Last night I was flipping through my copy of Euler: The Master of Us All. It’s a book about Leonhard Euler’s mathematical accomplishments. It’s interesting! I’d recommend it, despite the pretentious title. So Euler is playing with infinite series, which Euler loves to do, and the author inserts this bit of commentary: “By this time the reader must have noticed a number of symbolic manipulations that require careful handling.” That put me off a little bit. I hadn’t noticed, actually, Mr. William Dunham.
But this type of language, making assumption about one’s audience, is common in writing about mathematics. Here’s another one from a book I was reading about abstract algebra:
“The theorem we have just proved has several obvious but important corollaries:”
Obvious to who?
I find myself falling into this language in class. “It is simple to…” “You’ll notice that…”
This language reflects an ugly part of the culture of mathematics. For a long time, math has acted as a gatekeeper, labeling some students as “smart” and others as “not smart.” We tell ourselves that math is sequential and missing one day can cause a student to fall behind for a year. The way we talk about math reinforces these stories, and they function to recreate patterns of who has been successful learning math.
Here’s another fun quote I stumbled across last night:
One of the stories we tell ourselves about math is that, once you fall behind, it’s hard to catch up. For instance, yesterday I was teaching about rational functions. It’s easy to play with this chain of logic. I assume that first students need to understand fractions, variables, the order of operations, polynomials, intercepts, asymptotes, limits, and more. We could spend weeks searching for misconceptions in students’ prior knowledge, assuming they won’t be able to access the content until they’re fluent with every little piece. But is this always true? Is it possible to drive a car without being able to build an engine? What would mathematics look like if we chose to ask a similar question: Is it possible to engage in mathematical thinking without understanding what we assume is prerequisite knowledge? How might we restructure math class to make it more likely that every student can engage with key mathematical ideas every day?