Why teach calculus in the age of AI(mappingignorance.org) |
Why teach calculus in the age of AI(mappingignorance.org) |
Of course, I've never had to use any of that knowledge since, but I'm glad I went through the process to acquire it.
I've also used that knowledge quite a lot since, both in reasoning about problems and its also been beneficial in the confidence of accuracy in the methods of problem solving.
Bad malicious teachers nearly tortured math out of me. Without a substitute teacher who retired due to malicious politicking of his pears; but who had a phd in Mathematics Education, and was able to narrow in on exactly what I was taught that was incorrect (3 classes prior to the one being taken); and the patience and technique he used to destructively correct the false teachings while alleviating the operant conditioned anxiety; I wouldn't be where I am today.
There's quite a lot of malevolence in the world based in blindness, fortunately there are also some good souls out there helping elevate others.
Calculus is an elective course in most schools.
There may be good reasons to learn or not to learn calculus, or literary theory, or anything else, but the existence of some related technology isn't it. I'd go so far as to suggest that perhaps calculus is even more important for some folks to learn in the age of AI (e.g. applications in neural networks), and we don't know who those folks will be in advance.
Pharmacy school teaches Calculus. Why would that be? Do you need to run derivatives and integrals to fill prescriptions?
No. Teaching maths, particularly calculus, teaches people how to 1) not make mistakes and 2) catch your own mistakes quickly. Vitally important skills for someone filling out live-saving medicine.
A calculus class should ideally be making someone think much harder than that. Calculus is about understanding the relationships involved with continuous quantities and modeling the way things move and change. It is a basic prerequisite for understanding biochemistry and statistics, essential background for understanding pharmaceuticals.
Learning calculus achieves the same effect, though. It is not the teaching that is important (although some may find it useful).
Article archive: https://archive.is/2AdUJ
It's very short.
Mathematica can do calculus and linear algebra better than most if not all college graduates since decades ago, but colleges are still teaching those courses. That should explain enough.
One of the few moments in university was learning how so much was actually Calculus.
Whether it was physics, chemistry, etc, the formulas I was given ften had a calculus version.
It helped me open up to taking math and stats courses I never would have as a comp sci student, which in turn gave me a different perspective than just taking cs courses alone.
This feels like a "tech bro" idea from someone who has never touched a SEM field (STEM minus the T).
Honestly, the trend toward anti-intellectualism in the world is very disturbing to me, and it seems AI is enabling this kind of contempt for knowledge even more.
So many people never stop think about why we do the things that we are simply expected to do, instead of doing other things.
Learning calculus is very valuable to a relatively small number of people. We should absolutely make calculus available to any student, regardless in advances in AI. But a lot more people, particularly non-technical or non-math type people, would benefit more from experience with statistics than calculus. Statistics, combined with an introduction to simple programming, should be part of the basic high-school curriculum.
Math teaches you how to think internally in an interesting way like algorithms of thought. Physics teaches you how the world works reading it is one thing. Understanding the language is where the beauty lies
I've never done an integral by hand as part of any productive activity. Monte carlo integration and loops for multiply-and-add have proven incredibly useful. Why not teach those directly?
But when things break and you have to understand why, or you have to fix it, or even describe the problem in some detail that someone else is able to guide you to fix it, it sure helps to have an idea of how it works.
I trig, and calc 1 yet I hardly remember how to solve most problems because its not something I use regularly.
Same with subnetting or remembering certain programming languages.
I literally don't use these things 99% of the time and so I forget them. Sure I understand some of the basics and I could probably pick it back up way faster than someone who knows nothing but I am human.
So I don't think forgotten knowledge is wasted, it's still valuable.
You don't use your muscles? They atrophy. You don't make an effort to travel without a gps regularily, to force your brain to remember your way around naturally? Your spatial memory atrophies and becomes useless [here: https://www.nature.com/articles/s41598-020-62877-0 ]
People don't need to learn math anymore, hence, no more calculus lessons? People are literally becoming idiots who can't calculate simple change at the cash register without pulling out their calculators.
It's exercise. It keeps the brain itself from atrophying. It stops you from becoming a "wetware LLM" that's just parroting whatever echo of a thought (natural or otherwise) goes through it.
I don't use Calculus in my day job, but it was still valuable for me to learn to hone my ability to think
Sounds like a job for AI.
You can’t “brush up” on something you never learned
For the [current] layperson, each of those things I mentioned I might as well be speaking in Martian.
https://dibeos.net/wp-content/uploads/2025/08/what_happens_t...
You'll (probably) never apply the ability out the kinetic energy vs. time of a ball rolling down a hill, but these exercises build understanding of the tools. Derivatives are everywhere in a fundamental electric circuits course, you need to have an intuitive understanding of basic calculus. The relationship between current through and voltage across ideal inductors and capacitors are directly described in the language of calculus, even if you're not "using" the calculus substitutions you learned each time you analyse a circuit.
And good luck getting through a couple weeks of an introductory quantum mechanics course without using calculus as a fundamental building block. You can solve many of these problems with computers, but it's not going to build intuition on how to approach future problems. (I don't mean this as a joke or picking an arbitrary complicated-sounding topic; this is a core course in some engineering programs.)
Many engineering problems have nice closed-form equations (at least to get approximations). Obtaining those equations often involves calculus, and someone has to do that in the first place.
(I'm giving examples from the lens of my education, but each field of science, engineering, and mathematics will have their own context, and will vary from little-to-no calculus to being all-calculus.)
It's pretty clear to me as I work through problem sets that I'm never going to do any of this hand-computation in reality, in the same way that nobody computes eigenvectors by finding the roots of a characteristic equation. It's still fine by me, for 2 reasons: (1) because I'm doing this to replace the New York Times Crossword with something productive, and it's great for that, and (2) because every time I get annoyed at like messy trig derivatives with double-angle substitutions and stuff, I instead pivot to learning how to solve it with Sage Math, and so I get better at that instead.
But if there's a smarter sequence, I'm super interested!
For something more traditional, take a look at textbooks by Piskunov, Courant, or Apostol. Spivak's Calculus has excellent problems if you are looking for something more abstract and rigorous (probably better after a first course). https://archive.org/details/n.-piskunov-differential-and-int... ; https://archive.org/details/ost-math-courant-differentialint... ; https://archive.org/details/calculus-tom-m.-apostol-calculus... ; https://archive.org/details/introductory-calculus-book-colle...
Finally, if you want a strategy for those tricky integrals, per se, take a look at Schoenfeld's "Integration: Getting it All Together", https://files.eric.ed.gov/fulltext/ED214787.pdf ; some results of teaching the solution of integrals by this method were presented in https://www.jstor.org/stable/2320344
Not only will you be even more capable of picking up solving things numerically, but you'll also have the prerequisites for studying physics or probability or machine learning or Knuth's Concrete Mathematics. It opens doors to new intellectual vistas.
Solving things analytically (when possible) also can reveal more about the nature of the problem than doing so numerically, and can give the same satisfaction as finding an elegant solution in code.
You can definitely go an entire programming career without ever using it, but if you ever do run into a problem it solves, having this tool available to you is only a benefit.
Also College Math with APL: https://archive.org/details/APL_books/Introduction%20to%20Co...
Actually, Iverson et al. wrote dozens of math textbooks using array languages!
You CAN model predator-prey dynamics or disease spread using Monte Carlo techniques, but you can't read the historical literature without some grasp of differential equations.
There will always be some brilliant people that breeze through a hard course, and effortlessly acquire the intuition, and then wrongly attribute their understanding to the course. That doesn't help the rest of us, who are much more likely to build intuition by guessing about the outcome of a program and then checking if we are right.
I'm extremely optimistic about the future of mathematical literacy beyond the requirements for high school graduation.
I would feel real weird if there were things (in Calc II problem sets) I could solve in Sage that I simply couldn't do by hand.
I don't feel weird that there are things I can do, but will get wrong a bunch of times if I try to do them, and can quickly bang out in Sage. That seems fine to me. For a lot of these subjects, I don't care about automaticity, just intuition.
It's like a lot of linear algebra: being able to quickly do things by hand is kind of silly, because for real world problems (at least in data science) hand solves aren't even really feasible. But learning to do it by hand is important for building intuition.
I've been going through Math Academy with a IPython REPL open, too, and I've noticed that I need to avoid using it unless the problem specifically tells me to use a calculator or the implicit skill review in the problem gets skipped. Even writing little functions for myself to one-shot a problem means I'm missing chances to actively recall the steps.
Actually, given what you've said, you'd probably enjoy working through Sanjoy Mahajan's books on Fermi problems and estimations (the books are CC-licensed you can just download them): https://mitpress.mit.edu/author/sanjoy-mahajan-9006/