Construction of an operational view of fractions

Abstract

The purpose of this study was to investigate a middle school teacher’s multiplicative fractional reasoning. One middle school teacher participated in a fifteen-week-long teaching experiment, one session per week, in two ways. The first way was observing videotaped thexcerpts of a 7-Grade student solving fractional tasks, and the second way was engaging in fractional problems of the same general types that the student solved. Each teaching episode lasted fifty minutes and was videotaped. By observing the student’s ways of thinking and by engaging in fractional problems, the teacher developed insight into the fractional operations of dividing and repeating. An ability to produce fractional amounts of quantities using these operations enabled the teacher to relate two quantities multiplicatively and to establish three levels of units. However, the fractional operations were not sufficient to develop a fractional multiplicative scheme. Differentiating between the operations and the anticipated results of operating was critical in establishing a fractional multiplicative relationship between two quantities. Based on these results, an operational view of fractions was defined as a reflective level of fractional reasoning that permits differentiating the anticipated results of fractional operations from the operations.