At #TMC13, my session is about designing problem-based courses. But all math courses involve solving math problems; most do little else.

So what makes a course “problem-based”? R.A.C.S.V.P. (Répondez aux commentaires, s’il vous plaît.)

I’d love to get this fleshed out before we all get to Philadelphia!


Guess what? Poverty really sucks. It’s incredibly hard. All the lifespan studies going back to the 1920s show that poverty in youth is a very hard force. We need to build fault-tolerant schools and systems if we’re actually going to address equity. —Uri Treisman, Iris M. Carl Equity Address, NCTM 2013, at 35min, 10sec.

Every year at my school we try to improve the math program, and I think the quotation above captures what we’ve been going for. Our challenges:

Students in the wrong course

We do our best to place incoming freshmen from the information we have, but we’re finding that middle school grades don’t really communicate what students are able to do. So next year we’re hoping to up our game on intake, placing them from our own assessment within a week of their arrival. This also means offering, for freshmen, not only Algebra 1, but also an accelerated pre-algebra-plus-Algebra-1 course, as well as an Algebra 2.

Students below grade level

Next year we also hope to offer a couple options for kids to double up on their mathematics so they can catch up to a level that would allow them to take AP Stats, AP Calc or Pre-Calc before they graduate, even if they’ve previously gotten behind.

Intermittent attendance

This one is harder. Most of our courses are still structured around the expectation that every kid is there every day. That is a legitimate expectation to which to hold most of our students, but we don’t want their prospects to end if they miss some school. We’re all working on how to create structures to provide more individualized instruction to help fill gaps. A few of us have also been kicking around the idea of a master standards list spanning all four years (or at least three years), so students wouldn’t necessarily be tied to learning a particular blob of material in a particular 9-week term, but it’s unclear how to make this work without a lower student-faculty ratio.

Repeat repeaters

Fewer and fewer students are failing a course repeatedly. This is clearly a case where prevention is the best cure (see items above). Still, as long as the number of these students is more than zero, we need to find some way in that works for them.

Speeding up students’ applying new concepts

Throughout the program we’re increasing the amount of time that students spend solving problems alone and in groups, and articulating their solutions verbally and in writing. I’ve seen students’ abilities to think, write, speak, and critique improve dramatically. In my room, though, when students solve problems they tend to rely on concepts they are fluent in (like adding) and avoid concepts that are new to them (like exponentiation, or modeling with equations). Helping them master particular skills and provide copious opportunities for them to practice applying them in a meaningful context is an ongoing challenge.

The overall goal

We want to make sure that each student has a clear path to get from his or her current level of ability to, at a minimum, success in college coursework, and for most students, an AP or similar college-prep experience prior to graduation. We’re working to build a system where a student isn’t permanently derailed if something goes wrong for a while.

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.

I’m really excited: two teachers in my district are taking the leap to start from scratch, redesigning their teaching from first principles of what they really want to accomplish. They’re both experienced, excellent teachers and I suspect it will be inspiring to see what they come up with. Fortunately for all of us, they’re writing about it at langer.kogut.math. Drop by and say hi.

I feel like an explorer—there’s a whole universe of numbers!

-A student, who last week fell into trying to calculate the frequency distribution of the totals one can roll on four dice.

So I”ve never used anything but Google Reader to read math blogs, the only blogs I read. Now that Reader is dying, I’m looking for advice. Is anyone using anything instead of Reader?

UPDATE: Never mind. Elizabeth has got it covered:

If you are interested in teaching next year at a Boston high school just coming out of turnaround, with an excellent, fairly green faculty, inventing ways to accelerate children who may be several years behind up to deep understanding and creative application of secondary mathematics, write to me for more info (dangoldner, gmail).

I kind of feel like running up and down right now.

-A student, successful after 3 days of trying to prove the equivalence of the unit circle and right triangle definitions of sine.

Stuff I don’t know how to do, I don’t want to do.

-A generally productive student, explaining why she hadn’t even looked at today’s classwork.

Just after I pushed “publish” on the last post, I realized what I really wanted to say.

In our system, once you’re over four feet tall, it’s not okay to be a beginner at math. It’s not okay to come late to the party. There’s what seniors in high school should know, and if you don’t know that, then you’re behind. And if you’re my students, you’re not just behind, you’re the Achievement Gap.

And it’s not okay. Because students who are on track in math and who score well have more choices and get more encouragement and support. They do. So the ones that don’t, don’t get the same shot. Whatever the causes of that, there are not good reasons for that.

At the same time, it has to be okay. It has to be okay, in my room, to be where you are, and to take your time thinking about the things that interest and challenge you now, at whatever level of intuition you’ve developed. Because for me to hand you everything you should have by now and expect you to carry it forward without dropping it, just doesn’t work. If you’re a beginner, why would you want to be involved in anything where you are not welcome to be a beginner?