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My understanding of the first-year calculus/linear algebra in the U.S. is that: students basically only need to memorize all kinds of math formulas instead of theories (even for theorems, for example Newton-Lebnitz theorem or L'Hospital's rule, they only need to know those final form formulas without having to understand anything else like conditions in the theorem in depth). For most schools (not places like Harvard/MIT/Princeton), we do not teach any proofs at all to non-math majors.

I understand that perhaps most of the science/engineering/business/medical students don't need to know any proofs, but in the era when lots of math computational software and A.I. tools available, do students also need to know plugging in formulas and do computations by hand? At least from my teaching experience, most students still have really superficial understanding of the formulas they use, and the reason behind this is largely due to that they don't know why they are true.

On the other hand, as part of higher education, I don't see the point of not teaching any proofs in calculus/linear algebra. Certainly, you don't have to teach them the long proofs of the change of variables formula in high dimensions or the existence of Jordan normal forms, but at least the basic things like epsilon-delta language, formal definition of limits and spectral theorem for symmetric matrices. Rigorous mathematical thinking, at least in my opinion, is something that could benefit college students' overall development much more than just memorizing formulas and will surely make them better thinkers.

I understand that freshmen in the U.S. colleges may have little math maturity in understanding math proofs. Well, then we simply integrate rigorous math proofs into their education from scratch.

With computational software (and now AI) available for so many years, why are we still not teaching proofs in the first-year calculus/linear algebra, but instead still simply teaching students to plug functions/equations into formulas and do calculations that can already be done with modern computation tools?


Update: I removed AI from the title. My mail point is why we are not teaching deeper things like proofs in calculus/linear algebra given computation is no longer a challenge, rather than A.I. is going to replace blablabla...

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    I assume you are a graduate student in math. Why don't you go to your advisor and ask this question? Incidentally, the situation depends heavily on the country. In much of Europe teaching proofs to undergraduates is a norm. Commented Apr 7 at 21:03
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    The answer is simple. A university with a 10% graduation rate would go out of business very quickly. Commented Apr 7 at 21:07
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    Then I am surprised by the question. Maybe you were teaching exceptionally strong undergraduate students but many of the students I teach come from high school unprepared and have trouble with basic arithmetic and do not know how to think logically (and feel offended by attempts to teach them this skill). Still, in all my undergraduate classes I require students to know at least some of the proofs. Commented Apr 7 at 21:16
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    @NoOne: If you ever get a chance to teach logic to average undergraduate students, then you will find that most of them are simply unable to learn to do logic (as opposed to parroting and applying logical formulas) in any reasonable amount of time with any reasonable amount of effort. Commented Apr 7 at 21:36
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    @NoOne I downvote anything that's predicated on "We have AI now, why are we still doing XXX???" Commented Apr 7 at 21:54

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I'm a professor of mathematics who teaches a lot of linear algebra and calculus. I did not downvote, but I suspect downvoters disagree with the premise of the question, that access to AI tools should mean that we teach more proofs in these courses.

Even ten years ago when I started as a professor, students had apps on their phones that they could point at a classic calculus problem like "Compute $\int_{0}^\pi \sin(x) dx$" and get an immediate answer. So, like you, I decided to de-emphasize rote computation. For sure, my students still learn the basics in calculus, like the limit quotient definition of the derivative, power rule, product rule, quotient rule, chain rule, Riemann sums, fundamental theorem of calculus, u-substitution, and integration by parts. But, instead of doing dozens of examples where we compute such derivatives and integrals by hand, I only do a couple, then show students how to compute them in Wolfram Alpha.

This frees up a LOT of time, which I use to focus on the concepts of calculus. Sometimes we do go through an actual proof, but much more important to me is that they understand that calculus is about the study of limits, and what happens when $x$ undergoes a very small change. I focus a lot on secant lines limiting to tangent lines, what goes wrong for $f(x) = |x|$, and the geometry of the fundamental theorem of calculus. I also teach students how to apply these concepts, and what the derivative and integral are good for. We talk about sensitivity analysis, linear approximation, Newton's method, lots of real-world optimization problems, and lots of real-world integration problems, e.g., from probability theory. I'm now doing sequences and series, still focusing on the main tests for convergence and divergence but also showing students how to answer such questions using Wolfram Alpha, and then focusing on what sequences and series are good for, e.g., applications of Taylor series (to solving differential equations, or physics, Bessel functions, etc.), interest on bank accounts, build-up of medicine in a body due to dosing and half life, etc.

I teach linear algebra the same way. We certainly do some proofs, but you can get at the concepts of linear algebra very nicely using just a bunch of true/false problems. We focus a lot on the geometry of linear transformations, applications of eigenstuff, etc.

I try to remember that my university teaches about 100 students calc 1 and calc 2 every year, and probably 60 students linear algebra, but only ends up with 20 math majors per year. So in both classes a large majority of students will not be math majors. It would not be sensible to teach a course consisting entirely of proofs. Instead, I emphasize the concepts, applications, and the importance of rigorous definitions and theorems, even if students lack the mathematical maturity to prove those theorems. More important, to me, than students knowing how to prove a theorem, is knowing that it's essential to check the hypotheses before blindly applying the theorem, and knowing what to do if the hypotheses are not satisfied.

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  • Thanks for sharing your experiences! Very interesting and helpful! Just curious, what proof did you do in linear algebra? I tried to prove the associative law of matrix multiplication in front of students and feel like it was abstruse to them unless I seriously spent lots of time explaining...
    – No One
    Commented Apr 7 at 21:54
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    We prove things like the Spanning Subspace Theorem, which says that the span of some set of vectors forms a subspace. Similarly for the Distinct Eigenvalues Theorem. But, we have a designated proofs course, and students don't formally learn induction till then, so usually we just do the case for $n=2$ and $n=3$, then waive hands about how "the same kind of argument works for any $n$." We've also proved things like "similarity of matrices is an equivalence relation" or "the space of polynomials of dimension $\leq 2$ is a vector space", etc. Commented Apr 7 at 22:01
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    Readers should keep in mind that David White teaches at a fairly selective college that the average student would not be admitted to. I suspect what he attempts to do would be a failure at most universities. Commented Apr 7 at 22:20
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at least the basic things like epsilon-delta language

This is one of those things where, once you understand it, it seems trivial. It probably seems to you like you should be able to explain it in about 10 minutes and there's no reason your students shouldn't understand it immediately.

However, for 99% of students, epsilon-delta language is very confusing. I remember spending weeks trying to get intuition for it -- and this was after I had a good fluency with the computational aspects of limits and derivatives. Now it seems easy, of course...but that doesn't mean the next generation will find it any easier than I did.

In a larger sense, this is a common failure mode for instructors -- as we refine our courses, we tend to add more detail that we find interesting, or make them more rigorous in a way that we find satisfying. However, such changes are often counter-productive in that we shift our focus away from the fundamentals, and students end up with worse outcomes.

at least in my opinion, [this] is something that could benefit college students' overall development...and will surely make them better thinkers.

A lot of people feel this way about their pet subject. Philosophers in particular often say that the world would be a better place if everyone had a philosophy minor. I've heard similar claims for foreign languages, computer programming, literature, art, math, physics, etc. And I don't necessarily disagree; I personally have found great value and joy in all of these fields (except maybe philosophy). Indeed, as discussed here, this is the whole premise of the liberal arts experience in the US.

But there is a limit to what students can reasonably be expected to learn in four years. Is there room for improvement? Almost certainly. But trying to force novice students to cover more advanced material in a class that is (for many) already quite difficult and fast-paced is probably not the way to go about it.

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  • Thanks for the great answer! This makes me think a bit more about the limitation of what student could learn. I previously only thought about that background and since they are teens, they should be able to learn fast and grasp a lot of things even with weak background. Now it doesn't seem to be the case.
    – No One
    Commented Apr 7 at 22:04
  • @NoOne I was an excellent math student in high school. In a small urban HS, about 10-12 kids make it into "Math 5" which is supposed to be pre-calculus, I think. The usual teacher died so another teacher took over. I don't think she was as good nor as charismatic and friendly. She tried to teach us about epsilon-delta and I just could not get it, and nobody else seemed to either. I would have rather she focused on the equations with less theory so I'd have those memorized before I took uni Calc I and II so I could focus on the theory and proofs. (ended up in geodetic science--not too much math
    – mkennedy
    Commented Apr 8 at 14:42

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