数学学習において,「規則に従う」とは,どういうことか : 「言語ゲーム」理論からの示唆

抄録

Wittgenstein pointed out that the idea "the rule is understood by the interpretation" is misunderstanding. And, he insisted that we should admit the existence of the grasp of the rule which is not interpretation. So we can do some acts by following the rule not because of consisting of specified rule of the game, but just because of doing the practice customary based on no ground, that is without hitting other possibilities. It is not the agreement of the interpretation, but the agreement of the judgement that is essential to play the language game and to talk with each other. From this point of view, the customary practices is essential to discussion and it is necessary to form the mathematical customs before the mathematical discussion. So we should reconsider the discussion in mathematical class from this respect, and ask "Is the mathematical discussion; the interpretation game that should be based on the agreement of the judgement, actually made in the mathematical class ?". It is most important educational task for mathematics teachers of the lower classes in primary school to form some customary practices or techniques that are more useful for children to learn further mathematical activity. Therefore activities of concrete operation of matters or drawing pictures or figures are given priority to busy discussion or calculation drill or detailed explanation, especially in the primary school lower classes.