Have you ever told a joke to someone who "doesn't get it"? If you patiently explain the referents you may get them to "understand" the joke, but they will probably respond something like "So, why is that funny?"
In the simple example above, you probably found this funny if (1) you are familiar with the maxim "Time flies like an arrow", (2) your knowledge of the English language allows you to understand that "flies" can be a verb meaning "passes swiftly" or a plural noun referring to a type of insect and that "like" can mean both "as" and "enjoy" (3) your knowledge of writing style leads you to expect that when the same word appears in two successive short sentences, it will usually have the same meaning in both sentences.
I think we face the same problem when we try to teach mathematical understanding. A proof is most memorable to us when, like in getting a pun, we make a connection between two or more apparently unconnected thoughts, what is often called an "Aha!" moment. Without previous deep knowledge of the constituent thoughts, the student may be able to follow the step-by-step logic, and may be able to remember the proof for tomorrow's test, but the proof will not be memorable, and both the theorem and the proof will soon be forgotten. One implication for pedagogy is that the curriculum must be carefully planned so that, when a mathematical topic is introduced, the students will understand the constituent parts and be able to appreciate their connection. Otherwise, we are mostly wasting our time.
I recently came across a proof of the Pythagorean Theorem that was new to me that gave me an aha! moment. This was given in Sanjay Gulati's excellent "Mathematics Academy" blog as a Geogebra demonstration. He does not indicate the original source of the proof. The aha! moment comes for the connection between the Pythagorean Theorem and an apparently unrelated theorem that I always teach in my elementary geometry class, the "crossed chords" theorem. The aha! moment occurs from looking at the following picture.
4 comments:
Oh, that proof is beautiful. I have been using Pythagoras Theorem for more than ten years, and I couldn't imagine the proof is that simple.
Although I don't think there is humor in that picture though. Haha.
I think the problem with this being a proof of the Pyth Thm is that you need the Pyth Thm to prove the theorem of intersecting chords..
I don't know anything about the intersecting chords theorem, and I do know and love the Pythagorean Theorem. So to me this looks like a proof of the intersecting chords theorem.
Lovely.
Sue,
Actually the intersecting chords theorem holds for any two intersecting chords, not necessarily ones that are perpendicular to one another, so this method does not prove the intersecting chords theorem in general. Look up Crossed Chords Theorem in Google if you want a proof.
Pat B, you do not need the Pythagorean Theorem to prove the intersecting chords theorem. It has a very nice proof using simply similar triangles and the fact that inscribed angles that cut off the same segment on a circle are congruent.
Peter
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