> (the one exception is that I always found proving things difficult)

Proofs are at the heart of math. Everything else is something different.

Not in the American school system, at least. AFAICT grade school "math education" is mostly about lodging a particular calculation algorithm implementation into students' heads and making them repeatedly run it, on a variety of test inputs. A large majority of Americans will never do a single bit of mathematics in their lives, cradle to grave.

To be fair, there are lots of efforts to change it. But (being pretty harsh here) a significant number of math teachers really don't have the ability to understand math themselves, let alone teach it.

When you apply that to proofs its awful. To use a gaming analogy the real world is a sandbox game where you can build it any way you want as long as you follow the rules, but in K-12 school, proofs are either for exact memorization, or crazy contrived things on rails that you're only supposed to solve the one correct way.

Sort of a perl "there is more than one way to do it" vs a python "there should only be one way to do it". Nothing inherently wrong with either other than if you and your educational system philosophically disagree, its going to go extremely badly for you.

I don't care about K-12, since I am an unAmerican. But I easily believe you if you say that the subject called `mathematics' in school is horrible. (And has nothing to do with `mathematics' proper.)

V.I. Arnold:

“Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters.”

It's true, but telling this to someone who thinks that doing some additions is doing mathematics will not help.

You cn doa prety hrrible job speeling amd stiil grt the piont across

I've always loved the idea that the academic disciplines form a cycle.

Arts -> Social Sciences -> Life Sciences -> Physical Sciences -> Mathematics -> Arts

Mathematics was/is a very creative process for me.

I very much like the related point Paul Lockhart makes in "A Mathematician's Lament"[1]: that mathematics is an art form and ought to be taught like one.

[1]: http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf

> Up through my first differential equations class I found almost everything trivial to learn

Mankind hasn't yet figured out a good way to teach the first Differential Equations course.

The first serious Calculus course introduces some level of rigor and formalism, but this alone is not sufficient for Diff Eqs. And the DE courses are typically leaning towards applications, so they tend to succumb to the "fiddle these dx's and dy's around like it's magic, in the end it might work out" approach.

The business is well founded, which you will discover later. But no one has figured out how to massage that into the first course.

It's been a while but I think this is true of a lot of introductory courses in general, like Organic Chemistry. "Trust us and memorize these rules the basis of which will become clearer down the road."

This point is more thoroughly explained under section 7 here: https://web.williams.edu/Mathematics/lg5/Rota.pdf

Thanks for that. My engineering DiffEqs course is where my math foundation started to fail me.

During my PhD I found I had to occasionally lock myself in a conference room with a bunch of papers and all the whiteboards to do a sort of "deep dive", in order to get everything into my head and push things forward. Things that were a bit rusty quickly came back after a little bit of effort.

At the end I'd emerge with a bunch of photos of whiteboards with clear intermediate steps, and then I'd write it up as a short report in LaTeX. Of course, as the report turned into a paper, I took out all the useful 'trivial' steps - which always feels a bit wrong. I think it's also useful at this stage to drop everything for a month or two and come back and see if you still understand what you wrote.

I think the "deep dive", putting prolonged effort in, and writing it up are all key aspects.

When I was studying, I went to two schools, one using semester system and one using quarter system.

With the semester system I had plenty of time to read the material and really explore the mathematics behind what I was learning. With the quarter system I was so busy running around and catching up in different classes that I never really had the time available to sit down and do that. It sucked. I think the quarter system kind of 'beat' the interest out of me because now it is hard to get back into that state of mind.

semester > quarter :(

The quarter system is pretty non-optimal, I think. The college I went to ran on ten-week quarters, and even though it was normal to only take three courses, when each of those three is trying to jam an entire semester's worth of material into ten weeks, it's hard to even keep up with a surface level coverage of the material, much less dive deeply into it. It was pretty standard to have about 500 pages worth of assigned reading, plus extra research and writing, studying for exams, writing and debugging code. Fortunately, I was not an engineer or a hard-science major, so I didn't have the additional overhead of weekly lab work. When people took organic chemistry, it was a running joke that they were going on the "campus foreign-study program", since you'd see them about as often as you'd see friends that had gone off to Argentina or France for the term.

I can totally relate and to this day I describe one of my greatest failures as a manager as analogous to how "easy" math was to me. Until Calc II I really found nothing about math challenging at all. The result of this was that everyone wanted me to tutor them in math. The problem was that I didn't solve math problems like other people did, in fact I had no idea how I did what I did. I just did it. As a result when I tried to teach someone else it was an unmitigated disaster.

Fast forward to the real world, one of my biggest challenges is helping people figure out how to get from A-Z. A is obvious to me. Z is obvious to me (although I'm probably wrong as much as I'm right), but getting someone to understand my reasoning is almost impossible.

This is why I have always found proofs difficult. I just don't know what steps people think are important to spell out but.

Study "false proofs", try to see where they go wrong. This will teach you what details need to be spelled out in detail to easily tell the difference between a false proof and a true proof. You will learn how to demonstrate you have nothing up your sleeves so to speak.

Then spell out all of them, and have someone else tell you when it's become too tedious.

I can relate to this. My major was EE and I had similar experiences, but kind of liked diffeq. Fast forward some 25+ years later, I spend all day coding and look to my old math and some new comp sci books for fun. I keep an engineering notepad with different sections in it so I can just open it up and attack problems such as concrete math problems from graham knuth and patashnik or physics problems. Even though I code through business problems on a daily basis using X framework and Y API, I find the rigor involved is not as satisfying as struggling through finding the closed form of some equation. I want to solve a deeper problem, if that makes any sense.

It isn't the answer that I seek in doing this; it's the desire to know why the answer is what it is. I think I have become more interested in learning far after graduating from university.

General skill of problem solving seems to stick with you forever.

It's probably because most of your problem solving knowledge is your own invention. Whereas math and programming is your own invention if you solve all the problems by yourself, if you code everything from the start. This gives you the understanding of the problems, and the reasoning/tools to solve it.

I've been dropping in and out of problem solving fields through the years, and I've noticed that my knowledge practically disappears, but my problem solvings skills can adapt to any framework (mathematics, physics, programming) it just takes a little bit of time to re/learn the terminology.

Reminds me of the quote "What I cannot create, I do not understand." by Feynman.

Giambattista Vico.

The same is indeed of true of programming, at least in my experience. I took a management/consulting detour that lasted about 18 months and getting back into programming was quite though. I could not focus, could not hold the problem in my head, could not go from design concepts to implementation details. It took me a solid 6 months to get back to my former level.

> I think my point is that you can lose competence in math very very quickly through lack of constant exposure.

This is not that important in my opinion. If you're lacking exposure, it must be because you don't apply math every day, and it won't be a big deal. If you apply math every day? Then you will quickly catch up with whatever field you're using.

If you're in pure math, then none of that matters as well, only proofs do. And lack of exposure will often not harm any of the tricks you learned while constructing/understanding them.

> type theory

Not sure if related to your username, but I have a book on it (that was recommended to me here, in fact) on this topic, which I was curious about due to its implications in the programming field

TAPL by Pierce is as far as I can tell the book everyone recommends, but if you have another you think is good I'm interested.

>(the one exception is that I always found proving things difficult)

Same here and I have developed a kind of a complex around it. I used to spend hours practicing proofs just so nobody comes to know that I'm terrible at them. I used to find numerical proofs to problems that had simple general proofs.

However, looking at the other replies it seems we both haven't the faintest idea what Math is ;)

The same is true of creative writing in general. I've found pieces from my years as a student, which seem to have been written by an alien.