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# Revision tips for remembering definitions and proofs watch

1. I'm doing an undergraduate maths degree and I'm currently revising for upcoming second year exams. I typed this all out and realised it's mostly a rant so if you want to skip all that, the most relevant bit is in bold at the bottom.

I remember flicking through past papers during term time and thinking "ooh that's nice, lots of definitions and proofs and statement questions" (basically, memory work) and I was sort of comforted that it isn't all application because definitions and set proofs just meant guaranteed marks like if you know all of them, you know you got the question right and there's no second-guessing yourself about making an error somewhere but now that I am actually revising, I hate it, hate it, hate it so much. I don't even know why. I've never minded memory work before (I've always been kinda good at it) and before, writing down a statement a couple of times or staring at it long enough meant, it would just register and done. My memory with regards to rote-learning things here and there for exams was good but now it just feels like a chore and I'd so much rather just apply a theorem/formula/technique and be tested that way. It's so much more fluid and enough practice usually means being good at it.

Whereas writing out definitions, proofs, statements and whatnot is boring and really really demotivating having to do it over and over again and still not having it sink in enough for you to remember for the time when the exam comes. And the most silly thing is, sometimes it's not even that I don't have a clue as to what the thing is. It's just I don't know how to word it properly in technical terms. Like a simple example of this would be that.. I know what a minor of a matrix is. Give me a matrix and I'll work it out for you. But why do you want me to define it?!?!?! Isn't the fact that I can use it in context enough? My brain simplifies things for me so I literally remember it as "you cross out THAT row and THAT column, and work out the determinant and then that's the minor for THAT number". Obviously, I know what "THAT" means in my head but conveying THAT is hard in words and harder in mathematical form (for me, anyway). Which is why I'm really OCD about remembering the exact TEXTBOOK definition because left to my own devices, I WILL sound retarded to the examiner. So it just makes my life that much harder because I feel like its "wrong" if I don't write it in exactly the same way.

Looking at a question and being able to work it out in the exam is just so much better. I'm working hard to know how and why to do things and generally feel good(ish) about the modules that are more practical or the practical side to the modules which have what feels like a gazillion different memory based questions. With other questions, it's like I should know it before the exam but even if I draw a blank, atleast I can sit there and have a decent crack at it if everything goes horribly wrong but memory based questions... if you know it: GREAT, if you don't or even if you did know it but then go blank, you're sort of screwed and there's no "working it out" then and there either. I just have this horrible feeling when it comes to the memory work, I am going to sit there in my exam, looks upwards (half-seeking for God's guidance and half because it's what I do when I'm trying to remember something as if literally trying to search my brain for the answer ) and just wasting time like that when really there isn't time to waste because of the amount there is to do in two hours.

I'm honestly surprised about the amount of memory work required. Maybe it isn't all that much but because I'm dreading it, I've over-exaggerated it in my own head? Highly possible. I think I've gone a bit mad as it is anyway with all this revision.

I'm just annoyed at all this memory related business involved in exams but I genuinely need advice on how to remember the stuff that it is JUST about remembering. Sometimes I go over these things and remember them at the time but then proceed to forget afterwards. For example, I learnt quite a few definitions and proofs like two weeks ago for this particular module and I was like "yerr ". I understood most(ish) parts to the proofs, admittedly not all, but I did KNOW it but now I've just forgotten.. completely, or atleast significant amounts. This happens with nearly all the memory related things.

I don't want to leave it till last minute and then cram all that stuff in the night before the exam. I'd just feel too under prepared that way and at the same time, nor do I want to keep going over stuff only to forget it. It's just a waste of my time when I could be learning actual theory that relates to the application questions.

All my usual techniques are feeling a bit pants and like they're doing no good. I usually just write stuff and it registers but I've done that soooo many times and now I just feel like I'm murdering trees. Other times, I just try to picture the definition as it was written on a page but some of the things I have to learn are pretty bulky. Breaking it down into bullet points only makes it seem bulkier and trying to remember stuff photographically only works when the amount is reasonable in size for me. Advice would really be appreciated.
2. thank god someone else is suffering! there is so much to remember, I simply cannot get my head around it. in first year, so i have that nice feeling of dread that next year will be so much harder too!
I've been sticking up proofs/definitions etc on doors around the house, so that hopefully they will slowly become ingrained into my brain. And just as many past papers as possible so I can get a feel for the papers...
It's so annoying, I can remember them for a day or two but then it's back to square one...
3. (Original post by blestrange_)
thank god someone else is suffering! there is so much to remember, I simply cannot get my head around it. in first year, so i have that nice feeling of dread that next year will be so much harder too!
I've been sticking up proofs/definitions etc on doors around the house, so that hopefully they will slowly become ingrained into my brain. And just as many past papers as possible so I can get a feel for the papers...
It's so annoying, I can remember them for a day or two but then it's back to square one...
The last sentence you've written really hits home with me and its SOOOO depressing. From that, it's just plain obvious that we understand/know what is being asked but the exam is only testing our retaining ability which I would really not even mind one percent if it just stuck in my brain. It's just stupid going over the same thing again and again only to forget. It's a bit more justifiable with questions with big working out and blah but for a definition?!?! Not to mention how boring it is. It makes me want to cry! I keep doing exam papers and doing some of the questions but whenever it comes to a memory type question, I can't remember stuff of the top of my head so I refer to my notes and end up just copying it out thinking it might register THIS time round but it doesn't.

Sticking stuff around the house, is it working? Are you gonna leave it up to that method or try anything else?
4. eerrrm. well. it's working a little. i'm going to try to write 3 proofs/definitions from memory each day, random ones. rather than going over the theory each time i try and learn them, because i can remember that!
5. (Original post by blestrange_)
eerrrm. well. it's working a little. i'm going to try to write 3 proofs/definitions from memory each day, random ones. rather than going over the theory each time i try and learn them, because i can remember that!

Mnemosyne, its a flash card based programme that (after you created the cards) you rank how well you remembered it and then it will give you the cards you forgot/ remember less more often, with regards to sticking math formulas I would just use the selective print screen facility on the mac (or download equivalent for pc) then insert a scr shot of a proof from your lecture notes into the programme

6. For proofs, if you cant do it from theory, write them out loads of times until you can do it without looking at the book. Eventually it will stick. You must have persistance.
7. (Original post by blestrange_)
eerrrm. well. it's working a little. i'm going to try to write 3 proofs/definitions from memory each day, random ones. rather than going over the theory each time i try and learn them, because i can remember that!
I've finally found a system that seems to work really well for me. Say you have;

Definition 1
Definition 2
Definition 3
Definition 4
.................
Definition n

Read the first definition, write it down from memory, check it's correct/make any corrections. Do the same with definition two and then go back and write definition one from memory again. Then write definition three, check it and correct it. Then write definition 1, 2 again and continue like that for however many definitions you set yourself to learn in that revision session. and hopefully there won't be errors when you're doing it the second time/third time around for each definition.

Once you've sort of done that iteration a couple of times and feel fairly confident that you remember them start to pick definitions that you are weak on, followed by one that you are strong on and oscillate like that. And then repeat the iteration process followed by the weak/strong process.

It sounds time-consuming but in reality doing that will take you about an hour (say if you aim to learn ~ten definitions) and for some reason, its really working with me. I've managed to learn 4/5 long(ish) proofs yesterday and I seem to have retained them because I was able to write them out today and today I've learnt about 12 definitions like that. Obviously, I will go over them tomorrow (probably in a more brief manner than what I've initially written) but the definitions still feel fresh in my head even though I learnt them in the morning whereas before I'd pretty much forget half an hour later.
8. I suggest this technique:
Write down the theorem.
Try to prove it yourself. If you fail, read the first couple of lines of the proof, then try to prove it yourself again, repeat until you have a proof.
Get a blank piece of paper and repeat until you can prove it without reading any lines.

With a definition, find a theorem that relates to the definition and then try to prove it with weakened conditions in the definition. Doing that reminds you why all the conditions in the definition are necessary and can help you remember it.
9. I do my revision weekly (one week per module, give or take a day). While I'm revising a certain module, I'll fill the house with pages of definitions or important theorems (colourful + diagrams if needed) - put one/two on the side of my bed, outside the bathroom, on top of the TV, on top of my table and on any doors I use frequently.

It works since most of the time the definition/proof is literally the first thing I see in the morning and the last thing I see at night. Works wonders (although it's easier if you know what the proof is doing and the main steps)
10. (Original post by IrrationalNumber)
I suggest this technique:
Write down the theorem.
Try to prove it yourself. If you fail, read the first couple of lines of the proof, then try to prove it yourself again, repeat until you have a proof.
Get a blank piece of paper and repeat until you can prove it without reading any lines.

With a definition, find a theorem that relates to the definition and then try to prove it with weakened conditions in the definition. Doing that reminds you why all the conditions in the definition are necessary and can help you remember it.

(Original post by Preeka)
I understood most(ish) parts to the proofs, admittedly not all [/B]
I agree with the above technique. You admit yourself that there are parts of proofs you don't understand. This is what you should be concentrating on because then it's not a matter of memory, it's just understanding. If you understand the material well, then stating theorems will become easy because you'll recognise how the preceeding propositions and definitions all lead on. And then the proof of a theorem will be easy because you understand just what the assumptions in the statement of the theorem mean. And the best way I can suggest to help you for this is exactly what IrrationalNumber suggests. It's what I've been doing for the past few weeks in preparation for my exams.
11. For proofs and memorising I (as gay as it sounds) explain it out beautifully with lots of colour and look at it everyday trying to visually memorise it. I will then sporadically get a blank piece of paper and try and explain proofs out without looking and then compare them to my beautiful copy. I find this helps me perfect and remember the proofs until a point where it's mentally imbedded for ever, the more you do it, the more practice you get and the better you understand it.

And for really long ones or ones that I simply have no hope of understaning andi'm struggling with: i'll literally be staring at it minutes before the exams attempting to mentally take a photo of it, saving it to my short term memory and as soon as i'm in the exam i'll jot it down asap on scrap paper. Often the adreniline fueled exam forces it all to fall into place or I I just end up ising the model drawn out to write anything that can scrape me a mark on that question.

Sorry if that is of no help, I do economics.
12. I think that's the biggest rant I've seen in the maths forum and I've been using it since 2006...

Anyway, the trick is to do as many questions as possible. What did you do at school? Chapter 3 exercise 1 (10 questions), then exercise 2 (10 questions), and so on until the mixed exercise (20 questions) right? Then chapter 4, chapter 5, etc...

Do that for your university subjects.
13. Making sure you understand what you're doing; why each step is there, and what it does, and how it fits into the proof usually helps.

Knowing how you remember things is also useful; we don't all have the same methodology; NLP or memory techniques may help.

One proof that always bugged me, and still does, is the Cauchy-Schwartz inequality; basic, I know, but I just have no feel for it; and that's how I tend to remember, a combination of feeling, and imagery (backed up by razor sharp logic).
14. (Original post by ghostwalker)

One proof that always bugged me, and still does, is the Cauchy-Schwartz inequality; basic, I know, but I just have no feel for it; and that's how I tend to remember, a combination of feeling, and imagery (backed up by razor sharp logic).
That one is easy to remember if you remember
a) Consider (x+ty,x+ty)>=0
b) Differentiate wrt to t to find stationary value.
15. I prefer: <x+ty, x+ty> >= 0, so t^2 <y, y> + 2t<x, y> + <x, x> >= 0. Consider as a quadratic in t, we must have "B^2-4AC <=0". That is, 4<x, y>^2 <= 4 <y, y> < x, x>. Done.
16. (Original post by IrrationalNumber)
...
(Original post by DFranklin)
...
Thanks to you both; I might get the feel of it yet!

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