Why teach calculus in the age of AI(mappingignorance.org)
mappingignorance.org
Why teach calculus in the age of AI
https://mappingignorance.org/2025/08/18/why-teach-calculus-in-the-age-of-ai/
86 comments
ryandv(2)
It is shocking to me that people would seriously discuss not teaching calculus just because LLM tools exist. Computational math engines didn't make understanding how you solve an integral obsolete, but they can make certain tasks faster and less error-prone.
This feels like a "tech bro" idea from someone who has never touched a SEM field (STEM minus the T).
This feels like a "tech bro" idea from someone who has never touched a SEM field (STEM minus the T).
How many people remember how to solve an integral?
Not remembering U substitution of the top of your head is different than not being exposed to it. I remind you that Int(x^3 * 5x^2 + 7) == x^4/4 + 5x^3/3 +7x + C. Just from that alone I bet a lot of memories of integration slot back into your head, and you would know where to look up the parts you forgot.
You can’t “brush up” on something you never learned
You can’t “brush up” on something you never learned
It doesn't matter if you can't solve a randomly-appearing-in-your-newsreel integral; it matters that you have the background knowledge of what an integral is, that there are rules to solving it, and you can read up on the rules and understand them.
For the [current] layperson, each of those things I mentioned I might as well be speaking in Martian.
For the [current] layperson, each of those things I mentioned I might as well be speaking in Martian.
It's not necessarily just about remembering every rule and trick you can use to simplify and solve integrals. Calculus is fundamental to understanding problems, from basic exercises in a first-year undergraduate physics course to entire fields.
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.)
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.)
if you actually spent good amount of time in mathematics during academia, you have developed neural networks for logical reasoning and problem solving but they get activated in life situations giving an edge compared to others.
https://dibeos.net/wp-content/uploads/2025/08/what_happens_t...
https://dibeos.net/wp-content/uploads/2025/08/what_happens_t...
The question I keep wondering about is why teach anything, or what exactly is worth teaching.
How to use an LLM
Reading Pump Six will provide a partial answer.
Anything where you care about getting the right answer?
Because, nature in its infinite wisdom, gets rid of what's not used.
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.
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.
Why learn anything? I for one want my brain to be completely empty, devoid of any thought or knowledge. Then I can simply pay OpenAI to think for me.
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.
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.
It's not anti-intellectualism so much as people forgetting the reason we learn in the first place. It's one of the biggest downsides of going from a world where education was largely optional to one where people are largely shoved onto a treadmill from birth to college.
So many people never stop think about why we do the things that we are simply expected to do, instead of doing other things.
So many people never stop think about why we do the things that we are simply expected to do, instead of doing other things.
you literally cant do regression analysis without calculus.
[deleted]
Not teaching calculus does not imply not learning calculus.
No? You literally can't derive regression procedures without it, but you don't need to. You can just do least-squares.
Along the happy path yes.
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.
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.
Like, on the backend? You can't run code without electricity, but FAANG doesn't teach that in their onboarding
Yeah electricity is generally thought at school, because most pupils don't even do their homework, without electricity (light). Your employee isn't responsible for your education.
I genuinely wonder what even is worth learning in the age of AI. It just feels like learning stuff doesn't really matter anymore. Unless you're an expert in some field, a novice with access to an LLM will usually produce better results than you.
The most important skill with using AI is critical thinking. And that's really only honed by learning and questioning a lot of subjects.
I don't use Calculus in my day job, but it was still valuable for me to learn to hone my ability to think
I don't use Calculus in my day job, but it was still valuable for me to learn to hone my ability to think
> And that's really only honed by learning and questioning a lot of subjects.
Sounds like a job for AI.
Sounds like a job for AI.
Where will the experts come from in the future if no one learns and we just accept whatever the Markov chain generates for us from the Bayesian database of old information?
Good question, but that isn't incentivized by capital. We're in an era of massive student loans and unaffordable housing. Unless there's a clear and direct incentive to learn, people will take the easiest path. Life is tough enough as-is.
The number of people that can actually dedicate their life to producing new knowledge is tiny. It's not really that motivating of a reason.
Article has been hugged so commenting more along the comments here.
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.
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.
The point of learning mathematics is not to "not make mistakes". More strongly: the misconception that mathematics or mathematics education is about getting "right answers" as quickly and accurately as possible is a disaster for learning.
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.
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.
I'm pretty sure this is a retcon. There are subject-matter-specific reasons PharmD's learn calculus. They don't use it day-to-day, but lots of STEM curricula (a) don't make sense without calculus and (b) don't lead to jobs where you're integrating by parts every day.
> 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.
Learning calculus achieves the same effect, though. It is not the teaching that is important (although some may find it useful).
Learning calculus achieves the same effect, though. It is not the teaching that is important (although some may find it useful).
You literally need calculus to understand dosage response curves.
Fortunately, medical researchers discovered Tai’s Formula for the area under a piecewise linear function. No calculus required! :-)
https://en.wikipedia.org/wiki/Tai%27s_model
https://en.wikipedia.org/wiki/Tai%27s_model
lmfao yeah I heard about this!
> Article has been hugged
Article archive: https://archive.is/2AdUJ
It's very short.
Article archive: https://archive.is/2AdUJ
It's very short.
I was lucky to have a high school math teacher who derived calculus with us. Understanding calculus made understanding physics so much easier. Being able to solve some physics problems using calculus seemed like magic.
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.
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 think its nuts anyone even bothers to teach physics without calculus.
I'd rather they be exposed to physics without calculus than never be exposed to it at all.
Calculus is an elective course in most schools.
Calculus is an elective course in most schools.
Calculus should be taught to 10 year olds and it should be mandatory. I can't believe how much people in US are afraid of math.
Exactly. There’s nothing about the basics that can’t be grasped early. Sure you won’t be solving weird indefinite integrals or differential equations, but the concepts should be delivered early.
With sites like Math Academy, anyone who can afford the monthly fee will be able to learn not just calculus, but linear algebra and discrete math and probability as well at whatever age they want so long as they're willing to put in the work.
I'm extremely optimistic about the future of mathematical literacy beyond the requirements for high school graduation.
I'm extremely optimistic about the future of mathematical literacy beyond the requirements for high school graduation.
I was lucky enough to have several teachers that were able to undo a lot of the damage bad math teachers imposed on me (on a lag) and continue on from Calculus to the foundations of mathematical systems (Abstract Algebra).
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.
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.
I mean I think the bigger issue is that, at least for me, unless I am regularly using a skill I learn I forget most if not all of it.
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.
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.
I was recently studying some basics of quantum computing and had to re-learn a bunch of linear algebra. I thought I had forgotten everything, but when I sat down to review the material I picked it up very quickly (much faster than when I was in undergrad). It was like riding a bike. I think we think we forget, but the knowledge is actually dormant and much easier to get back.
So I don't think forgotten knowledge is wasted, it's still valuable.
So I don't think forgotten knowledge is wasted, it's still valuable.
I wouldn't recommend a traditional calculus course to anyone. There's no reason to do derivatives or integrals by hand, and that's most of the course.
The practical applications of running differences and running sums can be taught to people with minimal programming experience and without algebra.
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?
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?
not to anyone? really?
Many times you can speed up your code and boost accuracy by using higher order methods. What you describe, if I understand it rightly, is 0th order.
I'm early into Calc II right now (MathAcademy's equivalent of it), having started 6 months ago at a D-student's level of Algebra II, and I'm curious what the "right" calculus to learn would be.
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!
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 a more conceptual introduction leaning on using computers, whose goal was getting STEM students up to speed to understand the context of work in their various fields, you might enjoy https://www.science.smith.edu/~callahan/intromine.html
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
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
Neat!
I'm gonna push back and claim that learning calculus the traditional way is still worth doing.
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.
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.
We're not at odds! I think all I'm saying is that when manually working out integrals gets frustrating, there's a learning-mode escape hatch to just figuring out how to solve them in something like Sage; when the frustration subsides you go just get back to doing the manual stuff. When I'm in Sage I'm still learning stuff. I'm never abandoning analytical work.
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 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.
Oh sorry I wasn't pushing back against you specifically haha. Just those pushing the idea that learning calculus the traditional way is useless since we have computers.
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/
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/
Weird. Calculus is fun.
So much mathematical modeling is based on systems of differential equations. It's difficult to understand the conceptual approach without some basic understanding of calculus.
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.
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.
Differential equations is kind of the same situation. The important intuitions are more likely to come from repeatedly applying a function to an input and seeing where that takes the simulation, than by doing DiffEQ practice problems.
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.
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.
Iverson's Calculus course (using J) does this: http://www.jsoftware.com/books/pdf/calculus.pdf
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!
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!
Anyone have a mirrored link? Looks like the site is timing out now.
The question makes as much sense to me as "why teach literature in the age of typewriters?" Not that the analogies are perfect, but the idea that it's not worth learning something because a related technology has advanced significantly is a non sequitur.
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.
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.
Calculus is critical.
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.
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.
...how do you plan to do gradient descent if not without calculus? Seems like OP is trying to lock an entire generation into proprietary tooling.
Is AI even relevant here?
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.
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.
I used Wolfram Alfa like 15 something years ago to explain book exercise solutions step by step in some algebra and differential equation courses.
I'd never recommend someone not learn calculus, but the average non-technical high school or college student would probably benefit more from learning statistics instead.
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.
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 is like language. You can have a translation book and sure it'll work if needed. But being fluent teaches you about an entire new world.
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
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
As somebody who aced calculus, what I was effectively doing was just being a human calculator mindlessly applying memorized differentiation and integration rules to get desired result. No real thinking or problem solving, which begs the question of why bother really teaching kids to be machines when machines can do such task. I would imagine the difficulty others had was more due to the way the topic was approached/framed rather than their executive ability to do predetermined tasks.