does doing math make you better at it?

December 25, 2007 7:49 PM

Does doing math improve one's aptitude at math?

Please note that the type of math I'm talking about is the open-ended variety, things like proofs, as opposed to, say, differentiation of single-variable functions.

I ask because I have an incredible amount of difficulty with this type of math, which is unfortunate as it makes up a non-trivial amount of the curriculum I study. When I am asked to prove something, I think back to (the few) previous proofs I have completed for some type of wisdom, but when I do this I feel as though all the last proof taught me was how to prove that particular postulate, not how to do proofs in general.

My sample size of proofs I've completed is small enough for me to be unable to judge whether or not this is true, so I bring the question before you, AskMeFi. Am I wrong? Is this, slowly and unbeknownst to me, improving my reasoning skills? If it is, do you have any recommendations as to exercises that specifically target that learning and speed its course? Thanks in advance!

(On a tangential note, after grueling but successful sessions with the stuff, I do notice a certain pleasurably calm, objective state of mind that doesn't really come for me with any other sort of intense mental work. Can anyone corroborate this?)

posted by invitapriore to education (28 comments total)

Study of math teaches you rigor. Practicing math and doing it a lot maked rigorous thinking easier and more routine.
posted by Steven C. Den Beste at 7:56 PM on December 25, 2007

Practice makes perfect, especially in activities which are learned skills like proofs.
practice,
practice,
practice
posted by caddis at 8:07 PM on December 25, 2007

I definitely think practice makes one better at proofs, if for no other reason than once you've done a few you're a lot better at intuitively guessing which technique to use for which situation, which claims to contradict, etc. And reading lots of proofs doesn't do nearly as much, because the experience of starting on your own and not knowing where to go--the "open-endedness"--is most of the hard part.

I don't know what field you're in, but if it's computer science, CLRS is an algorithms textbook with some great proof exercises.
posted by goingonit at 8:20 PM on December 25, 2007 [1 favorite]

Yes, I think so. I had a miserable time with proofs when I started doing them, mostly because I was confused about how much (or how little) I could assume in each case. After observing and doing many different types of proofs, the "mental rhythm" of the proof process became easier to discern.

The most helpful thing for me was indeed practice. I worked on A LOT of proofs, skipping around if need be to the ones that gave more easily since sitting and staring at a proof that wouldn't give was just frustrating. I looked at An Accompaniment to Higher Mathematics and How to Read and Do Proofs. Neither text was particularly great, but I liked having more examples to look at.
posted by tbastian at 8:24 PM on December 25, 2007

I found the first math theory course I ever took to be an extremely difficult slog through learning both new material and new methodologies. However after a couple of semesters of doing lots of proofs, and different sorts of proofs, I found that I got a feeling for how to approach doing them, even for fairly unfamiliar material. I'd say that the practice (and trial and error) of doing lots of proofs for a while made it a lot easier later on. It's a peculiar phenomenon that what seems to be a very structured kind of problem relies heavily on intuition to find the correct approach. I'd say stick with it for a while longer.
posted by frieze at 8:27 PM on December 25, 2007

When your brain does something over and over, it figures out what kind of thought patterns benefit you. It becomes innate. Also, your subconscious starts to work on it. If you find that your dreaming about proofs, then that might make you feel like you are losing your life to the course. However, it is great news, as you've begun to train your brain to make proofs easier.
posted by mccarty.tim at 8:51 PM on December 25, 2007

Math isn't always something that comes easily to people. But Practice makes perfect. It's kinda like learning another language. Heck, math is another language. It just takes practice to become fluent.
posted by speek at 8:55 PM on December 25, 2007

My non-expert opinion is that doing anything with your brain makes you better at it, and the trick to getting good at anything is to practice it. The wikipedia article for "Expert" sez: "Many accounts of the development of expertise emphasise that it comes about though long periods of deliberate practice." [Expertise section, 3rd paragraph]
posted by Burns Ave. at 9:02 PM on December 25, 2007

Purely anecdotal evidence, but I think almost definitely doing math makes you better at it. This is pretty much taken as a given in India and some other Asian countries. I learnt a tremendous amount of math from the 10th to 12th grades in India, including several different ways of proving theorems and statements. Understanding which technique to apply in which situation was definitely a matter of lots and lots of practice. It was a gruelling time, but the effort expended then is a huge reason why math isn't a huge stumbling block for me in grad school.
posted by peacheater at 9:27 PM on December 25, 2007

Regarding recommendations to improve your reasoning skills: Firstly, do lots and lots of problems. Try not to refer to your textbook a lot while doing these problems. Part of getting these methods to seem automatic is to have to recall them each time you do a problem.
Next, think about where the problem you're doing fits into a framework of similar problems. Think about the conditions given in the problem and what part of your solution you would have to change given a different set of conditions and whether you would be able to prove this statement at all given this new set of conditions. You're probably already doing this but consciously write down which fact(s) each step of your proof follows from.
Then, think about what type of proof this is: for example, is this a reductio ad absurdum or a proof by inductive reasoning. There are general principles you can get from each of your proofs that can be applied to a number of different problems. This can become a little game for you. It still gives me a little thrill every time I recognize the method being applied to a proof.
posted by peacheater at 9:39 PM on December 25, 2007

Oh and this is getting to be rather long, but I would also like to recommend a couple of books by Raymond Smullyan to help with general logical reasoning -- namely "What is the Name of this Book?" and "Alice in Puzzle-land" both of which present many different logical puzzles of the knights and knaves type. I hugely enjoyed both these books and they helped my reasoning like nothing else had.
posted by peacheater at 9:45 PM on December 25, 2007

Different minds think in different ways. As far as maths is concerned (and I studied a good deal of that) I found that my memory was't very reliable, so that to learn stuff I actually had to understand it: I was more inclined in a kind of "big picture", theoretical reasoning (which of course included a lot of proofs), while I was absolutely miserable in studying with teachers who would just dump on the blackboard a pile of algorhytms to be committed to memory in an automatic fashion.
Of course (and, if you're likeminded), committing to memory how to perform proofs - which is just another algorhytm) doesn't help (better yet, it helps just one proof at a time), you have to practice a lot of them to get a hold of how you can actually proof and demonstrate this or that theorem on the basis of what you have learned up to that point. You can go a long way from there. (And yes, after you've done that if gives a sense of a relaxed, objective state of mind, like having climbed and descended a mountain and being finally able to observe and understand it in its entirety).
posted by _dario at 9:46 PM on December 25, 2007

The way it was for me was that the first proof based course I took was almost impossible, the second just very difficult, and the third and fourth ones were easier and easier. But not easy. Proofs are killer because they can be three lines, and getting those three lines can take hours. I had a professor who quipped to a kid asking him "I had pages and you have three lines for that answer, don't I get credit for trying?" He replied, "It's getting the right three lines that counts." (in an endearing british accent) Keep doing them, and find some friends to help you learn the way of thinking.
posted by apathy0o0 at 10:18 PM on December 25, 2007

The way that people become good at something is by doing it. A lot. For all values of people, and all values of something. Even Mozart endured ten years of hard work before writing his best symphonies.
posted by panic at 11:06 PM on December 25, 2007

I think that the reason that doing lots of proofs will make you better at that process is because there are a bunch of abstract but standard patterns, like induction as peacheater observes. Textbook exercise proofs are trying to get you to recognize and apply these patterns, rather than trying to make you another Euclid or Fermat. So lots of practice may not necessarily make you innovative but you'll get better at the exercises and test questions you'll face in academics. In the long run the patterns do begin to stretch cross-topic and cross-subject.

For me it eventually got to be like putting a puzzle together: you look at the facts and conclusion given to you in the problem and mentally eyeball each proof pattern in your repertoire against them, rotating and rearranging things until you find one that fits. And the more difficult problems often require a hybrid of several patterns that were previously demonstrated individually, so you have to learn to do it piecemeal on a subset of facts too.

So my bit of advice would be to go to the library, grab a variety of old textbooks on the subject, flip them open to the same topic, and try to find example problems and exercises that are the most isomorphic to be better able to see the pattern apart from the individual problem. (Or if you've got a good textbook / good professor and notes, more examples might not be necessary, it may simply be a matter of learning to see the forest amongst the trees.)
posted by XMLicious at 11:12 PM on December 25, 2007

after grueling but successful sessions with the stuff, I do notice a certain pleasurably calm, objective state of mind that doesn't really come for me with any other sort of intense mental work

I get this kind of thing happening when debugging code, especially other people's code. Works better between about midnight and 2am. It's nice.
posted by flabdablet at 2:57 AM on December 26, 2007

I'm an undergrad math major, and I just finished my first real, intense proofs course. I agree with previous comments. I think almost everyone struggles mightily with proofs at first. The best way to improve is to practice and ask lots of questions, as others have said.

One very important thing, I think, is to work on proofs that you cannot readily find the solution for, should you get stumped. I know if I had the solutions stashed away someplace, after enough frustration, I might be tempted to consult them to see where my problem lay. But struggling vainly is a very important part of the process, I think.

For some conjectures, the first time I have a crack at the problem I immediately find an elegant, subtle proof. Other times, I might struggle for hours (and pages) getting nowhere. I think failure is actually a very important part of the 'learning-to-prove-things' process; if you don't allow yourself to fail by looking up the answer, you may spare yourself some frustration but then you miss out on letting the problem bounce around in your thoughts and dreams for days, and the sudden moment of clarity as you're doing the dishes, or whatnot.

I'm not trying to imply that this is your practice -- this is just something I've observed in myself! Anyway, I think math is often not so much about 'the answer', but the process of finding the answer.
posted by ZeroDivides at 3:48 AM on December 26, 2007

People often have trouble with proofs because they approach them backwards. Mathematics is really about calculation, and a proof — even a proof by contradiction — is usually an abstraction of a calculation. If I don't immediately see how to prove a theorem, I usually find it helpful to play with a few examples to strengthen my intuition. Devise concrete examples of objects which satisfy the theorem hypotheses, and try to establish the theorem in these cases by concrete calculation. Then abstract away the details to make your proof. Sometimes you need to iterate this process a bit, if the examples you initially come up don't capture the full generality of the theorem.

And yes, doing Math does make you better at it. I was quite bad at Math in early high school, and now I have a PhD in it.
posted by Coventry at 5:26 AM on December 26, 2007 [2 favorites]

You need a combination of doing proofs and reading those done by experts. Depending on your area of interest, those experts will be very different. Erdos-style Discrete Math proofs (see Conjecture and Prove) differ from Rudin-style Analysis proofs (see Real and Complex Analysis). My favorite proofs to teach in a proofs course are Cantor's proof of the innumerability of the reals ("Diagonalization") and the irrationality of the square root of 2, both of which can be found easily. They're both proofs by contradiction, but employ very different methods. Seeing how experts do it will help you develop some intuition. I'd get ahold of an intro to proofs text, try the examples on your own, and then go back to the solutions if you get stuck. Good luck. Contrary to popular opinion it's really fun, and empowering, to be able to prove well.
posted by monkeymadness at 5:36 AM on December 26, 2007

Yes, practice will help. A formalized course of study (class, or using a text on your own) will definitely help. I think the simplest thing that helped me the most was being able to name different types of proofs. It's easy to forget an angle of attack when they're all just vague ideas floating around in your head, going down a list helps you consider all of your options in a systematic manner.
posted by anaelith at 5:57 AM on December 26, 2007

Seconding Coventry: keep practicing, and do examples. They really do help you understand what's going on. As the professor I TA'd for last quarter likes to say, when you're proving something about matrices, for example, do the 2x2 case. Then do the 3x3 case, and then generalize. Work easy examples first.
posted by matematichica at 7:42 AM on December 26, 2007

A couple suggestions:

• George Polya's How To Solve It is good for learning some strategies to conceptualize problems.
• Work through an intro to proofs or classical algebra book if you're not already doing this in a course; Introduction To Mathematical Thinking by Gilbert/Vanstone is what my university usually uses for its first discrete course for math/CS majors and it's a great book. Oddly enough, Fundamentals of Mathematics for Linguistics by Partee is a good book for exploring proofs in groups/set theory/logic/automata theory if you haven't already covered these topics in depth.
• Once you've taken a first course in combinatorics or analysis, you can start taking a look at the Putnam contest problem archive (it's hosted somewhere on UNM's servers, I think). The problems are by no means easy, but they're good for developing intuition.

posted by thisjax at 12:44 PM on December 26, 2007

I believe actual reports from the front as they bear upon your question paint a little more of a mixed picture than the rosy hues of the generally excellent answers you have received so far might lead a person to expect. There is the case of Bertrand Russell, for example:

In 1890 Russell went up to Cambridge, choosing that institution over Oxford because of his desire to learn mathematics. He devoted his first three years there to it. These years of intense preparation for the Mathematical Tripos left him disgusted with the subject. After the examination he sold all of his books on mathematics and vowed, somewhat prematurely, that he would never read a mathematical book again.

Perhaps his vow merely did not extend to writing them, but, in any case, I recall reading something of his from later in his life complaining that he was never the same after the immense grueling labor of working out the details of the proofs for Principia Mathematica, and that it had, in essence, left him a bit of a burned out shell as far as math was concerned.

Nor are accounts like Russell's uncommon; one of my favorite professors told me it took two weeks of prep time for him to be able to understand a theorem he had himself proved, and which was the primary basis of his reputation in mathematics.

If you combine things like these with the notorious belief that many great mathematicians have done their best work very early and have been relative duds thereafter, and the fact that doing mathematics seems to be genuinely painful and miserable for many otherwise dedicated students, I think you must open the door at least a crack to the possibility that there is something really strange happening in the brains of some people when they do mathematics-- namely, that mathematics could conceivably have significant excitotoxic effects in the brains of some people, and that for those people, doing mathematics tends to make them worse at it instead of better, because it damages the parts of their brains that do mathematics as they are doing it.
posted by jamjam at 1:39 PM on December 26, 2007

jamjam: The strange thing that's going on in Mathematician's brains is that they are devoting immense energy to something which yields almost no conventional reward. As a result, the motives which drive Mathematical research are often fragile things. Russell spent years on a project which went nowhere. That is bound to sap one's motivation in the best of circumstances. Doing anything will make you worse at it, if you burn yourself out.

Anyway, I imagine the OP would be quite happy to reach Russell's post-burnout level of competence. He seems to be talking about taking the first steps in learning to do proofs rather than preparing to carve a Mathematical edifice for the ages.
posted by Coventry at 4:44 PM on December 26, 2007

I took 12 years of math and spent 12 years being beyond terrible at math. Hope this is not the case for you.
posted by thebrokenmuse at 8:16 PM on December 26, 2007

I hope you're joking, jamjam.
posted by monkeymadness at 5:23 PM on January 12, 2008

I hope it turns out to be ridiculous too, monkeymadness, but I really do suspect not.
posted by jamjam at 8:16 PM on January 12, 2008

jamjam, it's self-selection. Math doesn't make you crazy. Only a certain type of crazy does math. When you get into set theory, however, I'll give you that people who are only mildly bananas turn into complete nutters. It's the nature of that particular field. However, nobody is going to get into set theory unless they have a proclivity towards that sort of thing, anyway.
posted by monkeymadness at 7:36 AM on January 13, 2008