An empirically-based trajectory for fostering abstraction of equivalent-fraction concepts: A study of the Learning Through Activity research program

Melike Kara, Martin A. Simon, Nicora Placa

Research output: Contribution to journalArticle


Promoting deep understanding of equivalent-fractions has proved problematic. Using a one-on-one teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.

Original languageEnglish (US)
Pages (from-to)134-150
Number of pages17
JournalJournal of Mathematical Behavior
StatePublished - Dec 2018



  • Equivalent fractions
  • Guided reinvention
  • Hypothetical learning trajectories
  • Reflective abstraction
  • Reversibility
  • Task design

ASJC Scopus subject areas

  • Education
  • Applied Psychology
  • Applied Mathematics

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