Development of teaching unit of proportional reasoning from the perspective of learning process of individual pupils in the mathematics classroom

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The purpose of this study was to capture the quality and mechanism of change in pupils' proportional reasoning through classroom teaching in upper elementary grade level. The change was examined both in the long and short spans. During one school year, three written tests were conducted in three 5^<th>-grade classes of different elementary schools. Pupils' approaches for proportional problems became multiplication/division based ; their use of addition/subtraction became less and less. One of the classes where pupils developed more sensible and divergent strategies than the other two was examined further in micro level. Here, the focus was put on the interactions between the change and mathematical notations that were introduced and treated in class. Two research questions were set ; (i) What were the mathematical notations that were relevant to pupils' change in proportional reasoning? and (ii) In what ways did such mathematical notations emerge, and in what processes did they exert such relevance through classroom teaching? For this purpose, a framework of "internalization of mathematical notations" was used, which characterizes person's evolving use of notations as three phases : "use as a label," "forming criteria for use" and "purposeful use." It was adopted to analyze the cases of five focused pupils under the teaching of "Quantity Per Unit." Based on the results obtained by the study, implications were made for teaching, together with the development of teaching units of "Quantity Per Unit" and ''Proportion." Tasks for the future were also clarified.