A Descriptive Study on Progressof 8th Graders' Proof-Conception in School Mathematics of Japan

最后，本研究旨在描述日本八年级学生使用基于相对真概念的数学证明说明的证明概念的进展。目的包括以下目标。目标1：阐明基于数学真理的相关概念开发证明教学材料的理论背景。目标2：根据数学中真理的相关概念，开发证明教学材料。目标3：利用自然观察的方法，根据数学中相对真理的概念，在整个证明教学过程中收集八年级部分小组的数据。目标4：用社会学/心理学的方法分析数据，描述八年级学生概念证明的进展情况。关于目标1，我讨论了以下几点。讨论1-a：数学中相对真理概念产生的背景与意义。讨论1-b：日本学校数学课程的发展。讨论1-c：初中数学教材中证明的介绍。讨论1-d：日本初中数学教材证明教学中命题真值的概念。关于讨论1-c，我得到了以下结论。在日本的初中数学教科书中，一个证明定义中的论证规则与一组证明论证之间不存在一致性，或者我们无法判断是否存在一致性。作为不一致性的原因，一个正确性标准（与学生可以获得的事实相对应）被应用于8年级之前的几何领域的命题，而另一个标准（连贯性，什么假设）被有意地应用于该领域的同一命题。此外，这两个标准的应用不仅是由于将证明或解释作为证明的功能，而且是由于在八年级之前学习几何领域的命题的局部系统化。关于讨论1-d，我得到了以下结论。关于真理的意义，可以提出两个概念：命题在学生所能认识的范围内的适用性，以及命题的真理性是否依赖于已证明的命题。关于真理的标准，我们可以提出以下的概念：对于一个命题，应当采用哪一种标准是预先确定的；对于学生所能认识的事实的对应，或者根据预先确定的用法区别从假设中推演出来。少

The study finally aims to describe the progress of 8th graders' proof-conception by proof instructions based on relative concept of truth in mathematics in Japan. The aim involves the following goals.Goal 1 : Clarifying the theoretical background to developmaterials for proof instruction based on relative concept of truth in mathematics.Goal 2 : Developing materials for proof instruction based on relative concept of truth in mathematics.Goal 3 : Collecting data of some small groups of 8th grader throughout proof instruction based on relative concept of truth in mathematics by natural observation.Goal 4 : Describing the progress of 8th graders' proof-conception by analyzing the data by sociological / psychological method.Regarding Goal 1, I discussed the following points.Discussion 1-a : Background & significance of relative concept of truth in mathematics.Discussion 1-b : Movement of curriculum in school mathematics of Japan.Discussion 1-c : Introduction of proof in textbooks of junior … More high school mathematics in Japan.Discussion 1-d : Conceptions of truth of a proposition in proof instruction following tectbooks of junior high school mathematics in Japan.As to Discussion 1-c, I got the following conclusion. In textbooks of junior high school mathematics in Japan, there is no consistency between regulations of argument in a definition of proof and a set of arguments of proof, or we cannot judge whether there is a consistency or not. As a reason of no consistency, One criterion of correctness (correspondenceto facts students can acquire) is applied for a proposition before the field of geometry in 8th grade, and the other criterion (coherence to comewhat assumptions) is applied intentionally for the same proposition in the field. Furthermore, the application of two criterions is causedby adapting not only justification or explanation as functions of proof, but also local systemization of the propositions learned before the field of geometry in 8th grade.As to Discussion 1-d, I got the following conclusion. Regarding the meaning of truth, two conceptions is possible to brought up : the applicability of propositions in the range students can recognize, and the dependence of their truth on the proved propositions. Regarding the criterion of truth, the following conception is possible to be brought up : it is pre-established which criterion should be applied to a proposition, the correspondence to the facts students can recognize or the deduction from assumptions under the pre-established distinction of usage. Less