I rewired my brain to become fluent in math (2014)

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.

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?

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

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).

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.

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 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'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.

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.

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.

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 :)

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.

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?

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

- 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.

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.

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 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?

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

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.

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.

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.

Because you'll be teaching the awful wasteful rote math that everyone hates and can't use, instead of the nice universal stuff that lets you transfer intuition and see how ideas communicate between different knowledge domains far beyond what was traditionally seen as math.

The 20th century gave us real new ideas in math. But our primary math education is still 200 years out of date. Until that changes, math education will remain a deadening cargo cult that throws away far more human potential than it develops.

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.

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.

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

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.

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 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!

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

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.

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).

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.