I have an idea that being intimately familiar with mathematical operations and ideas, as like one would with vocabulary and grammatical in a language, would help one internalize math better.
I think a lot of people just cannot come to terms with the fact that mathematics is inherently difficult, and being good at it is not just a matter of memorization. For example, I often see people complain about mathematicians using arcane notation, as if it was the notation that was preventing them from understanding what’s going on, when in reality it is just lack of understanding of underlying concepts and experience with arguments being put forward.
I think part of the problem is that mathematics is so precise that subsequent understanding is tightly coupled to a comprehension of prerequisites. In so-called "softer" disciplines, the concepts are less precisely delineated, and their relationships are fuzzier. If you don't quite understand some prerequisite concept, you can sort of muddle your way through even the later notions that depend on it. Not so for mathematics. If you fail to get a complete and precise idea of the basics you'll eventually face complete doom or confusion when working through the subsequent portions of a theory.
This is exactly the correct attitude. There is no royal road to mathematics, it requires a lot of careful thinking to put everything together in your head. When I was studying mathematics seriously, getting through 4 pages of a textbook in an hour was a really good pace, but usually I only managed half or quarter that.
> I think part of the problem is that mathematics is so precise that subsequent understanding is tightly coupled to a comprehension of prerequisites.
100% agreed with this and the rest of your comment. I often see people pick up a research paper, try to read it with very little understanding of the prerequisite concepts it is built upon, try to look these up, which only uncovers prerequisites of the prerequisites they need to understand first, then despair, give up, and blame arcane notation, probably because the alternative is just too humbling to contemplate.
I seriously don't get why people keep separating the two
I have only vague recall of most CLI things, but I can get up to speed again very quickly (when a situation demands it) because I've outsourced remembering arcane command line options to "man" and just need a system with docs installed and remember(!) that there are commands called "cat", "ls", "awk", etc.
A similar thing applies with math... the analogy strains a bit, but "cat", "ls", "awk", etc. are the 'understanding' which underpins everything else. I did a recent thing with force calculations and symmetry which apparently impressed some of my engineer friends' colleagues... but the details are unimportant. Just knowing what a sin/cos curve looks like and what an integral fundamentally is got me there... but the fundamentals were enough to get there, is the point :)
You memorize so much stuff that at a certain point the language "clicks" and you can infer meaning and rules even if you don't know them yet.
Remembering is understanding, understanding is remembering.
People who have a knack for memorizing long lists of arbitrary if-then tables can excel in rote mathematics (up to say multivariable calculus) without needing a philosophically deep understanding of what's going on, for the same reasons.
A good example is velocity. Many people who passed math classes can't answer "how long does it take to drive 80 miles if you are going 80 miles an hour". Such people never understood velocities, they just memorized some rules. Memorizing rules wont help you solve that question well, you need to make your intuition understand the relationship between velocities and distances.
As you learn more you start to build a net of intuitive connections between all the things, that should be your goal when learning these things. That net will last you a lifetime. Word based or symbol based memorization is mostly worthless in comparison, doesn't help you apply it to other subjects and takes more effort to build.
These are the ones who always express a special dislike for word problems.
Another aspect of understanding is _why_ you are doing this, where does it fit into the programme of necessary compression of infinite information density (i.e. the number line is infinitely "dense") so it may be accessible despite the material confines of a human brain and its limited, discretizing capacity for dealing with multiple elements in a single operation. So, philosophically, different lower dimensional spaces integrate to form higher dimensional spaces, in order to facilitate the description of change from one thing into another thing. One linear dimension changing into another linear dimension requires a transition through a quadratic space, from which we get a curve.
I found this less and less true as the math I did got harder, hitting a breaking point when I took abstract algebra. The course just didn't make sense at all until I pulled a perfect score on my final by whipping out the Anki decks and drilling the theorem proofs and homework problems from class on a schedule. I would go right back to that method if I ever returned to mathematics for my master's degree.
Similarly, there are often underlying assumptions that can be tricky to remember in the moment, e.g. certain log laws only holding for the absolute value of the argument. There's a combination of both understanding a tool to begin with, and remembering various equivalences, representations, and underlying assumptions that makes math difficult.
Part of memorizing proofs is also just increasing your exposure to certain ideas, because maths is one of those subjects where there's no real substitute for time spent thinking about something (i.e. mathematical maturity).
I wasn't tested on the exact proofs and problems I memorized. I was tested on variations and novel combinations of them I haven't seen before. That sure sounds like learning to me.
Further you can't just memorize a proof straight through, there isn't enough space in your brain for that - rather the act of mentally walking through the theorem over and over via an Anki flashcard prompt will eventually just... Change your logic, invisibly, to be correct. Which, again, sounds a lot like learning to me.
>The point of learning mathematics is to be able to transfer this skill into new domains, not to just regurgitate it.
I am far more confident in both my intuition and conscious reasoning around e.g. Abelian groups or the enumerative combinatorics applications of group actions than whatever I learned in real analysis, where I studied in the "usual" way. Indeed going back to learn Haskell a few years after that AA course was much easier than earlier attempts because I had a considerably stronger background in what kinds of things to look for in that domain.
But more importantly homework problems are rigged [1] and transfer learning is close to non-existent in every domain we've seriously looked at [2], so this is awfully close to moving the goalposts on what "really learning" something is by setting an unreasonably high bar to start with. Math certainly can transfer to new domains, but I would never call that "the point" of math, and that's also a totally different endeavor to be performed in addition to learning the math itself.
[1] https://www.johndcook.com/blog/2023/10/12/homework-problems-...
[2] https://www.econlib.org/archives/2012/08/low_transfer_of.htm...
Indeed it is, but that’s because most people just learn to pass exam by redoing the same exact problems with different inputs! Bringing up this fact does not support your argument in favor of memorization, instead it is closer to my view, which is that most of schooling is just cargo culting education, people just memorize the exam problems, pass, move on and forget. What’s the point of the whole thing in the first place if you can’t transfer?
Besides brushing up on my math, starting from simple arithmetic, In my spare time I also study Japanese. And one of the things that has helped me the most in my fluency and understanding has been the memorization of vocabulary and of grammar patterns and their usage.
Of course, I read materials at my own level and listen to material at my level and above and practice writing. However, I noticed the biggest boost in my comprehension after I memorize a large amount of words or really internalize grammar patterns. And I do this mainly through flashcards.
I have to spend much less mental energy to catch on to what is being expressed allowing me the ability to potentially comprehend more.
And analogously to my language studies, I would like to approach math in a similar fashion.
My point here is rather that this approach will not work with mathematics, simply because unlike language learning, which is mostly about acquiring and memorizing large amounts of simple X to Y mapping, mathematics has much less to do with memorization and more to do with building mental framework and placing new knowledge in appropriate places with in.
To understand the proof you have to accumulate enough mathematical knowledge that the trivial, omitted, steps in it become trivial to you as well.
This is not an issue specific to mathematics. Listen to a couple of surgeons talking shop in a hospital cafeteria. They're going to be using all kinds of arcane anatomical and medical terms the average person would have no hope of understanding.
Same goes for a pair of programmers talking through a complex bug deep within a massive codebase.
Economy of language is critical for communication among experts in a field. If everyone was forced to use the vocabulary of a 6th grader there'd be a whole hell of a lot of repeated explanations of concepts the other person already knows and it would just be frustrating.
Such an exposition might help students, if there was such an analysis of their field of study. What are the most useful concepts, including proofs and facts to know and understand to be able to learn more "higher level" concepts.
[1] https://www.amazon.ca/All-Math-You-Missed-Graduate/dp/100900...
When I did undergrad level engineering math, I pretty much never had to memorize. Just solve a lot of problems, and it's ingrained in your head. It helped that I utilize that material all over engineering and physics courses. Even now, over a decade since solving problems, I can still recall most of it and use it if needed.
Once I got to upper and grad level math, the approach of simply understanding and solving problems failed me, because you are usually not provided enough problems compared to earlier courses. It may have been good enough for the course to get an A, but not good enough to retain the material beyond the course. I've studied complex variables (Cauchy Goursat, residues, etc) at least 3 times. It's always a breeze, and it's always forgotten soon after. The same with statistics. With the latter, what did finally help me was spaced repetition. Even though it's been a few years since I last reviewed/relearned statistics, I can still read some material involving it and understand it. This is entirely because of those flashcards.
Understanding definitely has to precede memorizing. Insisting that memorizing is a sign of poor understanding, however, is parochial thinking. It's simply not true for the majority of folks.
Calculus and linear algebra are good examples of courses taught in a way that gives understanding, lots of people understand those after taking the courses, so there you shouldn't use memorization techniques.
So the reason you needed spaced repetition for statistics was that you never understood statistics, you just memorized it, like basically everyone else. It isn't wrong to do that btw, statistics is useful even when you just memorize it which is why it is taught that way, but don't trick yourself into thinking that you really understand the material the same way you did calculus or linear algebra.
As was statistics. While this wasn't a measure theoretical approach, it was akin to how calculus was taught.
They are organized by category with a focus on fundamentals.