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.
The study of fractions in elementary school begins in the second grade and continues every year through the sixth grade. However, there is an interval of about a year before they learn fractions in ...the following school year, and there are concerns about entrenching and forgetting what they have learned. We conducted a survey on the recognition of unit fractions among fourth-grade children, about one year after learning fractions in the third grade, in order to identify learning characteristics in the third grade of children who were able to identify some of the fractions based on unit fractions, and to obtain suggestions for learning guidance. Only children who could write following the objectives, children who only added the numerator, and children who were biased toward the division operation were included in the actual analysis. We attempted to document how the children in their third year learned fractions. This implies that beginning in the second semester of third grade, the viewpoint of relying on division begins to shift to the viewpoint of unit fractions, and beginning in the fourth semester, the viewpoint converges to the viewpoint of unit fractions.
The objective of this paper is to identify, using case examples, how the composition process of revision and refinement of mathematical knowledge relates to changes in the intention behind statements ...present in mathematical negotiations. To this end, analysis and consideration were conducted to clarify the methodology for the analysis of mathematical negotiations against the theoretical background of agreement constructivism, as well as to plan and conduct experimental lessons on the “division of fraction” unit in the 6th grade of elementary school. In this paper the children’s opinions in that experimental lesson are analyzed and considered by using the analysis frame created by T. Shimomura (in press). The results of the analysis and consideration revealed the following two points. First, differences in the intention behind statements were observed in conjunction with changes in the direction of the process of adapting to the “world” and “oneself” in the composition stage of mathematical knowledge. Second, the intention behind statements used to reach a consensus in the form of mathematical knowledge differed depending on the implications of the existence of others.
