My classroom is a game that my students play. I set the rules by how I allow them to succeed or fail in my class. If I’ve done it right, then the rules I set should motivate genuine learning and reflect that knowledge in the form of a ‘grade’. –Daniel Schneider

I’ve proposed a session for #TMC13 on “A map of problem-based class designs”.

Last year in pre-calculus, all my students had passed algebra 2, but some had passed with flying colors while others had passed with a D- (and a D- in geometry before that, and in algebra 1 before that). Everyone needed a different course. So, the rules were:

  • Problems spanned a range of difficulty from pre-alg to pre-calc.
  • You got credit for solving and presenting a solution, if you convinced the class you were correct.
  • You got credit for writing up what you presented and revising until I accepted it.
  • You couldn’t present something someone else solved, or write up someone else’s solution.


  • Almost everyone participated wholeheartedly and solved problems on their own that were challenging for them.
  • They all got better at writing.
  • Not much pre-calc was learned, as even the strongest mathematicians dropped in and out of the main precalc problem sequence in favor of brainteasers.
  • Presentations were a little desultory, since most students were totally unfamiliar with the problem the current student was presenting.

This year, we reorganized sections so that all pre-calc students had done reasonably well in algebra 2 last year. To get everyone learning the pre-calc content, I reorganized the course. The new rules:

  • Problems almost entirely required grappling with and mastering precalc concepts.
  • You got credit for solving a problem and presenting it.
  • You got credit for writing up someone else’s solution—but half as much as presenting an original solution.


  • People are learning pre-calc; some of them are mastering it.
  • They are all getting better at writing.
  • Presentations are better than last year and the discussions are more interesting.
  • Some students have earned good grades by reporting their classmates results, without ever solving a problem on their own.

Compare these examples to Exeter’s nightly problem sets, or College Preparatory Math’s groupwork, or any other problem-based course. All make different choices about what the problems are, who solves what on what schedule, and how assessment is done. I’m convinced the rules defining a problem-based course are the main factor in determining what students actually do, and therefore, what they learn. I’d like to discuss the feasible and infeasible regions in this choice space. What choices do you go back and forth about, and why? What results are you trying to drive, or avoid? What choices are you excited about? Worried about? Why? If “it depends on the kids”, what exactly about them does it depend on, and how?

I’d love to start hearing people’s questions and answers about this now. We can summarize and continue the discussion in Philadelphia.