[griffinmichl]

I mostly aced my math degree. The keys for me were:

1. Read and get a basic understanding of the material before lecture.

2. Attend every lecture and take notes very few if any notes. Most of what you need is in the book and math is about understanding. You cannot grok and write at the same time, and it's better to try to grok while an expert is explaining it to you.

3. Do a shitload of problems / proofs, depending on the class. Be honest with yourself when you don't fully understand something and stick with it until you do.

Math is different from other subjects, and you need to treat it that way.

[ljw_dev]

It does depend on the student. I also aced a math degree.I did not do 1 or 2. I took good notes in all lectures that I attended and these were my primary reference when it came to revision time. I used very few other resources than the notes I took and problem sheets given out in class. I did have friends who used books and took no notes.

However, all of us who did well did 3.

[PeterWhittaker]

Same for me. #3 all the way. In second or third year PDEs (cannot remember which), I simply sat down with the book and did all the problems. Went to class only to write and collect midterms (much to the displeasure of a regular attendee friend), at least one of which I aced.

[scott_s]

I took good notes in my undergrad math and physics classes, but never computer science. My degree was in computer science (minors with math and physics), and now I'm a computer science researcher.

The reason for the discrepancy was that the lectures were quite different. Math and physics lectures tended to be focused on problems, and how to do them. The professor would walk through example problems, explaining as they went. I diligently wrote all of that down, and would refer to it when I did my homework. I found writing down the examples as we went a good way to make sure I was actually understanding each step.

I never took notes in computer science classes because the lectures were more conceptual - beyond the first semester, the professor was never walking us through how to code. They were explaining new concepts, not showing us techniques I would have to reproduce. I didn't see much reason in taking notes about concepts, I would rather think about those concepts during the lecture. I found taking notes a distraction. The only exception here was some algorithms lectures, where the professor walked us through how to prove particular things.

Basically, if I was being shown techniques, I wrote down the techniques. If I was being told new concepts, I just listened.

[baby]

3. Can't agree more with that. If math taught me one thing, is that to fully understand something you need to do a huge amount of exercise in the topic.

2. I have seen a lot of very good students in my math bachelor, and none of them took notes. Furthermore, I've seen students taking a lot of notes, even some writing theirs in LaTeX and sharing it to the rest of the class to later flunk the year.

1. I wish I would have done that. But depending on your school, this might be just impossible.

[semi-extrinsic]

As for 2: YMMV, I always took good notes and aced all my undergrad (and most of my graduate) math. I've seen several studies showing that retention of material covered in a lecture is significantly better if you take notes.

I think the key problem for most people is speed: if you need to spend so much effort just on the note taking that your brain can't process/understand what you're writing, then obviously taking notes is a bad idea. I don't think this is accounted for in the aforementioned studies.

[Davidp00]

Do you have a link to that study?

I remember seeing a study that showed that people who take notes on paper do better vs people who take notes on a laptop. People who didn't take notes(or take very few) weren't mentioned.

[semi-extrinsic]

You're correct that most new studies compare handwritten vs. laptop notes. The effect of handwritten notes vs. no notes was studied quite a bit earlier, and I believe this effect is now as uncontroversial as psychology can be.

A good review of the literature (from 1989) is found here:

http://link.springer.com/article/10.1007%2FBF01326640

[InclinedPlane]

In general the best way to learn anything is to start with a skeleton then add details (flesh) later. This is very true when it comes to math, especially at higher levels. You need to understand the concepts before memorizing anything will be worthwhile.

I'd add a few more things to your list. A lot of students hit a wall when they run into calculus. Partly this is the same old problem of mathematics being taught poorly in general, and there's also an annoying element of calculus classes being difficult on purpose (being as it is a topic which can involve an arbitrary amount of memorization and busywork) for stupid reasons. But additionally calculus relies heavily in having a firm grasp of advanced algebra, trig, and pre-calc, all of which equally suffer from the general problems of our educational system. Reviewing those subjects, probing for weaknesses in understanding, and shoring up any weaknesses can go a long way to making your life easier as you tackle more advanced subjects.

Also, as a general rule and a corollary to your third point: don't treat college as a race to get a valuable piece of paper, one that requires merely toil and busywork to acquire. Treat it as an opportunity to acquire new skills, understanding, and knowledge. The piece of paper might improve your earnings incrementally but the knowledge will not only change the way you look at the world it will unlock a great deal of potential in terms of what you can do (in your career and elsewhere).

P.S. since we're stating bona fides, here's mine: I got my math degree mostly by accident. It was something I was good at and a reasonable default choice, and I hardly had time to think about it before I graduated at 20. I don't "use" my math degree in my career typically but it's definitely made me a better developer and critical thinker.

[analog31]

Regarding memorization and busywork, maybe a certain amount of it is valuable. I don't know if it's a great analogy or not, but memorization and busywork have a central role in learning to play a musical instrument. You can't learn to play music, solely from attending lectures and reading books. You have to program that knowledge and understanding into your ears and hands. This is done through memorization and busywork. To get past the most basic beginner stage, you either have to force yourself to do it, or derive some pleasure from it. If you do get past that stage, you can "hear" something in your head, and it comes out of your instrument automatically, so you have some bandwidth in your brain left over for thinking about higher level things, such as: How do I want to interpret this music? What are the other musicians doing? Are drinks on the house?

Is there something like that in math? I'm thinking of getting from A to B in a proof or solution by introducing things like definitions, theorems, common algebraic manipulations, and so forth. If you can't just see those things in your head, and write them down, then you won't have a reserve of mental bandwidth to think about the higher level structure of the problem that you're working on, and you'll reach a level of complexity and abstraction where the cumulative effect of small mistakes prevents you from ever getting to B.

Is it poorly taught? That's certainly a possible problem. One thing I noticed when I taught math, was that the kids were never given a higher level explanation of what they were doing. Memorization and busywork are necessary but not sufficient. My students did not understand what "show your work" means. What it means is that math is a mixture of theory and performance art, like music. Math has a weird social role too... by the time they are in high school, most kids know that they are learning math with no expectation that they will ever use it. Parents treat it as some sort of obedience training.

[klibertp]

> If you can't just see those things in your head, and write them down, then you won't have a reserve of mental bandwidth to think about the higher level structure of the problem that you're working on, and you'll reach a level of complexity and abstraction where the cumulative effect of small mistakes prevents you from ever getting to B.

The big difference between math and music is that the latter is a real-time applied form of the former, with "real-time" being the key. You can waste as much time as you want thinking over things and research solutions to the given math problem; the solution you arrive at after a week is as valid as someone else's solution written down in minutes.

[analog31]

I think that's true to a certain extent. However, being able to write down a solution in minutes might be beneficial if it's a small part of a larger effort, because it doesn't interrupt your "flow," to borrow a popular term. That kind of facility is also beneficial if you're working with other people. "Hold on, give me a week to come up with an explanation for my idea" has much less impact than "here is how it works."

[tnecniv]

I agree with this, except for 2. I need to write things to internalize them, and I get a lot more out of lectures I take notes in because it keeps me focused.

[gizmo686]

I used to feel like this. Then (on advice of my high school history teacher) I stopped, and very quickly found I can keep my focus (and focus better) when I am not taking notes. Some people find that doodling helps. Personally, I find that I need an open notebook and a pencil in my hand, or I get really insecure and distracted.

[baby]

> I need to write things to internalize them.

Have you tried air writing? Or just "doodling" some notes when in a lecture? It worked for me.

[stdbrouw]

This. By not worrying about whether your notes are actually complete / sensible, you get the cognitive advantages of writing stuff down without the distraction. You can still make a proper summary after class if need be.

[soperj]

I'm the same way. I remember things a lot better when I've written them down and also find my attention doesn't stray as much when I am writing things down.

[slmyers]

> 2. Attend every lecture and take notes very few if any notes. Most of what you need is in the book and math is about understanding. You cannot grok and write at the same time, and it's better to try to grok while an expert is explaining it to you.

My highschool calc teacher gave us the same advice and I carried it through uni.

[hasbroslasher]

On the contrary, I took meticulous notes and also aced my math degree. Part of grokking (for me) is taking the words someone else is saying, jumbling them up in my head, and turning them into coherent sentences. Additionally, a lot of the math classes I took didn't have books (or if they did, the books were drab and useless), so each day was a race to scribble out the proofs we did in class that you needed to understand to do the HW.

I even had one class where the final was essentially a recitation of proofs we did all semester in lecture (notes, no book) in which you had 90 minutes to do 6 proofs. Things like Green's Theorem, proof of irrationality of pi (Niven's proof), proof of the Chain Rule, F.T. of Arithmetic, a bunch of other stuff.

[klibertp]

> Math is different from other subjects

Probably mostly in how bad are the materials used. Have you ever tried learning a completely new mathematical "thing" using just books, exercises, on-line courses and so on, but without having an "expert explain it to you"?

It's doable, but - being an auto-didact in many other areas too, so I can compare - it's so much harder than most other things you'd like to learn that it's not funny. I don't know the reason, but I doubt it's related to how "math is different" or "math is hard" because it's not: if you have enough intelligence to play with legos you can also do math.

[FullMtlAlcoholc]

> 1.Read and get a basic understanding of the material before lecture.

I folliwed this same advice and went from being a B- to A avg student in just two quarters. If you can, review the material the night before so you can sleep on it and process it unconsciously.

In my case, another essential key was to attend both the professor's and the TA's office hours. Most of the time it was to discuss the material outside of the scope of the class. This ensured that I truly understood the subject matter and wouldn't forget it after the final.

[vlasev]

One summer after I had already done a group theory class, I went through most of the exercises in the textbook, and there were many. And after that, I understood the material much better than before. One could probably just do number 3 if and only if the textbook has a good progression of exercises from easy to difficult.