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Comment by litoE | Hacker News Reader

My secret sauce as an undergraduate for all my math courses was solving problems. Solve all the problems at the end of the chapters in the textbook. Find other textbooks in the library and solve all the problems at the end of their chapters.

In graduate school that was expanded: take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."

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> take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."

I recently-ish had a read through some of my old fundamental/pure maths notes from Uni, including plenty of proofs. The damned things are littered with steps which were "obvious" to my smug self-satisfied 20 year old self but impenetrable to me reading without much context 20 years later.

Git.

I did this but faced enormous problems with transfer/recalling definitions/etc. what helped me improve several fold in approximately exponential value

1) Drill/spaced repetition basic definitions. The Cornell note taking method is convenient to do this while taking notes.

2) Keep a diary of thoughts, things you couldn't solve or did solve. Especially identifying problems, what works or doesn't, why something went wrong. Metacognitive thinking was really useful for transferring problems to solving new ones.

3) A study group involving a lot of us explaining to each other.

Brute force exposure to as many problems and solutions as possible is the same way AIs learn.

That strategy in my opinion is not optimal for humans.

What we need to do is develop math resources that can help students learn things analytically & conceptually.

Like how they learn biology.

It's not quite like that.

Maths content is the ne plus ultra of conceptual.

It's totally possible to slog through a chapter of a maths text and feel like we got it. But it turns out our 'understanding' was a facade. We can't apply the concepts in a new situation. Exposing ourselves to feedback via problem sets reveals this.

> Like how they learn biology.

How do they learn biology?

An lot of brute force memorizing details that everyone else would look up. Biology feeds to medical fields where you have to have details you rarely used memories because you might be doing surgery on someone who is slightly different from typical and have to figure out what that thing in the way is - can you safely move/remove it or not. (I'm not a doctor, real doctors can come up with plenty of much more likely real world examples of situations where you don't have time to look up some detail)

I’ll add: go to the TA study sessions. And try to answer questions in class. Even if you have nothing to ask, use that time to do homework, listen to other people’s questions.

Yes, you’ll learn more. Also, TAs will recognize you put in the effort so if you’re arguing for partial points or you’re really close to a cutoff grade they will be more likely to bump you up versus someone they’ve not seen or noticed all semester.

> Solve all the problems at the end of the chapters in the textbook

It really depends on the textbook, isn't it? I find it impossible to solve all the problems in CLRS, for example. Our professor assigned one of the problems about universal hashing, and it took me hours to get the key insights to find the correct proof. I can't imagine how one can solve all the problems given so many competing priorities, except for a few truly talented.

The point is that you gradually get better at it. And by gradually I mean exponentially. If you start with a problem at then end of the book, it'll be very difficult.

Did you do the few hundred problems before that build up the techniques and theorems?

> given so many competing priorities,

Undoubtedly, the best time to do this was when you were young. The second best time is now. Pick a book and work on a problem or 2 every day. It will likely take 6 months or so but you will learn the material. This is an incredible way to level up in a technical area.

Imagine how much knowledge is in CLRS.

I agree with this. In the end it is beneficial with repetition for learning anything, and I think subjects with calculations are partly a craft where you need to get it in your fingers. I teach a subject, it not maths but a (different) engineering subject and what I see is that the students don't buy text books any more (for several understandable reasons), but one of the things they miss out on is a lot of practice problems. This becomes evident in the types of errors they do at the exam, and an apparent lack of system for laying out their problem and solution in an understandable matter for themselves (and me).

More often than the "at the end of the chapter" math problems are not about calculations, but are essentialy mini-problems requiring mini proofs. It's not about 1-1 applying the knowledge that was in the chapter, but rather about creative thinking. Some problems can take hours or days to solve. There isn't much to "get in your fingers".

You need to get creativity in your fingers. You can get good at solving problems. As those who did much better than me in school proved.

I find that in any creative fields it helps being really good at the fundamental skills so they don't get in the way of your creativity.

My issue during my undergraduate days was that the textbook didn't come with solutions, and I really don't know how to unstuck myself when I got stuck (except asking the TA in the office hours).

Ask your peers. As an ex-prof, one of the most prominent sources of missed potential I see in undergrads is not working with their peers. Like serious 2-6 hours sessions of collaborative problem solving.

You're right, but it depends on your peers. If you're the only one motivated to do extra work, your peers are unlikely to want to spend hours on verifying the solution. Guess it shows the importance of good peers.

Yep, I hated that. Spending an hour on a problem and get a solution you can't verify. It very much affected my motivation on starting on the problem knowing that I wouldn't know if I was right or wrong.

When I was an undergraduate we had Schaum's Outline Series (do they still exist?). Every chapter came with a bunch of completely solved problems plus another bunch of unsolved problems with the answers. The optimal strategy was still to try and do the solved problems by yourself and then verifying your solution.

An undergraduate text like Pugh's Real Mathematical Analysis can have over 500 exercises. Trying to do them all on top of your regular workload seems excessive, unless maybe it's an area you really want to specialise in. I also tend to find certain kinds of exercises boring because they appear terribly unmotivated.

On the other hand, I second the suggestion to engage more deeply with the subject material itself: Modify assumptions and see what happens. Can certain proofs be simplified? Try to reconstruct proofs by only memorising certain key details. Try to draw a mental map of a subject and how the different theorems and definitions relate together. Try to implement some proofs in an automated theorem prover, if that's your thing.

500 exercises over 10 weeks is 50 a week, or 10 a day. This leaves you weekends free, and you get the final weeks (typically 5) of the semester left for all those projects that are due at the end of the semester but you can't start early. 500 a day is too much, but you should be able to do 50-100 problems a day in your study time. And others report that when you do this you get good at doing math and so it takes less time.

Sadly I didn't do that. I graduated and do okay, but I encourage everyone to do better than me. As I get close to retirement I need a few people who are still working to build things (and medical treatments) that makes my life better (and in turn take some of that money I saved up over the years for your own life)

Pugh's exercises aren't trivial calculations, they're the proof kind of exercise. If you manage to do 10 a day consistently and still have time for your other courses, congrats.

I also don't think that people who develop new medical treatments necessarily did all the exercises in a pure maths textbook. Being able to prove that continuous functions on a compact set are uniformly continuous probably won't help you fight cancer.

Also, it's very unlikely that Calculus will be the only subject you're taking. Other subjects require homework too.

Someone correct me if I'm misremembering, but I think Donald Knuth wrote somewhere that when he'd be assigned the odd-numbered problems, he'd always do them, and do the even-numbered problems as well.

This is not applicable to every book. Try doing this with Casella and Berger and see how difficult it becomes.

Some books can take many many months to finish off like this, and most courses only cover a small percentage of the book.

Solving problems (and trying to invent proofs) is indeed the way to enjoy math.

po-shen loh?

“The proof is left as an exercise …”

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