I’ve spent some time this week with our Math teachers, who’re studying how to deploy a problem-based learning curriculum. It ended up not being too bad, since Math teachers tend to consider themselves a bit rogue and exceptional, which makes for a good atmosphere, I find.

For the layperson, maybe a way to explain what we’re doing is as follows: students and teachers work together to solve authentic, semi-open-ended problems by posing questions, proposing solutions, defending and dismantling ideas, and co-constructing rigorous solutions using mathematical symbology and concise English. It’s incredibly difficult to do, for a teacher, but it’s beautiful when you see it in action.

When this sort of class session is working, a few students emerge as protagonist-heroes and some others as foils; five students in back play the role of Greek chorus, emotionally echoing the action, at times with dread and at other times with exuberance. The teacher becomes an enigmatic Oracle, one whose goal is to draw out, with prompts and riddles, the students’ inner greatness. In pursuit of a solution, the class attacks the problem a dozen different ways, takes costly wrong turns, uncovers ingenious shortcuts, slogs through number-crunching together, makes glorious breakthroughs, arrives at an acceptable conclusion, and reflects together on the battle, licking wounds and celebrating one another’s bravery.

What I’m getting at is this: class feels more like a story you’re participating in (an extremely dramatic one, at that!), and less like a movie you’re watching.

Done right, students in a problem-based classroom have an experience of learning, and thereby learn deeper and learn more that will actually be remembered.

To do this well, math teachers must learn how to elicit ideas and questions from students, how to delay providing answers, how to provide hints at the right time, and how to build students’ capacity to think critically and argue effectively. The rewards of this approach do match the difficulty. You get to actually see students’ curiosity, creativity, generosity, fragility, and perseverance in action. Compare this with what a traditional math teacher hopes to see in students: endurance. Spending every day with students who are exhibiting only the ability to endure boredom, in order to do well on a test, is, for most of us, just no fun at all.

Shallow, ineffective teaching is what Richard Feynman railed against in a lecture to Brazilian university students and professors about the state of science education in Brazil. Here’s an excerpt of what he said to them:

One of the first things to strike me when I came to Brazil was to see elementary school kids in bookstores, buying physics books. There are so many kids learning physics in Brazil, beginning much earlier than kids do in the United States, that it’s amazing you don’t find many physicists in Brazil – why is that? So many kids are working so hard, and nothing comes of it.

[Consider this analogy of a] Greek scholar who loves the Greek language, who knows that in his own country there aren’t many children studying Greek. But he comes to another country, where he is delighted to find everybody studying Greek – even the smaller kids in the elementary schools. He goes to the examination of a student who is coming to get his degree in Greek, and asks him, “What were Socrates’ ideas on the relationship between Truth and Beauty?” – and the student can’t answer. Then he asks the student, “What did Socrates say to Plato in the Third Symposium?” the student lights up and goes, “Brrrrrrrrr-up” – he tells you everything, word for word, that Socrates said, in beautiful Greek.

But what Socrates was talking about in the Third Symposium was the relationship between Truth and Beauty! […]

The [students] can recite, word for word, what Socrates said, without realizing that those Greek words actually mean something.

from Surely You’re Joking, Mr Feynman

Students who have discussed, questioned, and argued about concepts and ideas in any discipline have a much better chance of knowing what they actually mean, and of applying them effectively.

We’re still working on balancing teacher-worked-examples with classroom discussions, as well as how to support all students in fully participating in their second language–challenges sufficiently worthy indeed of our group of rogue mathematicians.

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Coming soon: a Marxist, pro-Palestinian Samba party, Futebol in São Paulo.

Featured photo by Tim from Beco de Batman.