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

✍ Scribed by Tin Lam Toh (editor), Manuel Santos-Trigo (editor), Puay Huat Chua (editor), Nor Azura Abdullah (editor), Dan Zhang (editor)

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 263
• Category: Library

✦ Synopsis

This book presents both theoretical and empirical contributions from a global perspective on problem solving and posing (PS/PP) and their application, in relation to the teaching and learning of mathematics in schools. The chapters are derived from selected presentations in the PS/PP Topical Study Group in ICME14. Although mathematical problem posing is a much younger field of inquiry in mathematics education, this topic has grown rapidly. The mathematics curriculum frameworks in many parts of 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 this book addresses the needs of the mathematics education research and practice communities at the present day. In particular, this book aims to address the three key points: to present an overview of research and development regarding students’ mathematical problem solving and posing; to discuss new trends and developments in research and practice on these topics; and to provide insight into the future trends of mathematical problem solving and posing.

✦ Table of Contents

Contents
Editors and Contributors
1 Problem Posing and Problem-Solving in Mathematics Education: International Research and Practice Trends
References
2 Trends and Developments of Mathematical Problem-Solving Research to Update and Support the Use of Digital Technologies in Post-confinement Learning Spaces
1 Introduction and Background
2 The Role of Tools, Methods, and Results in Mathematics Developments
3 Conceptual Frameworks in Mathematical Problem-Solving Approaches
4 Problem-Solving Principles and the Use of Digital Technologies
5 Assessing and Monitoring Students’ Problem-Solving Competencies
6 Looking Back and Remarks
References
3 Problem Posing and Modeling: Confronting the Dilemma of Rigor or Relevance
1 Introduction
2 Positioning Students (and Teachers) with Mathematical Agency
3 Problem-Posing in Everyday Instruction
4 How Do Problems Relate to Topics in the Curriculum?
5 Interlude: Taming Messes and the Dilemma of Rigor or Relevance
5.1 Taming Messes
5.2 Rigor or Relevance
6 What is a Problem?
7 Mathematics is Interpretive: What is a Model?
8 The Reflective Role of Heuristics in This Approach
9 The Answer, Certainty, and Closure in This Approach
10 Connections Between Teachers’ and Students’ Work
11 The Dilemma of Rigor or Relevance in Mathematics Education
11.1 Complicated or Complex?
11.2 New Representational Tools and Approaches
12 Discussion and Conclusion
References
4 Types of Mathematical Reasoning Promoted in the Context of Problem-Solving Instruction in Geneva
1 Introduction
2 Theoretical Framework
2.1 Problem Solving
2.2 Mathematical Reasoning
3 Research Design
3.1 Context
3.2 Methodology
4 Results and Discussion
4.1 Types of Mathematical Reasoning Promoted in Official Resources for MSA Course
4.2 Types of Mathematical Reasoning Promoted by Teachers
5 Conclusion
Annex
Annex 1 Programme of “Mathematical and Scientific Approaches” Course (MSA)
References
5 Prospective Secondary School Mathematics Teachers’ Use of Digital Technologies to Represent, Explore and Solve Problems
1 Introduction
2 Conceptual Framework
3 Methodology
4 Data Analysis and Results
4.1 Episode 1: Understanding the Problem
4.2 Episode 2: Exploration
4.3 Episode 3: Search for Multiple Approaches
5 Conclusion
References
6 Primary School Teachers’ Behaviors, Beliefs, and Their Interplay in Teaching for Problem-Solving
1 Background of the Study
2 Theoretical Framework
3 Methodology
4 Results
4.1 Teachers’ Behavior
4.2 Teachers’ Beliefs
4.3 The Interplay of Teachers’ Behavior and Beliefs
5 Discussion and Conclusion
5.1 Limitations of the Study
5.2 Outlook
References
7 Movie Clips in the Enactment of Problem Solving in the Mathematics Classroom Within the Framework of Communication Model
1 Introduction
2 Movie Clips
3 Some Elements of Communication Theory
4 Mathematical Problem Solving with a Little Help from Communication Theory
4.1 Use of Movie Clip in Problem Solving
4.2 A Self-developed Movie Clip to Teach Problem Solving for Low Attaining Students
5 Common Features of the Two Movie Clips
5.1 Storyline Relatable to Students
5.2 Contextualized as a Decision-Making Task for Students
5.3 Inclusion of Humour
5.4 Reduction of “Noise” in the Movie Clip
5.5 Use of Simplified Language in the Movie Clips
5.6 Task Generalizable to New Problems
5.7 Suitable for Both Personal and Nonpersonal Communication
6 Use of Movie Clips for Mathematics Instruction
7 Final Remarks
References
8 On Teaching of Word Problems in the Context of Early Algebra
1 Introduction
2 Theoretical Frameworks
2.1 Problem Solving and Problem Posing Frameworks
2.2 Early Algebra Frameworks
2.3 Theoretical Aspects of Our Approach to Early Algebra
2.4 The Teacher’s Role
3 The SAP Project
3.1 The Texts of the Verbal Problems
3.2 Identification and Naming of the Variables
3.3 Construction and Argumentation of the Solving Process
3.4 The Intertwining Between Problem Posing and Problem Solving
3.5 The Structure of the SAP Project
4 Methodology
5 The Didactical Path
6 Some Key Episodes of the Experiment
6.1 The Library Problem
6.2 The Pinocchio Problem
6.3 The Problem of the Canaries
7 Analysis of the Results
7.1 Students’ Behaviours Within the Collective Discussions
7.2 Analysis of the Students’ Productions About the Individual Activities
8 Conclusion
References
9 Problem Posing by Mathematics Teachers: The Problems They Pose and the Challenges They Face in the Classroom
1 Introduction
2 The Characteristics of Mathematical Problems Posed by Teachers
3 Teacher Knowledge
4 The Report of the Two Studies
4.1 Study 1: Teachers Posing Word Problems of Multiplicative Structure
4.2 Study 2: Teachers’ Opinion About the Teaching of Mathematics Through Problem Posing
5 Concluding Remarks
5.1 How Important is Mathematical Problem Posing for Teaching?
5.2 Why is It Relevant to Investigate the Characteristics of Mathematical Problems Posed by Teachers?
5.3 What Difficulties Do Teachers Face When Teaching Mathematics Through Problem Formulation?
5.4 Why Should Teachers Be Problem Posers?
5.5 Teacher as Problem Poser and a Mediator of Teaching and Learning Situations Through the Formulation of Problems
References
10 Problem Posing Among Preservice and Inservice Mathematics Teachers
1 Introduction
2 Problem Solving
3 Problem Posing
4 Characterizing Word Problems in Mathematics
5 Conceptual Framework
6 Method
7 Results and Discussions
8 Conclusions and Recommendations
References
11 An Approach to Developing the Problem-Posing Skills of Prospective Mathematics Teachers: Focus on the “What if not” Heuristics
1 Introduction
2 Theoretical Background
2.1 Problem-Posing
2.2 Problem-Posing in Teacher Education
2.3 Evaluation of the Problem-Posing Products
2.4 Hungarian Traditions of Mathematical Competitions and Competition Preparation
3 Method
3.1 Research Question
3.2 Participants
3.3 Research Design
3.4 Evaluation of Problem-Posing Products
4 Results
4.1 The Cognitive Demand for the Product
4.2 The Product’s Attributes that Have Changed
4.3 Heuristic Strategies and the Mathematical Background
5 Discussion
5.1 Aptness
5.2 Fluency, Flexibility, and Novelty
5.3 The Look-Back Phase
5.4 The Role of the Control and Mathematics Resources
6 Conclusion
7 Limitations and Further Research
Appendix: Problems Applied During the Expert Group Intervention
References
12 Regulation of Cognition During Problem Posing: A Case Study
1 Problem Posing and Problem Solving
2 Research Background
3 Research Framework
3.1 Regulation of Cognition
3.2 Phases in Problem Posing
4 Research Design
4.1 Case Study
5 Analysis
5.1 Property Noticing
5.2 Problem Construction
5.3 Checking Solution
5.4 Looking Back
6 Concluding Remarks
References
13 Problem Posing in Pósa Problem Threads
1 Introduction
2 Background: Problem Threads and Problem Posing
2.1 Problem Threads
2.2 Question Posing Heuristics
3 Problem Thread on Symmetry
3.1 Phase 1: Problem Solving
3.2 Phase 2: Problem Posing (Free Exploration & Learning a New Concept)
3.3 Phase 3: Problem Solving
3.4 Phase 4: Problem Posing (Using an Analogous Concept & Question Aesthetics)
3.5 Phase 5: Problem Solving
3.6 Phase 6: Problem Posing (Arising from the Need for Precision)
3.7 Phase 7: Problem Solving
3.8 Phase 8: Problem Posing (Restricting the Topic)
3.9 Phase 9: Problem Solving
3.10 Phase 10: Problem Posing (Looking for Complexity)
3.11 Phase 11: Problem Solving
3.12 Phase 12: Problem Posing (as a Means of Assessment)
3.13 Phase 13: Problem Solving
4 Conclusion
References
14 Conclusion: Mathematics Problem Posing and Problem Solving: Some Reflections on Recent Advances and New Opportunities
1 Diverse Perspectives on Mathematics Problem Posing and Problem Solving
2 Teaching Mathematics Through and with Problem Posing and Problem Solving
3 Students’ Mathematical Disposition and Agency
4 Teachers’ Mathematics Problem Posing and Problem Solving
5 Concluding Thoughts
References