Proof in Mathematics: Origin, Practice, Crisis
Proof – What, Why and How?
By Adam P. DeutschHow do we know that what we know is, in fact, true? And how do we communicate what we know so that others will be convinced that it is true? These questions are central to the idea of mathematical proof and to the motivating force for teaching students of mathematics about proof and how to do proof. The ability to communicate mathematically, more generally, the ability to communicate logically, forcefully and convincingly is an invaluable skill to which all students should be exposed. Mathematics is an excellent vehicle for teaching students this vital skill.
The curriculum unit described in this document is designed to lead students to an understanding of what proof is, why it is important and how to construct good proofs. Essentially it is a unit about argument and communication and in this way crosses curricular lines by promoting skills which are universal and useful in the sciences, social sciences and humanities.
The Death of Common Sense?
By John B. SnodgrassMost would agree that to possess common sense is a good thing. As a science educator, however, I have found that common sense can be inadequate in explaining or in understanding of many topics. This paper will examine instances where common sense can be misleading and an inadequate explanatory tool. Explanations and proofs that go beyond common sense are presented to explain topics. Lessons and instructional strategies that support the explanations are suggested that are suitable for middle school students. Topics covered include freely falling bodies, determining the shape and circumference of the earth, floating and sinking, vacuums, and logic and mathematic problems. In every case, the topic will be thoroughly discussed and analyzed and, hopefully, interested teachers can find useful information for their own classes.
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