Last spring, Mathematics Teacher published my paper on mathematical modeling. In this month’s issue, they’ve published a response from Albert Goetz [$].
Goetz worries that our collective interest in mathematical modeling risks granting the premise of the question, “When will we use this?” Math doesn’t have to be useful, argues Goetz. It’s beautiful on its own terms.
An emphasis on modeling–seeing mathematics as a tool to help us understand the real world–needs to be tempered by an approach that gives some prominence to the beauty that abounds in our subject. I want my students to understand how mathematics can explain the world–there is beauty in that notion itself–but also to see the inherent beauty and magic that is mathematics.
Agreed. But I no longer find adjectives helpful in planning classroom experiences, whether the adjective is “beautiful” or “useful,” “real” or “fake,” each of which is only in the eye of the beholder. Instead I focus on the verbs.
Mathematical modeling comprises a huge set of verbs that range from the very informal (noticing, questioning, estimating, comparing, describing the solution space, thinking about useful information, etc.) to the very formal (recalling, calculating, solving, validating, generalizing, etc.). One of the most productive realizations I’ve ever had in this job is that all of those verbs are always available to us, whether we’re in the real world or the math world.
“Math world” is the only adjective you could use to describe these experiences. When students find them interesting it’s because the verbs are varied and run the entire field from informal to formal.
- Which One Doesn’t Belong?. Students notice, name, argue, etc.
- Malcolm Swan’s Area v. Perimeter Problem. Students choose their own rectangle, draw it, graph it, imagine other solutions, imagine impossible solutions, construct an argument about them, generalize to other shapes, etc.
- Polygraph. Students notice, name, communicate, eliminate, etc.
Trick your brain into ignoring adjectives like “real-world” and “math-world.” Those adjectives may not be completely meaningless, but they’re close, and they mean so much less than the mental work your students do in those worlds. Focus on those verbs instead.
We shouldn’t overlook the usefulness of using this part of math to model that part of math. I see calculus as a way of describing and analyzing curves, including their curvature. I see analytical geometry as a way of representing “pure” geometry. I even see algebra as a way of modeling numerical patterns. Modeling is not just about the real world.