Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

Do you resent proofs?

The main reason to present a proof in class is to explain a mathematical fact. "Explaining" means reducing to simpler mathematics. There may be additional purposes, for example to illustrate mathematical techniques. Depending on the level and purpose of a class a proof need not be called such. The teacher might ask " now why is this true?" and proceed to give the answer. Depending on the class, also different levels of rigor may be required. So your teacher does not present a proof because it's a bizarre ritual that for arcane reasons is required of a mathematician! Rather she is trying to help you understand the new mathematics!

While we are on the subject of proofs: there's a disease among students that I call the "Far Out Theorem Syndrome". Many exercises require the student to "prove" or "verify" something. The purpose of this is to build facility with the new mathematics and to expand your understanding. I see it over and over again that students go to the literature, find a theorem that may (but often doesn't) imply the desired result, and quote that theorem! That's useless, because it relates the new as yet ununderstood mathematics to some other even less understood mathematics, and just adds to the prevailing haze and confusion. The theorem isn't understood because the student just takes it on somebody's authority rather than making it his or her own by coming up with a proof or at least reading and digesting one. Don't fall prey to this disease!

Often it's useful to simplify the proof by adding simplifying assumptions or looking at special cases, and then assigning it to the students to fill in the gaps.

Students often complain that there's too much emphasis on proofs and too little on problem solving. If you encounter that situation in my class then that's because I firmly believe that problem solving is more efficient and more powerful if it's based on understanding math rather than on following recipes.

Fine print, your comments, more links, Peter Alfeld, PA1UM