Last year at Match and this year at Opus, my creative works have been math problems. Like Ira, I have enough taste to know that they're not that good. But I also worry that taste won't be enough to close the gap. Taste is too vague. I need something definite to aspire to. So, in the past few days, I buckled down and wrote out my criteria. Hopefully they'll be helpful to you, and to me.
Is it educational?
An educational problem deals with the right concepts and practices at the right level of difficulty.
Educational problems – problems that cover the appropriate skill at the appropriate difficulty – are worth solving. The issue is that students do not believe that they are worth solving. Educational problems make it possible for students to move forward – with effort – but they don't motivate students to make the effort. Most students need something exra to push them up the slope.
Is it motivational?
This is the human heart of the problem, the part that convinces us the problem is worth solving. Good problems motivate action by presenting a situation that is dramatic and simple.
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Dramatic and simple have implications not only for problems, but also for answer formats. Obtrusive answer formats (e.g. TF, multiple choice) increase the amount of noise on the page (less simple), and they interfere with the drama (no suspense). In narrative terms, obtrusive answer formats take a story about a hero resolving a conflict and turn it into a story about a minion completing an exercise. The best answer formats get out the way, leaving plenty of white space for the student to wonder and to work.
Is it useful?
The problem isn't over when the student finishes her first attempt. If the student has really invested, then she will want some feedback. A good problem empowers teachers, tutors, parents and computers to provide quality feedback by revealing the student's (mis)understandings. A useful problem is accurate, precise and clear.
When it comes to answer formats, accuracy, precision and clarity have contradictory implications. Accuracy and precision favor thorough answers (e.g. open response), but clarity favors minimal answers (e.g. multiple choice). My preference is for thorough – but structured – answer formats. Structure (organized work, boxed answers) allows us to maximize clarity in the context of an accurate and precise answer.
How we define and execute on our criteria matters. Right now, too many students are spending too much time on problems whose only merit is clarity. In the problems that I write for Opus, I want to make sure I do better. If I measure up to my criteria, then I will be setting teachers up to educate and motivate students (skill, difficulty, drama and simplicity), check for understanding (clarity and accuracy), and identify misunderstandings (precision).