Two proofs of the fact that 1+2+ … + n = n(n+1)/2. One proof uses induction; the other organizes the terms of twice the sum so each of n pairs sums to n+1. These proofs are used to start a class discussion about elegance.
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This proof of the Cauchy-Schwarz Inequality is used to start a discussion about proof elegance. The class compares this proof with the proof of the Cauchy-Schwarz inequality given in Proofs from the Book by Aigner and Ziegler. The class discusses which proof one would discover first and how it’s a good idea, after having proved something, to think about rewriting it. This writing sample was developed by Mohammed Abouzaid and Peter Speh.
Read more →Context: This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s Real Analysis (18.100C). This week students learn about the Taylor series and the Stieltjes integral (Rudin pp. 120-127). This recitation is often combined with Recitation 9 on proof structure. Authors: This recitation was developed by Craig Desjardins, Joel B. Lewis, Todd Kemp, Mohammed Abouzaid, Peter Speh, Kyle Ormsby, and Susan Ruff Communication objectives: Students should begin to develop an awareness of proof elegance as well as an appreciation for revision (even correct proofs can be improved). Recitation: This recitation varies each term depending on instructor inclinations. Some
Read more →A proof must, first and foremost, be correct. But even among correct proofs, some proofs are more satisfying, or “elegant,” than others. Even experienced mathematicians aren’t sure quite how to define elegance: elegance is, to some extent, in the eye of the beholder. Although teaching students to write elegant proofs may not be feasible, we can begin to raise student awareness of elegance. To raise student awareness of elegance, the following strategies have been used in M.I.T.’s communication-intensive offering of Real Analysis. Discussing examples, which can be found, for example, in Proofs from the Book. Discussing quotes from Hardy’s A
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