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Comment by baq | Hacker News Reader

> set her nose to the grindstone

Worth calling out the key observation: it takes practice, and lots of it (for most people, anyway) to get good at math. Just reading about it does zilch.

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It does a little more than zilch. Exercise is important. Reading is also important.

If you want to learn fast, you need a balance of both that works for you. Some people will spend months grinding hard problems on their own in a single chapter to make sure they really understand. Some people will read too fast and have to go back because they don't have solid foundations, and only thought they understood.

Reading is about as important as doing exercises. You're not going to reinvent all of math on your own. You get insights from both.

This is just not true. Reading is necessary but hardly about as important. If you are lucky and get a good text, the exercises guide you to "invent" the important parts of the theory. Otherwise its just definitions and theorems, and its up to you to make your own examples to gain an intuition, which to be fair is closer to life outside of a classroom.

It really depends on the text, it doesn't have to be just a list of theorems and definitions.

Some intuition can be conveyed through text, or geometry and visualization. A good explanation can make things click.

I think you may be right about your own experience, that you get most of your intuition from exercises, but I'm confident this varies. Some people get much more than zilch from a good text, in addition to doing exercises.

The real world is harder than a classroom, but we don't have to make classroom learning as hard as research. It's okay to start with help and increase the difficulty gradually. You don't have to do everything on your own!

The "if you are lucky and get a good text" is carrying a lot of weight here. My school identified certain young students as having high mathematical aptitude. While other students were learning the basics of addition and multiplication, the advanced students spent hours practicing multiplying nine digit numbers by hand.

The school was then always disappointed by by their performance in mathematics competitions. After all, the other teams were "wasting" their times unimportant reading about algebra, geometry, and combinatorics while our team was "practicing" math with hours of manual long division nightly.

The practice is certainly vital, but it's useless without a good text to guide you. Unless you're lucky and grab a good one on the first go, you'll need to read a few texts to find the good one.

I'd say that reading is as important to learning mathematics as breathing. You'll be a lousy mathematician if you spend all your time focused on your breathing, but you'll be worse one if you skip breathing entirely.

Could you elaborate on this please: "the exercises guide you to "invent" the important parts of the theory."

BTW: I think time not doing exercises is just as important; it's when your mind tries to piece together the data. Coincidentally(?) time resting, after physically exercising, is when your muscles strengthen.

Sure, texts like "linear algebra done right" or "Understanding Analysis" do a unusually good job of integrating large multi part examples where the reader works through them and proves the theorems themselves before they are explained in the book. The nominal case for most math texts is definition, lemmas, theorems, and sometimes they provide examples, and other times, readers are expected to make their own examples. For example, basic topology really only needs 10-20 pages to completely define, but to really understand why the axioms are chosen and the implications, you must work through examples, it is the only way. In fact there is a good book on topology specifically due to this " Counterexamples in Topology"

Thanks! I think I now recognize this from Spivak (Calculus), where much of the teaching is literally in the exercises. You are guided along, deriving/proving many things along the way, some incidental, some cumulative. (There's also important exposition in the exercises.)

A downside is you lose the thread if you skip exercises (e.g. do alternate ones) - the exercises are an integrated whole. But it's a lot to do all of them.

I hadn't gotten the impression that these helped show why exactly the axioms were choosen - though could well be there and I just didn't see it.

It depends a bit on what you call "reading" (i.e. whether you are reading passively or actively). If you stop and think through each proof yourself before reading the one in the book, that more-or-less turns the main content into a series of exercises.

Exactly. I'm not a programmer because I don't program. I can't play guitar because I don't play guitar. I can't speak another language because I don't practice one. It sounds obvious but doing the thing you want to do is critical to doing it.

It seems not to be entirely obvious to the students in my classroom. Today second day of the semester, job one is to explain that they need to have a notebook. They need to write in the notebook. And they need to do the homework in the notebook. They seem not to know that, many of them.

OP described a specific type of practice to get good test grades using 'Anki cards' . Standardized tests test for grinding and dedication not expertise or 'getting good'.

It depends entirely on the cards and what's done when a card is missed. If the cards are straight definitions then probably not the best way to "get good at math". But if the cards (you can do this with plugins) are things like generated problems using specific mathematical techniques (solve a system of linear equations, for instance) and after failing a card you can go back and study the material, then it's effective. You can also throw textbook problems into cards if you don't want to use a generator, but that's a lot more effort than most people are willing to put in. You don't even need to put in the solutions if you're willing to keep the book around. It does mean cards take longer than a few seconds, but it still has the effect of taking advantage of spaced repetition to develop mastery of a technique.

There is no (multiple choice) standardized testing in the Finnish school system and I cannot imagine passing the math tests based on rote learning. I have to assume she took the test somewhere else.

EDIT: The exams at the end of high school can be considered standardized tests, but they are taken at your own school, graded by your own teacher and only verified by the national test organisation. They are not multiple-choice tests.

> I cannot imagine passing the math tests based on rote learning.

why not. Its not exactly rote learning like a parrot. You just learn tricks and patterns in problems. Tests usually have a limited amount of patterns.

These are not tests generated from patterns. You may be asked e.g. to solve an applied problem you've never heard of or to sketch a proof to a given (simple) lemma you've never proved before.

OTOH, you are not supposed to solve every problem in the exam, so perhaps you can get the best grade even if you skip all the problems where there's no pattern to apply. In that case, that's a loop hole which the exam creators should plug.

> solve an applied problem you've never heard of or to sketch a proof to a given (simple) lemma you've never proved before

Can you link me to finnish high school question that is asking for 'sketch a proof to a given (simple) lemma you've never proved before '

I agree that this cannot be rote learnt .

I hear these problem types are not as common as they used to be, but here's some recent examples in this direction.

Fall 2023: "Prove that 2^12345678910 - 1 is divisible by 1023."

Spring 2022: "Using induction, show that the sum of the numbers on the line n of Pascal's triangle equals 2^n."

Here's the full exam from fall 2023: https://yle.fi/plus/abitreenit/2023/syksy/matematiikka_pitka...

I cannot seem to access that link outside finland.

That is indeed a tough proof to solve sight unseen. Any reason you say that students are seeing that question for the first time in the test. Seems like a famous questions, even chatgpt got the proof correctly .

I think the pressing question here is what the Anki cards contained w.r.t. these problems :-)

The high school curriculum and the text books are not focused on proofs or famous questions. Further, the goal of the exam is to find out your position in the normal distribution of your peers' math skills. You are not supposed to be able to beat everyone else by rote learning (so if you can do it, it's quite a hack!).

I don't think ChatGPT is a good comparison, because it has obviously memorized much more than a human could, and a test to poke its strengths and weaknesses would look different.

I had great success using Anki cards during my undergrad to fully internalize important definitions (i.e. trig identities, etc.) for utilization within practice problems.

Sometimes, you just know a thing or don't, and for me, spaced repetition cards helped a ton with the static things you just have to memorize (or derive). Knowing definitions is important—you're absolutely right that it's just one piece of the puzzle though.

Thinking about how to learn is fascinating. A few short things that helped me:

Essentially we need to get fundamentals first.

Then apply the knowledge and get challenged (feedback loop) -> build a project, play in front of others, speak the language.

Improve - not by force, but by understanding (remembering something and understanding something are two different things).

Synthesise - learn about a topic in a different manner or try to find similar concepts in completely different context /

Use mentors to amplify the knowledge and get feedback quicker.

Immerse yourself in practicality (work in a field, live in a country etc.)

This is a great summary. Thank you!

yes. learning to doing maths is a bit like learning to play an instrument in that you have to spend lots of times practicing scales or other simple stuff (doing the exercises) before you can really be creative on it.

or like learning to write - you got to spend lots of time practicing and memorizing your letters before you can start getting to words and sentences and novels.

It's also a bit different from learning to play music in that almost everyone appreciates music even without being able to create it themselves. There's a much more direct connection between the craft and its outcomes than with math.

please tell me that how can I get question for practice in math of university level? most of textbook even don't contain answers of practice in the book...

That's mostly because proofs are too long to write out easily in the back of a textbook, imo. Your best bet here is to learn the proofs of the theorems proved in the book first - they're usually much harder to come up with at first glance than the problems one bases on them. Or, if you're using a commonly used textbook, you can usually find people's own proofs online. Sourcing an easy to grok and easy to check proof is a very good skill to build in any case, it minimizes wheel reinvention at the neural level.

If you have a textbook you're interested in studying from, you can usually find courses from searching on Google that use that text. If you're lucky, they will make the homework and answers available online.

If you search long enough and look for older courses (before all of these online learning platforms like Canvas became popular), you can usually find lots of worked examples. There is no central location to find what you're looking for, but you should be able to find supporting resources.

Read the book and whenever you come to a theorem, stop and try to prove it on your own. If you can't, read some of the proof, then try to continue on your own.

Hop on the mathematics discord and ask in their book channel. It's the big server with a picture of a torus as the icon. There are some very knowledgeable people there

You just find books with the solutions.

Some like Hubbard and Hubbard's Vector Calculus come with a solutions manual.

Some like Knuth's Concrete Mathematics contain full solutions.

There's also whole genre of books like Schaum's Problem books and their Outlines which contain thousands of solved problems. And they're quite cheap.

For any given math subject X, you can probably search for an "X problem book".

Very true.

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