How does one create "good" math problems?

Question

As lifelong students of mathematics, we find problem-solving to be absolutely essential to enhance our understanding of the subject. Teaching others what we know serves to reinforce our existing knowledge and disseminate information to learners.

However, how does one go about creating "good" problems?

By "good", I mean thought-provoking, inspiring problems with solutions that are extensible to other domains. Also, this builds up to the level of olympiad problems, for which problem writers seem to have a remarkable degree of ingenuity and creativity in devising novel problems.

Answer 1

Since your question is very broad, here is a somewhat broad answer: Read about problem posing.

Three key pieces are:

Silver, E. A. (1994). On Mathematical Problem Posing. For the learning of mathematics, 14(1), 19-28.

and the book

Brown, S. I., & Walter, M. I. (2005). The art of problem posing. Psychology Press.

The latter is a re-print of a book that first came out in 1983. You can also find a related book edited by Brown and Walter; a citation for the most recent version is:

Brown, S. I., & Walter, M. I. (Eds.). (2014). Problem posing: Reflections and applications. Psychology Press.

Start with these three documents, their references, and (searching on google scholar) other papers and articles that cited them.

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To sketch out very roughly Brown and Walter's suggestion: Start with a mathematical scenario, list assumptions, vary constraints (in their terms: "What-if-not-ing"), and then ask questions. You can even "cycle" through this process repeatedly in order to produce problems of increasing complexity.

Of course, problem posing brings with it the danger of not knowing the answer to what you are asking.

For example, your starting scenario might use the Pythagorean Theorem:

Find all integer solutions for $x^2 + y^2 = z^2$.

This particular example is explored in Brown and Walter's book, but it seems to me a reasonable assumption to list is that the exponent everywhere is $2$, and to ask for integer solutions when the exponent is $3$.... or, if one feels particularly daring, to generalize and ask for exponent $k \geq 3$.

At a glance, this might seem like a reasonable question; but, if you are familiar with Fermat's Last Theorem, then you will realize that this is not an appropriate problem for most students.

You can find some of my brief remarks about problem posing and creativity in part $4b$ here, and a couple of other examples relating problem posing and intuition in the concrete example section here.

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A final remark: You start off by mentioning the "essential" role of problem solving in enhancing our understanding of mathematics. It may be worth noting that problem posing plays an important role in solving; consider Polya's list of heuristics and how many of them are questions: What's a related problem? What's a simpler problem? How can I generalize this problem? Etc. (Historically, both Silver, in the first piece cited above, and Kilpatrick, on problem formulation, trace this observation, i.e., that problem posing is an integral part of problem solving, at least back to a 1945 paper by Karl Duncker.)

As Cantor (1867) wrote in his doctoral thesis:

“In re mathematica ars proponendi pluris facienda est quam solvendi”

(“In mathematics the art of asking questions is more valuable than solving problems”).

Answer 2

For me there are perhaps three main types of problems which I assign:

1. Routine skill building: either are modeled on a computation which I have shown similar problems solved, or, are a proof problem which is just a natural consequence of definition with little extra technique required. For a proof course, many problems are little more than an invitation to care about what the notation actually means.
2. Breadth discovery: in every course there are certain topics for which we do not have enough time in lecture. It is a very rewarding experience for students to be guided through a short module of problems where they discover the essential features of a topic which is not covered in depth by lecture and other materials.
3. Challenge: here there are no rails, no box, no expectation anyone in the course solves it. Sometimes these are used to show the limitations of a current family of techniques to solve problems, sometimes these involve some fuzzy intuition which guides a creative leap.

I suspect most of the problems I write and/or assign fit into either 1 or 2, but students often accuse me of 3. Honestly, one of the reasons I try to surf the MSE a fair amount is to assess what is covered in my courses at other universities. Also, the international flavor of the MSE helps me get some cross-section of what is happening at schools all over the world.

Answer 3

Two suggestions:

1) Attend workshops and conferences and seek out problem solving sessions or presenters who are sharing their "favorite problems." When the problems and solutions are discussed unique methods and approaches appear.

2) Build a library and make time to read. Collect books, pdfs and sources. A textbook not suitable for the students can be a great source of problems. (Use Amazon and eBay to get used versions that are much cheaper.) Modify the textbook version as needed. Creativity in creation of problems comes from flipping through sources.

Answer 4

You didn't specify a specific level, but I think your question has merit in any case. I will take it at the K-8 level. First I want to address your specific requirement:

By "good", I mean thought-provoking, inspiring problems with solutions that are extensible to other domains.

I will interpret "inspiring" to mean that students will have a motivation to engage in the mathematics of the problem. For "thought-provoking" I will assume that you mean that the problems have a high likelihood to require students to engage in productive mathematical reasoning. These are essential characteristics of good investigations in a curriculum. That is to say, a good curriculum should contain activities and investigations that satisfy these.

I once asked the a well-known high-quality curriculum developer how she knew her curriculum problems fit the requirements of "realistic mathematics education" (which was the approach that inspired her curriculum. She replied that they had to try each activity with real students many times in the research and development process. While the first drafts may have been based on theory, in reality the finished curriculum was heavily tested.

Therefore, find and collect problems developed by good curriculum designers. If necessary, build your own library of such problems.

One final note: you suggested you wanted problems whose solutions were extensible to other domains. I suggest you be careful with this sort of assumption in looking for problems. What they come to understand in the process of problem posing and solving may help them form connections between contexts. However, you may find it difficult to support the notion of "domain transferrable solutions" in good math education literature. Focus more on what sort of mathematical reasoning the students will have the opportunity and resources to engage in.