A Taxonomy for Categorizing Generalizations: Generalizing Actions and Reflection Generalizations
Abstract
This article presents a cohesive, empirically grounded categorization system differentiating the types of generalizations students constructed when reasoning mathematically. The generalization taxonomy developed out of an empirical study conducted during a 3-week teaching experiment and a series of individual interviews. Qualitative analysis of data from teaching sessions with 7 seventh-graders and individual interviews with 7 eighth-graders resulted in a taxonomy that distinguishes between students' activity as they generalize, or generalizing actions, and students' final statements of generalization, or reflection generalizations. The three major generalizing action categories that emerged from analysis are (a) relating, in which one forms an association between two or more problems or objects, (b) searching, in which one repeats an action to locate an element of similarity, and (c) extending, in which one expands a pattern or relation into a more general structure. Reflection generalizations took the form of identifications or statements, definitions, and the influence of prior ideas or strategies. By locating generalization within the learner's viewpoint, the taxonomy moves beyond casting it as an activity at which students either fail or succeed to allow researchers to identify what students see as general, and how they engage in the act of generalizing.
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“Traditionally, generalization process has been linked to the use of algebraic symbolism and much of the previous research has focused on the students’ difficulties in generalizing (Dienes, 1961; Ellis, 2007).”
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- Authors
- Ellis, A. B.
- Affiliations
- University of Wisconsin–Madison (US)
- Corresponding author
- Ellis, Amy B.
- ORCID
- Ellis, Amy B.
- Venue
- Journal of the Learning Sciences
- Publisher
- Informa UK Limited
- Published
- 2007-03-28
- Volume
- 16
- Issue
- 2
- Pages
- 221-262
- ISSN
- 1050-8406 (ISSN-L)
- 1532-7809
- OpenAlex
- W1996850046
- Semantic Scholar
- 6f95faaad5e6155ce2660d8cc939eec9fb53f4d9
- Cited by
- 127
- References
- 62
- Citations by year
- 2026: 2 · 2025: 4 · 2024: 8 · 2023: 6 · 2022: 6 · 2021: 19 · 2020: 15 · 2019: 8 · 2018: 8 · 2017: 9 · 2016: 3 · 2015: 8 · 2014: 5 · 2013: 3 · 2012: 9
- FWCI
- 17.41
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of the Learning Sciences (publishedVersion)
- Topics
- Social Sciences › Education › Mathematics Education and Teaching Techniques
- Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
- Social Sciences › Education › Science Education and Pedagogy
- Keywords
- mathematical generalization, mathematical reasoning, seventh grade students, eighth grade students, patterns, teaching experiment
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