The Mind of a Mathematician

A while back, I had a conversation with a biology professor, about a speculative alternative approach to teaching a topic, which might yield deeper understanding. Their reply was roughly, "Nifty, but... My students will soon be taking the MCAT. It will determine whether their years of effort and dreams will succeed or fail. The MCAT doesn't test for that deeper understanding. It asks for . I would be doing my students a disservice, to spend our limited time together, on anything else."

Which is completely reasonable. One encounters similar constraints down into middle-school.

Similarly, imagine you are talking with a teacher in a country whose educational system prioritizes memorization and superficial understanding even more than our own. India perhaps.

You emphasize the importance of moving beyond rote whole-classroom call-and-response memorize-and-regurgitate. Of developing skills of system modeling, equational reasoning, and problem solving. More like the USA. And the teacher replies to you, similarly to the biology professor above.

Which is completely reasonable.

But surely, that's not all there is. One can step back, and deal with a larger picture. At least within the education research community of that country, there seems value in awareness and discussion, that there might be different, better ways of doing things. And an exploration of what might become possible under unfamiliar sets of deployment constraints. To inform long-term funding and effort. And system-level tweaks. But which might also turn up bits of overlooked opportunities for nearer-term improvement.

I suggest there is value in similar awareness, discussion, and exploration, within our own country/system.

And that there's value, possibly great value, in doing more of it than we have been. Examples of existent efforts might be life-sciences degree curriculum rewrites, and quantitative introductory biology.

For a concrete illustration (illustrative examples is what those "hyper-focus" points earlier were also intended to be ;) of opportunity, somewhat analogous to F=ma, is a second-year college biology top-of-the-wish-list that incoming students have a firmer grasp of central dogma. Which it turns out, with appropriate tooling, is accessible and useful down into middle-school. And so there's now community and institutional support for spreading that tooling and content.

Another is, there's some work on engineering early science education content, to "immunize" against later misconceptions. Early as in K and even pre-K, for primary/middle-school misconceptions. Some of it almost as side effect - content you might use anyway, but tweaked to have nice long-term properties. Rather than struggling to overwrite naive understandings later, you nudge them in the right direction while they're still being formed. I don't recall whether anyone has explored doing this for motion misconceptions, but they seem the kind of thing it might work for.

When people are doing desperate triage in production hell, they understandably have limited attention for someone saying "you know, if we had resources that we don't, and time that we don't, it might be possible to do much better here, and hmm... what might that look like...?" But a healthy tool ecosystem will have some people discussing and exploring that.

When covering the principle of least action: often, applying it will give infinitely many solutions, or none at all. An example with no solutions would be the harmonic oscillator with x(0) = 0, x(t) = 0, and t not a multiple of half the period. An example with infinitely many solutions would be the principle of least time applied between the two foci of an ellipse with reflecting walls. So what happens in these cases? Does Lagrangian mechanics just not work?