# Ellis (2007) JRME

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

The article *Connections Between Generalizing and Justifying: Students' Reasoning with Linear Relationships* was written by Amy Ellis and published in the *Journal for Research in Mathematics Education* in 2007. The article is available from JSTOR at http://www.jstor.org/stable/30034866.

## 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.

## Outline of Headings

- The Actor-Oriented Approach to Generalization and Justification
- Generalization Taxonomy
- Proof Schemes

- Methods
- Participants
- The Teaching Experiment
- Data Analysis

- Results
- The Four Mechanisms for Change
- Mechanism 1: Action/Reflection
- Mechanism 2: Focus
- Mechanism 3: Generalizations That Promote Deductive Reasoning
- Mechanism 4: Influence of Deductive Reasoning on Generalizing

- Episodes on Equivalent Speed Ratios
- Episode 1: The Struggle to Explain Equivalent Ratios
- Mechanisms for Change for Episode 1
- Mechanism 1: Action/Reflection
- Mechanism 2: Focus

- Mechanisms for Change for Episode 1
- Episode 2: Timothy Makes Connections to the Quantities
- Mechanisms for Change for Episode 2
- Mechanism 1: Action/Reflection
- Mechanism 2: Focus
- Mechanism 3: Generalizations that promote deductive reasoning
- Mechanism 4: Influence of deductive reasoning on generalizing

- Mechanisms for Change for Episode 2
- Episode 3: Larissa and Maria Connect to Quantities
- Mechanisms for Change for Episode 3 and General Discussion
- Mechanism 1: Action/Reflection
- Mechanism 2: Focus
- Mechanism 3: Generalizations that promote deductive reasoning
- Mechanism 4: Influence of deductive reasoning on generalizing

- Mechanisms for Change for Episode 3 and General Discussion
- Final Remarks on the Episodes

- The Four Mechanisms for Change
- Discussion and Conclusion

## Also

- APA
- Ellis, A. B. (2007). Connections between generalizing and justifying: Students' reasoning with linear relationships.
*Journal for Research in Mathematics Education*, 38(3), 194–229. - BibTeX

@article{Ellis2007, author = {Ellis, Amy B.}, journal = {Journal for Research in Mathematics Education}, number = {3}, pages = {194--229}, title = {{Connections between generalizing and justifying: Students' reasoning with linear relationships}}, volume = {38}, year = {2007} }