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Chewy Problems
I received a direct message from a friend this week who was feeling uneasy about the new curriculum approach his school had taken on. He sent me a link to the homepage of the company and it read something like this:
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Improved test scores
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Improved grades
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Improved teacher morale
Do I want all of those things? Absolutely. What bothered me wasn’t what was there, but what was missing. The specific company zeroes in on a pedagogical model that emphasizes the dispersion of knowledge from the teacher to the students with some worrying infographics to support the teacher-centric nature of it (I say “worrying” because of the easy, click-and-fit vision of teaching it is promoting). He asked me if I had anything I could send him, something to read, which might help discredit the notion that there is a one-size fits all approach to teaching, where teachers will tell and students will listen and learn. I sent him two recommendations, not necessarily to discredit, but rather tap into what I felt was missing, what the approach leaves out, and the subsequent “benefits” it omits. The two readings were:
Both are not new or unknown by many in the field, but both promote a model that is varied and unfixed, adaptable and flexible, and connected to the genuine curiosity that students and teachers bring to the classroom. Lockhart captures the passion many teachers have for mathematics but don’t feel empowered to instill into their students, restrained by imposed structures and rigid frameworks. Lampert presents a case for instruction that makes room for students (and teachers) to zig-zag between ideas, make connections between the known and unknown, and grow to understand the notion of “cross-country” mathematics.
They both include what I call, “chewy problems.” Chewy problems always have a little bit more to find out about, the type of problems that become even more interesting when you consider it from another angle or approach it in a different way. They require significant chewing. These come in a few forms, many of which you wouldn’t expect to see from a curriculum provider for some understandable reasons; they depend on teacher expertise, they are often best done on paper (or scraps of paper), students tend to take their own path and sometimes end up tackling a completely different problem, and (here’s the real kicker) students might not even “solve” them at all. These problems are, however, just opportunities for students to experience mathematics in the ways described by Lampert and Lockhart. They require students to make sense of the problem, form and refute conjectures, or even completely walk away from it for an idea to surface from their subconscious. I’ve picked out three different ones that popped up on my feed over the past week or so, which I believe get better the more you chew.
Eddie Woo (@misterwootube) came right out of the gates with the more interesting question of “well, yes – we know it can be solved but how many solutions exist?”
AMSI Schools (@AMSIschools and @cass_lowry) presented this problem which invites students’ imagination when you look at it at a different angle ?
Tierney Kennedy (@kennedy_tierney) has been on fire with the problems she’s been sharing in the Twitter-sphere. She just needs to remember the hashtag #MTBOz, of course!
Lastly, here’s a problem. A BIG problem, which shows the power of mathematical models for presenting real-world problems and might leave students with more questions than answers, but maybe the questions worth asking.
Written by John Rowe, @MrJohnRowe
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