Reflective abstraction, uniframes and the formulation of generalizations

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Abstract

In mathematics, generalizations are the end result of an inductive zigzag path of trial and error, that begin with the construction of examples, within which plausible patterns are detected and lead to the formulation of theorems. This paper examines whether it is possible for high school students to discover and formulate generalizations similar to ways professional mathematicians do. What are the experiences that allow students to become adept at generalization? In this paper, the mathematical experiences of a ninth grade student, which lead to the discovery and the formulation of a mathematical generalization are described, qualitatively analyzed and interpreted using the notion of uniframes. It is found that reflecting on the solutions of a class of seemingly different problem-situations over a prolonged time period facilitates the abstraction of structural similarities in the problems and results in the formulation of mathematical generalizations.

Original languageEnglish
Pages (from-to)205-222
Number of pages18
JournalJournal of Mathematical Behavior
Volume23
Issue number2
DOIs
StatePublished - 2004

Keywords

  • Abstraction
  • Combinatorics
  • Generalization
  • Mathematical structures
  • Problem solving
  • Reflection
  • Uniframes

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