Too Much Calculus – Gilbert Strang (2001) [pdf]
www-math.mit.edu
www-math.mit.edu
I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's free textbook:
http://joshua.smcvt.edu/linearalgebra/ (GFDL license, LaTeX source available)
(I get nothing out of recommending this open source textbook other than the joy of sharing a book I loved.)
What I wish I had done a lot more of is statistics. Being able to reason about probabilities clearly, getting an idea of how a system performs given various variables, and most importantly, being able to communicate this to others has been very useful to me. I think anyone (front end, backend, etc) can benefit from learning about statistics.
And optimization is really just differentiation (+ some geometry).
I'd say the problem is too little calculus and not too much.
Linear Algebra is very useful too, but it's not really fair to say one is more valuable than another. It just depends on what kinds of applications you end up working on. Universities should just teach both.
In practice, both have wonderful presence in math both applied and pure. Linear algebra is one form of abstract algebra which is easily one of the most important mathematical disciplines. It's also the characterization and method of manipulating large arrays of numbers in interesting and practical ways. Statistics is the mechanism of interpreting live data under knowledge of uncertainty toward generalization. It's also probably well characterized as a particularly well-behavior substudy of measure theory... the pure mathematical notion of generalized "substance"
Sorry, I meant my college calculus courses, although incidentally the statement applies to both.
I think colleges would be much better off teaching introductory math as "applied". I got significantly more out of statistics, because that class taught us how to spot problems where that branch of mathematics is applicable. I really liked how Udacity used real-life orbit calculations from various space missions as assignments. It gives the whole thing some context.
The weird thing is that I can't predict which things are going to be sufficiently unpopular amongst people impolite enough to downvote instead of coming up with a good counterargument (or even better: fact). I had /never/ expected something like this to get a single downvote (and I was down at -3).
Was extremely gentle and mind blowing.
I had a similar experience where my college linear algebra class was not good, taught by a grad student who seemed to have no enthusiasm for the material or for teaching, plus his accent was hard to understand at times. It was also the first exposure to proofs in the math curriculum so from the beginning I remember being overwhelmed by those and totally losing sight of the forest for the trees.
I sat through the whole semester thinking it was a pretty useless subject. My senior year I happened to take a machine learning and a quantum mechanics class at the same time and had a big ah-ha moment when I realized how big a role linear algebra plays in such a wide range of subjects. I went back through my old textbook and notes at that point and that new perspective made a huge difference, it was like it was completely different material and it was a lot easier to understand.
I had a somewhat old-fashioned education for an American, and learned proof via Euclid when I was 12. And again, but poorly, when I was 16, but that's another story. Proof is supposed to be learned a) thoroughly b) early and c) as a part of geometry. What's the point in even studying Euclid if you don't learn the method of proof?
I still prefer reading textbooks in paper, and one interesting note: it looks like a couple of companies and individuals are offering paper copies on Amazon, but none of them appear to be Henderson himself.
And the companies or individual offering paper copies still charge $20 – $30!
$20 for the official version (see the book's web page for a link). That is, essentially, the price of printing and binding.
I wonder why the imbalance exists. Too much focus on preparing people for careers in Engineering? Or preparing them for more abstract concepts like Analysis and Analytic Geometry?
Yes. Computer Science curricula tend to focus overmuch on preparing students to work as software engineers, and Mathematics curricula tend to focus overmuch on analysis (to the point that once you get past Calc 1+2+3, Linear Algebra, Differential Equations, Mathematical Analysis/Epsilon-Delta Calculus-with-proofs, and Real Analysis, things like Algebra and Topology are senior-year electives).
On the other hand, I'm prepared to defend to the death the statement that programming is applied formal logic. (Or possibly abstract algebra; that is likely more appropriate for some more modern things. On the other hand, I grew up in the land of Dijkstra, so....)
Also, I know it's not free (the coursera course is free) but it's still a great read and seriously applicable http://codingthematrix.com/
As for the average STEM student, in my experience, most freshmen wouldn't be ready for linear algebra (unless by "linear algebra" you mean "matrix arithmetic"). They have enough trouble understanding functions in Calculus I: linear algebra's "functions are basically matrix multiplication" feature would NOT help.
Another thing to keep in mind is calculus has had much more impact on history/society than linear algebra. The fundamental theorem of calculus is a legitimately earth-shattering breakthrough: none of the theorems in linear algebra come anywhere near it in cultural importance.
Pragmatically, calculus is the most suitable mathematics for the type of testing-oriented, uniformizable/transferable pedagogy that a university really (in cold practical terms) must depend on for first-year students. It would be extremely difficult to force the kind of uniformity onto linear algebra that would be necessary if you seriously want to have millions of transfer students moving around between thousands of universities every quarter, gossiping about which linear algebra teacher gives the easiest grades, etc.
That joke comes from the analytic calculus tradition of physics. I took a lot of physics while an undergrad, and I came out unable to do any useful physics. This is because as soon as you try even very simple problems the partial differential equations that result have either no, or extremely difficult to find solutions, and it is usually the former.
I spent years learning analytical tricks that I just don't apply.
There has been a lot of push back on this in education recently. There are Giordano's and Sherwood&Chabay's textbooks in physics now which emphasize computation over analytics. The reality is that if you want to do the vast majority of work in physics you will be doing things like reducing your problem to Ax=b, and throwing the linear algebra machinery against it.
I type this while taking a break from Strang's "Linear Algebra, Geodesy, and GPS" textbook, which I need for work right now. There is an endless need for this sort of math.
Sure, it is a matter of balance. But I flip through my diff eq books and I don't see a lot in there that I actually need. There is nothing in Strang that I don't know, and ditto for a good numerical methods book.
Even the work that Gauss did with orbits did was linear algebra and computation. He did not analytically derive the motion of Ceres, for example, but used linear algebra to compute it.
I, along with many other posters on this thread, and everyone that I have talked to, wish that there STEM education had been much more about Ax=b, the only problem we really solve, and a lot less about using huge tables of analytic integrations to try to find answers to equations that we will never encounter in the real world because they are so specially crafted as to have an analytic solution.
There are a lot of programs that are experimenting with different approaches though, and I think this will only get better with time.
I don't see any of the points you mention as particularly strong. There would be nothing wrong with presenting a more hands-on linear algebra course as a first course for freshmen. (Calling it "matrix arithmetic" does not do the notion service.) The material seems very immediate, and you can use a REPL like octave or matlab to reinforce concepts. You could deepen it as you wished by introducing more abstract concepts and by doing proofs, but I don't think that's necessary. I learned a lot from Golub and van Loan, for instance.
Second, the notion that calculus has had more impact on history/society than linear algebra is very hard to defend. Think of all the numerical modeling that has been done using matrices. Think of all the large-scale computations/optimizations that are done first by finding a linear system, or approximating one. I'm not saying there is a clear answer to this question, just that it can't be the foundation of an argument, because it's so undecidable. Another thing that makes it undecidable is cases where you need both (e.g., weather forecasting).
And saying it would be hard to teach the subject in a uniform way -- really, this is a very bureaucratic objection.
It is ideal (and indeed, not uncommon) to have linear algebra and calc 1 in parallel --- for special honors classes. It wouldn't be realistic to do it for the general freshman class. Do you realize just how many students that includes, in a big state U.? "a REPL like octave or matlab" would be wonderful, amazing, if it weren't about as realistic as the space elevator. Do you know how many endless headaches there are with software-augmented courses, even when it's just software you run in the browser? I applaud your ambitiousness but let's just say I don't want to be in your shoes when you've just told a thousand 18-year-olds to install Matlab!
>the notion that calculus has had more impact on history/society
Linear algebra is hard to even put your hands on, historically. There was no Isaac Newton figure putting the finishing touches on an amazing breakthrough "Fundamental Theorem of Linear Algebra". It is the book-keeping field of mathematics. "All the numerical modeling that has been done using matrices" almost entirely arises by discretizing differential equations so linear algebra doesn't win any points there.
The different history does not bear on the ultimate impact and general utility of the tools offered by calculus or by linear algebra.
Eigenvalues and orthogonality are not just "book-keeping". Why give forth empty slurs like this?
If you can do a decent treatment of eigenvectors in a non-honors freshman course, my hat goes off to you. Even in a 2nd-year course where students are more mature, that's hard (first time I took LA, eigenvectors were rushed at breakneck speed in the very last lecture). And while they're very important, eigenvectors are still a shadow of the historico-cultural impact of calculus.
The main use of linear algebra to me is describe local behavior of functions/dynamical systems. Sure, calculus is the setting of functions in full generality, but understanding the algebra of local interactions is an essential part of the whole -- how can you go far into calculus without understanding derivatives? Linearization is practical the way to solve 99% of engineering problems.
And I wasn't playing dumb. For one thing, even if I assumed the word cliché was used incorrectly, I still couldn't easily guess exactly what the OP really meant. Second, I don't like assuming people are using words incorrectly, it is safer to ask what is meant.
Can you substantiate "millions of transfer students between thousands of universities every quarter"? That seems like outlandish hyperbole. Irrespective of the precise number, a system with momentum is not sufficient justification for its continuation.
Linear algebra has really only come into its own since humans have had powerful computational systems at their disposal, as a large part of linear algebra's usefulness (including its relevance to the examples you describe) is efficiently organizing and performing calculations in higher order systems. That is not to say it is irrelevant or non-impactful - far from it! I'm just saying linear algebra has taken hundreds of years for its real power to be leveraged, and in the intervening time, calculus has reigned supreme as a system for modeling and describing real systems.
(FWIW, I agree with the author and the paper in question, but I think OP you replied to is absolutely correct that teaching both gives perspective that neither alone would impart.)
Studying any kind of engine (or for that matter, anything that moves), solid-state electronic device (from transistors to antennae) or power distribution network and doing any kind of design about them is literally impossible without calculus. Even algebraic approaches to these are formulated through discretization of a continuous model (which -- unsurprisingly -- you need calculus to study).
We have such a thing as databases, devices that can manipulate graphics and high-complexity economic models that are analyzed through linear algebra because Newton and Leibniz found the mathematical tools we need in order to model (continuous) change. We could mostly get along fine-ish without linear algebra, but the industrial revolution that spawned our society would be impossible without calculus, because without it you cannot design anything non-trivial that moves or runs on electricity.
Both are incredibly important. I would say that linear algebra and discrete methods are more important for computing, but calculus has had a larger impact on science especially physics. Newtonian physics - a revolution in thinking and unification of mathematics and science - is intricately tied to calculus.
Even in the example of economics that you cite, calculus is used heavily for the theoretical underpinnings for finding equilibria and deriving min/max.
I'm not. Consider all of physics. Physics is written down in terms of differential equations. You need calculus to get these equations. In terms of graphics, for example, are you magically creating only static images that never change in time? How do you think people came up with the algorithms to do things like refraction or reflection? These all involve calculus. Calculus is the most important tool in applying mathematics to the real world. It is how we derive the models for pretty much everything I can think of. Any time you want to model something where anything changes, that is calculus. Any time you want to find an optimal solutions, that is calculus. I can't think of a single area of applied mathematics that doesn't use calculus in some form. Even mathematics in general, calculus is used in some form pretty much everywhere in some form, except maybe in the foundations of mathematics like set theory or logic.
Solving them, on the other hand, is a different problem. This is where linear algebra is important. When solving an ODE or PDE, when I discretise it, all I am doing is re-writing it as a giant linear algebra problem. Quantum mechanics, for example, is dedicated to finding the eigenvalues and eigenvectors of the Hamiltonian. Linear algebra provides the tools to solve the problems posed by calculus. But it isn't always required.
In fact, if you look at the history of linear algebra, it only really became wide-spread when quantum mechanics was developing since there is a deep connection between linear algebra and quantum mechanics. Heisenberg had never heard of a matrix before despite discovering Heisenberg matrix mechanics. He was told by Max Born that what he had been doing is actually this thing mathematicians called "matrices". Now linear algebra is a required course for physicists (incidentally, linear algebra was the first lecture I ever attended at university).
So I would argue that learning calculus is the most important thing someone should know mathematically, since it is the tool we use to build models. The second most is linear algebra since it provides us the tools to solve these problems. But in my mind, knowing how to derive these models is a lot more important than being able to solve them. That and linear algebra doesn't give you any insight into why we are solving the problem that way, it is just a tool to solve the problem.
Also in Calculus 1,2,3 and so on, you learn increasingly more complicated techniques of calculus to solve more problems. But when you start numerically solving them, linear algebra doesn't care. It doesn't care if your equation came from the Einstein field equations or Newton's second law. It's all the same to it. So in a sense, I can teach you everything you'll need to know about linear algebra in a single course but the same is not true of calculus, which requires multiple courses.
I think:
Okay, then go back to pre-calculus times and somehow still invent the modern world without it and then get back to me on how irrelevant it is.
Also, you don't get to wear this shirt: http://www.topatoco.com/merchant.mvc?Screen=PROD&Store_Code=...
My feeling is that it would take a few hundred years extra (or maybe more!) to get to the point where you had a computer without calculus. Might even take a thousand. You'd only be able to do discrete stuff, so while Babbage's difference engine might be up for grabs (and mechanical calculators definitely would be) things like the Norden bomb sight or naval firing computers wouldn't be, since they're continuous not discrete systems.
You might not EVER get to computers because things like Nyquist's theorem and stuff like that doesn't even exist in a discrete-only world. Things just sometimes work and sometimes don't and it's sorta related to the frequency, but how is unclear. Aliasing is just something that randomly bites you in the ass and there's no way to develop a rigorous understanding of it and to engineer yourself out of the morass.
I would not volunteer for that mission. I like the modern world.
"The fundamental theorem of calculus is a legitimately earth-shattering breakthrough: none of the theorems in linear algebra come anywhere near it in cultural importance."
Linear Algebra has the Fundamental Theorem of Algebra (i.e. every polynomial has a solution in the complex numbers) and also the Isomorphism Theorems that say all vector spaces of n-dimensions are the same. For example polynomials of the form c+bx are equivalent to the complex numbers (over the reals) which are equivalent to two-vectors (x,y). Quantum physics is basically linear algebra, with quantum states defined as vectors in a quantum vector-space.
Don't dismiss the impact of Linear Algebra.
What do they have trouble understanding about functions?
Actually, it took me until just a few minutes ago to realize that one of the major differences between my CS-centric way of thinking and the proper mathematician's way of thinking: mathematicians treat all coextensive representations of the same object as identical, while a computer scientist tends to treat only provable intensional equality as actual identity.
Yes, it was amazingly important during all the wars and conflicts of the last century.
I don't think it was for the memory of Newton that they choose Calculus, but because it was more useful in a war machine than the alternatives.
For example, I'm currently studying probability theory at a fairly rigorous level (building up from measure theory) and my relative lack of skill with calculus is the main stumbling block. And I was recently talking with one of my professors who said that the biggest obstacle he sees students facing in trying to understand advanced mathematics isn't the difficulty of the math itself, but in their lack of mastery of the fundamental calculations that are necessary to follow the ideas and to come up with them on your own. Fluency with calculus is an enormous part of this. That said, I agree with the point, which is that for most students linear algebra is more useful.
Actually, I might go a bit farther and say that most students don't need to learn much math at all, at least in the way it is taught. Math is pretty poorly taught for a subject that is actually quite interesting once you start to understand it better (part of the problem is that in order to get to the interesting parts you need to first master all the boring stuff), and most people won't use any nontrivial math in their future lives.
While studying mathematics in college, once I had finished my Real Analysis requirement, I jumped headfirst into the algebraic side of things and never found myself using any sort of calculus. Even in my Topology course, we focused much more on using techniques from Real Analysis than specifically calculus topics (the former just being a generalization of the latter).
Probability theory is somewhat deceptive in its classification, since much of it "feels" a lot like a discrete mathematics course; however, much of the concepts, like you say, are underpinned by measure-theoretic principles, which is heavily analytic. It makes sense that calculus would come in handy in a much deeper study of probability theory.
You might make a discrete model for a problem and end up using calculus for approximations, equivalent calculations, or reasoning.
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-...
As someone who hadn't studied it for 20 years, it was invaluable to revisit the subject as an adult with the ability to reflect "ah, I can use that for... " instead of "Ok but when am I ever going to use this"
Bonus: when you switch him back to normal speed, he sounds stoned out of his mind.
I ended up scoring one of the highest grades in the class.
http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr...
I was a bit confused by your statement. You mean that the FFT can be done by matrix multiplication? I tried to interpret your statement the other way around, that you can somehow transform two matrices before you multiply them, trying to fit it into the convolution theorem, but I couldn't make sense of it that way.
That was the first online lecture series I ever watched, and I've been hooked ever since. I am overflowing with gratitude that so many professors at the world's best schools have put their lectures online. It is so good for a student like me, who went to a liberal arts school that didn't offer many advanced math or CS courses, and learns better from lectures than books. Strang's Linear Algebra course was one of the first way back in 2002. You're right, he is a wonderful teacher.
But it seems MIT took them down
http://www.reddit.com/r/Physics/comments/2ope15/mit_indefini...
This borders on criminal and is certainly a tragedy.
I'm quite struck and emotional about it.
I also had to go through a lot of heavy calculus courses which were very proof-centric and built from first principles. Rarely we skipped a proof of a theorem and we were expected to be able to reproduce and understand all of them at exams. I hated it often back then. However, these days I can appreciate the rigor it has taught me and I would wish I could send some of the programmers I met to similar courses so that they would learn to think clearly and precisely and be able to spot unmet or missing assumptions.
Our teachers and the creators of the curriculum were not stupid or naive. They knew well how much of hands on calculus we would use in practice vs. algebra or linear algebra. One of their main reasons for including calculus was to teach us rigor. I know that once someone overheard two teachers talking - the calculus teacher was asking someone from a CS department what to teach the CS graduates and the reply was that he can teach us whatever he wants, he just should make sure to teach us to think. I think they were quite successful at it even if it wasn't a very pleasant experience for most.
That also reminds me of this very relevant and true comic: http://savethepenguins.net/image/237344573 ("Wtf, man. I just wanted to learn how to program video games.")
Incidentally, for those who want to learn linear algebra for CS in a mooc setting there are 3 classes running at this very moment:
https://www.edx.org/course/linear-algebra-foundations-fronti... (from UT Austin)
https://www.edx.org/course/applications-linear-algebra-part-... (from Davidson)
http://coursera.org/course/matrix (from Brown)
The first 2 use matlab (and come with a free subscription to it for 6 months or so), the last python. One interesting part of the UT Austin class is that it teaches you an induction-tinged method for dealing with matrices that let you auto-generate code for manipulating them: http://edx-org-utaustinx.s3.amazonaws.com/UT501x/Spark/index... .
And of course there are Strang's lectures too, but those are sufficiently linked to elsewhere.
There's a fair point to say that one shouldn't learn so much calculus as to lose your linear algebra, but the opposite is just as true. There are a lot of comments in this thread along the lines of saying that linear algebra is far more useful than calculus in CS. This statement feel not exactly unfair but certainly like it's a reactionary measure.
Whenever you have a notion of change you have some kind of calculus underlying it. Even if you end up having to view that notion of change algebraically. For instance, there's a famous functional programming result that there is a well-defined "derivative of an algebraic data type" which allows you to talk about changes in data "over time".
I'd also feel very bad for anyone who tried to learn too much probability without studying quite a bit of calculus first.
[0] Sure there's discrete calculus and algebra underlies the mechanism of most calculus computation. Likewise there are continuous groups and you can't avoid how topology mixes the two concepts.
It was a revelation for me when I've tried to get into game programming, and half of the math taught in primary and secondary school became immediately useful, also linear algebra (which was only started at the final year of high school). Made linear algebra courses on the university a lot more interesting. There's a difference if you look at a class "what I need to pass an exam", and "what I could use in my next game".
Also tinkering with 3d graphics makes thinking in vectors, planes, matrices, dot and cross products intuitive after a few weeks.
BTW I wonder hwo many kids learnt basic linear algebra from Denthor's Asphyxia tutorials :)
Of course you can go far with "turn towards player and shoot" ai, but where's the fun in that.
Actually it would be fun to give student's assigment like:
here's the AI interface, make a game using it, and write your bot. Then we will run all the bots in free for all mode on all the games (except your bot on your game), and we will grade by sum of the points.
Send a raytrace from the bot to the player and see if it hits something other than the player, which is just one line
Pathtaking beyond simply moving towards the player is a lot harder, yeah.
You can accept that, use many rays, or do proper view frustum/body model(simplified probably) intersection, after cutting the view frustum with the terrain.
You can also consider lighting (so enemy can't see you hiding in a dark corner), or even dynamic lighting (and give enemies light sources).
Everything becomes hard after you think about it enough :)
But I agree simple solutions + some tinkering with parameters works most of the time (and gamers probably won't notice).
I took two semesters of Linear Algebra in college, the first course was an overview for physicists and engineers and was little more than the mechanics of matrix operations, and find eigen vectors and eigen values. The second course focused on Linear Algebra as an abstract algebra, with near zero application. Subsequently, reading this paper[1] and the way the author used Linear Algebra blew my mind, why wasn't I taught about THAT kind of Linear Algebra?
I suppose my examples illustrates: great teach Linear Algebra, but what facets to concentrate on? Abstract or concrete applications?
[1] http://www.geometrictools.com/Documentation/IntersectionSphe...
So yeah, maybe I'm just an outlier, but part of it may be a naming issue also. "I already took Algebra way back in 8th grade and it was a breeze! I don't need to take another Algebra class!"
Its not something that comes up every day at work, but it is still pretty often. For example, I was on a large team developing a complicated distributed system. The performance requirements were uncertain (like 500000 clients +- 200000, sending 10 +- 30 messages per second, etc). The system itself was in early development, hadn't been tuned, and we had only rough performance numbers for it.
Given only that, I needed to purchase the correct amount of hardware for the test lab, keeping in mind that the lead time for hardware was several months. I needed both statistics and some basic calculus to build a model based on what we knew, calculating variances for each intermediate and output, and then getting 50%, 90%, 99% confidence level estimates.
In the end, we got pretty close to a bullseye. On the other hand, I learned the important lesson to hide all the math from the client, lest they get awfully confused.
I took the discrete emphasis when I studied and I think I actually ended up with fewer discrete classes (5, IIRC) than calculus classes (6).
> Hamming, in discussion, concluded his life thinking "the best tool to teach thinking was to teach the calculus."
I'm just thinking here that Strang is saying Linear Algebra is more useful for _applying_, where Hamming would say Calculus is more useful for reasons that go beyond applications (not that it doesn't also have applications).
[1] http://www.cs.virginia.edu/~robins/YouAndYourResearch.html
[2] http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm...
There is a really good book in two volumes that I feel shows how methods of linear algebra can be extended using calculus to solve a wide range of problems: A Course in Mathematics for Students of Physics:
http://www.goodreads.com/book/show/2685577-a-course-in-mathe...
Volume 2 is more important. It starts demonstrating how methods of Linear Algebra can be used to solve circuit problem, then generalizes this to higher dimensions and from discrete to continuous spaces using the exterior derivative. Seeing this demonstrated gave me a much clearer, more intuitive understanding for both calculus and linear algebra.
Strang, Calculus, 2nd edition: http://www.amazon.com/Calculus-Second-Edition-Gilbert-Strang...
The page also links to another short tutorial which explains more of the geometric intuition behind linear transformations.
Neither tutorial can serve as a full course, but reading them will give you a good overview of what linear algebra is about.
can't image teaching point-set topology, teaching uryshons proof to undergrads, gee.
Without a firm understanding of probability then good luck interpreting or fitting a model. Without a firm understanding of linear algebra good luck implementing or using an algorithm.
And hell, if you have a streaming algorithm you probably no longer need linear algebra but to grasp either the probabilistic or geometric interpretations you'll probably need calculus.
I wish it weren't this way! Mathematics is a wide and fascinating field, and I think more students could find interest if our schooling reflected that. But moving some of the detail and drudgery that is the calculus sequence to subject matter that requires it, in order to make way for linear algebra, is a great start.
(Honest question, no snark.)
I think my first topology course, like in many intro analysis courses, shared concepts from calculus without requiring any previous expertise from our calculus courses. A rigorous approach to real analysis or point-set topology will be very set-theoretic, and you build the general definitions of continuity, compactness, etc from the ground up. So, I would have used concepts from my analysis and topology courses in calculus, if only it had been taught in the other order.
I guess it feels like limits are a rigorous foundation for calculus, but they are not calculus. In any case, I didn't need anything from differential or integral calculus (or that entire required course developing the theoretical basis for Stokes...), which is what I mean to say.
I think starting with discrete math first, where you teach what sets are, how proofs work and so on would be a good start. Or make your textbooks usable and not just containers of problem sets with unreadable extra paper inserted to make your back hurt more. Or make the target students and not their professors.
[1]: https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-wha...
Not for me! Heck, I never took freshman calculus or pre-calculus and, instead read a book and started on sophomore calculus then continued with, right, ordinary differential equations, advanced calculus for applications, and advanced calculus, topology, modern analysis, measure theory, functional analysis, exterior algebra, etc. Not enough calculus for me!
Strang, of course, is emphasizing linear algebra. Great! It's terrific stuff. I had an abstract algebra course that started on vector spaces and matrix theory. Then I got famous book from Princeton on multi-variable calculus and read the first chapters on linear algebra, e.g., learned Gram-Schmidt. Then I got a linear algebra book and read a few chapters and applied them to some problems in classical mechanics. Then I wrote my math honors paper in group representation theory, right, more linear algebra. The out of school, I wanted to know linear algebra better and read a good book cover to cover, carefully. Then I got Halmos, Finite Dimensional Vector Spaces and read it cover to cover. Wrote Halmos about his proof of the Hamelton-Cayley theorem and got back a nice answwer!
Then I got another famous text on multi-variable calculus and read it cover to cover, including the parts on exterior algebra. Then I did that again from Spivak's Calculus on Manifolds -- right, a lot of both calculus and linear algebra. Then I got Forsythe and Moler, Computer Solution of Linear Algebraic Systems, read it cover to cover, directed a project in interval arithmetic for linear algebra, took a course in numerical analysis with a lot of linear algebra for solution of partial differential equations, e.g., iterative solutions (Gauss-Seidel) and wrote corresponding code.
I also used linear algebra in work with the fast Fourier transform, multi-variate statistics, and more.
For more on linear algebra, I read the first part, linear algebra, of von Neumann, Quantum Mechanics.
From a world class guy in linear algebra, I took an advanced course in linear algebra -- by then the course was beneath me and I led the class by a wide margin. Favorite theorem -- the polar decomposition.
Read Coddington's book on differential equations -- gorgeous book.
The combination of linear algebra and calculus, with other math, continued.
At one time a little calculus did a lot: The initial value problem for the little first order, linear ordinary differential equation
y'(t) = k y(t) ( b - y(t) )
saved FedEx, that is, kept it from going out of business. So, useful calculus!
So, sure, I like linear algebra and like the combination!
I definitely do not see that I had too much calculus!
Here's what I saw, and still fume about: All I could get in chemistry was one too simple course in high school and one too simple course in college. All I could get in physics was one too simple course. I college, by the time I got to Maxwell's equations, I didn't know enough calculus to do really well. In college I never got math enough to do well with quantum mechanics.
But in grades 9-12 and then in the first two years of college, I was force fed like a goose with six years of English literature, with each year yet another play by Shakespeare. I agree that there was a good writer in England in the 1600s -- Newton! So it was also Chaucer, Milton, Wordsworth, ..., Dickens, the great natural order (all corresponding theorems and proofs omitted!), etc. With good information about people? Nope.
Then there was history: It never got to the 20th century. It never touched on technology, economics, or any other causes. There was plenty of time in history courses; it's just that the courses didn't have much content.
If my startup works, then I'll get back to more in calculus, functional analysis, stochastic processes, and mathematical physics. "More calculus, Ma!".