A Pragmatic Considerations of Collective Mathematical Understanding Process in a Small Group: Focusing on the Transformation of Interpretation
2020 Volume 26 Issue 1 Pages 13-27
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2020 Volume 26 Issue 1 Pages 13-27
The purpose of this paper is to construct the framework grasping collective mathematical understanding process (MUP). Then, analysis and consideration of practical example is shown by them. For this purpose, two big theoretical ideas are used; one is the transcendental recursive theory and other is the classification of mathematical representation register (MMR). The framework capturing collective MUP is constructed by them. In addition, learner thinking can be divided into three semiotic ones. They are semantic thinking, syntactic thinking, and pragmatic thinking. The semiotics refined by Morris have been classified into three categories to analyze the semiotic process. However, these are not only used to analyze of the symbolic process, but it is also possible to assume that humans think from a similar perspective when interpreting the sign. On the other hand, when learners in a small group interpret, there are various types. There are five main cases when classifying learner situations that appear during group learning; Individual-type, absorption-type, confrontation-type, agreement-type, idling-type.
Taking into account a number of factors as described above, a framework describing a collective MUP was constructed. Until now, PK model (Pirie and Kieren, 1994b) was able to capture only one-dimensional MUP of an individual, but the framework created in this paper can visualize two-dimensional MUP of a group. By analyzing the practical case described, it was visualized the transition of the registers used in the group and the appearance of folding back. From this aspect, the following features were found. First, learners tend to use syntactic thinking to raise their level of mathematical understanding. Second, if the group produces many agreement-type or absorption-type interpretations, it is possible to judge that a good group mind is working. Third, semantic thinking tended to be heavily used when the conversion between registers occurred.
