Fostering Collateral Creativity in School Mathematics: Paying Attention to Students’ Emerging Ideas in the Age of Technology (Mathematics Education in the Digital Era, 23)

This book explores the topic of using technology, both physical and digital, to motivate creative mathematical thinking among students who are not considered ‘mathematically advanced.’ The book reflects the authors’ experience of teaching mathematics to Canadian and American teacher candidates and supervising several field-based activities by the candidates. It consists of eight chapters and an Appendix which includes details of constructing computational learning environments. Specifically, the book demonstrates how the appropriate use of technology in the teaching of mathematics can create conditions for the emergence of what may be called ‘collateral creativity,’ a notion similar to Dewey’s notion of collateral learning. Just as collateral learning does not result from the immediate goal of the traditional curriculum, collateral creativity does not result from the immediate goal of traditional problem solving. Rather, mathematical creativity emerges as a collateral outcomeof thinking afforded by the use of technology. Furthermore, collateral creativity is an educative outcome of one’s learning experience with pedagogy that motivates students to ask questions about computer-generated or tactile-derived information and assists them in finding answers to their own or the teacher’s questions. This book intends to provide guidance to teachers for fostering collateral creativity in their classrooms. Preface References Contents 1 Theoretical Foundation and Examples of Collateral Creativity 1.1 Introduction 1.2 Theories Associated with Collateral Creativity 1.3 Collateral Creativity and the Instrumental Act 1.4 Three More Examples of Collateral Creativity 1.4.1 A Second Grade Example of Collateral Creativity 1.4.2 A Fourth Grade Example of Collateral Creativity 1.4.3 Collateral Creativity in a Classroom of Secondary Mathematics Teacher Candidates 1.5 Collateral Creativity as Problem Posing in the Zone of Proximal Development 1.6 Forthcoming Examples of Collateral Creativity Included in the Book References 2 From Additive Decompositions of Integers to Probability Experiments 2.1 Introduction 2.2 Artificial Creatures as a Context Inspiring Collateral Creativity 2.3 Iterative Nature of Questions and Investigations Supported by the Instrumental Act 2.4 The Joint Use of Tactile and Digital Tools Within the Instrumental Act 2.5 Tactile Activities as a Window to the Basic Ideas of Number Theory 2.6 Historical Account Connecting Decomposition of Integers to Challenges of Gambling References 3 From Number Sieves to Difference Equations 3.1 Introduction 3.2 On the Notion of a Number Sieve 3.3 Theoretical Value of Practical Outcome of the Instrumental Act 3.4 On the Equivalence of Two Approaches to Even and Odd Numbers 3.5 Developing New Sieves from Even and Odd Numbers 3.6 Polygonal Number Sieves 3.7 Connecting Arithmetic to Geometry Explains Mathematical Terminology 3.8 Polygonal Numbers and Collateral Creativity References 4 Explorations with the Sums of Digits 4.1 Introduction 4.2 About the Sums of Digits 4.3 Years with the Difference Nine 4.4 Calculating the Century Number to Which a Year Belongs 4.5 Finding the Number of Years with the Given Sum of Digits Throughout Centuries 4.6 Partitioning n into Ordered Sums of Two Positive Integers 4.7 Interpreting the Results of Spreadsheet Modeling References 5 Collateral Creativity and Prime Numbers 5.1 ‘Low-Level’ Questions Require ‘High-Level’ Thinking 5.2 Twin Primes Explorations Motivated by Activities with the Number 2021 5.3 Students’ Confusion as a Teaching Moment and a Source of Collateral Creativity 5.4 Different Definitions of a Prime Number 5.5 Tests of Divisibility and Collateral Creativity 5.6 Historically Significant Contributions to the Theory of Prime Numbers 5.6.1 The Sieve of Eratosthenes 5.6.2 Is There a Formula for Prime Numbers? References 6 From Square Tiles to Algebraic Inequalities 6.1 Introduction 6.2 Comparing Fractions Using Parts-Within-Whole Scheme 6.3 Collateral Creativity: Calls for Generalization 6.4 Collaterally Creative Question Leads to the Discovery of “Jumping Fractions” 6.5 Algebraic Generalization 6.6 Seeking New Algorithms for the Development of “Jumping Fractions” References 7 Collateral Creativity and Exploring Unsolved Problems 7.1 Exploring Palindromic Number Conjecture in the Middle Grades 7.2 The 196-Problem as a Possible Counterexample to Palindromic Number Conjecture 7.3 Formulation of Collatz Conjecture and Its History 7.4 Introducing the Rule: Initial Steps in Conjecturing 7.5 Deepening Investigation: Some Possible Paths Using a Spreadsheet 7.6 Fibonacci Numbers Emerge 7.7 More on the Role of a Teacher in Supporting Problem Posing by Students References 8 Egyptian Fractions: From Pragmatic Uses of Technology to Epistemic Development and Collateral Creativity 8.1 Introduction 8.2 Brief History of Egyptian Fractions 8.3 Egyptian Fractions as a Context for Conceptualizations of Fractional Arithmetic 8.4 The Greedy Algorithm and Some Practice in Proving Fractional Inequalities 8.5 Pizza as a Context for Introducing Egyptian Fractions 8.6 Semi-fair Division of Pizzas 8.7 The Joint Use of Wolfram Alpha and a Spreadsheet References Appendix Part 1: Answers to Challenging Questions Introduced in the Chapters of the Book Part 2 (for Chap. 8): Constructing Fraction Circles Using the Geometer’s Sketchpad Part 3: Programming of Spreadsheets