数学教育における一般化とその妥当性判断に関する考察 ─ 図の具体性を捨象することに着目して ―

抄録

　　In mathematics education the process of generalization is one of most important issue.　We define generalization as knowing process that has epistemological direction from particular to general.　It is distinguished from the process of extension.　Then, we focus on the process of validation in generalization. Because requisites for the validity of generalization and extension are completely different.　In this article, our interest is peculiar validation of generalization; that is, what particulars are involved in the general?

　　Proof and/or proving are necessary conditions for the validation, but they’re not sufficient conditions.　We focus on the“abstract from some things”and“abstract something”(Kant, 1781) in the process of abstraction．The former intends to leave from one’s perception; for this reason, previous studies have paid attention to the former.　By contrast, the latter is intended to stay one’s perception and to lead abstracted concepts to one’s perception; that is, how to use concepts.　So, the latter rejects the notion of“general concept”; generality of any concepts are derived from only how to use concepts．Every“general”contains infinite particulars, therefore “general concept”will be self-existent and limitless generality.　On the contrary, any generality derived from “how to use concepts”must relate to concrete somethings. Thus, we can clarify“ what particulars are involved in the general”．

　　As a result of this study, the following didactical suggestions are concluded: we must be seeing students’s abstraction as two different knowing, and focus on each of them in a didactical situation; when students ask about“what particulars are involved in the general?”in a specific situation, their generalization will be improved．