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

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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.