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The Art of Problem Solving: Exploring George Polya's Heuristics

We will explore the art of problem solving via George Pólya's exquisite 1957 book "How To Solve It: A New Aspect of Mathematical Method, Second Edition". What do you think of Pólya's book and its methods? Do they make sense to you? Does the Pólya method work? Is this the approach you use in solving problems? What do you like (and dislike) about this approach? What is missing in Pólya's approach? With what other systems of heuristical problem-solving could we compare Pólya's methods? What do you think of "the list"? Would you change the list in any way to make a poster to put in your work area?

Did reading Pólya's book improve your problem-solving skills? Are there problem-solving heuristics that you use which are not discussed in the book? Were you able to solve the 20 problems in Part IV of the book without looking at the solutions? Please bring some problems to the meetup that we can use to test out Pólya's heuristics.

More information is available on the Wikipedia page "How To Solve It".

You can find a PDF copy of "How To Solve It" on Helga Ingimundardottir's site. The book is still widely available in print, you can find it at your favorite bookseller. Amazon has two printings available: 2004 Printing (with a forward by John Conway) and a 2009 Printing; there is also a Kindle edition.

Here are two video resources that are interesting:

• A very good 60m 1965 lecture by George Pólya about Guessing

https://vimeo.com/48768091


• A nice 8m video summarizing Polya's problem solving method with some very nice examples

http://www.youtube.com/watch?v=uCS7GO0fkc4

John Conway's forward to Pólya's book mentions Alan Schoenfeld's December 1987 article in Mathematics Magazine on "Pólya, Problem Solving, and Education" which is interesting and can be browsed on-line for free (after registration). Schoenfeld wrote another relevant 37 page article "Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics" (1992).

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  • CJ F.

    I enjoyed last week's discussion on Polya's heuristics.

    We were unable to find much in the distinction of "analysis" (assume what is sought & derive consequences that are admitted; devising a plan) and "synthesis" (assume what is admitted and derive what is sought; carrying out a plan). I think we missed some interesting aspects in Polya's article on "Pappus" in "How To Solve It". instinctively I tend to prefer synthesis (valuing wholeness & concreteness); rejecting analysis as reductionism (cf. our discussion on Chaos by James Gleick). Could it be that analysis involves imagination & creativity whereas synthesis is mere validation & verification? Why does synthesis feel more wholistic to me? Does Euclid bias us toward synthesis? How does this distinction apply to "synthetic geometry" (a la Euclid) and "analytic geometry" (a la Descartes)?

    Do you see any value in the distinction between analysis & synthesis? Are they so neatly separable?

    January 4, 2014

  • CJ F.

    In the John Conway forward to Polya's book, he cites the interesting article "Polya, Problem Solving, and Education" by Alan H. Schoenfeld in Mathematics Magazine
    Vol. 60, No. 5 (Dec., 1987), pp. [masked] at http://www.jstor.org/stable/2690409 (free registration required).

    Schoenfeld writes that coaches for math problem-solving exams concluded "Polya was of no use for budding young problem-solvers. Students don't learn to solve problems by reading Polya's books". Schoenfeld also cites an AI researcher: "We tried to write problem-solving programs using Polya's heuristics, and they failed". Apparently "As of the mid-to-late 1970s, there was empirical reason, both in AI and in mathematics education, to doubt the solidity of the heuristical foundations established by Polya."

    Schoenfeld concludes "empirical evidence gained over the past decade indicates that Polya's intuitions may have been right". He suggests that metacognition (& social construction) are also necessary for heuristics.

    December 27, 2013

  • CJ F.

    In this exquisite 60 minute video from 1965, George Polya demonstrates that "mathematics in the making consists of guesses". Watch "Polya Guessing" at https://vimeo.com/48768091

    Although the video starts off slowly (the first 8 minutes are introductory), it quickly draws you in as you want to participate too and guess the answer to Polya's interesting elementary problem in solid geometry: how many parts do five (5) planes cut space into?

    I love the teacher-student interaction. Polya is masterful. What a group of intelligent students!

    Main ideas in "reasonable guessing": look at extreme cases, induction (observe, detect a pattern, leading to the courage of generalization), use analogy ("a sort of similarity" see pp. 37-46 in "How To Solve It"), test your guess.

    I highly recommend this 60 minute film with Polya himself demonstrating some of the key ideas from his heuristical methods for problem-solving: https://vimeo.com/48768091

    December 26, 2013

  • CJ F.

    On Saturday we will be discussing Polya's method of "Heuristics" for problem-solving. This short 8m video recapitualates the Polya method then gives 7 nice elementary examples for practice. Watch it here: http://www.youtube.com/watch?v=uCS7GO0fkc4

    You will need to pause the video to have enough time to solve the puzzles before the answer appears.

    I like the idea of renaming "trial and error" to be "trial and success" (Amy Edmondson in her recent book on "Teaming" called it "trial and failure").

    The video ends with some nice quotes from Polya.

    In the video, Polya's "The List" (pp. xvi - xvii) is refactored for presentation in the video. How would you rewrite "The List" to make it into a more useful "cheat sheet" for your own problem-solving praxis?

    Watch the short 8m video summarizing Polya's problem-solving method: http://www.youtube.com/watch?v=uCS7GO0fkc4

    December 25, 2013

  • CJ F.

    I was fascinated by the discussion in Polya's essay "Pappus" (How To Solve It pp. [masked]). Polya mentions that his translation is "a free rendering, a paraphrase". He cites a more scholarly translation in "The Thirteen Books of Euclid's Elements" by T. L. Heath, Vol 1, p. 138. I have that on my shelf and enjoyed reading Heath's translation and comparing it to Polya's.

    What is fascinating to me is how analysis is "working backward" (which is the name of another relevant Polya essay on pp. [masked]) and synthesis is working forward. In Pappus' vision analysis and synthesis are inverse operations of the mind. Analysis is more about puzzling it out and synthesis is more about validation (proof). Is that how you read the distinction?

    This view of analysis differs from what I read in Wikipedia (https://en.wikipedia.org/wiki/Analysis). It might be fun to discuss analysis & synthesis vis-a-vis the two Polya essays and our "traditional" view of analysis. Is the Pappus view incisive? Why?

    December 24, 2013

  • Sharon

    Ok can't resist. Need coffee.

    1 · December 23, 2013

  • CJ F.

    If you have not gotten a copy of Polya's "How To Solve It", you can look at the photocopy that Helga Ingimundardottir has on-line at https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf

    It is not searchable, but you can read some of it on-line.

    I am really enjoying the book. There are so many paths for our discussion. What aspects of the book do you want to discuss?

    December 23, 2013

  • CJ F.

    Next Saturday we will discuss "The Art of Problem Solving: Exploring George Polya's Heuristics". I'm thoroughly enjoying my re-reading of "How To Solve It" (I last read it about 20 years ago). It is exquisite!

    Although we will spend a lot of time covering "The List" (pp. xvi-xvii, 1-36), there are some wonderful articles in the "Dictionary of Heuristic" (part 3). The article on Modern Heuristic (p. [masked]) is an overview. It situates "the 12 principal articles of the Dictionary": 1) progress and achievement, 2) variation, 3) decomposing and recombining (which I particularly enjoyed), 4) definition, 5) generalization, 6) specialization, 7) analogy, 8) auxiliary elements, 9) auxiliary problem, 10) problems to find, problems to prove, 11) notation, and 12) figures. If you are running short of time, those articles will give substantial insight into Polya's Heuristic without tediously reading the whole dictionary. If you can follow some cross-references, serendipity will enhance the book!

    December 21, 2013

  • Elijah B.

    Hi, I have an epub version of Polya's "How to Solve it" in case anyone is interested. It is very nicely formatted.

    I'm not sure if I'll attend the next meetup (I'd like to), because I don't know how far I'll get in the book by the 28th. However, if none of you mind...

    December 12, 2013

    • CJ F.

      Elijah, You are welcome to attend even if you haven't read Polya's book "How to Solve It". The book is there to focus the discussion. It is recommended so we can have a richer discussion, but it is not required.

      The book is only about 36 pages. It includes about 190 pages of a "Dictionary of Heuristic". Followed by 20 pages of problems, hints and solutions. If you read the first 36 pages, you will be able to follow the discussion.

      Since it is a highly influential book, I'm planning to carefully read it. During the discussion, in addition to discussing the whole text, I'm hoping we can discuss the 20 problems from the problems section.

      Check out the description. In addition to an outline for the discussion, there are links to two videos and some supplementary readings. All optional, but worth checking out. The description/RSVP page is at http://www.meetup.com...­

      December 13, 2013

    • Elijah B.

      Great, thanks. 36 pages is doable.

      I plan on bringing a friend/colleague from the school that employs me. He is a 7th grade math teacher.

      1 · December 13, 2013

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