An Attempt at improving some verbal Expressions in Mathematics Education

笔者最近准备的实验表明，90%以上的实验对象认为自己对日语中给出的“必要条件和充分条件”的理解不满意。这些材料在日本通常在十年级教。我们的大多数科目是位于两所私立大学的一年级和/或二年级理科课程的学生。大都市区。我们的实验还显示，超过70%的受试者对陈述的记忆是正确的，然而，超过80%的受试者对自己记忆的有效性没有信心。这意味着他们不知道如何检查他们的记忆。大多数数学教科书的教师或作者似乎都意识到这种描述给学生带来的困惑。然而，作者认为他们的兴趣很大一部分是让学生习惯用法，而不是试图给出另一种好的表达方式。他提出了一个具体的步骤，即提供更好的表达，并打算减少这类学生的困难。他讨论了下列命题成立的典型情况：(1)q是p的必要条件。(2)p是q的充分条件，并像往常一样引入了某大集合U的子集p和q，它们分别由条件p和q表征，因而假定它们满足关系PCQ。我们的实验指出，学生对日语对应句(1)和(2)的主要抱怨是，即使他们清楚地理解了包含关系PCQ，也很难区分必要条件和充分条件。我们为这些条件的定义提出的文本已经过测试，表明它们帮助超过40%的受试者掌握条件的含义并区分它们。

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.