further questions. Students suggested restricting Q15 and Q16 with boundedness, which was similar to Q4 and Q9 in using the heuristic H2 by observing that all answers to a question are unbounded and thus adding boundedness as a restriction. With Q4 and Q9, this occurred with only one or two students. This time, however, it was suggested by many of them, so the teacher saw that students internalized the problem posing techniques used earlier. This episode illustrates how the teacher can use the problems posed by the students as a means of assessing their learning.

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Fig. 2 Example for Q17

Q18. Is there a bounded shape with infinitely many axes, but no POS? (H1) Q19. Is there a bounded shape with infinitely many POS, but no axis? (H1) Students immediately observed that Q19 is not possible based on Q5. At this point the teacher noted that this is a logical question to ask, but mathematicians usually look for questions that are not immediately evident, and hence are challenging to solve.

3.13 Phase 13: Problem Solving Q18 is a very difficult question. We show two different constructions, and both start in the same manner. A circle C and a point P on the circumference of C are given. Both constructions are subsets of C. Let the circumference of C be 1. Let P(x) be the unique point on C such that the distance between P and P(x) measured along the circumference in the positive direction is x. It means that P = P(0) = P(1) = P(n) for any integer n. And P(0.2), P(3.2), P(101.2), P(–0.8), and P(–7.8) are the same point. In general P(a) = P(b) if and only if a − b is an integer. Construction 1: An equilateral triangle has three axes and no POS, a regular 9-gon has nine axes and no POS, a regular 27-gon has 27 axes and no POS, etc. This is a useful hint for the first construction:   n S1 = P k : n, k ∈ N, n < 3k . 3

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S 1 is the union of vertices of well positioned regular 3 k -gons. S 1 does not have a POS, because the center of C is the only candidate for this, and the center is not a POS. The axes of the regular 3 k -gons are axes of the entire point set S 1 , and thus there are infinitely many axes altogether. Construction 2: Let q be an irrational number such that q < 1. Let S2 = {P(nq): n ∈ N} be a subset of C. We show that S 2 has infinitely many axes, but no POS, thus it is a solution to Q18. First we prove that S 2 does not have a POS. The center of C is the only candidate for being a POS of S 2 , but if R = P(x) belongs to S 2 , then the opposite point on the circle does not. This is true because the opposite point is P(x + m + 1/2), where m is a nonzero integer, which implies that the opposite point does not belong to S 2 (which follows from q being irrational). On the other hand, S 2 has infinitely many axes: the lines connecting the center of C and any point of S 2 . Students arrived at the solution in many different ways, with varying levels of teacher support, but mostly with a greater amount of teacher guidance than usual. In the Pósa method, teachers typically prefer their students to think about problems with less teacher guidance. This time, however, the teacher wanted students to experience the joy of finding a multi-step complex solution to a question they posed themselves.

4 Conclusion In our experience, problem posing by students enhances their learning in several ways: • It fosters creativity, courage, and self-confidence in mathematics, which then enables them to apply these skills in situations outside of mathematics. • Students become motivated to explore a new topic when it is built on their own questions. • Differentiation is built into problem posing, since students can ask questions at their own mathematical comfort level. • Students deepen their existing understanding when they look for questions to pose. • Problem posing fosters the need for precise mathematical language. Above all, posing questions is at the core of what mathematics is about. Thus, explicitly teaching students to pose their own questions is well-aligned with the Hungarian tradition of teaching students how to think like mathematicians (Connelly, 2010). We give the final word to Pólya who wrote, “Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you

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solve it by your own means, you may experience the tension and enjoy the triumph of discovery” (Pólya, 1945). Acknowledgements This work was funded by the Content Pedagogy Research Program of the Hungarian Academy of Sciences and supported by Budapest Semesters in Mathematics Education.

References Barabé, G., & Proulx, J. (2015). Problem posing: A review of sorts. Proceedings of 37th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1277–1284). East Lansing, US. Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83, 57–69. Connelly, J. (2010). A tradition of excellence transitions to the 21st century: Hungarian mathematics education, 1988–2008. Doctoral dissertation, Columbia University, http://gradworks.umi.com/ 3420877.pdf Gordon Gy˝ori, J., & Juhász, P. (2017). An extra-curricular gifted support programme in Hungary for exceptional students in mathematics, teaching gifted learners in stem subjects: Developing talent in science, technology, engineering and mathematics (pp. 89–106). Routledge. Gosztonyi, K. (2016). Mathematical culture and mathematics education in Hungary in the XXth century. In B. Larvor (Ed.), Mathematical cultures. Springer Birkhauser. Headrick, L., Wiezel, A., Tarr, G., Zhang, X., Cullicott, C., Middleton, J., & Jansen, A. (2020). Engagement and affect patterns in high school mathematics classrooms that exhibit spontaneous problem posing: An exploratory framework and study. Educational Studies in Mathematics, 105, 435–456. Juhász, P. (2019). Talent nurturing in Hungary: The Pósa weekend camps. Notices of the American Mathematical Society, 66(6), 898–900. Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Erlbaum. Maaβ, K., & Artigue, M. (2013). Implementation of inquiry-based learning in day-to-day teaching: A synthesis. ZDM—The International Journal on Mathematics Education, 45(6), 779–795. Mason, J. (2010). Effective questioning and responding in the mathematics classroom. http://mcs. open.ac.uk/jhm3/SelectedPublications/EffectiveQuestioning&Responding.pdf National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Author. Pólya, G. (1945). How to solve it. Princeton University Press. Silver, E. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. Silver, E., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27.https://doi.org/10.2307/749846 Silver, E., Kilpatrick, J., & Schlesinger, B. (1990). Thinking through mathematics: Fostering inquiry and communication in mathematics classrooms. The College Board. Silver, E., Mamona-Downs, J., Leung, S., & Kenney, P. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309. Singer, F., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7. Stockton, J. (2010). Education of mathematically talented students in Hungary. Journal of Mathematics Education at Teachers College, 1(2), 1–6.

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Van Harpen, X., & 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.https://doi.org/10.1007/s10649-012-9419-5 Voica, C., & Singer, F. M. (2013). Problem modification as a tool for detecting cognitive flexibility in school children. ZDM Mathematics Education, 45, 267–279. Winograd, K. (1991). Writing, solving and sharing original math story problems. Case studies of fifth grade children’s cognitive behaviour. The Annual Meeting of the American Educational Research Association, Chicago, April 1991, 3–7.

Chapter 14

Conclusion: Mathematics Problem Posing and Problem Solving: Some Reflections on Recent Advances and New Opportunities Edward A. Silver

In a paper on the nature of mathematics, the mathematician Paul Halmos asks, “What does mathematics really consist of?” He then mentions several “essential ingredients,” including axioms, concepts, methods, theorems, and proofs. But he concludes that “none of them is at the heart of the subject,” and “what mathematics really consists of is problems and solutions” (Halmos, 1980, p. 519). It appears that many agree with Halmos on this point. Mathematics problem solving has long been a major topic of interest for all segments of the mathematics education community: teachers, teacher educators, curriculum developers, and researchers. A search of Google Scholar for the topic “mathematics problem solving” yields more than 5 million citations! Roughly half of the papers in this volume address mathematics problem solving; the other half address the closely related topic of mathematics problem posing. This distribution appears to reflect a trend on these topics in the broader field. A search of Google Scholar for the topic “mathematics problem posing” yields fewer than 400,000 citations—less than 10% of the citations for mathematics problem solving— but the number of citations for each topic since 2000 is almost identical (about 18,000). Thus, mathematics problem posing may be a relative newcomer to the conversation about mathematics problems, but it is currently capturing an equal amount of attention when compared to the more established topic of mathematics problem solving. Halmos also noted the importance of problem posing: “I do believe that problems are the heart of mathematics, and I hope that as teachers (…) we will train our students to be better problem posers and problem solvers than we are” (Halmos, 1980, p. 524). In discussing how mathematical problem solving should be taught (to college students), Halmos argued: E. A. Silver (B) University of Michigan, Ann Arbor, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. L. Toh et al. (eds.), Problem Posing and Problem Solving in Mathematics Education, https://doi.org/10.1007/978-981-99-7205-0_14

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“The best way to conduct a problem seminar is, of course, to present problems, but it is just as bad for an omniscient teacher to do all the asking in a problem seminar as it is for an omniscient teacher to do all the talking in a lecture course. I strongly recommend that students in a problem seminar be encouraged to discover problems on their own.” (1980, p. 524) [emphasis added]

These statements by Halmos resonate with ideas found in a paper on problem posing by Kilpatrick (1987), “Problem formulating: Where do good problems come from?” Kilpatrick answered the question posed in his title by pointing out that most students would say that virtually all mathematics problems come from teachers and textbooks. Yet, contrary to students’ typical school experience, Kilpatrick reminds us that in life outside of school “many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (1987, p. 124). Problem posing was the topic of a plenary lecture I presented about 30 years ago, at the 17th annual conference of the International Group for the Psychology of Mathematics Education (PME). In that talk, subsequently published as a paper in For the Learning of Mathematics (Silver, 1994), I suggested some reasons why mathematicians and mathematics educators have been and should continue to be interested in problem posing, described several different activities and cognitive processes that have been referred to as problem posing, and summarized some research studies related to students’ mathematical problem posing. Because the research base at that time was rather thin, I hoped that a PME plenary lecture focused on mathematical problem posing would elevate its visibility among mathematics education scholars and stimulate more research on this topic. Since then, interest in mathematics problem posing has increased, as has research attention to this topic. I was delighted to be asked to comment on the papers published in this volume, which represent a select sample of work related to mathematics problem solving and problem posing that was presented at the 14th International Congress on Mathematical Education (ICME-14). Given the international scope of ICME-14, it is probably not surprising to see the diversity of authors from across the globe. The chapters in this volume were prepared by authors from 11 different countries, with representation from Asia, Europe, South America, and North America. Several are seasoned contributors to the mathematics education research literature (e.g., Lesh, Malara, and Toh), but it is refreshing to see that many authors are more recent entrants to the field. Their fresh perspectives and new insights provide good reason to be hopeful about the future of research on mathematics problem solving and problem posing. In this regard I note that the reference lists of the papers in this volume suggest both a strong connection to prior work of major contributors to the mathematics problem-solving and problem-posing research literatures (e.g., Cai, English, Kilpatrick, Schoenfeld, Singer, and Stoyanova) and a willingness to consider these topics from fresh perspectives. I find this encouraging because it suggests that robust lines of inquiry are likely to continue in ways that not only build on prior work but also take the work in new directions.

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1 Diverse Perspectives on Mathematics Problem Posing and Problem Solving Problems and problem solving have long been recognized by mathematicians and educators alike as a primary focus for mathematics education (Stanic & Kilpatrick, 1988). Problem tasks often form the backbone of classroom mathematics instruction, and the generation of problem solutions is a common focus of student activity in mathematics classrooms around the world. Contemporary interest in mathematical problem posing and problem solving across the globe continues to be strong in relation to school mathematics curriculum and pedagogy. Curriculum documents in most nations feature problem solving as a prominent aspect of the mathematics students should learn, and many also suggest an important role for problem posing. Looking across these documents, one can see that helping students learn to solve a wide variety of problems is an important goal in the study of mathematics, as is learning to use a variety of problem-solving strategies. Problem solving provides a context in which some mathematics concepts and skills can be learned and in which many can be applied to solve problems both within and outside mathematics. In the US, for example, Principles and Standards for School Mathematics (PSSM) stated that “Instructional programs from prekindergarten through grade 12 should enable all students to (a) build new mathematical knowledge through problem solving, (b) solve problems that arise in mathematics and in other contexts, (c) apply and adapt a variety of appropriate strategies to solve problems, and (d) monitor and reflect on the process of mathematical problem solving.” (NCTM, 2000, p. 52). PSSM also stated that the school curriculum should provide students with opportunities to “formulate interesting problems based on a wide variety of situations, both within and outside mathematics” (NCTM, 2000, p. 258). This document also recommended that students should make and investigate mathematical conjectures and learn how to generalize and extend problems by posing follow-up questions, very much in the spirit of Brown and Walter’s (1983) seminal writings about mathematical problem posing using the “What if not” technique or Pólya’s (1954, 1957) writings about the value of “looking back” at one’s solution to a mathematical problem in order to generate new insights. Similar suggestions are found in the National Statement on Mathematics for Australian Schools (Australian Education Council, 1991), which expressed strong support for the use of open-ended problems in mathematics classrooms: “Students should engage in extended mathematical activities which encourage problem posing, divergent thinking, reflection, and persistence. They should be expected to… pose and attempt to answer their own mathematical questions” (p. 39). One can find similar enthusiasm for treating both mathematical problem solving and problem posing not only as an important goal itself but also as a means to accomplish other important instructional goals, in curriculum documents in countries with mathematics teaching traditions as diverse as China, Cuba, England, and Hungary. The papers in this volume suggest that the curricular perspectives on mathematics problem solving and problem posing are also evident in contemporary research. Although the research literature on mathematics problem solving over

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several decades is replete with detailed examinations and cognitive analyses of individual problem-solving processes, that perspective has become less dominant, as researchers have turned their attention increasingly to a more socially situated view of problem solving. A similar, albeit more rapid, transition can be seen in the research on mathematics problem posing. When considering problem posing as a cognitive activity, researchers have sought to understand how students construct personal interpretations of specific situations and then pose problems based on those situations. For example, problem posing has been examined as “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). 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). One of the papers in this volume reflects this approach. Chua (this volume, Chap. 12) presents a close examination of the regulatory activity of a student engaged in geometric problem posing. About 40 years ago, Garofalo and Lester (1985) argued that successful mathematics problem solving involved more than adequate knowledge, and they called for more attention to the examination of regulatory aspects. Chua’s paper demonstrates that there may also be value in adopting this perspective in regard to mathematics problem posing. The other papers in this volume reflect a contemporary turn from research exclusively focused on the nature of problem solving and problem posing processes with a shift toward broadening the scope of investigation into topics that might pertain to the teaching of mathematics through and with problem solving and problem posing.

2 Teaching Mathematics Through and with Problem Posing and Problem Solving Teaching mathematics through problem solving or problem posing is not currently common practice, and this creates a need for examples of this kind of teaching (e.g., Zhang & Cai, 2021). Examples can help us identify its key features and characteristics, and they can suggest potential models for teachers to adopt or adapt. Welldocumented examples can offer guidance regarding teaching mathematics through problem posing and problem solving, including (a) how to design instructional tasks and activities, (b) how to utilize the learning opportunities they provide, and (c) how problem posing and problem solving can be incorporated into existing lessons to enhance learning opportunities.

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One key claim that has been made about the potential value of engaging students with mathematics problem posing is that it might assist them to become better problem solvers (Brown & Walter, 1983; English, 1997, 1998; Kilpatrick, 1987; Silver, 1994). Some studies have found a relationship between students’ abilities to pose and solve problems (e.g., Cai & Hwang, 2002; Ellerton, 1986; Silver & Cai, 1996; Verschaffel, et al., 2009). A few papers in this volume contribute to this research theme by examining examples of mathematics instruction that utilize the interconnectedness of mathematics problem solving and problem posing with each other and with other mathematical ideas and processes. For example, Malara and Telloni (this volume, Chap. 8) describe an ambitious approach to early algebra instruction that blends problem solving, problem posing and argumentation. They report multi-dimensional growth on the part of students in relation to problem solving, early algebra proficiency, and mathematical argumentation and relational reasoning. Promoting growth in mathematical reasoning derived from experience with mathematics problem solving is also a focus of Chanudet (this volume, Chap. 4). She carefully considers not only: what students might learn from problem solving when it constitutes the learning goal but also the ways in which students might grow in mathematical reasoning proficiency due to problem-solving instruction. Her analysis of the learning opportunities afforded by the problems made available to students in a particular problem-solving course shines a light on the possibilities for such learning to occur, but also draws attention to a need for careful choice of problems and experiences to allow access to a range of types of mathematical reasoning. A deliberate and detailed focus on the affordances and constraints of problems used in instruction is also evident in the paper by Pósa and colleagues (this volume, Chap. 13). Continuing a long tradition of examining the effect of task variables in mathematics problem solving (e.g., Goldin & McClintock, 1979; Lester, 1994), they describe and illustrate the use of problem threads (Juhász, 2019), which refer to a series of tasks with a specific structure and purpose, that provide students with opportunities to struggle productively towards understanding a mathematical idea, and show how the embedded problem-posing work reaps multiple benefits and enhances students’ learning. In my reading of these three papers, as well as the one by Chua, I see them directly or indirectly touching on a theme that is taken up explicitly by Brady and colleagues (this volume, Chap. 3)—students’ mathematical agency. As Brady and colleagues note in their paper, early proponents of problem posing in mathematics education often referred to increased student agency as a possible benefit. Drawing on Schön’s (1983) distinction between rigor and relevance, they argue for the value of a situation-based approach to problem posing to augment students’ mathematical agency. In addition to the approach they endorse, my reading of the chapters by Chua, Malara and Telloni, Chanudet, and Pósa and colleagues suggests that there may be multiple ways to achieve the objective of increasing students’ mathematical agency. In fact, viewing students’ mathematical agency as a goal intertwined with mathematics problem solving, problem posing, reasoning, modeling, and the learning of mathematical ideas resonates with the notion of mathematical disposition.

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3 Students’ Mathematical Disposition and Agency In my 1994 paper on mathematics problem posing, I argued that providing students with problem-posing experiences was likely to improve their disposition toward mathematics. I based my argument at that time on some anecdotal accounts of problem-posing interventions (e.g., Healy, 1993; Moses et al., 1990; van den Brink, 1987) and a few dissertation studies (e.g., Keil, 1965; Leung, 1993). Research conducted since then has generated a more compelling evidence base. For example, in a study conducted with prospective elementary school teachers, Akay and Boz (2010) found that problem-posing instruction had a positive influence on attitudes toward mathematics and perceived self-efficacy in mathematical understanding. A meta-analysis of research on the effects of problem-posing instruction on a variety of outcomes found a strong positive effect on attitudes/beliefs for precollege students in grades 4–12 (Rosli et al., 2014). And a more recent meta-analysis of research conducted between 1990 and 2019 with students across grades 1–12 on the effects of problem-posing instruction on a variety of outcomes confirmed a strong positive effect on disposition, with longer interventions having greater effects (Wang et al., 2022). Taken collectively these research findings suggest a positive association between students’ experiences with problem posing and their dispositions toward mathematics, which includes beliefs about, attitudes toward and interest in mathematics. Though it has received far less emphasis than specific cognitive outcomes associated with mathematics instruction, many view students’ disposition to be important as an outcome itself as well as a key influence on students’ motivation and persistence. Linking students’ disposition and their mathematical agency, and viewing mathematics problem solving, problem posing, reasoning, modeling, and the learning of mathematical ideas as intertwined goals resonates with the perspective taken in Adding it up, an influential report on mathematics learning, published in 2001 by the U.S. National Research Council (NRC) The authors used the construct of mathematical proficiency to characterize successful learning of mathematics. Mathematical proficiency was conceptualized as the interrelationship among five components or strands: (1) procedural fluency (knowing how and when to apply procedures), (2) conceptual understanding (holding deep and rich connections among ideas), (3) adaptive reasoning (the capacity to reason logically and to justify one’s reasoning), (4) strategic competence (formulating, representing, and solving problems), and (5) productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy) (Kilpatrick et al., 2001, p. 116). The first four strands were not new to the field when proposed in 2001, though placing them side-by-side within an overarching view of mathematical proficiency elevated each and emphasized the relationships among them. But the characterization of productive mathematical disposition as a component of mathematical proficiency was novel and suggested an important role for a set of factors that were not purely

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cognitive in nature. And in this characterization, we can see an opportunity to use problem solving and problem posing to promote students’ mathematical agency. As many of the papers in this volume suggest, including not only the ones discussed thus far but also those describing various uses of technological tools to engage students in problem posing or problem solving [e.g., Hernández and colleagues (this volume, Chap. 5), Santos-Trigo (this volume, Chap. 2), and Toh and Tay (this volume, Chap. 7)], classroom instruction rich in problem-solving and problem-posing experiences can help students develop a tendency to see mathematics as sensible, useful, and worthwhile; to believe that mathematics can be learned with steady effort; and, perhaps most important, to see themselves as an effective learners and doers of mathematics. Collectively, these papers illustrate how instruction can be organized to open the door to students gaining a greater sense of agency and autonomy with respect to mathematics. More opportunities for students to take control of the processes and outcomes involved with solving problems in mathematics class can lead to a greater sense of autonomy More opportunities for students to exercise choice related to problems they solve in mathematics class can lead to greater sense of mathematical agency. Actualizing the potential of mathematics classroom lessons that provide students with opportunities to engage in problem posing and problem solving will depend to a great extent on teachers. Several papers in this volume report studies that focused on teachers and mathematics problem solving or problem posing.

4 Teachers’ Mathematics Problem Posing and Problem Solving Despite nearly universal agreement on the importance of problem solving in the mathematics classroom and nearly 60 years of research on problem solving, its enactment has been a persistent challenge for teachers (Chapman, 2016), and enacting problem posing is no less daunting for teachers. As many research studies have found, problem solving in mathematics classrooms is often a ritualized performance by teachers and students. In this ritual, teachers provide a problem, and students expect that the problem will require them to use orpractice some method the teacher has recently taught. If that method does not quickly result in a solution, then the students ask the teacher for help. Commenting on this tendency, Schoenfeld (1989) observed a belief among students that they should solve mathematics problems in a relatively short time—a belief that may lead students to place little value on persistence, and consequently, exert less effort in solving challenging nonroutine problems. Students too often appear to lack a sense of personal agency or autonomy to persist in trying to solve problems when the method is not obvious, and teachers are often unsure how to assist in some way other than giving the answer or demonstrating the method.

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Promoting students’ mathematical agency requires the disruption of this classroom problem-solving ritual practice. But the challenges teachers face as they try to engage students in cognitively demanding activity in classrooms has been extensively documented (e.g., Henningsen & Stein, 1997; Silver et al, 2005; Stein et al., 1996). Toward this end, it is important that we understand how teachers view mathematics problem solving and problem posing and how to support their teaching mathematics through problem posing and problem solving. Several papers in this volume add to our knowledge base about teachers in relation to problem posing and problem solving. Hernández, Perdomo-Díaz, and CamachoMachín (this volume, Chap. 5) report an analysis of the activity of a dozen preservice secondary school teachers solving problems using GeoGebra. They report evidence of mathematical processes and activities that include extending and posing new problems and finding novel paths to solve the tasks with technology. Their findings suggest not only a link between problem solving and problem posing, but also the value of attending carefully to problem-posing activity that arises spontaneously during problem solving. Teachers’ problem posing has been a topic of research interest over the past several decades (e.g., Crespo, 2003; Koichu & Kontorovich, 2013; Leung & Silver, 1997; Silver et al., 1996). Several papers in this volume report examinations of teachers’ problem posing. Betran-Joaquin (this volume, Chap. 10) analyzed the problems posed by groups of preservice and inservice secondary school teachers. Though some variation was noted with respect to problem context and content domain, both groups posed routine tasks that could be solved in a straightforward manner. The author concludes with a call for the development of appropriate professional development and training to help teachers learn to craft good, mathematics problems. Because the mathematics tasks students engage with in class shape their learning opportunities, the selection or posing of tasks is a key aspect of teachers’ work. Crespo (2003) demonstrated that preservice teachers could 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 in his paper Kovács (this volume, Chap. 11) offers an interesting case of teacher professional learning related to problem posing. Preservice secondary school teachers enrolled in a problem-solving course who also received instruction in the use of Brown and Walters’ (1983) “What-if-not” technique generated more problems of greater cognitive complexity than a comparison group of prospective teachers also enrolled in the problem-solving course who were not exposed to the “what-if-not” training. These results augment the of findings of other researchers (e.g., Crespo & Harper, 2020; Crespo & Sinclair, 2008; Ellerton, 2013) who have studied ways to augment teachers’ proficiency in problem posing. Spinillo and colleagues (this volume, Chap. 9) probe not only the mathematics problems posed by a group of pre-secondary level teachers but also the opinions of another group of pre-secondary teachers about the teaching of mathematics through problem posing and inquiry approaches. Though different samples of teachers were used in the two parts of this study, this may serve as an example of how future research might link teachers’ problem posing, their beliefs about problem posing as

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a curricular goal or an instructional approach toward other goals, and their actual classroom practice. The paper by Rott (this volume, Chap. 6) offers another example of research attempting to link teachers’ beliefs about mathematics and their problemsolving teaching practice. His analysis of videotaped lessons in primary teachers’ classrooms suggests a methodology that might be fruitfully employed or adapted to examine actual teaching practice related to problem solving or problem posing.

5 Concluding Thoughts Emulating the example of Stacey (2016), in her discussion of a different set of papers on mathematics problem solving and problem posing, after reading the papers in this volume, I looked back at the Proceedings of the Problem-Solving Theme Group at ICME 5 held Australia in 1984 (Burkhardt et al., 1988). Though some observations found in papers in this volume echo themes found in reports 40 years ago, there are a few notable differences. In my opinion some of the differences can be considered advances. For example, the amount of attention given to problem posing, as well as to problem solving, is noteworthy. Given the importance of problem posing in mathematics curriculum recommendations across the globe, it is a positive sign that researchers are examining this topic with more vigor than was the case 40 years ago. Another advance is the inclusion of reports that utilize innovative technological tools, such as the papers by Santos-Trigo (this volume, Chap. 2), and Toh & Tay (this volume, Chap. 7). As technology has become ubiquitous in students’ out-of-school experiences, it is critical to attend to it in our thinking about mathematics problem posing and problem solving. Despite some advances, I also note a difference that might be a retreat. The report from ICME-5 contained more studies of teaching for, about or through problemsolving and examinations of the mathematical and pedagogical challenges contained therein. The authors of papers in this volume tried to make their investigations authentic and relevant to classroom instruction. Yet, with a few notable exceptions (e.g., Malara & Telloni, Chap. 8; Pósa et al., Chap. 12), most were conducted outside the context of student classroom instruction. Given the turn toward problem posing in the research community, the advances made to date in understanding problem posing, and the high level of frequent appearance of problem posing in official national curriculum guidance, the time seems ripe for the research community to cross the threshold once again into the classroom. We have detailed accounts of teacher professional development to support largescale initiatives aimed at mathematics problem solving and problem posing (e.g., Felmer & Perdomo-Diaz, 2016; Leung, 2016), and these reports provide a basis for considerable optimism. In addition to more research ON teachers, the field would be well-served by more research WITH teachers.

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The careful examination of well-designed classroom instruction aimed at teaching mathematics through problem solving or problem posing is likely to yield important insights. There are many possible bases of designing such instruction, including seminal sources (e.g., Brown & Walter, 1983; Pólya, 1954, 1957) related to mathematical problem posing and problem posing or other models, such as English’s multi-year instructional intervention (1997, 1998), Healy’s “build a book” technique (1993), Skinner’s “What’s my problem?” technique (1991), van den Brink’s (1987) use of children’s self-authored books, or Winograd’s “mathematician’s chair” (1997). Alternatively, studies of naturally occurring mathematics classroom instruction through problem posing pr problem solving would also be likely to enhance our knowledge. Mathematics classroom instruction infused with problem-posing activity should also be a fertile site for the examination of impact on students’ mathematical disposition, agency, and autonomy, topics identified above as important targets of investigation. The findings regarding growth in self-efficacy for prospective elementary school teachers (Akay & Boz, 2010) and for elementary school students (Bicer et al., 2020) offer models for such research and suggest that a focus on impact related to disposition, autonomy, or agency is likely to be fruitful. In his seminal 1970 book, Pedagogy of the Oppressed, Brazilian educator Paulo Freire (1970) contrasts two types of education that he calls “banking” and “problemposing”. The “banking” system consists of teachers depositing information into their students, like one deposits money in a bank. In contrast, Freire proposes an alternative type of learning experience that he calls “problem-posing.” In his vision of problemposing education, students are agents of their own learning, working in partnership with teachers to pose, select, and explore problems that interest and motivate them. Although Freire’s notion of problem posing is not specific to mathematics, nor is his construct identical to how we view problem posing in mathematics education, I think they spring from the same fundamental desire. The banking model describes a form of educational practice that we too often find in mathematics classrooms: an all-knowing teacher dispenses mathematical knowledge, and the students (usually) listen and (occasionally) learn. Building on the work reported in this book and other work discussed herein, I hope the readers of this paper will endeavor to generate the theoretical and pedagogical foundation on which to create educational settings in which students have opportunities to express agency and autonomy in relation to the mathematics they are expected to learn.

References Akay, H., & Boz, N. (2010). The effect of problem posing oriented analyses-II course on the attitudes toward mathematics and mathematics self-efficacy of elementary prospective mathematics teachers. Australian Journal of Teacher Education, 35(1), 59–65.

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