Mathematical Teaching and Learning: Perspectives on Mathematical Minds in the Elementary and Middle School Years

Contents
Contributors
Abbreviations
Chapter 1: An Introduction to Mathematics Teaching and Learning in the Elementary and Middle School Years
	1.1 An Introduction to Mathematics Teaching and Learning in the Elementary and Middle School Years
	References
Part I: Pedogical Approaches to Teaching
	Chapter 2: Instructional Supports for Mathematical Problem Solving and Learning: Visual Representations and Teacher Gesture
		2.1 Introduction
		2.2 Teacher Gesture as an External Support for Attending to Instructionally Relevant Information
		2.3 Do Teachers’ Gestures Help Students Encode Instructionally Relevant Information?
		2.4 Diagrams as External Supports for Discerning Structure
		2.5 Implications for Educational Practice
		References
	Chapter 3: Equilibrated Development Approach to Word Problem Solving in Elementary Grades: Fostering Relational Thinking
		3.1 Introduction
		3.2 Theoretical Background
			3.2.1 Operational Paradigm
			3.2.2 Insights from Neuro-education and the Developmental Aspects of Learning
			3.2.3 Relational Paradigm
		3.3 Equilibrated Development Approach
			3.3.1 Activities to Promote Relational Thinking and Modeling
				3.3.1.1 Activity 1. Communicating the Mathematical Structure of a Problem: The Captain’s Game
				3.3.1.2 Activity 2. Mathematically Impossible Situations (MIS)
				3.3.1.3 Activity 3. Working in a Computer Environment
				3.3.1.4 Activity 4. Differentiation Between Additive and Multiplicative Relationships
		3.4 How It Works in Class
		3.5 Conclusion
		References
	Chapter 4: Experiences of Tension in Teaching Mathematics for Social Justice
		4.1 Introduction
		4.2 Background and Context
			4.2.1 The Colegio Context
			4.2.2 Introducing Nora
		4.3 Methodology and Data
			4.3.1 Nora’s Orientation to Mathematics Teaching and Learning
		4.4 Teaching Math for Social Justice
			4.4.1 Tensions in Teaching Math for Social Justice
		4.5 Nora’s Tensions
			4.5.1 Student Tensions: Fostering Success and Changing Orientation
			4.5.2 Collegial Tensions: Changing Orientation and Mathematical Being
		4.6 Implications for Teaching
			4.6.1 Elicited and Eliciting Tensions
			4.6.2 Preparing for Sticky Situations
		References
	Chapter 5: Designing Inclusive Educational Activities in Mathematics: The Case of Algebraic Proof
		5.1 Introduction
		5.2 Theoretical Framework
			5.2.1 Multimodal Approach
			5.2.2 Universal Design for Learning
			5.2.3 Formative Assessment
			5.2.4 Algebraic Proof
		5.3 Method: A Design-Based Approach
		5.4 From Theoretical Tools to Design: An Educational Sequence on Isoperimetric Rectangles
		5.5 The Teaching and Learning Sequence: An Overview
		5.6 Analysis
		5.7 Discussion
		5.8 Implications and Conclusions
		References
	Chapter 6: A Sustained Board Level Approach to Elementary School Teacher Mathematics Professional Development
		6.1 Introduction
		6.2 Professional Development (PD) Programming
		6.3 Assessing Mathematics Professional Development
		6.4 Provincial Context
		6.5 The Professional Development Sessions
		6.6 Results
			6.6.1 Changes in Mathematical Achievement
			6.6.2 Teacher Perceptions
		6.7 Discussion
			6.7.1 Success of the Professional Development Programming
			6.7.2 Limitations and Next Steps
		6.8 Implications for School Boards
		References
Part II: Mathematical Learning
	Chapter 7: A Digital Home Numeracy Practice (DHNP) Model to Understand the Digital Factors Affecting Elementary and Middle School Children’s Mathematics Practice
		7.1 Introduction
		7.2 Digital Home Numeracy Practice (DHNP)
			7.2.1 Theoretical Underpinnings for DHNP
				7.2.1.1 Cognitive-Communicative Model (CCM)
				7.2.1.2 Affordance Theory
				7.2.1.3 Embodied Cognition
			7.2.2 Design Features of Effective Educational Media
				7.2.2.1 Virtual Manipulatives
				7.2.2.2 Digital Feedback
				7.2.2.3 Digital Scaffolding
		7.3 The DHNP Model
			7.3.1 DHNP Outer Model
				7.3.1.1 Home Learning Environment (HLE)
				7.3.1.2 Home Numeracy Environment (HNE)
				7.3.1.3 Implicit Components of the HNE: Parental Factors, Child Factors and Parent-Child Relationships
				7.3.1.4 Explicit Components of the HNE
			7.3.2 DHNP Inner Model
				7.3.2.1 DHNP Components
		7.4 How Does the Proposed DHNP Model Contribute to Middle School Mathematics Education?
		7.5 Potential Avenues for Practical Implications on DHNP
		7.6 Summary
		References
	Chapter 8: How Number Talks Assist Students in Becoming Doers of Mathematics
		8.1 Introduction
		8.2 Conceptual Framework
			8.2.1 A Situative Perspective on Knowing and Learning
			8.2.2 The Role of Mental Computation During Number Talks
			8.2.3 Developing Sociomathematical Norms for Doing Mathematics
		8.3 Study Context
		8.4 How Did Number Talks Assist Ms. Jones’ Students in Becoming Doers of Mathematics?
			8.4.1 Building Agency by Establishing Sociomathematical Norms
			8.4.2 Shifting Authority Through Small Group Number Talks
			8.4.3 Learning How to Share Mathematical Reasoning During Whole Group Number Talks
		8.5 Discussion
		8.6 Implications for Teaching and Learning
		References
	Chapter 9: Language Matters: Mathematical Learning and Cognition in Bilingual Children
		9.1 Introduction
		9.2 Biological and Cultural Evolution of Mathematical Skills
		9.3 Bilingual Brains Process Information Differently
		9.4 Bilingual Mathematical Development
		9.5 Insights from Bilingual Mathematical Education in USA and Other Countries
			9.5.1 Frequency of Language Use
		9.6 Evidence-Based Recommendations
			9.6.1 Allowing Code-Switching
			9.6.2 Allowing Other ‘Off-loading’ Strategies
			9.6.3 Strengthen Retrieval of Mathematical Facts in Both Languages
			9.6.4 Incorporating Home and Cultural Contexts
			9.6.5 Mathematics Instruction in the Home Language
			9.6.6 Immersive Bilingual Programs or Structured Immersive Sessions
			9.6.7 Feedback and Culturally Relevant Mathematics Instruction
			9.6.8 Discussions About Mathematics and Culture
			9.6.9 Making Connections Between Mathematics and Aspects of Children’s Lives
			9.6.10 Confirmations from Non-USA Contexts
			9.6.11 Need for More Innovation and Research
		9.7 Conclusion
		References
	Chapter 10: Mathematical Creativity of Learning in 5th Grade Students
		10.1 Introduction
		10.2 Mathematical Creativity
		10.3 Creativity Techniques
		10.4 Workshop Model to Stimulate Creative Thinking in Mathematics
		10.5 Application of Mathematical Creativity
		10.6 Data Sources
		10.7 Results and Discussion
		10.8 Considerations and Implications
		References
	Chapter 11: Symbolic Mathematics Language Literacy: A Framework and Evidence from a Mixed Methods Analysis
		11.1 Symbolic Mathematics in Curriculum
		11.2 Symbolic Mathematics Language Literacy (SMaLL)
		11.3 Empirical Exploration of SMaLL Variations
			11.3.1 Quantitative Strand: Cognitive Evidence of SMaLL
			11.3.2 Qualitative Strand: Metacognitive Evidence of SMaLL
			11.3.3 Mixed Methods Integration: Multilevel Evidence of SMaLL
		11.4 Discussion and Conclusions
		References
	Chapter 12: Grasping Patterns of Algebraic Understanding: Dynamic Technology Facilitates Learning, Research, and Teaching in Mathematics Education
		12.1 Theories of Perceptual Learning and Embodied Cognition
		12.2 Leveraging Perceptual and Embodied Learning Within Graspable Math
			12.2.1 Perceptual Features Guide Students’ Attention to Notational Structures
			12.2.2 Transforming Abstract Symbols into Objects Makes Algebra Concrete for Learners
			12.2.3 Immediate Visual Feedback Informs Students’ Problem Solving
		12.3 Graspable Math: A Tool to Advance Theory, Research, and Practice
		12.4 Research on Mathematical Cognition and Student Learning
			12.4.1 Evidence of Student Learning in Graspable Math and From Here to There!
			12.4.2 Analyzing Students’ Problem-Solving Processes in Graspable Math
		12.5 Implications for Research and Education
		12.6 Conclusion
		References
Index