Problem Posing and Solving for Mathematically Gifted and Interested Students: Best Practices, Research and Enrichment

✍ Scribed by Deniz Sarikaya (editor), Lukas Baumanns (editor), Karl Heuer (editor), Benjamin Rott (editor)

• Publisher: Springer Spektrum
• Year: 2023
• Tongue: English
• Leaves: 234
• Category: Library

✦ Synopsis

Mathematics and mathematics education research have an ongoing interest in improving our understanding of mathematical problem posing and solving. This book focuses on problem posing in a context of mathematical giftedness. The contributions particularly address where such problems come from, what properties they should have, and which differences between school mathematics and more complex kinds of mathematics exist. These perspectives are examined internationally, allowing for cross-national insights.

✦ Table of Contents

Foreword
Contents
Contributors
Perspectives from Research
Four Mathematical Miniatures on Problem Posing
1 Introduction
2 Problem Posing in Mathematics Education—Definitions and Conceptions
3 Four Mathematical Miniatures
3.1 Problem Posing as Generating New Problems
3.2 Problem Posing as Reformulating a Given Problem for Problem Solving
3.3 Problem Posing as Reformulating a Given Problem for Investigation
3.4 Problem Posing as Constructing Tasks for Others
4 Conclusion
Many Chefs in the Kitchen—a Collaborative Model for Problem-Posing
1 Introduction
2 The Project—Raising the Bar in Mathematics Classrooms
3 Problem 1—Jaime’s grade
3.1 The Initial Version of the Problem and What Triggered Its Inception
3.2 Revisions in Design and in Implementation
3.3 Retrospective Reflection on the Problem and Its Development
4 Problem 2—Positive or False Positive?
4.1 The Initial Version of the Problem and What Triggered Its Inception
4.2 Revisions in Design and in Implementation
4.3 Retrospective Reflection on the Problem and Its Development
5 Problem 3—The Relative Location of Median, Altitude, and Angle Bisector in Triangles
5.1 Impetus for Initial Version
5.2 Revisions in Design and in Implementation
5.3 Retrospective Reflection on the Problem and Its Development
6 Discussion
6.1 Eliciting Tensions and Conflicts in Perspectives On Problems and On Problem Solving
6.2 A Cumulative Notion of “Attractive” Problem
6.3 Risk-Taking in Developing Problems—the Role of Confidence and Ownership
7 In Summary
References
Would Specialist Problem Posers Endorse Problem-Posing Situations that We Design for Learners? Does It Matter?
1 Introduction
2 The Caveats of Research into Educational and Specialist PP
3 PP-Requiring Tasks and Situations in Educational PP
4 Where EPPMCs’ Problems Come from
4.1 Research and Its Problems
4.2 EPPMCs’ Triggers
5 Juxtaposition
6 From Research on Specialist PP to Educational PP
References
The Impact of an Enrichment Course in Mathematics on Students’ Problem-Solving Skills, Creativity, and Attitudes Towards Learning Mathematics
1 Introduction
2 Theoretical Background
2.1 Problem Solving
2.2 Creativity
3 The Study
3.1 Research Goal and Questions
3.2 The Enrichment Course
3.3 Research Participants
3.4 Research Tools
4 Findings
References
Questions About Fostering and Identifying of Mathematically Promising Students in Times of Covid-19 Pandemic
1 Introduction
2 About Our Fostering Concept
2.1 Taken Together
3 About Our Talent Search Process
4 What we do in the Prima-Project to Identify Mathematically Promising Students?
4.1 How can we Identify Mathematically Promising Students in Times of Corona Pandemic?
4.2 Instead of Trial Lessons
4.3 Problem 1
4.4 Problem 2
4.5 Some Examples for Solutions
4.6 About the Mathematics Test
4.7 Considerations About the Process
5 Taken Together
References
Problem Posing as an Integral Part for the Support of Mathematically Highly Gifted Teenagers Within the PriSMa Math Circles
1 Enrichment Programmes for Highly Gifted Students in the German Federal State of Hamburg
2 The PriSMa Math Circles
2.1 Basic Aspects of the Concept of PriMa and PriSMa Math Circles
2.2 Tasks for the PriSMa Math Circles
3 Problem Posing as an Integral Part of the PriSMa Math Circles
3.1 Classifications of Problem Posing
3.2 Problem Posing Within the PriSMa Math Circles
4 Example Tasks and Associated Problems Posed by Teenagers
4.1 Description of Task: “Eels”
4.2 Problems Posed by Students
5 Summary and Implications
References
Perspectives from Enrichment Programs: Best Practices
How to Use Motion as a Problem-Solving Tool? Problems from the Pósa Camps
1 Introduction
1.1 Brief Overview of the Tradition of the Hungarian Gifted Education
1.2 About the Pósa Camps
2 The Motion Problem-Thread
2.1 Introduction
2.2 Initial Problem—Introducing the Idea of Motion as a Tool for Solving Problems
2.3 Problem Group “The Halving of 100 Points”
2.4 Where Should the School be Built?
2.5 Twelve Couples at a Round Table 4/4
2.6 Further Selected Problems Without Discussion
2.7 Motion as a Subject, a Different Problem-Thread
3 Discussion
3.1 General Remarks About the Method
3.2 Future Direction of the Pósa Method
References
Using Maths Competition Problems at Camps
1 Introduction
1.1 The Georg Mohr Contest
1.2 Science Talenter’s Camps
1.3 Structure of the Paper
2 Background
2.1 Camps and Students
3 Learners’ Reactions to Competition Problems
4 Teaching Goals and Their Relation to Learners’ Reactions
4.1 Teaching Mathematical Language
5 Sense-Making and Ideation in Georg Mohr Problems
5.1 Sense-Making by Seeing the Answer
5.2 Getting an Idea by Changing Perspective or Form of Representation
5.3 Getting an Idea by Doing Examples, Being Systematic, Spotting a Pattern
5.4 Getting an Idea by Interpreting Dense, Mathematical Text
6 Teaching with Competition Problems
7 Maths Fairs: Making the Learners Pose the Problems
7.1 Riddle Market
7.2 Explain a Number Trick
8 Problem Posing for Ideas and Explanations
9 Conclusion
References
Teaching Units for Mathematical Enrichment Activities
1 Mathematical Enrichment Activities at the University of Flensburg
2 Discoveries in a 10-Adic Number World
2.1 Introduction
2.2 Concise Mathematical Presentation of the Topic
2.3 Teaching Implementation
2.4 Experiences and Notes from the Perspective of Mathematics Education
2.5 Final Remarks
3 Combinatorial Checkerboard Problems for Kids
3.1 Introduction
3.2 Implementation for Pupils
3.3 Discussion
4 Elementary Length Formulas for Triangles and Quadrilaterals
4.1 Introduction
4.2 Length Formulas for Triangles
4.3 Length Formulas for Quadrilaterals
4.4 Discussion