Thursday, May 11, 2017

How to Solve It

Today a guest post from Periklis Papakonstantinou, coincidentally not unrelated to Bill's post earlier this week. I'll be back with a special post on Sunday.

I'm teaching in an undergrad program that is half computer science and half business at Rutgers, but the CS part taught there is the real thing (I assume for Business too). This term I taught a very theoretical course in cryptography and I realized that (1) the students enjoyed it and (2) that they were lacking basic reasoning skills. I ended up teaching for a few weeks how one can structure basic logic arguments. I am not sure if they appreciated things like the hybrid argument but I believe I convinced them that without rigorous thinking one cannot think clearly.

So, I decided to teach a much more fun class (hopefully next year) titled "How to solve it" -- à la Pólya. The goal is students to develop rigorous problem-solving skills. At the same time, I'd like to use this course as an excuse to introduce basic concepts in combinatorics, linear algebra, and theoretical stats. I'm not sure whether the original book by Polya is appropriate for this and that's why I thought of reaching out to my peers for suggestions. Any ideas and thoughts on possible texts, topics, or notes would be greatly appreciated.


  1. Two highly recommended books:
    Daniel Velleman's "How to prove it: a structured approach", and Daniel Solow's "How to Read and Do Proofs".

  2. I had to prepare a course for a recent application to a teaching position, and I had a similar idea. I based it partly on a course by Mike Saks called "Introduction to Mathematical Reasoning".

    Here is the webpage including lecture notes:

  3. A few years ago I also gave a set of lectures aimed at filling basic gaps, and wrote an accompanying set of notes here:

    Unfortunately they are neither complete nor polished, but they do address several of the basic gaps that I have noticed, and contain some examples that are relevant for computer science (as opposed to math).

  4. There are several more books of Polya on this topic like "Mathematical Discovery" (2 Vol.) or "Mathematics and Plausible Resoning." (2 Vol.) You could also check out the many books written for training on Mathematcial Olympiads or Putnam Exams, like e.g. Arthur Engel's "Problem-Solving Strategies." Especially for combinatorics consult Lovasz' classic "Combinatorial Problems and Exercises" and many more! It may be fun to add various competitions to your course using these problem collections in the style of Loren C. Larson's "Problem-Solving Through Problems."

  5. Many thanks to all and many thanks to Lance as well.

    I want to give a class that is both fun and doable. Yes, I don't want the level of pain to go to zero.

    I think I'm going to add a couple of more challenging problems (as some of you suggested), here and there to spice things up. I'll mostly follow one of Polya's text and I'll write something complementary in the style of Emanuele's notes. I like these notes but the focus is slightly different than what I have in mind.

    If people want to comment in this blog-post later on, please do (I'll check it over later on as well), or send me a personal email.

    have fun,