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研究(2)学生猜想在几何问题证明中的应用研究(3)计算机辅助构造中的猜想与纸笔构造的比较研究(4)几何问题证明的课堂教学研究(5)几何问题证明新教材的研究。这些调查的结果如下：(1)科学理论被视为猜想，需要进行佐证或反驳。通过它。科学理论的可证伪性是可以表现的，而且科学理论的客观性和真实性也是可以追求的。(2)经验猜想比直觉猜想要多得多，学生对几何证明有多种信念需要证明。反例对修正他们的猜想起着重要的作用。资优学生有很多课程需要证明，有能力估计哪门课程容易，而非资优学生则不容易。(3)计算机有一些优点，如容易构造，但它也有一些缺点。例如，在显示器上测量图形中的线段长度是困难的。因此，不能得出哪种方法更适合于猜想的结论。(4)在教学证明之前，通过在图形中写符号来教授证明是一种有效的方法。(5)建议将欧几里得几何作为一门选修课，必要地利用平移、旋转、反射和伸缩等变换进行教学。

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.