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?

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.

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

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

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?

Ok can't resist. Need coffee.

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?

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!

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...