Proof: From the particular to the general

Posted by Kevin Houston on Apr 9, 2019 in How To Think Like A Mathematician, Mathematical thinking, Mathematics education | 0 comments

A common technique in teaching mathematics is to go from the particular to the general. We can look at a few examples and then state and prove the general result. We can do this for proofs as well. Instead of giving students a proof of a statement, give them the proof for a specific example of the statement and ask them to prove the general from that.

A classic example of this is to prove the irrationality of and then that of . Once the common features have been determined the proof for irrationality of for prime is (hopefully) straightforward.

For another example, consider a statement involving vectors in a space of dimension . Prove it for particular vectors in . Then, set the exercise of proving it for the general case. It is, of course, important to stress to students that this particular case proof is not a proof of the general case.

The benefit of this technique is that the students are usually able to understand the particular case (so feel good about themselves) and can then use it as a scaffold to prove the general.

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