# Can Compare and Contrast be used in Mathematics Teaching and Research?

I happen to be a fan of the "theme and variations" approach to problem solving. In certain cases, a certain question had drawn enough to attention to generate dozens of solutions.

There are some other examples of results having many different proofs but in the literature, I rarely read more than one proof of the same result. In many cases, nobody has taken the time to collect them into one place.

Is "compare and contrast" an acceptable methodology in Mathematics? Certainly it is in other subjects, but Mathematics is about proving new things rather than comparing what is known.

Put another way is collecting many proofs of an important result a meaningful way to spend your time? How do I explain the advantages to others.

• I'm pretty sure that compare and contrast is an acceptable methodology in mathematics education because I imagine that it aids in the learning process. Are you really asking if it is an acceptable methodology in mathematics (as in research mathematics)? I note that there are research journals that encourage multiple proofs of the same theorem, for example, the American Mathematical Monthly, but it may be because the MAA has an affinity for education (at least it seems that way to me).
– JRN
Commented Jul 16, 2015 at 23:50
• "In many cases, nobody has taken the time to collect them into one place" but sometimes they do, for example, this book on the Pythagorean Theorem, which I think contains 370 proofs.
– JRN
Commented Jul 16, 2015 at 23:59
• I second JRN's question about the context in which you are asking. Do you want mathematical examples, e.g., proving rationality of the zeta function in more than one way (Dwork did it using p-adic analysis... but surely the development of etale cohomology for an alternative proof that covered all of the Weil conjectures was very important!) or do you want examples in education, e.g., flexibility with multiple representations in the classroom? Commented Jul 17, 2015 at 1:42
• In regard to “is collecting many proofs of an important result a meaningful way to spend your time?”, remember the quote from Bruce Lee: “I fear not the man who has practiced ten thousand kicks one time. I fear the man who has practiced one kick ten thousand times.” Commented Oct 8, 2015 at 18:13