Abstract
In school mathematics, one aim of explaining a proposition should be to show the reason why the proposition has universal validity. However , that aim is not being sufficiently met. One reason seems to be that students cannot make use of the means necessary to achieve it. To achieve the aim of clear explanation, we can use as our means the content (What do we explain?) and the representation (How do we represent the content?). Our content is to reason the proposition from universal assumptions deductively. As to representation, I focus on "actions with material objects" in this paper. I set up the research problems as follows. What should a student do to show the reason why the proposition has universal validity by acting with material objects? I will offer the following conclusion in this paper. A student should do the following to show the reason why a proposition has universal validity by acting with material objects. i) Doing various actions with material objects to reach a conclusion from assumptions. ii) Finding invariant properties or relations with a possible suggestion as to common features within the action processes. iii) Producing the necessary actions forwardly and/or backwardly with a possible suggestion as to invariant properties or relations. iv) Organaizing actions from assumptions to conclusion on the basis of successive performance. v) Expanding the range to apply the organized actions.
