Teaching Practices Aiming“Explanatory Understanding of Proof”: On Classes which Students can Find Mathematically Interesting

Abstract

　　It is often attractive in mathematics to discover a regular pattern or a rule in mathematical objects.  And, especially in senior secondary mathematics, a discovery of a pattern should be followed by its proof.  However, it seems that most of students do not find it interesting to consider why the discovered rule holds, that is, how to prove it.  In light of Komatsu’s argument about “explanatory proof”, we propose the notion of “explanatory understanding of proof”, which can, we believe, induce students’ mathematical interest. “Explanatory understanding of proof” means the situation where students realize the discovered pattern depends on the specific properties of the objects involved in the statement and moreover students can have a perspective which can tell whether the pattern remains true or not in the similar relating statements.  Then, we propose a mathematics class design, in which students are expected to reach “explanatory understanding of proof”, so that students can find the lessons mathematically interesting.  The keys to this design are to include imitating activities in classes, and to prompt students to apply the understood principle to other situations.  We also practiced the designed class and analyzed the results.  These results indicate the partial effectiveness of our design and simultaneously imply suggestions for the future development of mathematical classes in the same line.