Learning algebra in my 60s

I feel the same joy when seeing a derivation of a novel algorithm where the effective procedure falls effortlessly out of the definitions. A good example is Dijkstra's derivation of Smoothsort[1]. It's worth noting that he was educated as a professional mathematician, not as a computing scientist.

[1] https://www.cs.utexas.edu/~EWD/transcriptions/EWD07xx/EWD796...

It's hard to explain to young kids what "out of scope for this class" means. And that the number of people who need these skills vastly the number of people who need to understand the derivation.

Lattice multiplication will probably take algebra to explain. Long division definitely will require algebra. In my school, there's a gap of 5 years between teaching the two. A lot more people in the world need arithmetic than they need algebra (easily over a factor of 10). We can't put off teaching arithmetic till they learn algebra.

A lot of people don't realize that this problem goes all the way to undergrad and grad education in engineering or science. Laplace transforms are very useful, but they require complex analysis to begin to understand. If you blindly apply the integration that is normally taught, the Fourier transforms of several simple functions have integrals that simply, clearly do not converge. Yet we're taught tricks to indirectly calculate them. How is that possible? How do we get a result from something that clearly diverges?

And don't even get me started on the Dirac Delta function.

Recently I picked up an introductory analysis book - it starts from Peano axioms and builds up natural numbers, sets, integers, rationals, and then finally reals. It requires a fair amount of mathematical maturity to explain simple concepts, like how multiplying a positive with a negative could result in a negative, or how multiplying two positive numbers can result in an even smaller number (something that I did get upset about in my school days).

While yes, it is convenient to cherry pick examples where it was taught poorly without intuition, the reality is that if you want to prepare someone to go into, say, engineering, there is a lot of math one needs to cover, and teachers just can't afford to spend time explaining things that are way out of scope.

Can't speak for the others, but you may come across as too one sided. I had your mindset, and one of the reasons I started getting left behind in grad school whereas others didn't was my stubborn refusal to memorize anything and insisting only on understanding and deriving as needed.

Once you go deep enough, you'll often find yourself relying on N random theorems you learned some courses ago to solve a problem, and those who had memorized them were more likely to solve the problem than someone like me who happened to forget the theorem existed in some book and is trying to rederive everything from scratch.

Of course, those who mostly or only memorize perform the worst.

The other issue is in my other comment: https://news.ycombinator.com/item?id=32388595

Basically, most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform. So some level of "take it on faith and memorize a few items" are needed. There are other examples of this.

Also, having studied under many top class physicists, I can tell you that a significant number of them cannot derive much of the mathematics they use, despite being wizards in applying the techniques.

> I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college.

Many mathematicians would disagree with your characterization. For them, the difference of squares is an abstract concept in algebra, and the geometric interpretation is merely a manifestation of it that just happens to work in some domain.

As an example, the formula is equally valid for complex numbers, but I doubt you'd get there from Euclid's. No doubt some geometric interpretation can be found for that as well, but then I'd pick some other algebraic field where it's true and you'd have to search yet again for a geometric interpretation.

1. none of the exams at Caltech required memorization. They were open book and open note. Memorizing simply wouldn't have helped. For example, one physics exam question was: "Assume magnetic monopoles exist. Derive how Maxwell's Equations would then look." If you didn't understand the ME derivation, you'd be completely lost. The same for FFTs, where the exam question was derive the hyberbolic transforms.

2. one winds up inadvertently memorizing things used often, like I knew all the trig identities from excessive use. I never attempted to memorize any of them. Just like I know a lot of the hex opcodes for the x86 :-)

3. we were not expected to use the Laplace transform until its derivation was demonstrated. This applied to all the formulas used. FFTs too.

4. I've forgotten an awful lot in the 40 years since. But I took an online MIT course and was pleasantly surprised that it was still there, it just needed a bit of oiling.

> most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform

Which is why some engineering schools are better than others. I applied to the USN&WR list of "top 10 engineering schools in the US". Caltech was #2.

I'm sorry you were left behind because of your insistence on learning it thoroughly. You were doing it right. You just were in a school that didn't value doing it right.

BTW, I did not mean deriving a formula from scratch every time you used it. That would be silly. Just that at some point you did, and thereby understood where it came from, hence understood its limitations, and knew how to adapt it to a situation not in the book.