| Research Abstract |
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
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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
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