Very similar experience for a lot of math programs in France. For about a year we did a lot of repetitive and uninteresting stuff (integration by parts, compute Taylor series on the whiteboard, ...).
The only concept from that time I used in my day job is the binomial coefficients. Yet I don't regret taking the class, in some inexplicable way I feel it has made be better (at what I have no idea).
Mostly it was learning about variables, plotting, some polynomials later. In the end the majority of it was just memorization without a reason for using it.
I don't really remember most of the calc stuff I had to learn but I can at least remember why it's useful but after 3 years I barely recall how to get a derivative and would have to look up integrals.
Personally whenever someone can give me a percentage quickly (sometimes at all!) I am impressed. I tend to feel math education is lacking (in America/general). I especially like articles that surface on HN with math and even more when they have illustrations/examples [1]. It's amazing how many times I had read about it before it just clicked.
I actually graduated in math, but I did struggle with these things. I also graduated in France where we have many more hours of class and the math program is pretty intense (did a year in Canada and it was a piece of cake in comparison). There’s also the fact that at that time you’re discovering yourself, partying, dating, etc. which takes a lot of time...
I took, we called it Diff-e-q, my freshman year in college and the professor died a couple weeks into the course and the new guy was not pleased about having to teach it. We did the applied aspects with the springs and the little ants running away from a candle heating the corner of a plate.
What I didn't like about it was just all the memorization. I had no desire to memorize a bunch of formulas that I knew full well in the real world I'd look up in a table or type into a computer. What I wanted to learn how to do was solve problems using math, not memorize patterns of formulas to apply to problems.
So I didn't memorize them and instead went to work and earned money to pay for college. Still passed the class but it was one of my lowest grades. It's a hard class even without the memorization.
It would have saved me a ton of frustration and wasted effort if it were a lot less than currently. In high school I remember hating math - it was obvious some people were far better at grinding through computations than I could ever hope to be, and it made me resentful being forced to go through what seemed a pointless exercise.
In university I made a late switch to computer science, and took real analysis thinking it would be very painful, but would make up for having no calculus. I actually enjoyed so much I switched to math, and so did sort of a math crash course over a year to prepare for grad school. Some of the math was hell. Studying differential equations, complex variables (computation oriented), and bits of differential geometry, etc. would leave me with so frustrated, wondering why I was putting myself through this. On the other hand, measure theory, complex analysis, functional analysis, algebra, and galois theory were so illuminating. I could just sit there for hours working through proofs, and not burn out.
However, this would not work for everyone (or most even). It shocked me that everyone else didn't see it my way, so it was very interesting observing how others studied/thought about math. I had a friend in all my pure math classes whom I worked on assignments with. He was a computational wizard who could flawlessly plow through pages of computations. My rate of errors - flipping a sign, carelessly misapplying a rule, etc. - was so much higher than his I had to conclude our brains were just wired in a totally different way. I noticed, when trying to prove something, we'd proceed very differently. He'd take what he knew to be true, and just begin enumerating some logical consequences of that, and go in the direction which seemed to have the smallest blowup in data. I'd usually assume the statement was false, and think and think and think about why that would be so absurd. Then, when I came up with the abstract explanation in my mind, in my mind I'd shape it into something concrete enough to write down (or even describe).
I identify a lot with the mathematician Alexander Grothendieck (at least before he became a little eccentric), because he's one of the rare examples (I know of) of an outstanding mathematician who seems to have approached math the way I do, and derives value from it for similar reasons. AG was reknowned for thinking about math in an extremely abstract way: many people learn by taking specific examples and then playing around with them mechanically until they get a general 'feel' for what's going on. Instead AG would describe the phenomenon being observed in the most abstract and general way possible, sometimes building an entire new theory of which the solution to the original problem was merely a trivial consequence.
Here is a two part piece biographical essay that, regardless of your interest in math, you'll probably find very interesting. He led an extraordinary life:
EDIT: Hmm, I sort of went off on a general rant instead of making the point I intended (this tiny text field makes that so easy, haha). I wanted to point out it's a common distinction in math made between the "problem solvers" and the "theory builders" (AG, mentioned above, epitomizes the latter). Most subjects in pure math are populated by people in one field or the other, and I believe that by forcing a computational approach on people early in their development, you're completely turning people off of math who could have fallen in love with the abstract theory. How prevalent this is, I don't know.
FWIW, when entering my CPGE [0] in France, our math teacher asked we forget everything about math except: natural numbers (0, 1, etc.), addition and multiplication. Everything needed to be scraped and "taught correctly", and it took only 2 years to get back up to speed (the entire program for the school year is available (in French): https://prepas.org/ups.php?entree=programmes).
Math is incredibly simple to build from scratch (as in: doesn't require a ton of knowledge) [1]. How long it takes for it to "click" though is another matter: I've had a very hard time with calculus and basic logic in first year, and thoroughly failed my second year.
I don't have a book to recommend though (everything was taught in class, no textbook); though I remember vaguely some books that others here do recommend.
I recently decided to start learning maths again. I stopped it after my penultimate year of secondary school because I found it such a joyless subject. I remember friends from the 'advanced maths' class lording it over me saying they were doing 'matrices' (no idea what they were) and 'second order differentiation' (differentiation was hard enough, so that must be way too tough, right?).
I figure i've really only missed out on a couple years of math education compared to the people I know who I consider 'good' at maths. And to my advantage I did do some statistics at university. With all the improved learning materials available to me, plus an alliance with programming, and my improved bullshit-detector for bad teaching and studying practices, it should be a breeze to catch up.
One problem I still have though is that maths just gets so incredibly boring... at least classes do. I used iTunes U and Khan Academy to study calculus and linear algebra. I had to start skipping past some of the really mechanical parts, because as the article said, as a programmer you just think 'put that in a function and never worry about it again.'
Breadth not depth is definitely what I'm after, although I do worry that it's the equivalent of being a musician who knows lots of diverse harmonic theories but still hasn't mastered some scales that would let him/her jam with other musicians.
Really? I had one sort of similar because I took as many cross listed math electives as possible and avoided any software type courses, but even then I learned a lot less analysis, algebra, geometry than a good math student would learn
I found math class stultifying and boring until the focus switched from learning algorithms for computation to doing proofs of relations and concepts. Also I began to discover the actual applications for advanced math, instead of learning it for its own sake.
I wish in hindsight that instead of taking endless years of calculus there was a high school version of real analysis, and exercise problems rooted in real-life to motivate the learning.
Sounds like she would have enjoyed a stats class more. Applied math without as much emphasis on first principles and formal proofs. I didn’t like university level math (first year lvl) because I preferred programming and doing math without computer assistance seemed to involve a lot of wasted effort. Maybe I’ll revisit it with Lean.
I'm old. There were no PCs then, so the analytic stuff was useful. What I remember most about the pure math classes were proofs about whether or not something was solvable. But never anything about how one might actually solve anything. And far too much number theory. All too abstract for me.
I thought I was good at it until I hit 17-18, doing advanced mathematics. I really didn't understand Taylor series, and I just froze up on the calculations. You fall behind, and then the class just moves forward and it stops being fun anymore
I don't know the solution, but as you said - I went back into it later and it was much easier. Catching up those years inbetween was hard though!
I had an almost identical experience with my ODE class. This is exactly when I started hating math, and moved out of computer science (which was part of the math department at the university I was at).
It took me a long time to return to learning higher maths, this time on my own according to the needs of my job, which is far less an ideal environment than University.
If ODE contains "a few basic concepts that they will remember for the rest of their lives" I never learned it, and I wonder what those concepts would be.
The math courses from my first university year where 90% a redo of the math from my last two years of high school. The difference was that in university they emphasized the theory over exercises, basically forced us to learn 1 - 2 hours demonstrations of some theorem.
I remember that class, IIRC it was just called Geometry at my school. It took me about six weeks to figure out that we weren't doing calculations and so I felt like I was eternally weeks behind that semester as I struggled to internally catch up with the theorems and principles being taught. I always felt like the teacher was doing a good job, I had just missed something (I may have missed a couple of crucial days at the beginning of the year).
We had a very rigurous program in highschool back in Eastern Europe, it was so good that taking clac in college in the US was redundant. However, I remember having a similar click moment when I visually or intuitively understood derivatives and integrals. Before that it juggling math mechanics, which in itself is also not bad and helps build a certain muscle that can be used later on. But I am familiar with the fumbling a bit through some math classes.
My conclusion is different though. I think there are different types of thinkers and different teaching styles and when the subject becomes very complex the disparity between the two is exacerbated. And luck plays the role in matching up with the right teacher/professor for you.
I simply could not appreciate mathematics until university. Memorizing times tables and repetitive drilling is not the way it should be taught at the earliest introduction. I don't know how it can be done properly, but the way it is done at the early phases bored the hell out of me. Later, much later, I learned to appreciate it's power and beauty.
I hated maths in school, had to persevere all the way through my Electronic Engineering degree, and still didn't "like" it, until much later, many years later, when it was no longer academic, but pure interest.
I do wonder what a difference it would have made had I had teachers and lecturers half as interesting, humerous, and engaging as this.
The only concept from that time I used in my day job is the binomial coefficients. Yet I don't regret taking the class, in some inexplicable way I feel it has made be better (at what I have no idea).
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