Ask: How do I memorize mathematical proofs for my calculus exam?

[DrorY]

Hi,

I studying computer science, taking a few courses of basic mathematics (Linear Algebra and Calculus II).

Our exams require us to mostly cite well known proofs, like the Bolzano–Weierstrass theorem etc. Altogether there are close to 50 such proofs to memorize.

Regardless of my understanding of the material, most of the students memorize these proofs, as there are very difficult, non trivial and are marked in a very harsh manner.

What do would be the best way to learn these theorems off by heart?

[mmq]

I don't think that memorizing is the perfect approach to learn mathematical proofs. Generally, you should be able to try the proof based on your understanding of the concepts, and hence if you can't proof them, it only means that you didn't get the concept well. When I used to be in the CPGE [1], we had one exam after two years intensive courses of maths, physics philosophy and engineering sciences. And we had pretty much everything demonstrated. The only way we, other students and I, dealt with it is by understanding the concept very well, rereading the courses and going through problems. It's been 6 years now that I left CPGE, And I am still able to proof the Bolzano-weirstrass theorem, and I am sure that most of the other students could also.

[1]I don't know if you are familiar with the french educational system, and specifically the CPGE( classes préparatoires aux grandes écoles). It consists of two very intensive years of mathematics, physics, philosophy and engineering science.

[tubbzor]

1. understand the basic components of each proof 2. practice and apply 3. goto 2

I recently got through my undergrad math courses and was also looking for a way to memorize all the formulas, proofs, ect. A few different methods for consideration: holistic learning methods mnemonics loci/memory palaces

Personally, my favorite was to use a mnemonic or image then continually using this when applying to practice problems really helped it stick in my mind.

For a simple example the integration by parts formula: integral[ udv ] = uv - integral[ v du ] I would remember the right hand side as "UV [uv] 'voodoo' [v * du]" and imagine a shaman lining his voodoo dolls up inside a tanning bed as to burn them with the UV rays! Once you have the basic parts like that, writing in where the minus and integral signs go is second nature from practicing and applying.

Hope this gets you started

[wmasson]

It's basically impossible (or inadvisable) to learn proofs without understanding. Break the long proofs into sections like you might with a long function. Try to write summaries of the higher order steps. Try to uncover the intuition behind the proof or theorem.

My personal method for studying maths was to have latex versions of my notes. This is great because it makes lookup very easy (just ctrl f). I have a program that gets all the theorems and quizzes me on them and records how much I can recall. When it asks a question I rewrite the proof in full and only check back when absolutely necessary, then compare to the notes. Your quizzing program can focus on theorems that you did badly on.

It's important to focus your efforts. Pay special attention to the proofs that seem important or that a lot of other theorems depend on.

[DrorY]

what's the name of the program?

[wmasson]

It's just something I wrote myself. Fairly adhoc - it just checks through the latex notes for theorems and keeps them in a list with a current score. The picking is done using softmax selection which picks randomly with a bias towards very low scores. When I complete a theorem I enter how well I did at recalling it and then it updates the score using temporal difference learning. I'm going to clean up the code and put something like it on github sometime.

[infinityetc]

What helped me to learn proofs was applying the logic and axioms they relied on over and over again. When working through problem sets (or doing extra problems for study), don't just stop when you can cite the theorem, but work through why it's applicable to that specific use case.

You can then start working through the general cases, until that's the way that you start viewing the problems in the first place. If you decide to go further in math, this process will really prove out (pun intended), as the basic logic will stick with you, even if you don't remember the name of the theorem you're trying to use.

[randomchars]

http://ankisrs.net/

I'm not sure what kind of cards you should make for this as I didn't use it to learn proofs (only starting uni this fall) but I've used it for everything from equations and syntax to poems.

The key is to break the info into the smallest pieces that still make sense. You'll also inadvertently learn a lot by making the cards.

[roderick3427]

Check out this guest post by Scott Young. He was able to complete an entire MIT computer science curriculum in less than a year. http://calnewport.com/blog/2012/10/26/mastering-linear-algeb...

[DrorY]

Very impressive, however I fear that he doesn't cover specifically the issue of learning by heart theorems.

[geebee]

What a bummer. When I discovered that the best way to get an A in my math theory classes was to memorize proofs, I pretty much guaranteed that I wasn't going to get good at math.

Before I discovered this, I did the homework and learned from it, and I tried to figure out proofs in the moment during exams. One of my proudest moments was on an advanced linear algebra test, where I proved a theorem in a way that was not anticipated by the professor. He wrote a nice note next to my proof. But I missed more than I got, and I received a B+ in the class.

Later, I figured out that most professors rarely present completely new theorems on undergraduate exams, so I copied everything down and especially attended the pre-exam review sessions. I learned everything by heart. If you asked me how to prove anything that the prof had reviewed, I could bounce it to you immediately. I think my abstract algebra teacher had been doing some of it on the spot, because when I regurgitated one of his proofs back to him on an exam, he wrote a nice note next to it as well. It didn't feel nearly as great it had when it was really my own creativity, I remember thinking, "don't you realize you did this in class?"

Now, of course you can't learn it by rote. It's impossible to learn something as complicated as a proof, even a short one, without understanding it. So my advice about learning the proofs by heart is to make sure you completely understand them. I found that I still did get stuck, and I came up with little memory tricks - not mathematical things, the kind of tricks you'd use to remember people's names.

There was a rare breed of student who didn't memorize the proofs, and only sometimes came to class. They'd get the questions right on the exams anyway. The professors loved them.

I doubt I'm smart enough, but I do feel that I might have eliminated my chance of become one of these students by making sure I was an A student who memorized proofs rather than a B student, for a while, who understood the underpinnings and used logic and deduction (gasp! mathematical reasoning).

The best solution, I'd say, is to go ahead and memorize those proofs, since grades do matter an you aren't getting into a top grad school with mainly B's (and if you're doing this for engineering or something, I guess to some extent it's just something to get through). But if you care about math theory and want to be good at it, make sure you spend a lot of time working independently on problems that don't have an answer you can memorize.

[ekm2]

Find a book that has well written proofs and then just write them out on a first reading while looking at them.On a second reading,write them out without looking at a book.It eventually just sticks.I suggest using calculus books by Thomas&Finney or Apostol.

[hdivider]

Memorisation sucks - I remember having to memorise a similar number of proofs, years ago. (And then again a year or so later, in electromagnetism.)

Here are some things that helped me, personally:

Proofs usually have two things that you can focus on if you're having trouble getting them internalised: intro formalism and some key twists.

The intro formalism is just the stuff that will be used in the proof - you've seen this: new definitions, maybe new symbols, and maybe some assumptions and 'wordy stuff'. Focus on understanding and recalling these first. Even if you can find it easier to recall other, more mathematical facts about the proof, it's possible you'll end up with a muddled argument, because the intro stuff usually makes things precise, and eliminates edge cases.

Once you've got the beginning formalism sorted out in your mind, the rest is the hard stuff: the key twists, the actual logic being performed. What I found useful is to avoid memorising every logical step in a derivation as a separate thing. Instead, memorise a few in-between steps and construct exercises for yourself by trying to go from the intro stuff and the in-between steps to the actual final proof.

I used to have a little notebook with my own exercises in them, all constructed from proofs that I had to memorise. I didn't do this for each and every proof, just for the ones that were conceptually troublesome.

The key is to make the memorisation of proofs more like regular mathematics with exercises - with the full proofs being the solution to the exercises.

Another point - and this is somewhat unorthodox: for very hard proofs, the key mathematical machinery being used can be set at a level that's actually too high for your current course. I found this quite frustrating, so (in these cases only) I decided to focus on (almost) rote memorisation first, and then understanding. I know that goes against the usual advice of understanding always trumping mere recall, but exams that force you to memorise lots of proofs are an artificial condition anyway; real-world maths isn't like this. So weigh this for yourself - maybe start with memorising hard stuff, then gradually chip away at it and try to understand it if you can.

All this obviously takes time, and for ~50 proofs this implies a substantial amount of work. I'm guessing you still have a fair amount of time left until your exam, so it's best to do this stuff early and consistently.

I've got a chapter on memorisation in my book about maths-heavy exams [1] - but it's not a free ebook. However, if you want it, send me an email and I'll send you the .mobi if you wish (or we can share via dropbox, rapidshare - we're spoilt for choice :)) - maybe some parts of it could help.

[1]: www.amazon.com/Exam-Mastery-excel-maths-heavy-ebook/dp/B00AMU20HE/ref=sr_1_4?ie=UTF8&qid=1355929591&sr=8-4&keywords=exam+mastery