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Connections Between Generalizing and Justifying: Students' Reasoning with Linear Relationships

Amy B. Ellis 

May 2007, Volume 38, Issue 3, Page 194

Abstract:
Research investigating algebra students' abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students’ errors, less is known about what students do understand to be general and convincing. This study examines the ways in which seven middle school students generalized and justified while exploring linear functions. Students' generalizations and proof schemes were identified and categorized in order to establish connections between types of generalizations and types of justifications. These connections led to the identification of four mechanisms for change that supported students' engagement in increasingly sophisticated forms of algebraic reasoning: (a) iterative action/reflection cycles, (b) mathematical focus, (c) generalizations that promote deductive reasoning, and (d) influence of deductive reasoning on generalizing. 

Keywords:
Grade 5
Grade 6
Grade 7
Grade 8
Algebra
Article
Conceptual Development
Connections / Applications
Math Coordinator / Coach
Teacher
Problem Solving / Problem Posing
Survey Studies
Skill and Fluency
Reasoning and Proof
Representation