An Attempt at improving some verbal Expressions in Mathematics Education

作者最近准备的实验表明，90%以上的实验对象认为他们自己对日语中的“必要条件和充分条件”没有满意的理解。在日本，这些材料通常在十年级教授。我们的受试者大多是位于北京的两所私立大学的一年级和/或二年级理科学生。都会区。我们的实验还显示，超过70%的受试者对这一陈述具有正确的记忆，但超过80%的受试者对他们自己的记忆的有效性没有信心。这意味着他们没有办法检查自己的记忆。大多数教师或数学教科书的作者似乎都意识到了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程度上是为了使学生习惯于这种用法，而不是试图给出另一种好的表达式。他提出了一个具体的过程，即提供改进的表达式，并试图减少这种学生的困难。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。像往常一样，引入了一些大集合U的子集P和Q，它们分别由条件p和Q刻画，我们的实验指出，学生对(1)和(2)句的日语对应句的主要抱怨是，即使他们清楚地理解了PCQ的包含关系，他们也很难区分必要条件和充分条件。我们提出的定义这些条件的文本经过测试表明，它们有助于40%以上的受试者理解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.