Quantitative Reasoning in Mathematics and Science Education

Introduction
Contents
Quantitative Reasoning as an Educational Lens
	1 Origins of a Theory of Quantitative Reasoning and Its Applicability
	2 Chapters in This Book
	3 Conceptualizing Units and Conceptualizing Quantification: Aspects of Quantitative Reasoning Needing Greater Attention
		3.1 Quantification of Interest Rate as a Rate of Change
		3.2 Quantification of Kinetic Energy
	4 Connections with Chapters in This Book
	5 Conclusion
	References
An Intellectual Need for Relationships: Engendering Students’ Quantitative and Covariational Reasoning
	1 Theorizing Quantitative and Covariational Reasoning
	2 Mathematizing as a Way of Thinking Emerging from Students’ Intellectual Need
	3 An Intellectual Need for Relationships
	4 Four Facets of an Intellectual Need for Relationships
	5 Attributes in a Situation: What Are the Things?
	6 Measurability of Attributes: How Can Things Be Measured?
	7 Variation in Attributes: How Do Things Change?
	8 Relationships Between Attributes: How Do Things Change Together?
	9 Task Design Considerations: A Ferris Wheel “Techtivity”
	10 Discussion
	11 Conclusion
	References
Abstracted Quantitative Structures: Using Quantitative Reasoning to Define Concept Construction
	1 Introducing the Abstracted Quantitative Structure Construct
		1.1 Quantitative Reasoning
		1.2 Covariational Reasoning
		1.3 Figurative and Operative Thought
		1.4 Three Forms of Re-presentation
	2 Further Defining and Illustrating the Abstracted Quantitative Structure Criteria
		2.1 Empirical Illustrations
	3 Discussion and Implications
		3.1 Research Implications
		3.2 Teaching Implications
	References
Number Systems as Models of Quantitative Relations
	1 Learning the Meanings of Words
	2 Giving Meaning to Number Words
	3 Rational Numbers and the Situation/Action-Schema Pair
		3.1 Part-Whole Situations and Relevant Action Schemas
		3.2 Ratio Situations and Relevant Action Schemas
		3.3 Intensive Quantities and Relevant Action Schemas
	4 Concluding Remarks
	References
Quantitative Reasoning as a Framework to Analyze Mathematics Textbooks
	1 Curriculum and Textbooks
	2 Quantitative Reasoning and Whole Number Multiplication and Division
	3 Quantitative Reasoning: A Theoretical Model for the Examination of Textbooks
	4 Analysis of Japanese Curricular Materials
	5 Whole Number Multiplication in Mathematics International Textbooks
	6 Whole Number Division in Mathematics International Textbooks
	7 Discussion
	References
Constructing Covariational Relationships and Distinguishing Nonlinear and Linear Relationships
	1 Introduction
	2 Theoretical Background
		2.1 Foundations of Covariational Reasoning
		2.2 Using Direction and Amounts of Change to Conceive the Basic Types of Covariational Relationships and Distinguish Between Nonlinear and Linear Relationships
		2.3 Representing Covariational Relationships Graphically
	3 Methods, Participants, and Analysis
		3.1 Subjects and Setting
		3.2 Data Analysis
	4 Building to Nonlinear and Linear Growth: A Task Sequence with Student Work
		4.1 The Faucet Task: Gross Covariational Reasoning and Emergent Thinking
		4.2 The Growing Triangle Task
		4.3 The Growing Trapezoid Task: An Increasing by Less Relationship
		4.4 The Triangle/Rectangle Task
	5 Discussion
		5.1 Middle School Students’ Covariational Reasoning
		5.2 Task Design in Relation to Our Theoretical Framework
		5.3 Implications for Developing Other Mathematical Ideas
		5.4 Concluding Remarks and Areas for Future Research
	References
A Conceptual Analysis of Early Function Through Quantitative Reasoning
	1 Introduction: Functions as Rates of Change
	2 The Case of Linear and Quadratic Growth: Conceptual Analysis
		2.1 Identify the Attribute to Be Measured
		2.2 Identify the Quantities Affecting the Relevant Attributes
		2.3 Imagine Gross Coordination and Coordination of Values
		2.4 Quantify Covariation
	3 Data Examples: Students’ Reasoning with Linear and Quadratic Growth
		3.1 Identifying the Attribute and the Quantities
		3.2 Imagining Gross Coordination and Coordination of Values
		3.3 Quantifying Covariation
	4 Task Design Principles for Supporting Function Reasoning Through Covariation
		4.1 Leverage Contexts with Continuously Covarying Quantities
		4.2 Develop Covariation Before Allowing Calculation
		4.3 Choose Exact Relationships
		4.4 Choose Genuine Contexts
		4.5 Provide Opportunities for Visualization, Manipulation, and Justification
	5 Conclusion
	References
Geometric Transformations Through Quantitative Reasoning
	1 Geometric Transformations Through Quantitative Reasoning
		1.1 Isometries from a Purely Mathematical Standpoint
		1.2 Importance of Geometric Transformations
		1.3 Difficulties in Understanding Transformations
		1.4 Quantitative Reasoning and Covariational Reasoning
		1.5 Understanding Points as Multiplicative Objects and Conceptualizing mathbbR2 Quantitatively
		1.6 Conceptualizing Translations Through Quantitative Reasoning and Covariational Reasoning
		1.7 Conceptualizing Rotations Through Quantitative Reasoning
		1.8 Conceptualizing Reflections Through Quantitative Reasoning
	2 Discussion
	References
Instructional Conventions for Conceptualizing, Graphing and Symbolizing Quantitative Relationships
	1 Orienting to a Problem
	2 Introduction
	3 The Need for Conventions to Facilitate Changes in Pedagogy and Student Success
	4 Elaborating Quantitative and Covariational Reasoning
	5 Speaking with Meaning: A Convention for Improving Instructors’ Communication
	6 Scaling the Convention of Speaking with Meaning Across the Pathways Project
		6.1 Speaking with Meaning in Instruction and Curriculum
		6.2 Emergent Shape Thinking and Conventions for Meaningful Graphing Activity
	7 Pathways Conventions for Graphing
	8 Implementing the Quantity Tracking Tool
	9 Emergent Symbol Meaning and Conventions for Meaningful Symbolization Activity
	10 Quantitative Reasoning and Algebra
		10.1 Emergent Symbolization
		10.2 Emergent Symbolization in Instruction and Curriculum
	11 An Example of Unproductive Beliefs in Action
		11.1 Quantitative Drawing and Building Imagery for Quantitative Relationships
	12 Discussion
	13 Concluding Remarks
	References
Mathematization: A Crosscutting Theme to Enhance the Curricular Coherence
	1 Defining Mathematization
	2 The Learning Progression for Mathematization of Science
	3 The LP for Mathematization of Science
	4 Evidence for Mathematization to Be Used as a Crosscutting Theme
	5 Conclusions
	References
Applying Quantitative and Covariational Reasoning to Think About Systems: The Example of Climate Change
	1 Introduction
	2 The Earth’s Energy Budget
	3 Conceptual Framework
		3.1 Systems Thinking Competencies
		3.2 Quantitative and Covariational Reasoning
	4 The Context of the Study
	5 Quantitative Reasoning and Understanding Climate Change
		5.1 Preliminary Work: Making Sense of (Unfamiliar) Quantities
		5.2 Making Sense of the Energy Budget as a System Quantitatively
		5.3 Conceptualizing Dynamic Relationships and Cyclical Processes
	6 Conclusions and Implications
	References
Operationalizing and Assessing Quantitative Reasoning in Introductory Physics
	1 Introduction
	2 Operationalizing Physics Quantitative Literacy
		2.1 Quantitative Modeling in Physics
		2.2 Facets of Quantitative Reasoning in Introductory Physics
	3 Assessable PQL Learning Objectives
		3.1 Methodology
		3.2 Sequence-Level Learning Objectives
	4 The Physics Inventory of Quantitative Literacy
	5 Conclusion
	References