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I know there is a fair bit of literature on mathematical problem solving (e.g., Polya, Schoenfeld). I am wondering if anyone can direct me toward good sources on mathematical problem posing.

More precisely, I'm looking for any studies, articles, or expository essays on what goes into writing a good math problem.

Ideally, I'd like literature that answers/tackles questions such as:

-How does one come up with a good math problem?

-How does one modify an existing math problem/result to pose a new problem?

-How does one sequence several related problems so as to be most effective pedagogically?

I'm particularly interested in problems at the gifted high school level or undergraduate math major level, but would welcome most any references. Thanks!

Edit: Without trying to define too precisely what I mean by "problem", think of questions akin to what you would see as A/B 1/2 on the Putnam.

Edit 2: I think now that the best way to answer this question is by taking well known literature on problem posing and seeing who cited it. For example, one could use google scholar to check who cited the book "The Art of Problem Posing" by Brown & Walter (see here) or the article "On Mathematical Problem Posing" by Ed Silver (see here). At some point I will build a fairly extensive bibliography on mathematical problem posing, creativity studies, and the literature lying at the nexus of the two; if there is interest, I will make that list available here on math.SE.

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You might find this interesting and useful: mathoverflow.net/questions/68546/teaching-a-pedagogy-course –  Amzoti Sep 14 '12 at 14:20
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2 Answers 2

One "classic" about problem posing is this: The Art of Problem Posing by Steven I. Brown and Marion Walter, The Franklin Institute Press, 1983; revised edition by Lawrence Erlbaum and Associates, Hillsboro, NJ, 1986; second edition by Lawrence Erlbaum and Associates, Hillsboro, NJ and London, 1990.

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I've read some of Brown and Walter's work before; this is a great suggestion. (PS: I believe Phil Smith distributed a chapter from this book in his course on Problem Solving at TC!) –  Benjamin Dickman Sep 11 '12 at 18:37
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Here are some items for you to consider:

  1. http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html (the basic idea is that the way you should pose a problem is to allow the students to learn to ask the questions themselves so they can explore and do problem solving rather than the typical mechanical approach)

  2. There are many problem books that have these sorts of problems, for example:

    • Problem-Solving Strategies (Problem Books in Mathematics) Arthur Engel (Author)

    • Problem Solving Through Problems Loren C. Larson (Author)

    • Putnam and Beyond [Paperback] Razvan Gelca (Author), Titu Andreescu (Author)

    • Mathematical Olympiad Challenges Titu Andreescu (Author), Razvan Gelca (Author)

  3. Learn to write better mathematics and have the students do the same, for example:

    • www.ohio.edu/people/mohlenka/goodproblems/goodproblem.pdf
  4. Let the students explore mathematics and problem solving using various means, for example:

These are just some suggestions to help you figure out how to proceed and there is MUCH more that can be said!

Explore!

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This post answers the question "How to write solutions to given problems" beautifully...but what about generating good problems? –  Alex Nelson Sep 11 '12 at 14:56
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Watch the TED talk link and then use the books referenced and Mathematica to complete the story. HTH ~A –  Amzoti Sep 11 '12 at 15:08
    
Great suggestions! $+\left(\frac 12 + \frac 12\right)$ –  amWhy May 19 '13 at 0:28
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