I love math and majored in it in college. The rest of my family is all scientifically inclined, but I think found/find math itself opaque and somewhat intimidating. I remember my brother asking me at one point how one would ever find, for example, the Pythagorean theorem intuitive. The author’s quote is the response I wish I had. The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind. Which happens after a lot of practice.

FWIW I think it’s appropriate that the author talks about herself a lot. She’s trying to explain the subjective, cognitive experience of going from math-phobia to math mastery over her career. She can’t explain that without talking about her background and her perception of the process from inside her head.

That talk is about something superficially different—ego in math—but on reflection, I think the desire to look smart actually really does set one up for success in math in the particular way that the OP article describes.

When you just want to look smart, you don’t care whether you know something because you thought of it or because you read it in a book. You just care that you can show off what you know and solve problems easily. So you voraciously read and memorize and try to accumulate a massive mental database of facts to show off. Then at the end you find you’re actually good at the thing.

What should one do instead, in order to avoid merely “looking”/“sounding” smart?

To be honest, this sounds like orienting one's self in the 'space of mathematics'. Is it not possible that, just like one can navigate by landmarks (proofs) or by the space itself (deep understanding), that there are in fact two roads to intuition in mathematics, of which ones is practice and fluency, and the other is deep insight and understanding?

I'm not sure I understand the question. Yes, it is often attributed to him. If you mean "is it not true that he said it?", I don't know, but my suspicion, as with many such famous quotes, is that he probably at best said something like, or reminiscent of, it.

My question: Do you think an approach like in the article is possible to learn Math B? I have tried several times, unsuccessfully. I'm proficient in most domains of Math A. (Differential equations, linear algebra etc, symbol manipulation, geometry, and how tho apply them to practical problems).

Math B seems, in contrast, beyond me. There is a programming analogy: Math B is like Haskell, or pure functional programming, which also is as ungraspable to me. I am wondering if maybe this is partially genetic, partially something you have to learn at an early age. Or maybe it takes a formal learning path.

I tie my poor mental arithmetic skills partly to never properly learning multiplication tables, at least not all of them and perhaps something lacking in my brain which also means I have a terrible sense of direction.

Yet, when it comes to symbol manipulation where the numbers don't matter until the very end, then I am good at that.

I thought this too.

When you're young the multiplication table seems like a daunting thing to memorize, but after graduating university, it doesn't seem so bad.

So I went back and memorized my times tables using Anki. It was pretty easy, but ultimately changed very little and I easily forget them if I stop practicing.

I've come to realize that not mastering the times tables were a symptom, not a cause, of my learning difficulties.

For whatever reason, 6x7, 6x8, 7x7 and 7x8 are a persistent hole in my ability to memorize. Sure I can temporarily memorize them, but they shortly evaporate back into the void and I’ll have return to quickly having to calculate them out again.

I’ve also got this thing where I get mixed up between verbal “eleven” and “twelve”. They sound different, but at the same time somehow sound just similar enough that the boundary that should exist around them as symbols never properly formed. I have to pause and manually match the number to the sound, every time. What’s especially funny to me is I have no such problem distinguishing between onze and douze from French, which I only know a few words of and certainly never hear in real life.

I’d like to think the first problem I’d eventually fix if I was using those multiples constantly, but I’m not so sure because the second problem definitely doesn’t improve.

I definitely relate to that for some things. I have a distinct memory of struggling to memorize 6x7, 7x7 and 7x8 in elementary school. What I settled on was just artificially making the numbers "stick out" in my head... It's hard to explain, but for example for 7x7=49, the way I pronounce it in my head is really distinct, and I'm also very "passionate" (as a mnemonic, but also genuinely) about how it "makes no sense" that the numbers 7 and 7 could make 49. Similarly with 42, I have this idea of 6 and 7 combining in such a clunky way that they somehow produce 42, and this image of them forming a kind of gnarled-up branch to "reach" 42.

Anyway, just an interesting thing I've never put into words about this point -- I'm realizing that when I can't remember something, I kind of toss around a concept in my head until one of these nonsense mnemonics has a match. I'm also reminded of "memory palaces," where people assign concepts to a mental location even when there's not necessarily a correlation, and it drastically improves their recall. Maybe I should try this more -- now I'm wondering why I apply this to some things and not others.

And my dad had me trained to reply 49 when asked "what's 77?" early, to pull off as a party trick. Though I couldn't answer and followup questions hahah.

That helped.

Nowadays I don't often have to use these tricks, the vast majority of the time I'm already typing into something that can give me the result (a browser, or a terminal), or am in no rush and can just use my phone or watch. At most it makes for an occasionally nice trick to show in conversations.

Been playing guitar for 20+ years, can't memorize the note names on some frets.

Studied music in college, I still need to count lines sometimes when reading sheet music, besides some reference points I can't seem to memorize the locations of notes on the staff.

Not like I have a general memorization problem. I am good with human languages, programming languages. Have very good working memory etc. But some things just stump me.

The way that our brains process symbols is fascinating. If anyone out there has any literature or reading on this, I'd be interested. Especially, as related to ADHD/Autism.

As for the others, I think remembering these in several different ways is stickiest. For example, you might think of 7·7 = (5 + 2)² = 25 + 2·10 + 4 or perhaps 7·7 = (6 + 1)² = 36 + 2·6 + 1 or 7·7 = 7·(10 − 3) = 70 − 21. If you already know 7·7, then 6·8 = (7 − 1)(7 + 1) = 49 − 1. You can try computing 7·8 by repeatedly doubling: 7, 14, 28, 56. Etc.

Because of this pointless mental exercise it also sticks in my mind that 11 squared is 121 and 13 squared is 169 (though the presence of 69 helps with that one).

1. 6⋅7 = (6⋅10) - (6⋅3) = 60 - 18 = 42

2. 7⋅7 = (7⋅10) - (7⋅3) = 70 - 21 = 49

3. 13⋅19 = (13⋅20) - (13⋅1) = 260 - 13 = 247

4. 58⋅61 = (58⋅60) + (58⋅1) = 3480 + 58 = 3538

If we go up another order of magnitude, then the system starts really grinding to a halt though tbh :)

Did you learn them when you were young? Those are completely ingrained for me.

https://www.smbc-comics.com/index.php?db=comics&id=1914#comi...

If you want to practice division as well, check out Zetamac or (if you don't want to be timed) my simpler tool: https://math.twilam.com/

I did learn my multiplication tables correctly, but I was always horrible at mental arithmetic—my elementary school periodically had formal exams for mental arithmetic, and I consistently failed them. It was the only type of task where I ever consistently scored less than 90%,

I am also terrible at directions, my parents used to be worried about what would happen to me when I lived on my own—thankfully, Google maps became a thing and nobody notices I am bad at directions anymore.

Like you, my learning difficulties are strictly compartmentalized to these two specific domains. I hold a PhD in engineering, and I even have a minor in math, lol.

When I spend too much time away from one, it really shows. The other day, I tried to do the trig approximations, and it was like getting up after sitting too long with legs crossed. The fluency just wasn't there.

https://en.m.wikipedia.org/wiki/Fermi_problem

(Folks with neurodivergencies like autism/ocd/adhd etc are particularly more likely to have these.)

Around that time, I went from noodling around with programming, to taking it seriously. I learned a bunch of programming languages, and landed a web development job straight out of school. I wasn't just done with math, I was done with school, too!

After a few years of that, I got bored with web dev, and decided I'd rather try my hand at engineering of some sort. I enrolled in community college, and quickly discovered that all of the engineering courses had... math prerequisites. So I bit the bullet, and for the first time, applied myself. Turns out that I wasn't intrinsically bad at math; I just hadn't been sufficiently motivated! I was paying my own way, so I ended up taking a job in the tutoring center. As I transferred to university, I found myself taking more and more of these math "prerequisites" and not following through on the engineering courses. I matriculated as a math major, and today I've got a PhD in math.

In my mid-20s, I didn't even believe that I could be Math A person. But I got good at that stuff, for the sake of engineering! And then I went straight through to Math B (and, almost amusingly, forgot most of those Math A skills -- watch out, unused skills get rusty!)

I actually credit my programming experience for the intermediate transition from my "bad at math" late teens to my "willing to try Math A" mid-20s. Programming taught me to think rigorously, and abstractly. So I must push back on the notion that this is intrinsic to a person, and must be learned at an early age: I wasn't doing Math B until after 25 when my brain was supposedly fully mature. And while I did have the benefit of a formal education, I would assert with some confidence that the relevant detail there was that I was in a cohort of students who were working together, beholden to homework deadlines and exams -- because math is hard and it's really easy to get demoralized without that external reinforcement.

This is what I would tell people, but just use a weight lifting analogy. If you're out of shape, of course you will struggle to do any sort of exercises. But if you keep working at it in a disciplined way, while being kind to yourself and praising your progress, eventually you can get good at it.

Calculus is a tiny bit of new material plus a shit ton of rote algebra. Even undergrad prob and stat was 80% algebraic manipulation.

Once you're fully ensconced in the major, it pivots into type B. And it turns out that I hate type B but slogged through it with medium-good grades.

looking back on it now, I've come to like type B and wish I could go retake those classes with my current perspective.

I think my original distaste was largely due to what felt like a bait-n-switch: start out majoring in something you like and are good at, but then pull the rug out and switch to something completely different

Due to internal changes in my uni, for the first time, my freshmen year, the math department taught proper proof-based Calculus 101 (from Apostle of all books) to all majors. Then the engineers and biologists complained so much, they had to cut out a lot of proofs from Calculus 102. There were even more complaints, so by second year, there were hardly any proofs in the core math courses. In a few years, the calculus courses had become devoid of proofs.

Some unis have separate intro courses for math majors, but it's very difficult to offer them in the current economic climate.

You can do proofs for calculus, probability, or logic and still feel like you’re working with the types of problems you do in type A math.

But once you start doing proofs in modern algebra or topology you’re doing things with abstract objects that seem to exist for the amusement of mathematicians that look down on “applied math”

Apostol?

https://en.m.wikipedia.org/wiki/Tom_M._Apostol

We had his book in school or college math.

Calculus, Volume 1, One-variable calculus, with an introduction to linear algebra,

Calculus, Volume 2, Multi-variable calculus and linear algebra with applications to differential equations and probability,

Damn good books.

Say, analog electronics mostly works in the pure functional domain. An amplifier does not try to change the input signal. Instead, it produces a more powerful output signal, following the shape of the input signal. A tone generator in a musical instrument does not try to make a key on the keyboard vibrate. Instead it produces a sound signal according to the key pressed (which note and what velocity).

The simplest way to try practical pure functional programming is to connect a few Unix processes via pipes:

  cat somefile.py | egrep '^def \w+' | wc -l

But how to achieve something like updating with that? By looping the output back to the input, and switching o the "next version" once it's computed. Conway's game of Life looks like an ultimate "update in place" thing. But it's purely functional, too: the new state of the map is completely computed based on the previous state if the map. Then the new map is seen as "the current map". Similarly, frames in a drawn animation do not change, but they are shown at the same place one after another, giving the impression of motion and change of "the same" picture.

In general, our Universe may be seen as a purely functional computation: its next state is a function of its past states, and the past is immutable.

Many won't but I agree in the purest (sorry) sense.

There is no scattered changing state. I think we all learned input-function-output as a construct in maths class?

Spreadsheets (sans-VBA) is arguably the most prolific programming language and simplest, being used by people who do not recognise they are programming. Felienne Hermans gave a good talk on this subject in GOTO 2016.

Spreadsheets have numerous shortfalls though, and "real" functional programming languages make it difficult to not feel intimidated: in my experience, but this is getting much better.

[1] is a game of life in calculang, functional language I'm developing, where for all it's verbosity at least I hope the rules and development over generation (g) can be reasoned with (sans-state!).

Not very practical but can show calculang computation/workings as it progresses and as parameters change - things that are easy for FP and otherwise intractable, and which further help with reasoning.

But, a big challenge is to be approachable (not intimidating), and I'm trying to make that better. I think it helps enormously to be focused on numbers as calculang is, and not general programming.

[1] https://6615bc99ad130f0008ecc588--calculang-editables.netlif...

Certainly knowing some category theory helps. Finding out mathematical underpinnings of e.g. monads is some effort, and a foray into a non-daily math territory. I'd compare it with learning about matrices and quaternions when doing 3D geometry stuff. In either case, you have to make an effort and acquire a bit of special knowledge. But once you've learned it, it becomes and remains "intuitive", you don't have to think deeply or write complex formulae to use it. You just write [int(x) for x in line.split(",") if x] and don't mentally split this chain of monadic operations into an isomorphism and a catamorphism.

I've not found a book that makes this point completely explicitly, but most of those which cover sequent calculus get you half way there.

The rest of type B math is intuition which lets you guess at new conjectures and how to get you from the assumptions that you've made and the conjecture that you want to prove efficiently.

In regards to the article, the course of this type that i took had frequent quizzes that required nothing but reproducing precise definitions or proofs we had learned. Of course the ideal would be for the student to be able to reproduce these from understanding. But in practice i was doing a lot of brute force memorization of definitions - i just hadn't internalized the language of mathematical logic well enough to reconstruct a concept's definition yet. however, it got my foot in the door and having those definitions in my head made the next courses easier, so if i retook that transition course a few years later on, i would not have needed to do so much memorization. i got better at learning those kinds of basic definitions.

So my answer to your question is yes to some extent - the memorization aspect of learning described in the article is useful for learning the first step to Math B as well. Also if you want to make another learning attempt, be sure and go back and start at that freshman/sophomore level transition course i was describing!

I thought these were well-known terms and thus that the dichotomy you describe was itself well-known, but I thought I’d add this comment on the chance that you weren’t familiar with them.

Then there is Math "Y". This is all the guys who use those things the X guys are selling, the proverbial hammers they have produced. They assume the X guys did their work correctly and that when they use the products they've bought i.e. the rules, theorems and strategies developed by the X guys, to solve a particular equation or problem, the answer is correct. For example, they assume the limit of the sum of two polynomial functions on the reals is equivalent to the sum of the limits of those functions - they don't care about all the nitty gritty details and justifications - the X guys figured all that out for them. They Y guys are much more concerned with figuring out how to get the rocket into space or ensure the skyscraper is soundly built.

I would say from my experience, very little of mathematics education is in the X camp, I'm not saying this is a bad thing though, perhaps it is similar to the fact that most programmers are not compiler programmers or programming language creators :)

Type B on the other hand far more about understanding. You will never understand the theory of a mathematical space and how to apply it, by simple repetition. That is a far more theoretical and creative endeavour. You need to learn it and apply it to understand it. I suppose you could call the process of applying it some kind of repetition, but in my opinion the insights comes from applying it to concepts you already know.

A formal learning path is a very good idea, because people with more knowledge know what order you can progress in, in such a way that you actually apply your knowledge in a natural way and build on previous learnings. And it is definitely a huge help that teachers can help you guide your learning when you are stuck.

The repetitive practice is in this manipulation of symbols. It takes a long time and deliberative practice to learn this skill. You just practice by doing symbol repetition in different contexts, instead of doing the same thing over and over again like multiplication tables, because your symbol manipulation abilities have to be general [1].

If you try to teach, you will quickly discover that there is a wide difference in this ability for math majors by their final years. And the students who have poor symbol manipulation abilities inevitably struggle at the higher level concept application, because they keep making mistakes in the symbol manipulations and having to redo it.

[1] Contrast the training of 100m sprinters (multiplication table), who only run 100m on a fixed track that they will eventually race on, and the training of cross country runners (symbol manipulation), who practice on a variety of routes, because their races are on different routes.

Granted, it was one brain (mine) studying both subjects, so it should not be shocking that I learned both in the same way. Of course I practiced lots of problems and derivations in my physics class, but I also practiced and memorized lots of proofs in my upper level (i.e., more theoretical) math classes.

And truth be told, maybe even in my liberal arts courses as well. Thanks to programming, I got really good at typing. Thanks to owning a personal computer (one of the first at my college) I started writing and re-writing a lot. Repetition and practice even got me through those courses.

It was simply mercenary at the time, not wanting to waste time during exams recalling the easy stuff gave me more time to think about the hard stuff. But I think it did help me in the long run. I still use a lot of that stuff today, at age 60, though it's certainly more computer-aided than it was back then.

As someone who is decent at Math B but mostly incompetent at Math A I suspect it comes down to the old analysis vs algebra opposition — being better at thinking about things visually/spatially vs linguistically. Both are trainable though.

I think the approach outlined here works well enough. When teaching us about adjunctions, our category theory lecturer used to have us recite the definition of a left adjoint at the start of every lesson, and he'd draw the diagram as we spoke. I can't say I can still recite it by heart but I do feel like I have a decent intuition for adjoint functors.

[1]: https://graphicallinearalgebra.net/

I think it's the key. Learning maths isn't something you can do on the side. It's countless hours of intensive active learning and problem solving. I don't see how it can be done outside an academic path. I remember my undergraduate program, we had 14 hours of class of maths alone a week, plus a 4-hours exam every other Saturday, and I was working several hours a day on top of this, most days of the week.

Maths can be fun, but who wants to do that kind of effort for the pure joy of learning?

This is not just an analogy – Haskell (or pure functional programming in general) is a lot closer to math than other ways of programming. Specifically, it is derived from category theory (which fits to your description of math B being things ending in "theory" as well).

It took me a while to buy in to high-volume memorization as a learning technique (especially coming from CS, where memorizing facts is not a huge emphasis). After a while though, I started recognizing how the quick recall encouraged by the system enhanced my understanding of concepts vs replacing it (I wrote about this a couple years ago [0]).

[0] https://samrawal.substack.com/p/on-the-relationship-between-...

Education as a discipline has no real knowledge or skills in the Anglosphere beyond basic essay writing. Historically teaching was a craft, taught in trade schools or on the job, but for various reasons it moved to university without universitiea having of the content developed, so universities took the low road and asked PhD candidates to essentially write essays on the philosophy of education so that's now the content.

Those who can, do. Those who can't, teach. And those who can't teach, teach teachers, and since the content is just writing essays on the philosophy of education their confidence in non-existent abilities is never pressure tested.

They're not mutually-exclusive sets. In professional schools (law, medicine, perhaps others), some of the best-regarded instructors are successful practitioners who teach on the side.

For older kids, it can be a bit tricky, as their goals and opinions will matter a lot more, but look into evidence based study habits for exams, and help them get assignments done with a decently efficient process I guess.

I memorized my multiplications tables like a good little boy, and I can do decent amounts of math in my head, but I am not a fast or as capable as the default calculator that comes with every smartphone on the planet.

In school I was tested on memorizing a map and drawing precise borders, rivers etc and various acronyms that were obsolete even at that time. They added 0 value to my education, except as unnecessary stress in having to memorize them. Similarly with being expected to memorize various physics and math theorems such that understanding was treated as having zero value (that is, if your wording isn't exactly the same as in the textbook, it doesn't matter if your answer is actually perfectly correct in terms of understanding). It was to the point of being prescribed several arbitrary essays to memorize, with one of them being randomly expected to be reproduced in an exam.

None of that added any value, most of it turned out to be hilariously divorced from real life expectations.

Even with your disingenuous example of language learning. People who argue that they don't need to learn the language because they can just use a translation app are making the point that they aren't interested in learning the language and have limited enough need to interact in it to get by with just a translator. Anyone who has actually interacted with a language they don't know understands that translation software doesn't obsolete knowing a language.

I do agree that memorization is important, by the way. I just think that it's useless if you can't also derive your way there. Nothing wrong with having checkpoints in your mind though.

There was a slang term "refrigerator thoughts" that described someone staring into the refrigerator while thinking of a show plot and realizing there is a plot hole, a disturbing implication, etc. In any case, that's an example of insight. Hopefully we have these spontaneous realizations about less trivial things as well.

The more stuff you've got in your head, the more insights you'll have, which drive more questions whose investigations puts yet more stuff in your head.

I don't think you can split it. The critical piece is actually getting information into your head in the first place.

https://www.youtube.com/watch?v=F3TG1AzBJYo

If you want to know how you can become better in math and rewire your brain to be math compatible I‘m afraid you will be none the wiser after reading this.

- Memorization and rote practice are important for learning, not just the current Zeitgeist (2014) of “understanding” without the former. This becomes the foundation that allows you to focus on higher-level things like understanding and applying formulas.

- Experts develop “memory chunks” which allows for example chess masters to draw on thousands of different past games, openings, variations.

For a primary school example, if you can solve basic arithmetic problems in service of a fun and challenging logic puzzle, that is more motivating than solving a page of arithmetic problems one after the other.

More generally, while mathematics certainly requires putting in time and actually doing the work of thinking a whole lot about a variety of hard things in the service of solving hard problems, very little of that is memorization per se.

> In the United States, the emphasis on understanding sometimes seems to have replaced rather than complemented older teaching methods that scientists are—and have been—telling us work with the brain’s natural process to learn complex subjects like math and science.

The older teaching method also sucked.

In my opinion, the single most important thing primary school math education should be teaching is how to attack and solve nontrivial word problems. Unfortunately we did not have any of that before, and still do not have any now. Cf. https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/trav...

I've spent plenty of time on math (PhD in algebraic geometry) and educating people, and for sure when I taught college freshmen and master's students I spent a lot of time challenging folks to engage their minds, spirits, and intellect. At the same time, we have to admit there is a stage of childhood where kids just love memorization and facts. Dino facts, shark facts, math facts, Pokemon facts, My Little Pony facts, whatever. Let's not force kids to reckon too much with meaning when they're in the facts for facts sake stage -- and once they've got their impressive facts list, they'll make sense of the meaning much more easily, as discussed in the article and here!

Cool. If I try that with my 5 and 7 year old, they shout "boring" and run to the far side of the house.

> math ability is outstripping the reading ability

Presumably you can read problems aloud to the 6 year old. But also, kids can (if taught; this is by no means a necessity) learn to read much earlier than they can learn to write numbers with a pen. My two kids both could read very well by about the time they turned 4.

Please no. Non-trivial problems that are expressed in natural language, sure: I'm in favour. But once you get past basic arithmetic, "word problems" are just algebra obfuscated with a prefabricated template. A non-trivial problem, written in natural language, will usually admit multiple solutions: cutting that down to something you can fit into a mark scheme, without making the problem convoluted or forced, is hard. Example:

> Jacob and Sally want to split a rectangular cake between them, but neither is very good at cutting. Jacob can cut precisely (wasting no cake), but will miss the middle by 10% of the distance to the edge of the cake. Sally can cut accurately (exactly in the middle), but will obliterate 10% of the cake in the process. (Neither Jacob nor Sally want to eat crumbs.) Who should cut the cake, and why?

This is a fun problem, but a rubbish exam question! Word problems completely sidestep this issue by starting with the algebra, and then replacing symbols with words until it's almost prose. For example, consider the Hannah's sweets problem (solution: http://www.murderousmaths.co.uk/hsweets.htm):

> There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow. Hannah takes at random a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3. What is the value of n?

While nominally a "good" exam question, it only makes sense within a rigid and rigorous context that's quite alien to an untrained person's understanding of English prose. And it requires bold assumptions about the fundamental nature of statistics (see https://plato.stanford.edu/entries/statistics/) that go completely unstated. Understanding English and Maths aren't enough to answer this question: you also have to understand Maths Exams.

This is absolutely not true. It's worth throwing some "trick questions" at kids from time to time to make sure they read carefully, but the best word problems, while non-obvious to solve, are not obfuscated in their setup.

Young children should not be taught to solve word problems using algebra, but should be helped to try a wide variety of their own methods (guessing and checking, making a table, drawing a picture, breaking the problem down into several steps, solving a simpler problem, working backwards, using physical props, etc. A rush to turn everything into algebra is harmful to children's mathematical development, and many if not most types of word problems can be more profitably attacked with a variety of other tools/ideas.

Word problems are infinitely varied, and can get as difficult as you like, from 1st grade arithmetic up through unsolved professional math research problems.

> cutting that down to something you can fit into a mark scheme

This has nothing to do with the fundamental purposes of word problems. We're talking here about learning mathematics per se, not busywork or arbitrary ranking systems done for some bureaucratic purpose.

In my experience, the first step to solving a "word problem" is always to reverse the find-and-replace performed by the question setter, to yield the original algebra (e.g. the first part of the Hannah's sweets question). This is what we're taught to do in schools. You're right to object from a pedagogical perspective, because this is a horrible thing to make children do, and completely defeats the purpose of word problems.

The trouble comes when schools teach people how to pass exams. But, given how hard exams are, and how little they reward understanding beyond that stage of the curriculum, the optimal strategy is to learn no more than your current stage (plus exam technique), then learn no more than the next stage (plus exam technique), and never get around to actually learning maths.

Kind of like what FizzBuzz can be like for testing for basic pseudocode programming. Where someone who isn't really thinking about the problem will mess up and go for what seems like the obvious solution on the surface, but someone who understands how to code will eventually realize that the obvious "clever" seeming solution is a trap and they have to do it the plain way.

I've found those kinds of problems to be the most fun to deal with. A math specific example I vaguely recall from middle school involved calculating the number of handshakes that would happen in total if everyone in the class shook everyone else's hand once. The path to the solution isn't explicit in the question, and for someone who doesn't already know of combinations/permutations, it takes a bit of abstract logic to construct the necessary expression. Yet it doesn't require particularly advanced math to figure out and tests the student's understanding of how to translate real problems into math.

Problems where the word problem is just a literal transcription of an equation are not that useful or fun, but also, I think that it takes getting to a fairly high level of education before the math gets advanced enough to more frequently cover meaningful word problem solving (probabilities and statistics, linear algebra, or differential equations), yet the tools for interpreting such problems need to be taught earlier.

Personally I only accept „understanding“ once I can explain it and reuse it in a different context. But I am not self centric enough to deny that there absolutely are plenty of people who love to memorize without having an abstract understanding. And they are doing just fine.

The cool thing about area of a rectangle is you can just turn it into a multiplication array. Which is something they learned to help them understand multiplication.

If she just memorized multiplication, she wouldn't actually understand the formula.

Thoughts flow through my brain like electrons through wires, both at a speed I cannot truly comprehend. The paths of my thoughts, the way words connect together, the emotions they evoke, the feelings that are associated with - these are all malleable. I have been working to rewire myself away from pessimism and towards optimism for years now. It's not easy, and sometimes I fall back into old patterns. As years have passed, though, I've found the new pathways easier, the new roads getting more familiar. My thoughts have previously wanted to go one way, and I spent time yanking at them to go a different way. I spent enough time at it that I now on good days more naturally go they way I want to go.

It's not just sitting and repeating a single thing over and over again, it's working with your own natural inclinations so that you can recognize when you experience X and naturally reach for Y that perhaps Z is what your preference really is, upon reflection. If you can practice that enough then, in the moment, you can sometimes find yourself not following the old pathways but the new.

Rewiring seems like a pretty good analogy as when you rewire a house you work hard -- it is dirty, dusty work. Pulling wire is hard and thankless because when you're done you cover it all up and, if you're lucky, it works! Then at the end you're . . . back in a state where nobody but you knows any different what is happening behind the walls. Things work, and others might have no idea anything changed at all. But you put in the hard work and you know how things actually work on the inside now and it's exactly how you want it, not how it was before.

If you've got a better analogy, I'd love to hear it.

I would think that would be obvious to anyone that merely memorizing something like 'f=ma' would be meaningless without deliberate attempts at application (both theoretical and practical).

There was a kludgy attempt at tying the study of foreign languages to STEM, but it just amounted to, everything is ultimately a craft. You have to practice to perfect it.

The author, Barbara Oakley, has a free Coursera course that is pretty good:

https://www.coursera.org/learn/learning-how-to-learn

... then you just have to do it. And keep working on it, even though it feels awful

Currently, the emphasis in training and education is on ensuring students understand the material, and rote memorization is viewed as a failure mode.

The author acknowledges this but introduces an argument that rote memorization is critical to achieving fluency.

I suspect that stating this position among typical education-focused circles will result in pushback.

To lend credibility, she adds her lived experience as a way to explain what she means—and to clarify that she isn’t saying everyone is wrong, just that we may be too harsh on memorization.

This is apparent to me because, frankly, if it weren't for the additional content she added, I wouldn’t have spent more than a few seconds before dismissing it.

The article even made me concerned about an internal project I am involved in, prompting me to verify that I hadn’t overlooked some issues.

If you want a TL;DR, the golf analogy matters: If you want to learn math, you need to understand it and then practice it so much that it becomes second nature.

I Rewired My Brain to Become Fluent in Math - https://news.ycombinator.com/item?id=33890921 - Dec 2022 (9 comments)

I Rewired My Brain to Become Fluent in Math (2014) - https://news.ycombinator.com/item?id=13674101 - Feb 2017 (46 comments)

The building blocks of understanding are memorization and repetition - https://news.ycombinator.com/item?id=12508776 - Sept 2016 (94 comments)

How I Rewired My Brain to Become Fluent in Math - https://news.ycombinator.com/item?id=8402859 - Oct 2014 (144 comments)

How I Rewired My Brain to Become Fluent in Math - https://news.ycombinator.com/item?id=8400837 - Oct 2014 (6 comments)

Years later, I’ve been slowly trying to “catch up” with my mathematical knowledge, and I find myself the most interested in topics that relate to the lives of mathematicians (and how events impacted their work) and to the philosophy of mathematics. I didn’t get any of this in school math classes, which focused purely on calculations and formulas.

I had the same experience with accounting, as well: boring in isolation but fascinating when connected to the history of double entry accounting in Italy, global trade from the 1500s onwards, and so on.

From an intellectual perspective (different than my engineering perspective), I'd label mathematics as quantitative philosophy. I like philosophy too.

The lack of freedom to pursue wny individual interests aside, major subjects like History, mathematics, biology are all reduced down to a bunch of facts you can fit in a textbook and test in a multiple choice format.

I think that people get slotted into the likes math or doesn't like math category quite early in life, often because they don't have an on-ramp to finding it interesting. They then spend the rest of their education avoiding it because "they're not a math person," when in many cases they probably just needed a bit more context/history/philosophy/something interesting to get them on the right track.

The reality is that every advance in mathematics evolved after decades of people crunching numbers (typically for some engineering application) with an earlier form of math until they worked out the advance. I think students would have more confidence in their own ability if they understood that the mathematical innovators were equally stumped for long periods of time.

I mean it can be a great story hook to start off with a subject, but in the end isn't math about the calculations, formulas and proofs?

> Sorry, education reformers, it’s still memorization and repetition we need.

It seems needlessly confrontational, and misses what the article is about. The article asserts that practice and repeated use of math is important. I don't think it's really suggesting that we should go back to how math was taught in the US 40 years ago.

But maybe I'm just out of touch with math education reform: In high school I was graded on how fast I could do matrix multiplication, and thought matrices were kinda stupid. Then I learned about linear algebra and coupled oscillators in college and thought they were awesome.

So I'd assumed the educational reform was about removing the busy work from math and focusing on what you'd actually use it for. Am I wrong?

She was successful because they emphasized drills, lots of drills, and more drills.

Teachers and students hate drills. Teachers, because they're tedious to grade, and students because they're boring. But they work. It's no different than doing the same Super Mario Bros. level again and again until you time your jumps just right.

I've often thought that gamification of drills would be a great way to get kids to learn their math facts or whatever, but there seems to be an allergy to doing this in the US education system. What the US education system seems to be addicted to is moving from one hype/fad to the next, as that's where the money trough seems to be.

While maybe not ideal, it’s something I like as it puts problems into the game as a required mechanic for success.

[0]: https://www.prodigygame.com/main-en/

My view on them changed as well. First I found them stupid and mind-numbing. Now my life is busy and chaotic and drills are one of the few easy, zen-like moments. Maybe students need harder lives to actually come to appreciate their repetitive and simple nature.

It's also worth noting that half of all teachers leave the profession leave within five years. Forty years thus represents about eight generations of teachers being trained with their own biases, refining their ideas in classroom practice, and then training the next generation. On top of this are the cycles of trendiness in the various schools of thought in educational psychology, as well as the varying policy platforms of governments. Educational practice from four decades is less historical and more archaeological.

On this forum, I will say, imagine your object-oriented programming language can build all sorts of abstractions but can't multiply numbers. So every time you have to multiply numbers in your algorithm, you have to instead write a few lines of assembly code that do the same thing. How much efficiency would you lose.

Just practice multiplying numbers.

Yes, you will almost certainly memorize anything with enough exposure, but targeted memorization is also useful, if the former's not going fast enough.

It’s useful daily. Pre-algebra is useful fairly often (even if I weren’t a programmer, plugging numbers into a formulas and basic graphing are very handy, quite often). Trig I think I managed to use once, but only because I didn’t know the right way (if you find yourself using trig on a minor home project you’re probably missing some trick or standard or something that lets you not do that—I suspect it was the case then)

That’s… about it. Stats, kinda, but mostly looking up the formula for the thing I want and plugging in the numbers, which barely counts.

My feeling is that we're producing graduates who don't understand abstract math or how to turn it into value in the real world. Category theory would only provide the abstraction part. I think we'd still have to work hard to instill the duty to translate abstract math into real human values. That's why we got to teach both the abstraction and the translation to people at a very early age. Otherwise they will simply never do it for themselves.

I don’t enjoy math as a hobby (recreational math puzzles can be fun, but may as well just be Sudoku—I don’t like doing real math as a hobby). I feel dyslexic reading proofs and formulas—algorithms feel natural, I have to translate everything into those to make any sense of it and that’s painfully tedious.

The jobs I’ve had have never really needed it. I’ve seen people who go casting about for reasons to MATH mostly be frustrated and make things worse. Maybe like once every five years I’ve seen something come along that benefits from some not very advanced college level math and everyone in the team gets a little thrill that we have an excuse to use any of that even a very little bit for a day or two, and that’s that.

I’ve been measured in the top 1% of spatial reasoning ability which you’d think would make me suited to math, but I kinda hate it and don’t actually seem to have a knack for it, so instead I use that ability to skate by doing pretty damn OK for myself but without seeking out reasons to use math more—and, for something allegedly so ultra-useful, such reasons rarely manifest on their own. I dunno, show me the job and a big raise first and I might brush back up, but I’m not gonna do something I find about as fun as pulling floor staples simply out of some sense of duty (duty to… do what? Do more math? Why? I want the motivation first, I’m not interested in forcing it)

For anyone interested in knowing what concepts are commonly learned in a first linear algebra course, check out this concept map from my book https://minireference.com/static/conceptmaps/linear_algebra_...

We could be teaching 8 year olds how to use crayon drawings to solve differential equations, graphically analyze back propagation, analyze electronic circuits, and validate quantum teleportation protocols. This stuff is fun and relevant to understanding the technological world all around us.

It would be about as easy and fun as common core. We have a lot of new math from just the last 15 years that shows us how to present these really complicated subjects in a simple way by doodling diagrams that are so simple that even a kid could learn it.

For an example, look up "Kindergarten Quantum Mechanics" by Abramsky and Coecke. They only joke about teaching it to kids. But I think that kind of stuff is definitely how we'll teach our kids 100 years from now, when quantum computers might be important for even kids to understand.

For (the few) students who actually understood the subject the problems are just busywork, for those who didn't it is the most important part of the learning process. There is exactly one way to understand mathematics, which is actually doing it. This can be many things, but actually solving problems is an important part. I believe that problems should be interesting, but repeated recall definitely is important as well.

2) I don’t make FAANG money but I’ve consistently earned 2.5x+ median for a person my age in my area (I’m in like a 3rd tier US city, couple million people, several tech headquarters, a couple of which are household names)

3) FAANG interviews are a game. The point of how awful it is and how prep-necessary is to selection-bias the pool so the vast majority of candidates are suitable, to select for only those who want it bad enough (and/or have lots of free time), to make jumping to peer companies difficult to keep wages down (if that last weren’t the case, they’d find a way to stop doing it for people who’d already passed it one or more times—the process is expensive) and finally to build esprit de corps via hazing (hazing is very effective at that). Don’t take it personally.

if you haven't very recently taken a class that uses CLRS as the textbook, then it makes sense you would have to do more memorization and practice with those concepts.

Basically, the things you need to get good at is remembering the runtime complexity of the main data structures, and to be familiar with the core structures that are available in whatever platform you'd be writing code in.

I was working on a Java-heavy team, so the important thing was to remember the various Map types, PriorityQueues, Stacks, arrays, and being familiar on how the references in Java work. The algorithm stuff wasn't too hard once you have a somewhat intuitive understanding of all these structures and when they're useful.

For example, one thing that they seemed to love in interviews was having you implement a least recently used cache, specifically an LRU cache where every operation is constant-time. Easiest way to do that is to build a doubly-linked list and have those point to a wrapper type inside a hashmap, it's not terribly hard but it does require familiarity on which data structures are useful. In this particular case, the rookie mistake it so try doing it with a minheap.

I think she does mean "use and practice" when she says "memorization", which is fine, but I think that phrasing could lead education in a direction that would be worse for people with memory issues like mine.

(Not that a kid can necessarily see that, it’s just funny because that stuff is about as useful as it gets)

1. First gain the intuition

2. Second develop rigorous understanding

3. Third gain fluency from repetition

Repetition without understanding, or even intuition, is not really going to help.

There are game-based learning that really gets to the fluency part.

Of course this probably does not help with very tough questions that require out of the box thoughts, but I think it still helps.

I learned math by blindly following algorithms. Over time i gained intuition from seeing how variations in the input changed the output. The deeper understanding kind've slipped in there...I don't know how. I don't think you can build real understanding without rigorous practiced repetition.

When I applied it to concepts I was learning on my own -- monoids, monads, semi-groups, semi-lattices, partial orders, etc. what I found was that I'm often overthinking the intuition. The intuition for a specific idea is very, very precise. It is exactly as it is, and yet something so precise and clear seems difficult to get.

It helps to approach things from a lot of different angles until you "get it". It's not always about repeitition on manipulating symbols. The Soviet method for teaching math (and remember, the Soviet system was intended to raise enough mathematicians to be able to work with a planned economy) was to let their kids manipulate things with their hands in a concrete way. It was in a way more like Common Core, but you're playing with toys with your hands.

I can tell you that I picked up being able to add and subtract things at an early age, but I didn't really get into the deeper stuff until I was exploring monoids and groups.

I've met people who tell me, subtraction and negative numbers are difficult. They know how to do the operation arithmatically, even fluently, but they don't "get it". No amount of repetition was going to change that. I had a similar block when I came across the idea of instantaneous rate of change. I get the proofs, the idea of limits, and how it defines instantaneous rate of change, but it was the first thing that I came across that I couldn't get over the idea that this is an abstract idea, not concrete. I didn't know how to handle it. It wasn't until I came across Azad's way of explaining the _intuition_ on instantaneous rate of change that I focused on getting the intuition first before trying to develop rigor.

I don’t think that from “dull” memorization and mindless practice alone (how it was often done in school) understanding at a deeper level somehow magically happens. What I do believe is that both (“smart”) memorization and explorative/playful interaction with the subject matter (turning it upside down, interrogating it, relating it in different ways, etc.) support each other, which, after some time, creates enough structure in ones neural pathways such that deeper insight can emerge from intuition.

When we try to learn math conceptually, we focus a lot on metaphorical or contextual understanding (“what is this like?”, “how does this apply to the real world?”, etc.), which may divert attention away from the actual subject matter, which is the symbols, formulas, equations, axioms, theorems, diagrams and so on. It is all one really needs to understand.

Sometimes I read texts where the author talks about mathematical ideas in a very vague and abstract language, mystifying and obscuring it or entangling it with other subjects. But then it turns out what they write makes no actual sense in the language of mathematics, which itself is very concrete and clear. Maybe they have fallen into the trap of thinking that they knew about the subject when all they have is a broad conceptual perspective without any grounding in the actual rules of the formal system.

These days I spend my free time reading books on higher mathematics and mathematical logic. Doing exercises certainly is the key to solidifying insight. I've also found adopting an old practice from my literary school days, writing essays, to be extremely helpful. There is a certain sense in which Wittgenstein's later "language is use" philosophy is deeply true when it comes to mastery of any intellectual material. The brain really does work just like the body—if you hit the gym each day you will build muscle and retain it, if you stop you won't.

Teachers were so utterly disparaging, it became an extremely stressful experience. It undermined our ability to focus in the first place, so not instantly getting whatever was being "taught" induced more fear, which made you lose even more focus. It was a terribly negative feedback loop.

Later on, I started reading math books on my own and realized that not only I wasn't bad at it, but what kind of motherfuckers were those so-called teachers, and how clueless they were in pedagogical terms.

I talk to a lot of people about math and many of them (adults) have intense math anxiety. I always wonder what kind of trauma could have led to this, because I assumed the I-suck-at-math is such a private feeling, at worst maybe your parents might see your grades and scold you about them. None of this is super traumatic, I thought...

But the perspective about public shaming by the teacher and other students piling on is much more intense, so I can see how some people really don't want anything to with math in later life. Those motherfucker teachers indeed!

Use scientific terms for science. Use normal words for your own experience unless you really were hooked up to a machine and were measured in some way.

This is not to say that object rotation is a proxy for mathematical ability, but just a demonstration that there are cognitive tasks which people vary widely in ability on when untrained and which we don't have an effective way to train, at least at an arbitrary age. Lots of Math seem to be this way.

Of course, there isn't a switch that gets flipped which suddenly makes you "good at math". But still, I think most people with an interest can study it, learn something, and make some progress. I'm happy playing a sport, but I know what separates me from an athlete and I know I won't ever be a professional athlete.

One is similar to a CPU, and provides the ability to follow procedures accurately and fast. This helps with arithmetic and a lot of cases where one can follow a memorized pattern for how to solve a problem.

The other is more like a GPU. This helps in a number of areas, such as visualization, geometric intuition, inductive reasoning and often creativity when facing novel problems. Most of these are a form of pattern recognition.

Any source for this claim? I'd like to read more about it.

Great. So where are some books, programs, or apps that will help us do exactly this? Not impressed that the article focused more on their personal journey and provided no recommendations on how to follow it.

Anyone here have recommendations on apps that might help? One HNer a few years ago posted a great little web app to practice rapid addition/subtraction, etc. which I used daily to noticeable benefit until it disappeared. Of course something working up in complexity from there would be good too.

If you want to practice addition/subtraction I suggest Zetamac(https://arithmetic.zetamac.com/), this is what most people use to train for HFT/MM interviews(although I heard there are more specialized tests now).

My mind embraces set and fractal theory readily and spatial awareness is decent. For me, math is applied using abstract logic. I'm still have math-phobia

I have to question the veracity of this sentence. How could they possible keep progressing to the next grade if they keep failing math and science? I'm sure it is hyperbole but this doesn't seem like a great way to start off an article like this.

You can fail math all year long, but achieve the bare minimum in the subject during your annual standardized test and pass to the next grade.

Course credits/grades do not ~~affect~~ limit? progression in many/most? US school districts before high school.

School year F -> STAR test minimum + intervention or summer school D -> graduates

At best you present a potential way that it might not be hyperbole.

I graduated high school in 1995. Was I too early? I don't remember high school math just being a bunch of discussions without problem solving.

This is the Wikipedia page for Barbara Oakley, which has more details about her:

https://en.m.wikipedia.org/wiki/Barbara_Oakley

Then I realized it was the same author!

Definitely worth reading if you've been avoiding deepening your math understanding.

"I re-wired my brain" offers no such hope.

Good find, thank you.

Isn't that just learning?