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

课堂数学教学的最重要目标之一是确保每个学生都能体验到新事实的演绎发展。在许多传统的数学课堂上，如果一些学生已经熟悉了一个命题及其证明，而他们的老师想教学生如何使用演绎来发展这个命题，那么那些已经熟悉了这个命题及其证明的学生就不太可能在教与学的过程中学到任何新的东西。像上面描述的学生这样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，在传统的几何课堂上，教师通常认为所有学生都已经知道了理解教师在课堂上的解释所必需的某些事实。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，他们也就不能从教学中受益。 ...更多信息 从他们老师的指导中。这些学生可能被称为准备不足的学生，如果教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往往很小。如果教师试图满足准备过度的学生的需求，那么准备不足的学生就无法理解教师的解释。准备不足和准备过度的学生在数学课堂上基本上相互抑制。这是传统几何课程中最严重的问题之一，这不是一个容易解决的问题。我们提出了一个数学课的原则，它满足以下条件：（1）准备不足和准备过度的学生都能以他们自己的方式在他们所知道的（旧）事实的基础上推导出新的事实;（2）每个学生都能通过将他/她的推理结果与其他学生的推理结果相结合来构建一条新的推理路线（或推理流程）。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