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.
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.