Abstract
In this research we approach to the understanding processes in generalization of mathematical concepts based on the "Optimizing Equilibration Model of Mathematical Understanding". In this paper, in particular, we will clarify the processes through which children understand inclusion relations between quadrilaterals and generalize their schema of parallelogram. Difficulties in understanding inclusion relations are not consisted in mathematical logic, but in such children's tacit property that each angle of rectangle is not 90°. We have discussed means that they overcome the difficulties and understand inclusion relations, and indicated that the following means are effective. 1. conceiving geometrical figures dynamically by manipulating on operative material and seeing continual transformations from parallelogram to rectangle or rhombus 2. understanding conservation of properties in the transformations 3. understanding negative properties in the transformations 4. analogy to the inclusion relation between parallelogram and rhombus 5. analogy to the inclusion relation between rhombus and square We also clarified that after understanding the content, children's schema of parallelogram becomes to be in the states of the followings. 1. Properties of a geometrical figure are organized, and children can define a geometrical figure by using those properties. 2. Children conceive properties of a geometrical figure as common properties between geometrical figures. 3. A geometrical figure is a bearer of their properties. These states are effective for children to learn proof in lower secondary school. Therefore, children should learn inclusion relations between quadrilaterals in primary school.
