Characteristics of Mathematical Negotiation in the Process of Building a Fractional Concept:Based on an Analysis of Small Group Learning in Quantity Fraction

Abstract

Fractions of quantity are considered difficult to understand due to confusion with division fractions that were learned before. Therefore, this study focused on group learning when studying fractions where existing mathematical knowledge is superior. The aim was to characterize the mathematical negotiations influencing the selection of an original unit that is necessary to capture a fraction of quantity as a fraction of subordinate units. We conducted a survey on learning fractions of quantity in the third grade of elementary school. Further, we analyzed, both quantitatively and qualitatively, results of the pretests and intentions behind the discussions observed during group learning. Specifically, we extracted three groups from the experimental lesson and examined aspects of the discussions using an analytical framework we developed based on Searle’s speech act theory. The results showed two types of mathematical negotiations—integrated and distributed—which are necessary to select an original unit in a fraction of quantity. We also considered differences from discussions in which mathematical negotiations did not occur, and examined implications to provide learning guidance.