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Comment by smabie | Hacker News Reader

Memorization is absolutely crucial to getting good at math. In fact, I don't think there's any other way to get good at math besides memorization and doing problems, over and over and over again. It's rote, sure, but it has tremendous value.

If you think you understand something, but can't actually solve problems without referencing anything, you don't actually understand it.

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I have a degree in math and I strongly disagree with this comment. I was always terrible at memorizing things, and I still regularly reach for Google to double-check formulas and theorems I should know well. In practice, I think a lot of math people end up memorizing things the same way programmers memorize the syntax and tools of their preferred languages; it's not necessarily something you explicitly set out to memorize, but you use them so often that you end up internalizing them anyway.

Not going to argue math, but that final statement is a poor way to frame understanding/learning. You can understand something well enough to solve it from memory a few times when in close proximity to the time you learnt it. On the other hand you can understand something well enough to know how to check the references and solve it for the rest of your life.

That is it is better to understand how to solve problems with references than without. You'll forget most you know but once recorded you can't forget. Then you just need to know it exists so you can find it.

This is absolutely backwards.

No amount of reference sheets will help you with mathematics if you don't understand something.

I think you have backwards: you only understand something after grinding out problems and memorization.

Anecdote time. Never have I ever understood concepts through repetition alone. Grinding problems is for building patterns via using different input sets, not unlike ML. It is true that it "clicks" sometimes after a threshold and you actually understand it, but usually what grinding does is reinforce the algorithm without actually understanding it - e.g. surely you know that c squared equals a squared plus b squared, you've probably grinded that enough to memorize that equation. Can you prove it or at least explain why that equation is correct? Because i can't, not even after repeating it a million times. That's the difference between memorization and understanding.

> Memorization is absolutely crucial to getting good at math.

I have a terrible memory and I excel at math (and majored in physics in undergrad) precisely because it doesn’t require memorization.

I don’t know anyone who is good at math who operates by memorization.

I think this is only half true. Of course, if you want to be good at maths, you will have to remember things because you can't always look up or re-derive every smallest detail (and because you have to become good at pattern-matching at some point). But the memory tends to work best if it's activated by deliberate practice. For example, instead of memorising a proof line-by-line it's much better to try to remember the key points and then practicing filling in the gaps. If necessary, repeat this process several times. When I was studying for ODEs, I tried to re-derive many of the standard techniques by trying to remember which trick to apply instead of trying to memorise the formula.

^ This. For my Mechanical Engineering undergraduate courses, almost all of the formulas (models, actually) could be re-derived from linear approximations on infinitesimal elements. Memorize and internalize how the model is derived, practice re-deriving the model, and you can re-derive the formula during the exam, and do a bit of hand-wavy intuitive double-check of your answers. If you've only memorized the formula, you're going to have a very tough time coming up with an estimate against which to check your formula's answer.

On a side note, I really wish we had more emphasis on the conditions under which the linear approximations broke down. I remember sitting at the front of the MIT 2.002 class, and there was a demonstration of metal fatigue using a hydraulic press at the back of the classroom. Professor Sanjay Sarma stepped up to the front row in order to better see the demonstration at the back, and so I asked him about my intuitions about which way the model diverged from reality under vibration frequencies high enough that the quasistatic assumptions built into the model broke down. He looked to both sides of us and told the students on either side of me not to listen because they might get confused, and then we had a little discussion about conditions beyond which the model applied and which way the model's error went under those conditions. It was simultaneously one of the best and most disappointing moments in my education. It was an exciting discussion, but I was sad that the world beyond the linear approximations was considered to be likely too deep a rabbit hole for most of the class. Sanjay (as he preferred to be addressed) was an excellent educator, and I'm sure his judgement was based on past experience... each semester has a given complexity budget, and the field of Mechanical Engineering is so broad (statics, dynamics, thermodynamics, fluid dynamics, mechanisms, control theory/sensors/OpAmps, manufacturing techniques, design for mass manufacturing, numerical process control, destructive/non-destructive testing, etc., etc.) that undergraduates need to spread a limited complexity budget across so many subjects that they can each only be covered relatively shallowly.

I'm very curious about the type of math you have learned this way. Every math class was heavy on repeated exercises and memorization, but real understanding didn't come until outside of class (or after the semester) where reflecting on larger ideas lead to mental connections and analogies. I'll freely admit that I have to look up the integral/derivative of simple trigonometric functions every time, despite every attempt from the instructor to hammer those in during Calc II. Doing hundreds of integrals on homework and study guides kind of worked to learn those rules, but only imperfectly and temporarily. Doing hundreds of exercises over and over again has been even less useful for my later math classes. If some proof doesn't make sense, the next step is not to go through 100 examples, it is to break down the pieces into logical structures that are recognizable. All real understanding of math fundamentally works this way. (disclaimer, my math past calc/linear algebra has been in CS classes, so I'm definitely open to hearing how those from other backgrounds or those with more formal math education disagree with me)

If you think you understand some mathematical rule, but can only show 100 problems from memory, you don't actually understand it that well. If you can convincingly show why no counterexample exists, then you understand it in the strongest possible way.

Doing the same problem again and again does not do much to learn math. Doing various problems again and again is necessary, but that is not memorization. Because you are not actually remembering things.

Also, it was not that unusual for exercised based university math tests to allow references. Precisely because it does not matter all that much for difficult exercises, remembering everything is not the point.

Is there not an in-between? I used to be very good at maths, but a year out of practice and I might struggle at some topics. I still retain the intuitive understanding and that grants me a rapid path back to being able to solve problems.

The intuitive understanding that you still retain was only built up by repetition and memorization in the first place, no?

Absolutely not. I honestly can’t identify with your comments in the slightest. The only part of math that I ever learned by memorization was addition and multiplication tables in elementary school. Every subsequent revelation was based on pure understanding - zero repetition required. Up to what level of math have you studied? I honestly can’t imagine anyone thinking they’re good at undergrad-level math if they do it by memorization rather than understanding.

The great grandparent to your post (the one who initially brought up the necessity of rote learning) is a quant at a market maker.

Anecdotally, my favorite professor from undergrad (born in China, PhD from the best department in his field [my personal opinion]) said he thought the reason for Russian/Chinese dominance in certain areas of math was due to how those areas benefitted very much from rote practice. He advised all of us (American undergrads) to drill and kill certain techniques in order to build up our pattern matching.

I don’t think they’re advocating doing hundreds of worksheets on the power rule or trig substitutions or memorizing line by line proofs. Our brains do follow formal rules when doing math, but the insight necessary to find a way to solve a problem that isn’t straightforward isn’t through application of rules, it’s through a tacit intuition that you build up by doing lots of math. There is no other way.

It’s like how everyone feels like they understand physics to a PhD level while watching the Feynman lectures, but if you were to hand them any of the problems afterwards, what seemed like such a natural stream of thought is just simply out of reach. It’s much easier to go over something and declare “this makes sense” than it is to come up with that something in the first place.

+1 on the drilling specific techniques. Any skill is reinforced by consistent targeted practice, and to think that math is an exception where you can break through with pure genius is just deluding yourself.

The only way I got through my undergrad math was by doing problem set after problem set until the concepts were second nature, and the courses that didn't have a sufficient breadth of exercises to drive home fundamental concepts ended up being the ones I struggled with the most.

> break through with pure genius

Applying derivations that someone else invented doesn’t require “pure genius”; it only requires you to be able to follow the person who discovered it, which is a lot easier than finding it yourself.

> The only way I got through my undergrad math was by doing problem set after problem set

It sounds like you didn’t really understand what you were doing then. It’s hard to phrase this without just sounding like I’m bragging, but I never had to practice doing anything I actually understood. If I felt like I needed practice, that was a sure sign I didn’t get it, which I always tried to fix with careful thinking instead of repetitive memorization

> It’s much easier to go over something and declare “this makes sense” than it is to come up with that something in the first place.

Obviously, but that’s not what we were talking about. We were comparing memorization to understanding, not inventing to learning.

If you’re good at math, you should be able to re-derive any formula or procedure quickly (up to, say, constant factors) without having to memorize it (after the derivation has been explained to you).

If you run into a problem that you can’t solve because you didn’t drill the steps hard enough, you don’t actually understand the problem. This isn’t necessarily your fault - many math courses teach by symbolic manipulation without the conceptual grounding required to actually re-derive the symbolic procedures yourself. Few students will seek that understanding on their own outside of class, in which case they’re stuck with memorization.

> the reason for Russian/Chinese dominance in certain areas of math was due to how those areas benefitted very much from rote practice.

I think this supports my point - these “certain areas” are small. There is a relative paucity of mathematical/physical innovation from China (especially per capita!). The west still dominates mathematical invention.

Repetition to wire up thought processes and logic (why do X instead of Y) != Arbitrary memorization of trivia (X formula or Y proof)

The difference between learning X and understanding X

Training the common tools and pattern recognition for things algebra. At higher level you really need to have a repertoire of various tricks to apply to integrals and derivatives... I never properly learned these for University level.

Often the problems had some trick that you had to apply to solve it. And without knowing those certain tricks from routine finding right solution was pretty hard.

I had a math teacher that explained math in two categories.

Either you will understand permutations and combinations almost instantly, or you will have to do tons of examples.

Likely not applicable to all kinds of math but relatable to some concepts for sure

Calculations are based on memory but everything else not so much in my opinion. Sure, some methods for solving common problems are memorized

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