by jerf | 2025-09-08 14:50:25+00:00
A lot of trig. Trig identities is my bête noire, though I would save off one day for them, after Taylor polynomials, to derive a few through practice that way. But I remember grinding on them for quite a while; I would just have one day and use them as Taylor fodder. (Taylor polynomials are not practically useful for most people but they are extremely important as "the simplest thing that converts transcendental functions into the world of the elementary arithmetic functions"; it is good to demystify how one might compute sine or cosine "from scratch".) But I wouldn't nuke trig out entirely as classical geometry is an almost unparalleled place to learn basic proof techniques and mathematical thinking in a playground where you are not distracted by a lot of arithmetic. I think I still like that better than graph theory, and simple trig/geometry is in fact practically useful for a chunk of the students we're pushing through these courses. It would just get toned down.

I would cut out a lot of integration grinding. The concept of integration is extremely important, and I want the students introduced to the ideas of doing it symbolically, but the details of all of the manipulations are much less important than the concepts. I would retain only things that are of mathematical interest, like integration by parts (useful as an exercise in how much fluidity you have in doing math and a good check you understand the concept).

Symbolic differential equations I'd cut down a lot. I think there's a sense in which they are useful but the utility is not revealed until a couple of semesters in. Even my college semester was frankly not that useful, you really need to dedicate yourself to them to get the value out of them. I'd put in some more work on numeric integration, and working with computers to get them done. The concepts of differential equations are super, super, super important; the exact knowledge of how to solve this super precise form of this exact differential equation is not. More practical experience to teach intuition, less grinding on symbolic details. The net time spent on them if I were writing the curriculum might even go up, because I'd tip in some (very) basic chaos mathematics here instead of all that grinding.

Matrices I'd have a hard think about. HN is at the epicenter of their utility so it might be hard to see that for most people it's not all that useful in any sense. I might like to move them on to a very explicitly STEM-focused track.

If that sounds horrible, remember that my whole point here is that there's a lot of stuff that should be in the curriculum that currently isn't, or is just glossed over very briefly. Like, I'd like more work spent on basic financial math, both for personal finances and doing things like a bit of running stochastic financial scenarios (integrate this into the stats curriculum, for instance). Which also brings up the idea of doing more simulations of somewhat larger statistical scenarios than you can solve as a closed form by doing Monte Carlo simulations; statistics and probability is very important but for most people it spins off into the symbolic weeds when in fact most problems people will have in real life are not cleanly amenable to such things. (This might also help erase the implicit belief that stats tends to teach that everything is uniformly distributed.)

The curriculum as it is is also structured as an unmotived series of solutions to problems the students don't have. I would try to give them the problems before the solutions; e.g., if I'm building a stats curriculum around simulations that go into symbolic math rather than just handing them symbolic math from the beginning, imagine covering the Central Limit Theorem from the point of view of me giving out a couple of different simulation assignments that all "happen" to converge to it despite very different scenarios, and we can discuss & teach why the seemingly totally different scenarios produced such a similar outcome, rather than just handing it down from On High as a Solution to a problem none of them have.

You can't have things like that if you aren't willing to cut something else because we are full up right now. There's a lot of room now for people to get more intuitive grasps of these things by virtue of working with them through practical numeric integration, Monte Carlo simulations, maybe we'll do a brief section on enough 3D geometry to do a useful introduction to 3D modelling/CAD/CNC/3D printing/the whole complex of modern tools available that use some form of 3D modelling, there's so much useful stuff that's just aching to be let in that the culling of the old needs to be a bit aggressive after a century.

To be honest I'd not cover this dimensionality stuff, except perhaps as a special-interest side day or something. It's not bad to do a few of those, just to expose students to the diversity in math. I got graph theory like that in my own high school, and it can stay that way; I wouldn't amp it up any. Dimensionality does pivot nicely into fractals and fractals have notoriously pretty pictures associated with them, so maybe I'd sneak it in with that.