Cross-Cultural Study on the Role of Conjectures in Problem-Solving in Geometry

本项目的研究分为以下几个方面。(1)猜想在数学和数学教育中的作用的基础性研究（2）学生猜想在几何问题证明中的实践研究（3）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（4）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（5）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（6）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（7）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（8）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（9）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（10）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（11）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（12）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（13）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（14）计算机辅助作图中猜想与计算机辅助作图中猜想的比较研究（15）计算机辅助作图中猜想与计算机辅助作图中猜想（4）几何解题课堂教学的研究（5）几何解题新教材的研究通过研究，得出以下结论：(1)科学理论被认为是一种猜想，应该加以证实或反驳。它不仅可以体现科学理论的可证伪性，而且可以追求科学理论的客观性和真理性。(2)学生对几何证明存在着一定的信念。反例在修正他们的猜想方面发挥着重要作用。有天赋的学生有很多课程要证明，并且有能力估计哪门课程容易，但没有天赋的学生没有。(3)计算机有一些优点，如结构简单，但它也有一些缺点。例如，难以测量显示器上的图中的段的长度。因此，它不能得出结论，这是更好的猜测。(4)将符号写进图中，然后再进行校样教学，是一种行之有效的教学方法。(5)本文提出了欧几里德几何应作为选修课讲授，并应充分利用平移、旋转、反射和伸缩等变换。

The researches of this project are separated into the following areas.(1) A basic study on the role of conjecture in mathematics and mathematics education(2) Practical researches on conjecture of student in proof of geometrical problem(3) Comparative researches on conjecture in computer assisted construction v.s. paper-pencil construction(4) A study of classroom-teaching on proof of geometrical problem(5) A study of new teaching-materials on proof of geometrical problemThe findings of these investigation are as follows.(1) A scientific theory is regarded as a conjecture which should be exposed to corroboration or refutation. By means of it. the falsifiability of scientific theories can be represented and moreover, the objectivity and truth of them can be pursued.(2) There are experiential conjectures much more than intuitions.Students have some kinds of beliefs about geometry proof to be proved. Counter-examples play important roles to modify their conjecture. Gifted students have many courses to prove and capacity to estimate which course is easy, but non-gifted don't.(3) Computer has some merits such as easiness of construction but it has some demerits. For example it is difficult to measure the length of segment in the figure on the display. So,it could not be concluded that which is better for conjecture.(4) It is a effective method to teach a proof by writing the signs into the figure before teaching a proof-writing.(5) It is Proposed that Euclidean geometry should be taught as an optional one by making essential use of transformations such as translation, rotation, reflection and dilatation.