### Abstract

We report on a teaching experiment intended to foster a concept of multiplication that would both subsume students’ multiple-groups concept of whole number multiplication and provide a conceptual basis for understanding multiplication of fractions. The teaching experiment, which used a rigorous single-subject methodology, began with an attempt to build on students’ multiple-groups concept by promoting generalizing assimilation. This was not totally successful and led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to several significant conclusions, which in turn led to a revised hypothetical learning trajectory. The revised trajectory aims to foster a concept of multiplication as a change in units.

Original language | English (US) |
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Pages (from-to) | 151-173 |

Number of pages | 23 |

Journal | Journal of Mathematical Behavior |

Volume | 52 |

DOIs | |

State | Published - Dec 2018 |

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### Keywords

- Fraction learning
- Guided reinvention
- Learning trajectory
- Mathematics learning
- Reflective abstraction
- Task design

### ASJC Scopus subject areas

- Education
- Applied Psychology
- Applied Mathematics

### Cite this

*Journal of Mathematical Behavior*,

*52*, 151-173. https://doi.org/10.1016/j.jmathb.2018.03.002