### Measurement Approach to Teaching Fractions: A Design Experiment in a Pre-service Course for Elementary Teachers

#### Abstract

In this paper, we present a design experiment in a “Teaching Mathematics” course for prospective elementary teachers where we sought to develop a *measurement approach* to fractions. We focus on the conceptualization of the mathematical content of the approach. We attribute our progress in the conceptualization to our efforts to overcome the challenges we had to face in bringing our students – prospective teachers – to thinking about fractions in a theoretical way. We describe some of these challenges in the paper. The approach was inspired by an approach under the same name, but addressed to children, developed by the psychologist V.V. Davydov and described in the paper by Davydov and Tsvetkovich (1991). Davydov’s measurement approach proposes that, in order to develop a concept of fraction with sources in reality, children’s attention should be directed to multiplicative relationships between quantities defined in terms of concrete units (such as kilos, inches or cups) rather than to indeterminate objects such as pizzas or cakes. Our original contribution is a systemic study of these relationships and operations on them, as a theoretical system, using definitions derived from measurement situations, mathematical reasoning based on these definitions and generalizations of observed patterns. It is intended to support a gradual process of abstraction of the notion of fraction as an abstract number that represents a measure of the relationship between two quantities. Furthermore, our proposal for conceptualizing fractions in this way is addressed to teachers, not to children. By its focus on relational reasoning about quantities and gradual construction of a theoretical system, the approach both requires and is expected to foster the development of quantitative reasoning and theoretical thinking.

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