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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 m

by kmill 10y ago

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!

saberlynx 10y ago
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.
- pbhjpbhj 10y ago
  >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.
LeanderK 10y ago
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.

by kmill 10y ago

saberlynx 10y ago

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.

pbhjpbhj 10y ago
>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.

LeanderK 10y ago

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.