The aim of this paper is to find theoretical foundations of teaching mathematical proof for the sake of curriculum development based on the mathematical activities throughout six years of secondary schools in Japan. To accomplish this aim, we first search for, through the review of related literatures, the principal aspects of proof and proving that should be taken into consideration for the curriculum development. In particular, we examine the distinction often made in Japan between “proof” (shoumei) and “demonstration” (ronshou), the idea of “local organization” introduced by Freudenthal (1971, 1973), and the idea of “mathematical theorem” proposed by Italian research group (Mariotti et al., 1997). This literature review shows that proof and proving are often discussed in relation to the propositions to be proven and the system of mathematics within which proof is carried out. And we consider that three elements―statement, proof, and theorem―that characterize a “mathematical theorem” are principal aspects that evolve throughout the learning in secondary schools. We then develop and propose a framework that allows us to design a curriculum by studying two questions: what kinds of teaching contents should be included in each aspect?; what kinds of evolution should be envisioned on the nature of each aspect? In this development, we lean on different perspectives such as the viewpoint of mathematical logic to identify different kinds of “statement”, Freudenthal’s idea of “local organization” and “global organization” to characterize different levels or natures of “theory”, etc. In this paper, we also discuss the relationship between the developed framework and the mathematical activities which is another crucial point to be considered in our curriculum development around mathematical proof.
