To help students learn to write proofs, Russell E. Goodman of Central College has developed Proof-Scrambling Activities. Students must correctly order the scrambled sentences of a proof. These activities help students identify when a proof is logically correct, to recognize how authors use words like “therefore,” “next,” etc., to indicate the direction of the logic, and to gain experience reading and comprehending proofs. Enclosed are two activities, a quiz, and notes for educators.

Read more →# Posts Tagged *Proof*

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.

Read more →This intentionally mediocre presentation of the proof that convergent implies Cauchy is used to begin a class discussion of when the motivation for a proof should be given before the proof and when it should be given after the proof. Written by Joel Lewis.

Read more →These three samples of proofs by contradiction are used to illustrate when contradiction should (and shouldn’t) be used as a proof strategy. Students identify which proofs shouldn’t use contradiction and suggest revisions of those proofs. Developed by Todd Kemp and Joel B. Lewis.

Read more →In this writing assignment from M.I.T.’s communication-intensive offering of Real Analysis, students choose 1 from among 3 (or so) proofs to write for their peers. The choice of problems varies each year depending on which problems have already been assigned for homework. We include the assignments from a few different years here to illustrate the range of problems assigned. It may be wise to warn students if some problems in an assignment are more challenging than others. For example, the Fall 11 assignment contains problems of different difficulty levels (“WritingAssignment2”). Many (but not all) of the problems come from Rudin.

Read more →This annotated proof illustrates how to format a theorem and proof and how to use guiding text to communicate the structure of the proof. Comments about formatting assume that students may not be using LaTeX. The text is an excerpt from the lecture notes for M.I.T.’s Principles of Applied Mathematics, on the topic of the pigeonhole principle.

Read more →These drafts of an article by Mark McLean illustrate how a proof can be improved by pulling out a lemma. Although the article is on an analysis topic beyond the understanding of Real Analysis students, Mohammed Abouzaid has drawn attention to the structure of the article by highlighting relevant guiding text, so the improvement caused by pulling out a lemma is clear.

Read more →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 compact subsets of Euclidean space (Rudin pp. 38-40). Likely trouble spots for students at this point in the term include how much detail to include in proofs; induction; Heine‐Borel is only true for Euclidean spaces. Students have been writing proofs in their problem sets since the beginning of the term. Authors: The proof-by contradiction handout was developed by Todd Kemp; the guiding-text handout was developed by Susan Ruff, the homework assignment is by Hans Christianson, Craig Desjardins, Joel Lewis,

Read more →Students who are new to proofs will need guidance for how to structure proofs and how to be sufficiently rigorous without going into too much detail. Perhaps the most helpful strategy is to provide individual feedback on assignments. It can also be helpful, however, to point out to the class peculiarities of particular kinds of proof and to discuss proof-writing strategies. Kinds of proof Texts for proof-writing courses teach the basics (see Resources below). The following materials can help students to discern when and how to apply various kinds of proof. This weblog by Gowers discusses the question “When is

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