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.
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.
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.
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.
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.
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.
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.
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:
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.
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.
> 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.
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."
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.
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.
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.
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.
> 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.
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.
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.
> 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.
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.
Haha, this is very similar to the list I came up with when I was flunking at Caltech:
1. attend all the lectures
2. attend all the recitation sessions
3. do 100% of the homework, and on time
4. make sure to understand every homework question and how to solve it.
5. write legible notes
This was good enough to get a baseline B. To get an A, you had to put in much more work.
I know this stuff seems obvious. But it took me a while to figure out I couldn't just coast and wing it like in high school, and many other students had the same experience.
I'll got a bit further if I may, but its mostly a tip for the instructors:
Start from the basics. This makes such a tremendous difference I think, because we have these pre-conceived ideas about math, numbers, calculus, etc. (due to usually shitty high school education) that must be shattered and rebuilt from the ground up.
Here are the example notes that my undergrad (in fact, first college math class) professor used:
I went to the UofA 20 years ago. The math profs at that time were so bad I got turned off math forever and went into business school for finance. I have many stories, but the worst was when one prof quit half way through the semester because he demanded silence in class but didn't get it. Then the math dean said we could learn the rest of the course by reading the textbook with no lectures and the entire grade would be from the final exam. Seems like they have improved since then.
This is true for more advanced topics as well. One piece of advice I give people is to not be afraid to re-learn topics. Even getting an A in the class doesn't mean you've really "got" it. I don't think I really "got" linear algebra until maybe halfway through the first year of my PhD.
>I know this stuff seems obvious. But it took me a while to figure out I couldn't just coast and wing it like in high school, and many other students had the same experience.
This.. So much this for me. It took a long time for me to realise that I couldn't just wing it, as I did for school, and once I got that I went from failing to near top of the year.
My math grades had some ups and downs, and the key variable was doing the problems / proofs over and over. Likewise for physics courses, which involved a lot of the same kind of stuff.
I think it's important to get the "mechanical" work, e.g., step by step derivations, so it can be done quickly and accurately. For me, this is also resulted in committing the definitions, axioms, theorems, etc., to memory. During exams, being able to speed through this work gave me the spare time I needed to sit back and think about each problem, especially the one "surprise" problem in the set.
Oddly enough, I kind of treated the humanities courses in a similar way. I signed up for courses that were known to be graded based on mostly written work -- essays, papers, etc. Because all of these courses involved the same "mechanical" work, e.g., writing a paragraph supporting a concept, I also got very quick at it, which compounded itself in terms of getting assignments done quickly and writing fairly lengthy, coherent answers, in blue-book exams.
Note: It also helps a lot to actually enjoy the stuff. Attending the lectures helped me later in life, as I observed the teaching styles -- good and bad -- of my teachers. That experience has helped me with presentations and other kinds of interactions in my regular job.
* Watch video's on Khan Academy/youtube
* Use Wolfram Alpha to confirm/deny yr solutions
* Get a tutor who can explictly show you how they approach solving any arbitrary problem you bring them
* Accumulate a list of math tricks that you can use and how to use them (example: how to long divide polynomials)
* Print out old exams and pretend to take them ~3 days before final
* Rewrite your lecture with each example problem on it's own page(s)
* Create an index on your lecture notes so that you can quickly identify what type of problems and topics were covered in each lecture.
* Identify yr strenghts - we can't all be good at everything - knowing yr strenghts will give you a foundation of confidence to build upon
* Queue cards for memorizing all those trig identities
* Expect to spend a lot of time - this is not always possible
True. As you've noted elsewhere, this is intended for code. It doesn't work very well for text. You have to break the lines yourself rather than relying on the browser. If you break wrong, those with small viewports end up with nasty side scrolling.
Strange. That was the case for my first couple years, in classes with hundreds of students, where the university hadn't yet shed its "look to your left, look to your right" mentality. Later classes had fewer students and engaging professors, since they'd weeded out many of the worse students.
Its more a side effect. Everyone knows everyone cheats on the TOEFL yet it would be incredibly awkward in academia to discuss that issue, so its kinda glossed over to avoid massive interpersonal drama. But, yeah, any prof or TA you have who doesn't know English is a known cheater, violation of honor code, etc. It does make you wonder if their CV or transcript or reference letters or published papers are as fabricated as their obviously fake TOEFL scores.
Toefl is a surprisingly easy test to pass even with a strong non-native accent. I know people with mediocre to bad command on English who achieved scores of over 100 on the online test. There is no reason to cheat the system.
That may be, but several of my classmates in grad school told me that they regularly wrote their papers in Chinese and sent them to a translating service for English. These were students who were conversational in English. It makes you wonder how much the translator changed the content.
I am not acquainted with the Chinese system, but here in India, you have to personally go to the designated test centers and take the test. At the center there is some Airport level frisking before you get to see your computer and you need to have your passport with you to prove your identity (if I remember correctly). You are also not allowed to wear jackets/overcoats/caps while taking the test. Really very little chance for gaming this system.
It may have disappeared from the internet, but there was a great satirical newspaper at Georgia Tech back in 2001 or 2002 called "Shaft News". They ran an article about a calculus professor complaining, "How can I be expected to teach when the students don't even know basic Chinese?!"
I have had incredible success with this in languages for grammar rules and vocabulary, as well as for literary-courses. But except for some basic integration rules, I have a hard time using Anki for math courses. What kind of cards did you create?
And since he comes up regularly on HN: Cal Newports Stealth Studying[1] is really efficient in an Anki deck
My calc anki cards have copyrighted material in them so not off hand no.
But generally what I would do is use them to memorize anything I needed to know while I was solving a problem but couldn't remember.
The important thing is to know how to write mathematical expressions in LaTeX syntax, because this lets you make non-crap cards. You might do something like:
\frac{d}{dx} \cos x = ?
On the front and then
- \sin x
On the back.
The important point is that between the front and the back you actively recall the information you're trying to remember. Undergrad mathematics can be thought of as a context free grammar with a set of production rules, memorizing those production rules is a major pain in the ass and this helps.
(Maybe other people don't have trouble with this, I nearly failed every math class in high school and am decidedly not a math prodigy. Your mileage may vary.)
Avoid cards where it takes a long time to state the back side of the card. Even if you know the answers, you'll get bogged down and it'll take too long to review. Answers should be short and easy.
Better to create 5 cards where you replace 5 parts of a formula with blanks and say "fill in the blanks", than one card that requires remembering 5 things.
There is some pretty obviously advice in here; basically take notes and do your homework for the sake of comprehension and not merely completion. But, "keep a list of hard problems" is pretty good advice that, I think, most people do not adhere to.
My breakthrough in math came when I started studying with other people. Best way to learn something is to explain it to somebody else. And you're a lot less likely to get stuck on something for half an hour if you have other people to help you. By studying with people I aced calc just by going to class and keeping up on homework.
Depends on who you are. I can't listen to other people talk about math without getting confused myself. Other people think about math differently and often throws my own thinking into a loop. I need to study alone and ask questions if truly stuck.
The issue I have with lists on how to be successful in a class is that they rarely provide much guidance on how to prioritize the work. Of course students are successful in a class when they read & understand the material before the lecture, follow the lecture attentively, redo all example problems, solve & understand all assignments and maybe read a few textbooks on the subject. The difficult part, at least for me, is how to decide what learning activity will help your understanding in the most any given moment and how to split your time between the different classes and other activities.
This list reads like my strategy to do well in a math class without learning the material (literally; this method got me through numerical methods and advanced calculus, both classes that I had no interest in and took only to satisfy the degree requirements).
My thoughts on the particular points:
>Keep a list of THINGS TO MEMORIZE.
If this list is anything but empty you should feel bad. "Memorizing" something in math is a way to act (and test) like you understand it without understanding it. The one exception is in computationally heavy classes, where you memorize the completed solution of common forms. However, if your goal is to understand the material, you should be able to derive everything you are memorizing without thinking. The memorization is only to save you time on the exam by skipping the computation. Having said that, as I mentioned above, if you do not understand something but still want a good grade; memorize it. The professor will never see the difference.
>Watch for example problems.
Because these will be the problems on the test.
>Know how to do every homework problem assigned!
Yes. Also, know how the book/teacher wants you to do the problem. It is often times hard to write a problem that can only be solved one way, so it is sometimes possible to avoid using a method you don't like or understand.
> Start the homework at least a few days before it is due.
Yes. Also, if possible, consider homework due at the last office hours before it is actually due. Otherwise, you can't go to office hours for help.
>Keep a running list of HARD PROBLEMS.
Good advise if you are going the memorization route; otherwise, this is just memorizing the solution to a particular form of questions. Also, in my experience, if you are going for the understanding route, this list just does not stay relevant long enough to justify keeping it.
>When you get your homework back: Look over the things you got wrong.
Always good advice. Having said that, if you got a problem wrong (for reasons other than computational/algebraic mistakes), the bigger issue is that you thought you got it right. This means that you not only did not know how to solve the problem, but that you have misunderstood some concept that you need to learn.
>Find a quiet place, set a timer for the amount of time you'll have in the exam, and take the practice test. Don't look at the practice test before you do this.
Good test prep advice in general.
>If a problem is hard, skip it and come back later.
I triage problems much more aggressively. If a problem looks time consuming I skip it. If a problem looks like it involves thinking, I do enough work to verify that it actually involves thinking then skip it.
>Do a quick check of each problem to be sure your solution is reasonable. E.g. if the problem asks for a distance, is your solution positive?
Do this check after you finished the test. If you got something wrong, but didn't finish the test, then knowing you got it wrong doesn't help; you still didn't have a chance to correct it. Also, you are more likely to notice an incorrect answer after spending time away from the problem.
Having said that, sometimes you answer "feels" wrong as you are solving it. If this happens and you see where you went wrong, correct it. Otherwise, complete the incorrect solution (if feasible), and mark it. This gives you the chance for partial credit; and sometimes your feeling is just wrong, and the answer is weird.
>Write SOMETHING on every problem. The grader really wants to be able to give you some partial credit.
If you have time. If you really have no idea how to approach a problem, then your time would be better spent doing better on the rest of the exam instead of producing a plausible looking solution. Mark these problems, and, if you have time at the end, come back and look at them again.
Having said that, this is still good advice. If you think you know how to solve part of the problem; do it. If the part you know how to do requires you having computed something that you do not know how to compute, then clearly write "let a = thing I can't compute". Graders don't have time to look closely at your answer; make it easy for them to give you partial credit.
If you are answering a proof based question, and cannot figure out how to prove a particular fact that you need for your proof, consider writing "it is clear that". You will be amazed how often this works.
>When you've tried everything, go back to the problems worth the most points first.
Triage. Go back to the problems that you think you got wrong and can improve first.
>Given time, double check your algebra carefully!
If possible, verify your answers using a different method. For example, if you are asked to find the integral, verify your answer by taking its derivative. You are less likely to make the same mistake, and for many problems it is easier to verify an answer is correct then to find the answer
>After the Exam
Write down what you can remember about the problems you could not solve (including ones where you put down something that might be correct, but were not sure about). Solve these problems (using book/notes/TAs/etc if needed).
When you get the exam back, compare it to the list of problems you knew you didn't get. You don't care about these problems at this point; you already knew you missed them and worked through them. You learn nothing new by the grader telling you that you missed these.
The problems that you did not know you got wrong are where you should focus your attention. If you just forgot about them, then work through them like you did the ones you already knew about. If it was a computational/algebraic error, don't worry about it (but do do more practice if these errors cost you a significant amount of points). Pay special attention to problems that you thought you got right but didn't. These highlight the areas where you have a misunderstanding of the material.
Final remarks:
HARD PROBLEMS list:
As you might have noticed, I don't like this. What I do like is a concepts list. When you go to study, read through the list and make sure you understand all the concepts. It should be small enough that there is no point in creating a separate list for hard concepts; and what you consider to be an easy/hard concept will change as you get more practice with some things, and never see other things between the first month of class and the final.
Studying for the test:
As you probably noticed, my opinion is that many of these points are techniques to study for the test. This is a valid thing to do in school (after all, you are graded on the test, not understanding), but be aware when this is what you are doing. If you plan on taking another class that builds on this one, then this method will come back to bite you then.
In class notes:
Do not make them. The only reason you should be writing during a math lecture is to use the paper as a scratchpad to think about what is being said. Anything else is a distraction from the lesson. All the material should be in the book. If it isn't, you can ask the professor for a copy of his notes. If you want your own notes (which can be a good idea), write them after class. If you do take notes, keep them in a separate notebook. Interspersing them with problems and scratchpads just makes it more difficult to use them.
LaTex:
If you are taking math as a gen-ed requirement, ignore this. If you are going into a math heavy field, learn LaTex as early as possible. Write you notes in LaTex. Do your proof based homework in LaTex. This will pay off down the road, when you A) know LaTex and B) have a digital record of you previous math classes. Plus, turning in proofs written in LaTex make them feel far more correct, so you might get graded easier.
Study groups:
Form a study group of people with similar skills as you. If you try studying with people far better than you, then you will end up just having them give you the answer; in which case you are better off talking to a TA (who is probably better at explaining things). In a study group, you want to be part of finding the answer. Along the same note, if part of your group just gets the answer (and you are in the part that does not), start by talking with the other people who do not know the answer. That way you can figure it out together, instead of just being told what the answer is.
I don't know, man. My grades always seemed to reflect my understanding pretty damn well. If it wasn't reflecting my understanding, I knew it wasn't.
To be fair to the writer, it'd be a much longer and more difficult article if it was called "Tips for understanding concepts in math." I don't even know how you could approach that.
Also, what the heck is wrong with memorizing definitions? You can't prove something using nothing. Eventually, after doing enough proofs employing the definition (or axiom or theorem or whatever else you memorized), you won't have to even think about it anymore - similar to how you or I could find the first derivative of a polynomial without thinking about it or know the definition of an even number.
Re: memorization: Don't tell people not to memorize.
What you meant to say is memorization is necessary but not sufficient. If you really didn't memorize, you couldn't answer "What is a derivative?" It would be as unrecognizable as "What is a Jabberwocky?"
The mind is great at synthesizing material and discovering patterns, but you have to feed it the material first.
Von Neumann said: "In mathematics you don't understand things. You just get used to them."
As a math PhD student who is supported through teaching undergraduates, I have to spend a lot of time convincing my students that they have misunderstood what math is and they do in fact have to memorize the definitions of the words. It used to be in high school that many of them could ride right through a course relying only on their native intelligence to derive or deduce everything, but this tends to bite them when they refuse to believe that each word has a specific definition in the subject --- and many times they are defined the way they are due to a few hundred years of effort by many smart people trying to figure out the best way of expressing some mathematical phenomena. For instance, "continuous" doesn't mean "can be drawn with a pencil" --- while that is a fine intuition, if you need to prove something is continuous from first principles you had better remember that it means the function equals its limit at each point. Another is "linearly independent," though the usual error students make is wishful thinking that you only need to check that no vector is a scale multiple of any of the others.
For my own studies, there are many many definitions and theorems I just have to memorize. It is true that I have familiarity with how the theory is all proved, but in the end I have to remember what all the main theorems say. For instance, there is no way you can figure out what "Alexander duality" is or the axioms of a "group" are by their names alone.
Re computational errors: a lot of times they tend to be hidden conceptual errors, since if you had understood the concepts better you would have detected the error!
Re hard problems list: I disagree that keeping such a list amounts to memorizing certain kinds of solutions. I try to keep one mentally, which I use to try to figure out what about the problems seem hard. The hope is that the hardness of the problems dissolves, leaving me with fewer problems and more insight. Even better is when the hard problems suggest other hard problems.
Re "it is clear that": I recommend you find a grading job on campus. The usual reasons this "works" is that the grader is overworked and forgets it is only clear to them, or the student didn't say any nonsense elsewhere so they've earned the benefit of the doubt. But don't delude yourself outside the exam. (The worst are students who make their work intentionally messy to try to throw off the grader. Believe me, the mess is transparent.)
I used to have a distaste for memorization, believing in an ideal of "real math" which was some fountain of pure understanding we might come to know, but over the years I realized that to even begin approximating that ideal you need fluency in the various languages of mathematics --- and to learn a language you need to memorize its vocabulary at the least!
As a math PhD student I concur. I got through most of my undergrad memorizing nothing. It was easy back then. The definitions were simple and intuitive. what's a group? I hold an intuitive picture in my head which can be translated into words if I want. What's the Taylor series of cosine? If you want to do serious work in applied math you'd better have that on the back of your hand. You can derive it every time of course. You'll just take WAY longer. What is are the homology groups of a group with integer coefficients? Ha. Good luck doing anything with it if you haven't memorized it. Not memorizing might work in undergrad. I've reluctantly come to the conclusion that it doesn't work anymore. In fact not memorizing anything is biting me in the ass. It seriously hampers my workflow to have to look up the definition of Calc iii stuff. If you don't have some things mmeorized you won't even understand lectures. They don't take the time to let you get used to a definition in grad school. They give it to you and just RUN with it. They expect that out of you. They don't remind you. They expect that they can just give you the definition and start using it to prove theorems right away. Most of the time they don't give you the intuition for it either. No examples. You need to get used to the language of math and do it fast.
>What is are the homology groups of a group with integer coefficients? Ha. Good luck doing anything with it if you haven't memorized it. //
I feel you're missing the point the parent was making. The point as I see it is not to try not to know the answer without deriving it but to not set out to simply memorise the answer.
Of course through use you'll commit certain proofs, formulae, corollaries or what have you to memory; but that's not what people generally mean by "memorising". When studying [undergrad honours, UK] I didn't set out to memorise things and often relied on working out formulae/results/theorems from first principles or from some other theorem. [Perhaps this is a physicist thing, I did joint honours in Phys/Maths]
Remembering and memorising are different approaches that lead to a similar result; however IME remembering when you didn't set out to is often much better. For me I could cram and memorise some results, but then that subject matter never stuck around long afterwards.
I second this. I did my first math exam (Linear algebra 1 and 2) while trying to minimise memorisation and it didn't work out so great (passed, but barely). You have to memorise, to really understand it you also should be able to do a proof, but you need to memorise it. It's too much and too complicated to do it on your own, especially in the exam. Normally you don't have a ton of time and event the most simple theorems take time to work out on your own (and sometimes you just get randomly stuck, it just happens). Don't underestimate the work that went into calc 1 or 2, just because you can follow the derivation and proof does not mean one can define the theorem in the equal amount of time.
The most useful thing that I did in any of my undergraduate courses was to make a point to go to all of the office hours offered. No matter whether the office hours were with the professor or a TA, I made a point of going, asking any questions that I had, and showing them the methodology that I was applying to a project/assignment/etc...
While I can't guarantee that this resulted in better grades, I know for a fact that the professors who I built relationships with during my undergrad have been incredibly helpful during my career!!
I'm curious how many of these are also on the list of 'tips for success as an aspiring mathematician'. I just finished reading Benoit Mandlebrot's memoir and he repeatedly talks about how poor academic standing is in predicting a person's ability as a researcher. As someone who is not a mathematician I'm genuinely curious.
Find a section of the math course that is being taught by a professor that is actually fluent. Math is rather infamous, particularly in introductory undergraduate courses, for having professors, brilliant though they may be, who speak broken, incomprehensible English.
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.