An Attempt at improving some verbal Expressions in Mathematics Education

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Recent experiments prepared by the author have shown that more than 90% of the subjects of the experiments think they themselves do not have satisfactory understanding about "the necessary condition and the sufficient condition" given in Japanese. These Materials are usually taught in the tenth grade in Japan. And most of our subjects are 1st and/or 2nd grade science course students of two private universities located in. The metropolitan area. Our experiments also revealed that more than 70% of the subjects bear a correct memory of the statement, however, more than 80% were not confident about the validity of their own memory. It means that they know no measure to check their memory.Most of teachers or authors of mathematical textbooks seem to be aware of this kind of students' perplexities caused by the description. Nevertheless, the present author thinks that a large part of their interest is to accustom students to the usage rather than to try to give alternative good expressions.He proposes a concrete procedure, namely to provide improved expressions, and intends to diminish this kind of students' hardship.He discusses the typical case where the following statements are valid:(1) q is a necessary condition for p.(2) p is a sufficient condition for q.and introduces as usual, subsets P and Q of some large set U, which are characterized by the conditions p and q, respectively, and are consequently supposed to satisfy the relation PCQ.Our experiments pointed out that students' central complaint about the Japanese counterpart of the sentences (1) and (2) is that it is hard for them to discern between the necessary condition and the sufficient condition even when they clearly understand the inclusion relation PCQ.Our proposed texts for the definition of these conditions have been tested to show that they help more than 40% of our subjects to grasp the meaning of the conditions and to distinguish them.