Mathematical Challenges For All

Contents
Chapter 1: Introduction to Mathematical Challenges for All Unraveling the Intricacy of Mathematical Challenge
	1.1 Introduction
	1.2 Challenge as a Springboard to Human Development
	1.3 Mathematical Challenge
	1.4 Model of Factors Influencing Mathematical Challenge
	1.5 Explaining the Model Using Works Presented in This Volume
	1.6 Concluding Notes and Questions for Future Research
	References
Part I: Mathematical Challenges in Curriculum and Instructional Design
Editors
	Chapter 2: Introduction to Part I of Mathematical Challenges For All: Mathematical Challenges in Curriculum and Instructional Design
		References
	Chapter 3: Development and Stimulation of Early Core Mathematical Competencies in Young Children: Results from the Leuven Wis & C Project
		3.1 Introduction
		3.2 A Research Project Consisting of Four Parts
		3.3 Early Mathematical Patterning
		3.4 Early Computational Estimation
		3.5 Early Proportional Reasoning
		3.6 Early Probabilistic Reasoning
		3.7 Conclusion
		References
	Chapter 4: Mathematical Modelling as a Stimulus for Curriculum and Instructional Reform in Secondary School Mathematics
		4.1 Mathematical Modelling in the School Curriculum
		4.2 Analytical Framework: Curriculum Policy, Design, and Enactment
		4.3 Case Study 1: Modelling as a Stimulus for Mathematics Curriculum Reform
			4.3.1 Background to Applied Mathematics Curriculum Reform
			4.3.2 Factors That Support or Hinder the Implementation of Modelling in the Applied Mathematics Curriculum
		4.4 Case Study 2: Modelling as a Stimulus for Mathematics Instructional Reform
			4.4.1 Background to the Young Modellers Transition Year Project
			4.4.2 Factors Supporting or Hindering the Implementation of Modelling in the Young Modellers Transition Year Project
		4.5 Discussion
		References
	Chapter 5: Personalized Mathematics and Mathematics Inquiry: A Design Framework for Mathematics Textbooks
		5.1 Introduction
		5.2 Role of Mathematics Textbooks
			5.2.1 The Cypriot Mathematic Textbooks
		5.3 Design Models
		5.4 The Structure of the PMMI Framework
			5.4.1 Personalized Mathematics
			5.4.2 Mathematics Inquiry
				5.4.2.1 Mathematics Inquiry Phases
				5.4.2.2 Exploration: Romance
				5.4.2.3 Investigations: Precision and Generalization
		5.5 The PMMI and the 4C’s
		5.6 The PMMI and the Role of the Teacher
		5.7 Conclusion
		References
	Chapter 6: Math-Key Program: Opening Mathematical Minds by Means of Open Tasks Supported by Dynamic Applets
		6.1 Introduction
		6.2 Openness of Math-Key Tasks
			6.2.1 Examples of Math-Key Tasks
				6.2.1.1 Task 1: Birthday Pa rty
				6.2.1.2 Task 2: Distance from School
				6.2.1.3 Task 3: Car Speed
				6.2.1.4 Task 4: Polygon from Two Squares
				6.2.1.5 Task 5: Average Test Grade
				6.2.1.6 Task 6: Tiles on a Sq uare
				6.2.1.7 Task 7: Expressions of Parabolas
			6.2.2 Solution Spaces and Mathematical Challenges Embedded in Math-Key Program Tasks
		6.3 Varying Mathematical Challenge with Dynamic Applets
			6.3.1 Math-Key Applets
			6.3.2 Examples of Math-Key Applets
				6.3.2.1 Applet for Task 1: Birthday Party Applet
				6.3.2.2 Applet for Task 2: Distance from School
				6.3.2.3 Applet for Task 3: Car Speed
				6.3.2.4 Applet for Task 4: Polygons from Two Squares
				6.3.2.5 Applet for Task 5: Average Grades
				6.3.2.6 Applet for Task 6: Tiles on a Square
				6.3.2.7 Applet 7: Expressions of Parabolas
			6.3.3 Math-Key Applets Characteristics
		6.4 Summary
		References
	Chapter 7: Making Mathematics Challenging Through Problem Posing in the Classroom
		7.1 Introduction
		7.2 Challenging and Worthwhile Instructional Tasks
		7.3 Posing Problems Properly: From Routine Problem Solving to Non-routine Problem Solving
		7.4 Students and Problems: From Solving to Posing
		7.5 Problem-Posing Tasks
			7.5.1 Problem Situations in Problem-Posing Tasks
				7.5.1.1 Real-Life Context Examples
				7.5.1.2 Purely Mathematical Context Examples
			7.5.2 Prompts in Problem-Posing Tasks
		7.6 Teachers Learning to Teach Mathematics through Problem Posing
			7.6.1 Changing Beliefs and Increasing Knowledge About Teaching Through Problem Posing
			7.6.2 Equipping Teachers to Develop Problem-Posing Tasks
			7.6.3 Supporting Teachers to Develop Teaching Cases
		7.7 Summary and Looking to the Future
			7.7.1 Problem-Posing Tasks
			7.7.2 Teaching Through Problem Posing
			7.7.3 Supporting Teachers to Learn to Teach Through Problem Posing
			7.7.4 Problem Posing and Mathematical Challenge
		References
	Chapter 8: Challenging Students to Develop Mathematical Reasoning
		8.1 Introduction
		8.2 Exploratory Approach and Teacher Actions
		8.3 Mathematics Reasoning
		8.4 Methodology
			8.4.1 Grade 8 Lesson: Edges of Pyramids and Prisms
				8.4.1.1 The Task
				8.4.1.2 Launching and Autonomous Work
				8.4.1.3 Whole-Class Discussion
			8.4.2 Grade 9 Lesson: Comparing Areas of Rectangles
				8.4.2.1 The Task
				8.4.2.2 Launching and Autonomous Work
				8.4.2.3 Whole-Class Discussion
		8.5 Conclusion
		References
	Chapter 9: Mathematical Argumentation in Small-Group Discussions of Complex Mathematical Tasks in Elementary Teacher Education Settings
		9.1 Introduction
		9.2 Mathematical Argumentation
			9.2.1 Components of Mathematical Arguments
			9.2.2 Collective Mathematical Argumentation in the Classroom
		9.3 Classroom Discussions
			9.3.1 The Quality Talk Model
			9.3.2 QT Professional Development Model
			9.3.3 Educator Professional Vision
			9.3.4 Discussion Model Components
		9.4 Adapting Quality Talk to Develop Mathematical Argumentation in Elementary Mathematics Teacher Education Settings
		9.5 Collective Mathematical Argumentation: Two Discussion Excerpts
			9.5.1 A Discussion in a Methods Course
			9.5.2 A Discussion in an Elementary Classroom
		9.6 Discussion and Conclusions
		References
	Chapter 10: Commentary to Part I of Mathematical Challenges For All: Commentary on ‘Challenge’ in Terms of Curriculum Materials and Tasks, the Teacher’s Role and the Curriculum
		10.1 Innovations in Textbooks, Curriculum Materials and Tasks to Promote the Mathematical Challenge
		10.2 The Teacher’s Role in Promoting Mathematical Challenge
		10.3 Understanding What Constitutes an Appropriately Challenging Curriculum for Learners
		10.4 Conclusion
		References
Part II: Kinds and Variation of Mathematically Challenging Tasks
Editor
	Chapter 11: Introduction to Part II of Mathematical Challenges For All: Many Faces of Mathematical Challenge
		11.1 What Is a Mathematical Challenge?
		11.2 In the Chapters
		11.3 Conclusion
		References
	Chapter 12: Probing Beneath the Surface of Resisting and Accepting Challenges in the Mathematics Classroom
		12.1 Introduction
		12.2 Human Psyche
			12.2.1 Six Aspects of the Human Psyche
			12.2.2 Initiating Action
			12.2.3 Psycho-Social-Coordinations
		12.3 Resisting, Accepting and Parking Challenges
			12.3.1 Recognising Challenge
			12.3.2 Responding to Challenge
				12.3.2.1 Accepting
				12.3.2.2 Rejecting
				12.3.2.3 Resisting
				12.3.2.4 Deferring or Parking: Letting-Go, Hanging-on, and Pausing
				12.3.2.5 Giving-Up
		12.4 An Indication of Pedagogical Issues
		12.5 Final Reflections
		References
	Chapter 13: Mathematical Challenge in Connecting Advanced and Secondary Mathematics: Recognizing Binary Operations as Functions
		13.1 Introduction
		13.2 Literature
			13.2.1 Connecting Advanced and Secondary Mathematics in Secondary Teacher Education
			13.2.2 Connections
			13.2.3 Binary Operations and Functions: A Mathematical Connection
		13.3 Methodology
			13.3.1 The Binary Operations Task
		13.4 Findings
			13.4.1 Four Conceptual Stages
			13.4.2 Mathematical Challenge as a Pedagogically Powerful Activity
		13.5 Discussion
		13.6 Conclusion
		References
	Chapter 14: Mathematical Challenge of Seeking Causality in Unexpected Results
		14.1 Introduction
		14.2 Intellectual Need
		14.3 Seeking Causality: Three Examples
			14.3.1 Example 1: Rope around Earth
			14.3.2 Example 2: Horizontal Translation of a Parabola
			14.3.3 Example 3: Division by a Fraction
			14.3.4 The Need for Causality as a Challenge
		14.4 Creating Challenge in Simple Tasks
		14.5 Teachers Responding to the Five-Digits Task
			14.5.1 The Five-Digits Task
			14.5.2 Determining the Largest Product
			14.5.3 Follow-Up Activity: The Challenge of Causality
		14.6 Mathematicians Responding to the Five-Digits Task
			14.6.1 Interview with Ada
			14.6.2 Interview with Ben
			14.6.3 Comments on the Mathematicians’ Approaches
		14.7 Proceeding with a Self-Challenge
			14.7.1 The Case of Five Consecutive Digits
			14.7.2 General Case
			14.7.3 Detour: On a Simpler but Similar Task
		14.8 Discussion
		References
	Chapter 15: Visualization: A Pathway to Mathematical Challenging Tasks
		15.1 Introduction
		15.2 The Mathematics Classroom
			15.2.1 The Teacher and the Tasks
			15.2.2 Challenging Tasks
		15.3 The Potential of Visualization
		15.4 Visual Contexts and Challenge in Mathematics
			15.4.1 Example 1: Visual Solutions in Problem-Solving
			15.4.2 Example 2: Visual Solutions in Proof
			15.4.3 Examples in Pre-service Teacher Training
			15.4.4 Symmetries
			15.4.5 The Vasarely Rhombus
			15.4.6 Rational Numbers
			15.4.7 Paper Folding: The Cube
			15.4.8 The Cup
		15.5 Concluding Remarks
		References
	Chapter 16: Towards a Socio-material Reframing of Mathematically Challenging Tasks
		16.1 Introduction
		16.2 Tasks in Digital Technology Environments
		16.3 Digital Technology-Inflected Mathematical Challenge
		16.4 Mathematical Challenging Tasks Using TouchCounts
			16.4.1 The Initial Task: Distributing Candies Amongst Children
			16.4.2 The Follow-Up Task: Comparing 12 and 18 as Multiples of 6
		16.5 Conclusion
		References
	Chapter 17: Creativity and Challenge: Task Complexity as a Function of Insight and Multiplicity of Solutions
		17.1 Introduction: Problem-Solving
		17.2 Creativity and Mathematical Ability
		17.3 Insightful Problem-Solving
		17.4 Divergent Problem-Solving
		17.5 Insight-Requiring or Insight-Allowing Problems
		17.6 Research Experiment
			17.6.1 The Tasks
			17.6.2 Findings
		17.7 Summary
		Appendix 1
		Appendix 2 Model and Scoring scheme for the evaluation of creativity (based on Leikin, 2009b)
		References
	Chapter 18: Challenging Undergraduate Students’ Mathematical and Pedagogical Discourses Through MathTASK Activities
		18.1 Welcoming Mathematics Undergraduates to Mathematics Education
		18.2 A Mathematics Education Course for Mathematics Undergraduates: Theoretical Foundations
		18.3 A Mathtask: Students Discuss How to Solve an Algebraic Inequality
		18.4 The Course: Context, Objectives, Structure, Activities and Assessment
		18.5 A Research Study of Student Responses to a Mathtask: Participants, Data Collection and Data Analysis Method
		18.6 Analysis of Student Responses to a Mathtask
			18.6.1 RMD in Responses to the Mathematical Challenge of the Mathtask
			18.6.2 Engaging with the RME Routine of Referencing Relevant Literature (Explicitly or Implicitly)
			18.6.3 Endorsing the RME Narrative of the Importance of Considering Social Interactions During Mathematical Activity
			18.6.4 Ritualized Engagement with RME Theory and Findings
			18.6.5 RME Theory as a Descriptor of Pedagogical Prescription
		18.7 How Facing the MC and PC in Mathtasks Works as a Boot-Camp Experience for Newcomers into RME Discourse
		References
	Chapter 19: Commentary on Part II of Mathematical Challenges For All: Making Mathematics Difficult? What Could Make a Mathematical Challenge Challenging?
		References
Part III: Collections of Mathematical Problems
Editor
	Chapter 20: Introduction to Part III of Mathematical Challenges For All: In Search of Effectiveness and Meaningfulness
		References
	Chapter 21: Problem Collections and “The Unity of Mathematics”
		21.1 Problem Collections: To What End?
		21.2 Problematized Curriculum Development
			21.2.1 Real Additive Groups
			21.2.2 Discrete Real Additive Groups (DA)
			21.2.3 Extensions
			21.2.4 Discussion
		21.3 Problems that Collectively Explore Diverse Aspects, Representations, and Applications of a Focal Mathematical Context/Space/Phenomenon
			21.3.1 The (ℝeal) Euclidean Algorithm (EA)
			21.3.2 The Euclidean Square-Tiling of an (a × b)-Rectangle
			21.3.3 Fair Distribution
			21.3.4 The Diagonal of a (c × s)-Rectangle
			21.3.5 Tony Gardener’s Game of Euclid (Gardiner, 2002)
		21.4 Problems that Are “Isomorphic,” i.e. Structurally the “Same” Problem, Even Though Their Contexts, Even Their Mathematical Domains, May Be Quite Distinct (See Bass, 2017)
			21.4.1 Isomorphic Problems
			21.4.2 An Example, and a Question
			21.4.3 Degrees of Sameness; Discernment Tasks
			21.4.4 A More Subtle Example of Common Structure
			21.4.5 The Measure Exchange Common Structure Problem Set
		21.5 Mathematically Distinct Problems Reducible to a Common Mathematical Model
			21.5.1 The Expanded Usiskin Problem Set
			21.5.2 Presentation to the Students
			21.5.3 The Student Presentations: Slowly Raising the Curtain
			21.5.4 Relation to the Classification of Platonic Solids
		21.6 Problems Unexpectedly with the Same Answer, for Deep Structural Reasons
			21.6.1 Walks on the Positive Half-Line
			21.6.2 Binary Rooted Trees
			21.6.3 Associations
			21.6.4 Triangulations
			21.6.5 Multi-Theorem
		21.7 Concluding Discussion
		References
	Chapter 22: Meeting the Challenge of Heterogeneity Through the Self-Differentiation Potential of Mathematical Modeling Problems
		22.1 Introduction
		22.2 Theoretical Background
			22.2.1 Mathematical Modeling—Modeling Problems and Modeling Processes
			22.2.2 Heterogeneity, Differentiation, and Self-Differentiation
		22.3 Theoretical Analysis of the Self-Differentiation Potential of a Modeling Problem
		22.4 Methodology and Design of the Study
		22.5 Results of the Study
			22.5.1 Analysis of the Groups
			22.5.2 Empirically Confirmed Self-Differentiation Potential in Setting Up the Real Models
			22.5.3 Empirically Confirmed Self-Differentiation Potential When Building the Mathematical Model
			22.5.4 Empirically Confirmed Self-Differentiation Potential in Mathematical Work
			22.5.5 Empirically Confirmed Self-Differentiation Potential When Interpreting the Mathematical Results
			22.5.6 Empirically Confirmed Self-Differentiation Potential in the Validation of the Solutions
			22.5.7 Reflection
		22.6 Summary and Discussion
		References
	Chapter 23: Taiwanese Teachers’ Collection of Geometry Tasks for Classroom Teaching: A Cognitive Complexity Perspective
		23.1 Introduction
		23.2 Analytical Framework
			23.2.1 Diagram Complexity Dimension
			23.2.2 Problem-Solving Complexity Dimension
			23.2.3 Analysis Examples of GCN Tasks
		23.3 Methodology
			23.3.1 Selection of Teachers
			23.3.2 Data Collection
			23.3.3 Data Analysis
		23.4 Results
			23.4.1 Collections of Types of Mathematical Instructional Tasks
			23.4.2 Cognitive Complexity of the Tasks Collected by Taiwanese Teachers
			23.4.3 Cognitive Complexity of Tasks Situated in Sources of Curricular/Instructional Materials
		23.5 Discussion
		References
	Chapter 24: Problem Sets in School Textbooks: Examples from the United States
		24.1 Introduction
		24.2 On Problem Solving in Schools: A Historical Observation
		24.3 Analysis of Problem Sets: Certain Methodological Considerations
		24.4 Robinson’s Geometry Textbook
		24.5 Algebra by Robinson
		24.6 Textbook by William Hart
		24.7 Textbook by Larson et al.
		24.8 Discussion
		24.9 Conclusion
		References
	Chapter 25: Exams in Russia as an Example of Problem Set Organization
		25.1 Introduction
		25.2 Certain General Considerations
		25.3 Russian Exit Exams in Algebra Before 1917
		25.4 Soviet Exit Exams in Algebra
		25.5 The Age of Changes and Experiments
			25.5.1 Changes in Ministry of Education Exams
			25.5.2 St. Petersburg Exams
			25.5.3 Centralized Testing
			25.5.4 Collection of Problems
		25.6 The Uniform State Exam
		25.7 Discussion and Conclusion
		References
	Chapter 26: Complexity of Geometry Problems as a Function of Field-Dependency and Asymmetry of a Diagram
		26.1 Introduction
		26.2 Background
			26.2.1 Field-Independence-Dependency (FID)
			26.2.2 The Need to Develop a New FID Instrument Specific to Geometry
			26.2.3 Symmetry
		26.3 Development of Geometry Field-Dependency-Symmetry (GFDS) Instrument
		26.4 Examining the GFDS Instruments and the Research Hypotheses
			26.4.1 Subjects, Setting, and Data Analysis
			26.4.2 Findings
		26.5 Discussion
		References
	Chapter 27: Structuring Complexity of Mathematical Problems: Drawing Connections Between Stepped Tasks and Problem Posing Through Investigations
		27.1 Introduction
		27.2 Problem Posing Through Investigations
			27.2.1 Characterization of PPI
			27.2.2 Example of the Space of Posed Problems
			27.2.3 Complexity of PPI Processes and Outcomes
			27.2.4 Investigation Strategies
			27.2.5 Using DGE for Investigation and Discovery
			27.2.6 On the Structure of the Set of Posed Problems
		27.3 Stepped Tasks
			27.3.1 Characterization of Stepped Tasks
		27.4 Concluding Notes
		References
	Chapter 28: Flow and Variation Theory: Powerful Allies in Creating and Maintaining Thinking in the Classroom
		28.1 Introduction
			28.1.1 Diamonds
			28.1.2 Answers
		28.2 Engagement and the Optimal Experience
		28.3 Maintaining Flow
		28.4 Building Thinking Classrooms
			28.4.1 Thinking Tasks
			28.4.2 Visibly Random Groups (VRG)
			28.4.3 Vertical Non-permanent Surfaces (VNPS)
			28.4.4 Using Hints and Extensions to Maintain Flow
		28.5 Research Questions and Methodology
		28.6 Case I: Factoring Quadratics
			28.6.1 Thinking
			28.6.2 Tasks
		28.7 Case II: Solving One and Two-Step Equations
			28.7.1 Thinking
			28.7.2 Tasks
		28.8 Case III: Radical Expressions
			28.8.1 Thinking
			28.8.2 Tasks
		28.9 Cross Case Analysis
			28.9.1 Brevity of Introduction
			28.9.2 Maintaining Flow
		28.10 Conclusion
		References
	Chapter 29: Commentary on Part III of Mathematical Challenges For All: On Problems, Problem-Solving, and Thinking Mathematically
		29.1 Framing a Major Point of This Chapter
		29.2 Learning to Implement Heuristic Strategies
		29.3 Metacognition: Monitoring and Self-Regulation
		29.4 Problems as a Mechanism for Countering Unproductive Student Beliefs
		29.5 Problems that Invite Multiple Solutions, an Antidote to “Answer Getting”
		29.6 Transfer of Authority: Who Determines What’s True?
		29.7 Problems that Are Generative: And a Bonus, for Developing a Sense of Mathematical Initiative
		29.8 This Year and the Years to Come
		29.9 Concluding Thoughts
		References
Index