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 Proof by Contradiction Necessary?” Examples from Analysis are included.
- This in-class activity demonstrates that sometimes proofs by contradiction would work better as direct proofs.
The activity’s sample proofs also illustrate how to clearly signal proof by contradiction (e.g., “Assume for the sake of contradiction that…”)
- Henry Cohn points out some dangers of proof by contradiction, and suggests strategies for avoiding those dangers.
- The first of these two proofs illustrates proof by induction.
- This assignment includes a flawed proof by induction that all cows are the same color (Exercise 3). Students are asked to identify the logical flaw in the proof.
- This lesson plan includes an example of how to use auxiliary functions in proofs.
- In analysis, epsilon-delta (and similar) proofs conventionally choose a constant so that a specific property holds without justifying the specific choice of that constant. To help students to understand and write these proofs, it can be helpful to give an example of how to choose the constant, and then to demonstrate how to write a conventionally concise proof based on the choice. (For example, prove that a real number x exists such that x2 = 2. This proof is explained on pages 41-42 of Bartle and Sherbert’s Introduction to Real Analysis, 3rd Edition.)
Teaching students to generate and write proofs
- Cathy O’Neil’s blog post How to teach someone how to prove something summarizes her proof-writing course, which is designed to avoid some common pitfalls of such courses. These common pitfalls include rewarding speed and “ostentatious displays of cleverness.” Cathy’s course avoids such pitfalls in part by permitting students to revise and resubmit their proofs as many times as they like.
- 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.
- The following recitations from M.I.T.’s communication-intensive offering of Real Analysis address proof writing:
- Recitation 4: Strategies for proof writing; guiding text
- Recitation 9: Proof-writing strategies and proof structure
- Recitation 9b: Proof elegance
- Susanna Epp’s “The Role of Logic in Teaching Proof” (The American Mathematical Monthly, 110, December 2003, pp. 886-899) presents teacher-tested “ideas for helping [undergraduates] learn to construct simple proofs and disproofs.” The article suggests accessible subject matter (e.g., puzzle problems) and how to use it to introduce logical principles, ways to guide students’ fledgling efforts, and ways to motivate the need for proof.
Please feel free to contribute teaching strategies for this page.
Textbooks, examples of elegant proofs, & other resources for students
Many texts are available to help students with proof writing. A few follow:
- How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, by Kevin Houston
- M. Aigner, G. Ziegler, Proofs from THE BOOK Springer 2004
A collection of elegant proofs. THE BOOK refers to God’s book of perfect proofs, an idea of Paul Erdös.
- Stephen Maurer’s Undergraduate Guide to Writing Mathematics contains a chapter about reasoning and proofs (Chapter 7).
- A. Cupillari The Nuts and Bolts of Proof Elsevier Academic Press 2005 (MAA Review)
- J. Brown, Philosophy of mathematics: A contemporary introduction to the world of proofs and pictures. Routledge/Taylor & Francis Group, 2008.
- Alsina, Claudi(E-UPB); Nelsen, Roger B. Charming proofs: A journey into elegant mathematics The Dolciani Mathematical Expositions, 42. Mathematical Association of America, Washington, DC, 2010.
- Reuben Hersh’s book Experiencing Mathematics: What do we do, when we do mathematics? (AMS 2014) includes a chapter on how mathematicians convince each other. The book makes the point that a proof is not a logically rigorous progression from certain facts to a conclusion, but rather it is a means by which mathematicians convince each other.
This 2-page handout illustrates the basic theorem-proof format:
- An Example of a Proof with Guiding Text, by Peter Shor. The proof uses the pigeonhole principle.