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, and Todd Kemp.
Communication objectives: Writing proofs and guiding readers. Because students have been writing proofs since the beginning of the term, students have an initial familiarity with proof writing. The purpose of this recitation is to help them to think critically about some of the choices involved with writing proofs (which proof strategy to use, how much detail to include, etc.). In some terms, an objective has been to prepare students for timed proof writing in exams.
The topics addressed in this recitation vary based on the inclinations of the instructor. In the past, topics have included
- When (not) to use proof by contradiction This worksheet includes three proofs by contradiction for students to critique. Students are directed to identify those proofs that could be rewritten as direct proofs and to construct the direct proofs. Recitation instructor Kyle Ormsby notes, “The students were rather adroit at picking out the unnecessary argument, so we didn’t spend much time going over it.”
- Guiding-text This annotated mathematics paper demonstrates how an author can use “guiding text” to guide the audience through the structure of a paper, section, or argument.
- Proof-writing strategies
- Students can gain insight into a proof by generating examples and trying to find a counterexample.
- If so inclined, students should go ahead and write a proof as a proof-by-contradiction. Once a draft is complete, the student can re-assess the proof to check whether it would work better as a direct proof.
- How much detail to include in a proof depends on the audience, why you are writing the proof, and how much confidence your audience has in you.
- Strategies for writing timed proofs on exams.
- Writing epsilon-delta proofs.
This recitation is usually followed by a proof-writing assignment.
Students choose from among three proof options because they will critique each other’s writing: students write the most helpful critiques on proofs they have not yet considered themselves. To ensure that the proof options are sufficiently evenly distributed among the students, students order the problems by preference, and the instructor chooses which problem will be solved by each student. For more information, see the page on peer critique.
Depending on the semester schedule, sometimes this recitation is followed by preparation for an exam review.
Kyle Ormsby writes, “I talked too much…An idea for making this more interactive (and useful!) in the future: Prepare a proof of something topical to the course, omitting all guiding text but providing blanks in which the students can insert their own guiding text. Give a mini-lecture (3 or 4 minutes?) on guiding text, have the students fill out the worksheet in teams, then conduct a class argument about what guides best.”
Peter Speh writes, “On the second writing assignment, a lot of students ended up having trouble solving the mathematical problems correctly, even after peer review. Although I didn’t do so, it would be valuable for instructors to give students feedback on mathematical errors at the same time as the peer review gives them information about ways to improve the exposition.” [Another option has been used successfully by Kyle Ormsby: the instructor gives quick feedback on the math before the peer review, with just enough time for the student to fix any math problems before the peer review.]
BTW, this weblog by Gowers discusses the question “When is Proof by Contradiction Necessary?” Examples from analysis are included.