Conceptions and Consequences of Mathematical Argumentation, Justification, and Proof (Research in Mathematics Education)

Foreword
Acknowledgments
Contents
Contributors
Introduction: Conceptualizing Argumentation, Justification, and Proof in Mathematics Education
	Conceptions of Argumentation, Justification, and Proof in the Literature
		Argumentation
		Justification
		Proof
	Charges Given to Authors/Synthesizers
	Organization of This Volume
	References
Part I: Argumentation, Justification and Proof in an Elementary Classroom
	Overview of the Elementary Level Data
		Ms. Kirk’s Lesson
		Description of the Lesson
		Conclusion
		References
	Argumentation in the Context of Elementary Grades: The Role of Participants, Tasks, and Tools
		Introduction
		Arguments and Argumentation
			Argumentation in the Classroom
			Mathematical Content for Argumentation
		Brief Summary of the Lesson
		Analyzing the Lesson Transcript
			The Role of Definitions, Established Facts, and Common Language
			The Role of Participants (Teacher and Students)
			The Role of the Task
			The Role of the Tool in Terms of Evidence and Potential Connections
		Discussion
			The Role of Definitions and Common Language
			The Role of Participants
			The Role of the Task
			The Role of Tools
		Conclusion
		References
	Justification in the Context of Elementary Grades: Justification to Develop and Provide Access to Mathematical Reasoning
		Role of Justification in the Mathematics Classroom
		Student Justifications and Justification Tasks
		Potential Relationships Between Justification and Equity
		Summary
		Methods
		The Nature of the Tasks
		Students’ Justifications
		Providing Access to Participation Through Justification
		Discussion
		References
	Proof in the Context of Elementary Grades: A Multimodal Approach to Generalization and Proof in Elementary Grades
		Introduction
		Theoretical Framework
			Definition of Proof
			Generalization and Proof
			The Multimodal Nature of Proof
			Modal-Specific Epistemological Commitments and Generalization
		Method
		Data Analysis and Findings
			Excerpt 1: Defining the Equals Sign
				Summary
				Multimodality
			Excerpt 2: Discussion of 14 + 3 = 15 + 2
				Summary
				Generalization
				Multimodality
			Excerpt 3: Jacob’s Proof of 40 + 9 = 27 + 22
				Summary
				Generalization
				Multimodality
		Discussion
		Conclusion
		References
	On the Meanings of Argumentation, Justification, and Proof: General Insights from Analyses of Elementary Classroom Episodes
		References
Part II: Argumentation, Justification and Proof in a Middle Grades Classroom
	Overview of Middle Grades Data
		References
	Argumentation in the Middle Grades: Exploring a Teacher’s Support of Collective Argumentation
		Exploring Collective Argumentation in the Mathematics Classroom
		Teacher Support for Collective Argumentation Framework
		Methodology
			Limitations
			Data Analysis Procedures
		Results
			Episode of Argumentation and Teacher Support
			Directly Contributing an Argumentation Component
			Asking Questions Prompting an Argumentation Component
			Other Supportive Actions
		What We Learned from Mr. MC’s Facilitation of Argumentation
		Conclusion: Larger Spheres of Argumentation
		Appendix A: Sample from Spreadsheet Deconstructing Argument
		References
	Justification in the Context of Middle Grades: A Process of Verification and Sense-Making
		Theoretical Framework
		Literature Review
			The Teacher’s Role in Promoting Justification
			Justification as a Learning Practice
		Method
		Results
			Justification Process
			Factors That Shaped the Justification Process
				Task as Written and Setup
				Task as Implemented
		Discussion
			What Might Students Be Learning About the Justification Process?
		Conclusion
		References
	Proof in the Context of Middle Grades: Can We Label Middle School Arguments as Proof with a Capital P?
		Introduction
		Reframing the Definition of Proof for the Current Context
		Methods
		Findings
			Classroom Episode
				Students’ Written Work
				Summary
			Is a Proof Accessible to This Classroom Community?
			Data from Another Project
		Conclusion
		References
	Argumentation, Justification, and Proof in Middle Grades: A Rose by Any Other Name
		Introduction
		The Task and Its Possibilities
			Potential Proving-Related Activities
			Potential Instructional Goals
		Framing Matters
		Capturing the Breadth of Proving-Related Activity
		References
Part III: Argumentation, Justification and Proof in High School Mathematics
	Overview of High School-Level Data
		Overview of the Data
		Part 1
		Part 2
	Argumentation in the Context of High School Mathematics: Examining Dialogic Aspects of Argumentation
		Introduction
		Conceptual Framework
			Students’ Dialogic and Justifying Moves
			Communicative Approaches and Structuring Argumentation
		Methods
		Results
			Students’ Dialogic and Justifying Moves
			Structuring Argumentation and Using Communicative Approaches
		Discussion
			Dialogic Argumentation and Teacher Support
			Dialogic Argumentation in Studying Mathematical Argumentation
		References
	Justification in the Context of High School: Co-constructing Content and Process
		Introduction
		Theoretical Perspectives
		Methods
		Findings
			How Were Norms for Justification Enacted and Co-constructed?
			Which Mathematical Ideas Were Justified, and Which Mathematical Ideas Were Left Unjustified?
			To What Extent Were the Members of the Classroom Community Convinced by the Justifications Provided?
		Discussion
		References
	Proof in the Context of High School: A First Approach Through Discussion, with Occasions and Missed Opportunities
		Introduction and Background
		Method
		Results
			Analysis of the Task
			Analysis of the Class Discussion
				Part 1: Discovering the Formula
				Part 2: Constructing Triangles
				Part 3: Towards a Formula (Again)
		Discussion
		Conclusions
		References
	Reasoning Is in the Eye of the Lens-Holder: Observations Made Through the Lenses of Justification, Argumentation, and Proof at the High School Level
		Introduction
		Defining and Framing Different Forms of Reasoning
		The Research Questions and Their Influence
		Framing of the Task and the Classroom Activities on Days 1 and 2
		A Series of Commonly Analyzed Episodes
			Key Episode 1: Discussion of Solution to the Day 1 Task (Lines 1–35)
			Key Episode 2: Dividing a Quadrilateral into Triangles (Lines 111–130)
			Key Episode 3: Returning to the Day 1 Function (Lines 247–272)
			Looking Across the Key Episodes
			Considering the Non-common (Gray) Episodes
		Concluding Discussion
		References
Part IV: Argumentation, Justification and Proof at the Tertiary Level
	Overview of Tertiary Level Data
		The Mathematics Involved in the Lesson
		Outline of the Class Discussion with Selected Excerpts of Transcript
		Conclusion
		References
	Argumentation in the Context of Tertiary Mathematics: A Case Study of Classroom Argumentation and the Role of Instructor Moves
		Theoretical Frameworks and Analysis Methods
		Results
			Vignette 1: How to Fill in the Cayley Table
			Vignette 2: Leveraging Specific Activity Toward General Discussion About Associativity
			Vignette 3: Instructor Solicits More Rules, but These Are Specific to the Relationships Between R and F as Symmetries of a Triangle
			Vignette 4: Demonstrating the New Rule and Moving Toward Incorporation of New Rules
		Discussion
		References
	Justification in the Context of Tertiary Mathematics: Undergraduate Students Exploring the Properties and Relations of the Dihedral Group
		Introduction
		Defining and Operationalizing Justification
		Context for the Data Provided to Us
		Research Questions
		Method of Analysis
		Results
			Category 1: Perceptual Pattern Recognition Justifications
			Category 2: Recounting of Process Justifications
			Category 3: Mathematically Rationalized Justifications
			A Note Regarding Source of Justifications in the Classroom
		Discussion
		References
	Proof in the Context of Tertiary Mathematics: Undergraduate Inquiry-Based Learning in Abstract Algebra as a Precursor to Mathematical Proof
		Methods
		Data and Results
		Discussion
		References
	Mathematics Educators as Polymaths, Brokers, and Learners: Commentary on the Tertiary Chapters on Argumentation, Justification, and Proof
		Mathematics Educators as Polymaths: Definitions Have Natural Histories
		Mathematics Educators as Brokers: Adapting Definitions to Balance Obligations
			Brokers Between Mathematics Instruction and Mathematics Research
			Brokers Between Mathematics Education and Psychology
			Brokers Between Researchers and Instructors
		Mathematics Educators as Learners: Accumulating Knowledge of Complex Phenomena
		References
Part V: Lenses on Researching Argumentation, Justification and Proof Across the Grade Levels
	Participation in Argumentation: Teacher and Student Roles Across the Grades
		Opening Spaces for Student Agency and Ownership
		Responding to Others’ Ideas by Building and Critiquing
		Multiple Frameworks for Teacher Actions
		Considering Argumentation as Larger than Mathematics
		Consequences of the Definition
		Concluding Thoughts
		References
	Justification Across the Grade Bands
		Introduction
		Using the Definition of Justification
		Frameworks and Classification Schemes: Capturing and Making Sense of Classroom Activity
		Justification as a Social Process That Serves a Purpose in the Community
		Unique Contributions of Justification as a Construct
		References
	Lens, Blinders, or Kaleidoscope? Using a Definition of Proof to Make Sense of Classroom Activity
		Introduction
		Operationalizing the Definition
			Elementary Grades Operationalization
			Middle Grades Operationalization
			Secondary Grades Operationalization
			Postsecondary Operationalization
			Two Tensions Reflected in These Operationalizations
				Community Tension
				Process/Product Tension
		Insights Drawn from the Analyses
			Elementary Grades Insights
			Middle Grades Insights
			High School Insights
			Postsecondary Insights
		Our Synthesis of the Four Chapters
			The Multifaceted Nature of Proof in the Classroom
			The Emergent Nature of Proof in the Classroom
			The Role of the Teacher
			Operationalizations Have a Massive Impact on Resulting Findings
			Lens, Blinder, or Kaleidoscope?
		Conclusions
		References
	Conclusion: Considering the Consequences of Our Conceptions of Argumentation, Justification, and Proof
		Consequences of Our Definitions of Argumentation, Justification, and Proof
			Argumentation
			Justification
			Proof
		Moving Forward
		References
Index