I’ve always dreaded being asked for my “teaching philosophy.”
For years, I gave nonsense or scattershot answers. “Logic and critical thinking are paramount.” “I care more about conceptual understanding than computational skill.” “A balanced, student-centered approach is always best.” “We buzzword to buzzword, not for the buzzword, but for the buzzword.” At best, each of my disjointed half-theories captured only a piece of the puzzle.
Worse still, none of my replies explained why I devote so much class time to plain old practice. If I was such an enlightened liberal educator, why did I assign repetitive computations for homework? On the other hand, if I was a traditionalist at heart, why did I fall head-over-heels for high-minded progressive rhetoric? Was I an old-school wolf, a new-school lamb, or some strange chimera?
Well, I’ve finally got my answer, and it only takes eleven words: Math is big ideas, approached from as many angles as possible.
What do I mean by “big ideas”? Well, here are a few examples:
And what do I mean by “many angles”? I mean that, in our best moments, my students and I come at these ideas like undergraduates approaching a dessert buffet: relentlessly, purposefully, and from all sides.
First: the historical angle. Even when the names-and-dates history doesn’t fit into my lesson plans, I try to contextualize each idea as part of a long lineage, to show how it answers a question, unlocks a door, fills a hole. I want my students to see each idea as one scene in a grand narrative of mathematical discovery.
Second: the verbal angle. English class isn’t the “opposite” of math class, as too many students think. Rather, good language skills empower us to discuss ideas of all types and stripes, especially mathematical ones. A precise and evocative vocabulary is beyond precious. Language allows us to debate productively, to learn as a collective, to think as a team.
Third: the scientific angle. Math’s most explosive ideas send shockwaves throughout the sciences. Physics, obviously—but also economics, biology, geology, chemistry, even psychology and sociology. Math has a symbiotic relationship with the sciences: it furnishes them with a powerful toolkit, and they provide it with concrete examples, a corporeal form for its abstracted soul.
Fourth: practice. Math without any computational practice is a mushy math, a math with no spine. To understand what makes, say, linear equations tick, you’ve got to solve ‘em, graph ‘em, play with ‘em in a hundred different ways. You can’t grasp patterns until you’ve worked through examples. Without multiplication facts at your fingertips, you’re unlikely ever to apprehend deep truths about the distributive property. If you’ve never spent a day multiplying out products of the form (ax + b)(cx + d), then you’ll never internalize the methods for factoring quadratics.
So there you have it. Big ideas from many angles.
I don’t always succeed from every angle—I might botch the history, or shortchange students on practice, or be just plain ignorant of the relevant applications. But that’s why I’m glad my students have other teachers—each with their own philosophy, each in their own distinctive way enriching our students’ understanding of those crucial big ideas.