Some of the other online modelling resources, such as Dan Meyer’s blog, don’t really fit what I would class as meaningful modelling, and can feel contrived, or of little relevance/import to students’ lives; if I am going to spend the time bringing modelling situations to my classroom, I want to address matters of importance, socially or politically.
Yes, Iâ€™m interested see how Dan Meyer promotes a sort of pseudo-modelling that seems to be quite popular among certain teachers. I think one aspect that appeals is that he suggests a narrative that is immediately accessible. On the other hand some of the questions are not particularly meaningfully tackled using mathematics seriously.
I see two tacit questions.
One, should math be important?
And by “important,” I’m using Danny’s definition: relevant to a student’s life, either socially or politically.
See, there isn’t any one agreed-upon definition of “mathematics.” They’re all arbitrary, personal, and cultural. And given finite hours in a school year to spend learning math, they’re all political. They create winners and losers. Class time spent how you’d prefer is time not spent how someone else prefers.
So I help students learn math for one reason alone, and it doesn’t have to be your reason also. I want to help students learn to puzzle and unpuzzle themselves. Math offers us the opportunity not just to solve puzzles, but to generate them from scratch â€“Â just you and your brain and maybe something to write with.
Those puzzles may have sociopolitical importance, but that’s a higher standard than I choose to set for myself. So it’d make more sense for Geoff and Danny to criticize my standard than to assume I’m aiming at theirs and missing. I’m not.
Two, should modeling be important?
I suspect Danny, Geoff, and I would agree more about the point of mathematics than the point of modeling. Their criticisms specifically concern modeling, and the fact that I ask questions like “How many pennies are in the pyramid?” and “How long will it take the water tank to fill?” rather than questions like (I’m guessing here) “Is capital punishment sentencing just or unjust?” or “How should California manage its water supply?”
But there is much more consensus around the definition of “modeling” than “mathematics,” and that definition doesn’t specify culture, context, or importance. Modeling is mental work, work of a certain character, work that I think we’d all agree is uncommon in many classrooms and unfamiliar to many students.
Modeling asks questions about a context. It works to make those questions more precise and tractable. It nourishes those questions with data where none exists. It sets reasonable bounds on an answer before finding a solution. It solves questions mathematically and then tests those answers against the world’s answer.
Basically, “modeling” is a verb and it doesn’t help our understanding of the verb to attach it a priori to adjectives (like “important” and “relevant”) or to nouns (like “capital punishment” and “water supply”). If you want to understand modeling, ignore the adjectives and the nouns. Watch the verbs.
Additionally, we have to remember (as math teachers) that we are not the only teachers and courses these students encounter. I teach mostly 11th and 12th graders, and they frequently tell me about the political conversations they are having in government class or the serious social topic they are writing about in English. I have observed that, although students seem to appreciate these connections to real-world problems, these topics are heavy, and at times students appreciate engaging in â€œlighterâ€ application problems and activities.
Except that when you watch students engaging with a task that they are motivated to understand they are doing all sorts of things that relate to the â€œway they view their place as a member of societyâ€œ. I canâ€™t imagine a situation in which a student isnâ€™t both learning something about their place in society and simultaneously asserting some version of their belief about their place in society. Itâ€™s happening all the time.
So working on socially relevant issues is valuable. But â€˜relevance to meâ€™ means, â€˜real to meâ€™, and as the RMP project has shown, well as it has confirmed what has been known for ever such a long time, what can be real to someone has to do with what they can imagine, can grow to imagine, and is not confined to what they already do every day.