Developing Learning Activities That All Students Enable to Construct Mathematical Thinking Flows Like "Story-Constructing"

课堂数学教学最重要的目标之一是确保每个学生都能体验到对新事实的演绎发展。在许多传统的数学课上，如果一些学生已经熟悉了一个命题及其证明，他们的老师想教学生如何使用演绎来发展这个命题，那些已经熟悉这个命题及其证明的学生不太可能在教学过程中学到任何新的东西。如上所述的学生可能不需要上课，他们可能会被称为过度准备。另一方面，在传统的几何课上，老师通常假设所有的学生都已经知道某些事实，这些事实对于理解老师在课堂上的解释是必要的。如果有一些学生不知道这些事实，这些学生就不能理解老师的解释，因此他们就不能从老师的指导中获益更多。这些学生可能被称为准备不足，如果老师试图满足准备不足的学生的需求，那么在课堂上实现课程目标的进展通常是微乎其微的。因此，教师在努力满足准备不足和准备充分的学生的需求方面面临着两难境地。如果老师试图满足准备充分的学生的需求，那么准备不足的学生就无法理解老师的解释。准备不足和准备过度的学生在数学课上基本上会互相阻碍对方取得进步。这是传统几何课程中最严重的问题之一，并且它不是一个容易解决的问题。我们已经发展出一种满足以下条件的数学课原则：(1)准备不足和准备充分的学生都能以自己的方式根据他们所知道的（旧的）事实推断出新的事实；(2)每个学生可以通过将自己的推理结果与其他学生的推理结果相结合，构建一条新的推理线（或推理流）。少

One of the most important goals of instruction in classroom mathematics is to insure that each student experiences the deductive development of new facts. In many traditional mathematics classes, if some students are already familiar with a proposition and its proof, and their teacher wants to teach the students how to use deduction to develop the proposition, those students who are already familiar with the proposition and its proof, are unlikely to learn anything new during the teaching-learning process.Students such as the ones described above, may not need to take the class, and they might be called over-prepared for the class. On the other hand, again in traditional geometry classes, teachers routinely assume that all students already know certain facts that are necessary for understanding the teacher's explanations in a lesson. If there are some students that don't know these facts, these students cannot understand the teacher's explanation, and they consequently cannot benefit f … More rom their teacher's instruction. These students might be called under-prepared, and if the teacher tries to meet the needs of under-prepared students, then there is often minimal progress in meeting the goals of the lesson in the classroom.So, the teacher faces a dilemma in trying to meet the needs of both under- and over-prepared students. If the teacher tries to meet the needs of over-prepared students, then the under-prepared students cannot understand the teacher's explanation.The under- and over-prepared students essentially inhibit each other from making progress in the math class.This is one of the most serious problems in traditional geometry classes, and it is not a problem with an easy solution. We have developed a principle of mathematics lesson which satisfies the following conditions :(1) Both under- and over-prepared students can deduce new facts on the basis of (old) facts they know in their own way ; and(2) Each student can construct a new line of reasoning (or flow of reasoning) by combining the results of his/her reasoning with those of other students. Less