Abstract
We have conducted a research project "School Support Project" in an elementary school for about four months from September, 2010 to December, based on the methodology of "Design Research". One of the purposes of the project was to reform the sixth graders' mathematics lessons on the topic of division of fractions. A main concern of the lessons was to discuss with the students the calculation methods. We tried to reform the mathematics lessons based on the viewpoint "Mathematics as the science of patterns" known for the name of Erich Ch. Wittmann in mathematics education. Based on the view point, the word problems for introducing the division of fraction were selected from some practical research reports carefully. The learning trajectory was outlined based on the "Epistemological Triangle" devised by Heinz Steinbring. The research focus of this paper is to examine the effectiveness of the view point "Mathematics as the science of patterns" with reporting what happened in the classroom. The lessons were recorded with video cameras and the detailed transcripts were prepared for the analysis. The results of the analysis were summarized as follows: ・We were able to find many students who found patterns, checked them and formulated as calculation methods of division of fractions in the class. ・We could infer that the viewpoint of "Mathematics as the science of patterns" played an important role to reform the sixth graders' mathematics lessons on the topic of division of fractions. ・Therefore, the viewpoint of "Mathematics as the science of patterns" could work as a viewpoint by which primary teachers reform their everyday mathematics teaching. ・The word problems for introducing the division of fraction and series of equations generated from them were important to conduct the lessons compatible with the view of "Mathematics as the science of patterns". Because they played an important role as a Reference Context for the students. ・Moreover, it was important for teachers to organize the classroom interaction carefully from the view of "Mathematics as the science of patterns", especially the five realized viewpoints which are compatible with the view of "Mathematics as the science of patterns". ・The most important role of the viewpoints of "Mathematics as the science of patterns" is that it gives an elementary teacher who does not specialize in mathematics a basis to evaluate what is the essential element of the teaching of mathematics, and it enable him/her to reform his/her everyday teaching by oneself.
