Mathematical Cognition and Understanding: Perspectives on Mathematical Minds in the Elementary and Middle School Years

Contents
Contributors
Abbreviations
Chapter 1: An Introduction to Mathematical Cognition and Understanding in the Elementary and Middle School Years
	1.1 Introduction
	References
Part I: Cognitive Factors
	Chapter 2: Infusing Spatial Thinking Into Elementary and Middle School Mathematics: What, Why, and How?
		2.1 Introduction
		2.2 What Is Spatial Thinking?
		2.3 Why Are Spatial Skills and Mathematics Related?
		2.4 Does Spatial Instruction Improve Mathematics Performance?
		2.5 How to Best Leverage the Space-Mathematics Association?
			2.5.1 Isolated Approaches to Spatial Training
			2.5.2 Embedded Approaches to Spatial Training
			2.5.3 Strengths, Limitations, and Theoretical Underpinnings
		2.6 Translating Theory to Practice: Infusing Spatial Training Into Mathematics Teaching
		2.7 Conclusion
		References
	Chapter 3: Understanding the Relationship Between Attention, Executive Functions, and Mathematics: Using a Function-Specific A...
		3.1 Introduction
		3.2 Attentional Abilities and Mathematics Proficiency
			3.2.1 Attention
				3.2.1.1 Attentional Abilities with Mathematics Learning Profiles
				3.2.1.2 Mathematical Abilities Within Attentional Profiles
			3.2.2 Executive Functioning
			3.2.3 Working Memory
			3.2.4 Processing Speed
			3.2.5 Cognitive Load Theory
		3.3 Intervention for Mathematics Remediation
			3.3.1 Task-Specific Intervention and Remediation
			3.3.2 Function-Specific Intervention and Remediation
				3.3.2.1 Indirect Intervention of Attention-Related Skills
				3.3.2.2 Direct Intervention on Attention
				3.3.2.3 Direct Intervention on Working Memory
		3.4 Discussion
			3.4.1 Future Research
			3.4.2 Implications
		3.5 Conclusion
		References
	Chapter 4: Instructional Support for Fact Fluency Among Students with Mathematics Difficulties
		4.1 Background
		4.2 Typical Developmental Trajectories
		4.3 Development of Fact Fluency Among Children with Mathematical Difficulties
		4.4 Importance of Developing Fact Fluency
		4.5 Evidence-Based Instructional Intervention Strategies for Fluency Building
		4.6 Incremental Rehearsal Strategy
			4.6.1 Research Supporting the Incremental Rehearsal Strategy
			4.6.2 Steps for Implementing Incremental Rehearsal
			4.6.3 Implications for Classroom Practice
		4.7 Summary
		References
	Chapter 5: The Development of Arithmetic Strategy Use in the Brain
		5.1 The Development of Arithmetic Strategies
		5.2 Arithmetic Strategies in the Developing Brain
			5.2.1 Brain Regions that Are Activated During Arithmetic Procedures
			5.2.2 Brain Regions that Are Activated During Arithmetic Fact Retrieval
		5.3 The Use of Educational Interventions and Experimental Paradigms to Study the Development of Arithmetic Strategies in the B...
			5.3.1 Arithmetic Educational Interventions
			5.3.2 Experimental Manipulations: Arithmetic Drill Studies
		5.4 Discussion
		References
	Chapter 6: The Role of Neuropsychological Processes in Mathematics: Implications for Assessment and Teaching
		6.1 The PASS Theory of Intelligence
		6.2 Operationalization of PASS Processes
		6.3 The Relation of PASS Processes with Mathematics Performance
		6.4 Clinical Use of CAS
			6.4.1 The Cognitive Profile of Children with Mathematics Giftedness
			6.4.2 The Cognitive Profile of Children with Mathematics Disabilities
		6.5 Interventions Based on PASS Theory
		6.6 Conclusion
		References
	Chapter 7: The Interplay Between Motivation and Cognition in Elementary and Middle School Mathematics
		7.1 Situated Expectancy-Value Theory
			7.1.1 Domain-Specific Cognitive Factors and Situated Expectancy-Value Theory
			7.1.2 Domain-General Cognitive Factors and Situated Expectancy-Value Theory
			7.1.3 Summary of Situated Expectancy-Value Findings
		7.2 Self-Determination Theory
			7.2.1 Domain-General Cognitive Factors and Self-Determination Theory
			7.2.2 Summary of Self-Determination Theory Findings
		7.3 Achievement Goal Theory
			7.3.1 Domain-General Cognitive Factors and Achievement Goal Theory
			7.3.2 Cognitive and Meta-cognitive Strategy Use and Achievement Goal Theory
			7.3.3 Summary of Achievement Goal Theory Findings
		7.4 Combining Cognition and Motivation: An Example Using Situated Expectancy-Value Theory
		7.5 Open Questions and Recommendations for Instructional Practice and Future Research
		Appendix A: Review Methodology
		References
	Chapter 8: Design Principles for Digital Mathematical Games that Promote Positive Achievement Emotions and Achievement
		8.1 Introduction
		8.2 Why Are Digital Mathematical Games Effective?
		8.3 Emotions
			8.3.1 Achievement Emotions
			8.3.2 Importance of Achievement Emotions
		8.4 Emotional Foundations of Digital Game Design
			8.4.1 Visual Aesthetic Design
			8.4.2 Musical Score
			8.4.3 Game Mechanics
			8.4.4 Narrative
			8.4.5 Incentive Systems
		8.5 Do Emotional Design Principles Promote Positive Achievement Emotions and Learning Outcomes?
			8.5.1 Review Process
			8.5.2 Which Emotional Design Principles Are Used in Mathematical Game Research?
			8.5.3 How Effective Are Emotional Design Principles at Improving Achievement Emotions and Learning Outcomes?
				8.5.3.1 Visual Aesthetic Design
				8.5.3.2 Musical Score
				8.5.3.3 Game Mechanics
				8.5.3.4 Narrative
				8.5.3.5 Incentive System
		8.6 Summary of Design Principles of Digital Mathematical Games´ Impact on Achievement Emotions and Achievement
		References
Part II: Mathematical Understanding
	Chapter 9: The Number Line in the Elementary Classroom as a Vehicle for Mathematical Thinking
		9.1 Introduction
		9.2 The Number Line
		9.3 Methodological Considerations
		9.4 The Instructional Sequence
		9.5 Results and Discussion
		9.6 Conclusion
		References
	Chapter 10: Longitudinal Approaches to Investigating Arithmetic Concepts Across the Elementary and Middle School Years
		10.1 Introduction
		10.2 The Importance of Conceptual Knowledge of Arithmetic
		10.3 How to Measure Conceptual Knowledge of Arithmetic
		10.4 The Development of Conceptual Knowledge of Arithmetic: Part I
		10.5 Study Designs for Assessing Conceptual Knowledge of Arithmetic
			10.5.1 The Cross-Sectional Design
			10.5.2 The Longitudinal Design
		10.6 Development of Conceptual Knowledge: Part II
		10.7 A Longitudinal Study of Additive and Multiplicative Inversion, Associativity, and Equivalence
		10.8 Future Directions and Practical Implications
		References
	Chapter 11: Obstacles in the Development of the Understanding of Fractions
		11.1 Introduction
		11.2 Conceptual and Procedural Knowledge of Fractions
		11.3 How Natural Number Knowledge Both Facilitates and Hinders Fraction Learning
		11.4 Educational Interventions and Implications for Teaching
			11.4.1 The Concrete-Representational-Abstract Sequence
			11.4.2 Playful and Game-Based Interventions
			11.4.3 Improving Pre-service Teachers Pedagogical Content Knowledge
			11.4.4 What Can Parents Do to Help Their Children Learn Fractions?
		11.5 Conclusion
		References
	Chapter 12: The Role of Groundedness and Attribute on Students´ Partitioning of Quantity
		12.1 Introduction
		12.2 The Role of Problem Characteristics in Word Problem Solving
		12.3 External Representations in Problem Solving
		12.4 Groundedness and Equal-Sharing
		12.5 Role of Object Attribute in Partitioning Strategies
		12.6 An Investigation of Children´s Partitioning Strategies as a Function of Problem Features
		12.7 Documenting Children´s Partitioning Strategies
			12.7.1 Equal-Sharing Problems
			12.7.2 Picture Perception Task
		12.8 Results
			12.8.1 Mental Representations
			12.8.2 Partitioning Strategies
		12.9 Discussion
			12.9.1 The Role of Object Groundedness
			12.9.2 The Role of Object Attribute
			12.9.3 The Role of Unit Type
		12.10 Practical Implications and Conclusion
		References
	Chapter 13: Designing Worked Examples to Teach Students Fractions
		13.1 Introduction
		13.2 Human Cognitive Architecture
		13.3 The Worked Example Effect
		13.4 The Worked Example Effect and Age Differences
		13.5 Empirical Evidence of the Effectiveness of Worked Examples with Lower Primary School Students
		13.6 Conclusion and Future Research
		References
	Chapter 14: Developing Fraction Sense in Students with Mathematics Learning Difficulties: From Research to Practice
		14.1 Domain Specific Concepts, Procedures, and Representations
			14.1.1 Why Are Fractions Hard for So Many Students?
			14.1.2 Fraction Magnitude and Equivalence
			14.1.3 Fraction Arithmetic
			14.1.4 Common and Persistent Fraction Arithmetic Errors
			14.1.5 Representations to Build Fraction Knowledge
		14.2 Techniques That Support Learning Across Domains
			14.2.1 Using Integrated Models
			14.2.2 Connecting Concrete and Abstract Representations of Concepts
			14.2.3 Using Gestures to Promote Learning
			14.2.4 Distributing and Interleaving Practice
			14.2.5 Providing Retrieval Practice with Corrective Feedback
			14.2.6 Presenting Side by Side Comparisons to Promote Relational Thinking
		14.3 Development of the FSI
			14.3.1 Description of the FSI
			14.3.2 Efficacy of the FSI
		14.4 Concluding Remarks
		References
Index