I'm not too sure if this is the right place for my question, but I couldn't think of anywhere more suited.

I am a first year university Math student, I am currently revising for six end of year examinations. My problem is that I am struggling to make the step up in time for exams. I would like to know how you suggest it is best to revise efficiently for Math exams. I have spent a lot of time making notes, but when it comes to tackling papers I don't seem to be able to get very far without them.

I appreciate your time to answer my question. Thanks.

  • $\begingroup$ A little more detail: Conceptually, I find I do not struggle to follow through proofs. I do however struggle to remember them, and the task of writing one from memory alone often leaves me confused and lost. Also, I struggle with applying theorems to questions without any direction as to how to approach it. $\endgroup$ – Mark Apr 17 '14 at 10:53
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    $\begingroup$ @Roland: Your argument is quite similar to one I gave for admitting questions from self-learners and why I also would like to see this question here. $\endgroup$ – Wrzlprmft Apr 17 '14 at 13:06
  • $\begingroup$ @Wrzlprmft: When I wrote my comment, I had our inclusion of self-learners in mind, indeed. Is there a way to polish the question a bit to make it fit better? There's also meta post on this kind of question now: meta.matheducators.stackexchange.com/questions/323/…. I've included my initial comment as an answer there and deleted it here. $\endgroup$ – Roland Apr 17 '14 at 14:56
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    $\begingroup$ I think the question fits fine topic-wise. It is however rather broad and the circumstances could be described in more detail. $\endgroup$ – Wrzlprmft Apr 17 '14 at 15:12
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    $\begingroup$ @user43290: How do your end-of-year exams look like? Are they oral or written? Do they focus on calculating, doing proofs or just recalling memorised knowledge? In the likely case that they all three of these aspects are required, which one is giving you trouble? What exactly do you refer to by papers? From what point are you starting, i.e., do you have the feeling that you have forgotten everything you did in that year or are there any aspects of the subjects which you expect to tackle well? (Keep in mind that education systems always differ more than you expect them to do.) $\endgroup$ – Wrzlprmft Apr 18 '14 at 8:34

My first suggestion is not to prepare for your exams alone. Prepare for your exams with some of your colleagues, but make sure each of you reflects on your own progress toward readiness independently; the role of you and your colleagues is to provide feedback for each other.

My next suggestion is to identify areas of concern and focus your efforts on these areas. Which topics are the most challenging for you, and which topics do you actually have evidence you understand (do not rely on "oh I remember how to do this, try practice problems and verify you can produce the same answers as your source of the problems").

You should also regularly quiz yourself on the topics. There is evidence to suggest that our brains memory emphasizes information which is regularly retrieved, which means that just taking a quiz helps you remember that information, since you essentially force your brain to recall it.

Next, you should ideally attempt non-standard problems in the domain of mathematics you are attempting. A recent randomized-control study suggests that if you follow a sequence of steps someone else provides in order to solve a problem, you are less successful at remember your attempts for later than if you just struggle with problems (presumably collaboration is helpful here) even if you do not get the answer to the problem. A non-standard problem without a known solution that lies within the domain of mathematics you are studying should help simulate this effect.

Find examples of work you did unsuccessfully in the past (or ideally, find a collection of such examples with you and your colleagues) and take the time to reflect on what worked in what you did, and what did not work.

Construct a 1 page summary of everything you learned in that content area, even if you do not get to use it during your exam. This summary should be clear and easy to read (don't use 2 point font!). Doing this will help you focus on the essential elements you need to understand for this unit, and creating it will help you recall those elements later.

Make sure in the days leading up to your exam that you relax. If you have taken the other suggestion from this post and spread our your preparation over the weeks in advance of your assessment, you will not need the last day to cram. You should take a break, get some rest, and be well hydrated (Hopefully this is obvious, but don't drink or do drugs during this day, both will negatively affect your performance) before your exam.

Good luck!

  • $\begingroup$ Yes, good advise. My math teacher at school told us to prepare a cheat sheet for exams. Not to use it, but to organize the subject matter, and distill what is essential. $\endgroup$ – vonbrand Apr 22 '14 at 12:19

The only advise I can give is to work on the subjects constantly. Don't have any "zero weeks," weeks where you didn't do something on some subject, getting (back) on track gets hard very rapidly. You have my permission to take a day off to do something totally different once a week (it does wonders for not ending up all stressed out).

Collect earlier homework and exams (the 'web teems with them), give them a shot. Look for lecture notes, they usually contain interesting examples and suggested problems. You learn by doing, not by reading/looking. Check for interesting questions around MSE, ask if you get stuck.

For other than this (very general) advice (might as well tell this your friend who's mayor is English...), you'll have to give a lot more detail.


My strategy for exam preparation:

Write increasingly short outlines of the course, then of the content/proof and check that you can get from each version to the longer version.

Let us say that you have a proof of the existence of an Eulerian circuit in a graph with appropriate conditions, then the roughest sketch of the proof might read at most something like this:

  • Start anywhere and go on as long as possible.
  • Never stuck until you return (even degrees).
  • Repeat with unsatisfied vertices and glue them to the first circuit (connected!).

Now, I check that "Proof of the existence of an Eulerian circuit" triggers this list of keywords. Then I check that I can convert each step into formal proof.

If the proof is more complicated, you can have one more intermediate stage of unfolding the proof.

Incidentally, this strategy corresponds to the way memory works. Good chess players are much better than normal people when asked to remember a chess position, but they are almost equally bad at memorising a random placement of chess figures on a chess board.

Why? Because they remember something like "In the left half, this is basically the game of Karpov vs. Kasparov in 1985 except for one pawn, in the right half, it is the classical position after ...", so they reduce the position to something like five items where each item again stands for several items.

It will be very, very hard to remember a proof if you cannot represent it in your head with 3 to 7 informal steps. I would go so far to guess that the problem is that you think that "following a proof" (in the sense of seeing that each step is correct) is enough. However, if you cannot partition the proof into steps, you have not understood it in any meaningful sense.

One could argue that the professor should do this structuring and I do usually provide it. However, you will remember it even better if you manage to find the structure yourself.


If I would give only one advice to focus on, it would be to find someone to discuss the problems with. Preferably someone on your own level of understanding like a classmate. The reason is simple:

Your brain has to form a mental model on how everything works. When you listen to a lecturer, or read a text, your brain can basically put itself on standby if it wants to. You don't ever have to prove that you actually understand the stuff.

But when discussing, you have to start with your mental model and then try to express it in words, making it impossible to not know your stuff. And the amazing thing is that you can start with the small bits and pieces you do know and merge them with what the other person knows to create a new mental model quite easily.

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    $\begingroup$ Yep. The best way to learn something is to teach it. On a tangent, in a computer lab they had a teddy bear, when somebody got stuck debugging they had to explain the problem to the teddy bear before asking one of the staff assistants. Most realized their mistakes when explaining to the bear. $\endgroup$ – vonbrand Apr 24 '14 at 13:00

The most helpful aid to my undergraduate exam preparation was writing 3 x 5 notecards. I usually wrote the name of a theorem/term/concept on the front and on the back gave the definition/description and/or an example. I think notecards were helpful to me for the following reasons:

  • The act of writing the cards helped me to remember my notes and look over them carefully without wasting time.
  • Notecards help condense concepts into a short space so that I was not overwhelmed by too many pages of notes.
  • Using the notecards before the test helped me see which areas I needed to spend the most time reviewing and which I already understood.
  • Notecards are also a useful reference once a class is over in case you want to go back for a definition, key concept, or another test.
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    $\begingroup$ I have made 3x5 notecards for all of my topics, unfortunately I have over 200 of them :( $\endgroup$ – Mark Apr 23 '14 at 11:21

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