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I sometimes spend inordinate amounts of time memorizing math articles or theorems/proofs or formulas. My question is "am I wasting time?" and will 'active thinking' or 'working out problems' be faster way to master mathematics?

I am absolutely a beginner. So at an apprentice stage sometimes I find that best way to grasp a subject is through verbatim scribing. Also, memorization seem to be my forte.

Mathematics is a language and just like when trying to learn the basics one has to memorize grammar, does the same theory apply in this field?

I used to browse MO, this website, wikipedia but since "mathematics is not a spectator sport" I imagine more fruitful way would to be to isolate small problems and work on it?

I am sorry if the question is very general.

user 85795
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    I would say your time is better served doing lots of problems. As you keep doing them, you'll eventually remember what you need to remember. – J. M. ain't a mathematician Nov 26 '11 at 01:41
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    What J.M. said. The point of Maths if you want to do anything with it beyond passing exams is that you would need to learn how to figure out it by yourself. The best bet would be to do lots of problems and a ton of problems. On memorization I find it's very helpful to write down stuff even if it's useless. Like if you are trying a problem just write down what coming into your mind. – simplicity Nov 26 '11 at 01:50
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    Making notes while reading is still essential, it helps you remember what you learned (and you gotta remember something to be able to do practice problems). Part of your jot notes should be trying to come up with examples, or doing some calculations as you go along, or putting theorems in your own words, modifying a proof etc. (not verbatim) – AnonymousCoward Nov 26 '11 at 01:50
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    Doing math is the best way to get good at math. But, I had a professor that said "Memorize every proof of a main theorem so that I could ask you ten years from now to reproduce it." Some proofs are so instructive, it is good to memorize. – J126 Nov 26 '11 at 01:52
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    @GottfriedLeibniz Making notes is sort of passive. Reminds me of people I see just copying the notes neater, which is good handwriting practice however I don't see how they are learning anything. – simplicity Nov 26 '11 at 01:53
  • Thank you. I understand many famous mathematicians had near photographic memory or eidetic imagery (for instance Poincare as mentioned by E.T.Bell) but that does NOT conversely entail that good memorization skill is a requirement. But my question was because since I am a beginner, if I memorize concepts and carry them around with me, will not insight follow when I am relaxing? –  Nov 26 '11 at 01:55
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    @ZeeshanMahmud E.T. Bell makes a lot of crap up(stuff he wrote on Cantor was mostly garbage). Also that is bad reasoning. You can't just pick one Mathematician and then generalize from it. Erdos was autistic, it doesn't imply you need autism to be a famous Mathematician. – simplicity Nov 26 '11 at 01:57
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    @simplicity: I met Erdős a number of times (no paper). I have known a couple of autistic people. Erdős was not. Singular, yes. But sociable, capable of having perfectly normal conversations. – André Nicolas Nov 26 '11 at 02:29
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    I would say that one of the most important skills for mathematics is to learn what needs to be memorized... – Arturo Magidin Nov 26 '11 at 02:38
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    I want to emphasize what J.M. said. Doing a problem and having that "Ah-HA!!!" moment, when you apply a theorem or use an important concept for the first time, will sear that new knowledge into your head. If you're working on memorizing something, you're probably just wasting time (or cramming for a test ;-) ). It's the difference between asking for someone's name and sitting down, introducing yourself, and getting to know them. – Dimitrije Kostic Nov 26 '11 at 03:00
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    In my opinion, mathematics student should memorize minimum primitive information, because only this information are not based or linked to your previous knowledge, therefore if it remains unlinked, we tend to forget it. In every topic try to see the big picture, try to link issues. Only LINKED information can be stored for a prolong period. For more motivation take a look at a great series of posts by professor Santo D’Agostino How much mathematics should a student memorize, part 1 – com Nov 26 '11 at 06:53
  • I find it next to impossible to answer this question in a useful way, because I have met both people who overestimated memorization and people who underestimated memorization. Some skills not only have to be memorized, they have to be automatized. Among students, there are those who think that it is enough to know that a differentiable function is "smooth" and those who would learn $1+1=2$ by memorizing "a vertical line, a cross, a vertical line and two parallel horizontal lines have to be followed by a squiggly line". – Phira Nov 26 '11 at 12:32
  • All mathematicians now a days just memorize the set theoretic axioms and any definitions from their field. Then, they will just derive the results they need and forget them when they don't. It is the most efficient in terms of brain space. – Christopher King Sep 05 '13 at 22:13
  • @ArturoMagidin: nice comment. Should be turned into an answer I think. – Taladris May 25 '14 at 16:17

6 Answers6

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As you say, mathematics is a language. In any language, memorizing vocabulary very quickly becomes useless, if it is not paired with active use. Certainly, memorizing an entire Spanish dictionary before attempting to formulate one's first Spanish sentence is not the right way to learn Spanish. You should learn a few basic words, and practice using them in all sorts of combinations until you are comfortable with their properties; perhaps you must take a few things on faith from a fluent speaker. Then learn a few more words, plus maybe a grammar rule, and practice them intensively as well, using them in sentences together with the words you learned earlier, looking at correct and incorrect instances of that grammar rule. Each time, you expand more and more, bringing in more intricate aspects of the language, learning more vocabulary as a natural part of expanding your general facility with the language.

The most important part is that you practice using the language, expressing your own thoughts with it. Memorizing a dictionary will never teach you the language, nor will memorizing books or articles that use it. Reading with understanding is much better. Reading with understanding, together with using the language in conversation and writing as often as you can, is ideal. It is the best way to practice the vocabulary and grammar you've learned from a teacher or book and checking whether you really understand it. After all, consciously trying to memorize anything will only go so far; if you want to reach the point where you can use the language fluently, you have to practice it, so that each time you learn something new, it becomes ingrained, second-nature. This unconscious memorization is ultimately much more important.

In mathematics, we do not use our mouths, but our minds. Think of exercises from a book as conversation prompts. You should answer in the language of mathematics, as best you can, using the vocabulary and grammar you've learned so far. Note that knowing lots of vocabulary and grammar do not, in and of themselves, let you participate in even the most rudimentary conversations - you have to have something to say first, and you have to practice expressing it in the language. Those are not things one can practice passively. So, without a doubt, learning mathematics requires an immense amount of active thinking.

You might find this post helpful: How Do You Go About Learning Mathematics?

Here are some relevant quotes:

  • "What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises." - G. C. Lichtenberg

  • "The only way to learn mathematics is to do mathematics." - Paul Halmos

  • "Keep in mind that there are millions of theorems but only thousands of proofs, hundreds of proof blocks, and dozens of ideas. Unfortunately, no one has figured out how to transfer the ideas directly yet, so you have to extract them from complicated arguments by yourself." - Fedja Nazarov

  • "Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?" - Paul Halmos

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Zev Chonoles
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  • Thank you for useful response. However- and I am not trying to argue for argument's sake- a misconception seems to be memorization cannot be active thinking or is a passive and blind activity. But consider memorizing Hasse diagram for Young lattice. One cannot not memorize it if one is not forced to see the pattern. I would be very curious to know two things 1) if pattern finding becomes better if one is forced to commit them to memory; 2) whether Eastern schools promote memorization in their curriculum and if it benefited them. –  Nov 26 '11 at 02:48
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    I do not deny that memorization is an active process, one that takes effort and skill, but it does not exercise your capacity for coming up with your own mathematical thoughts and arguments, and this is by far the most important aspect of learning mathematics. Memorization has a place, insofar as one must actively try to remember a mathematical definition or idea until continuous practice, use, and general engagement with it have ingrained it in you. Pattern finding becomes easier if you practice pattern finding, of which remembering patterns you've already found is only a small part. – Zev Chonoles Nov 26 '11 at 04:38
  • I don't have any actual knowledge of "Eastern schools" - this term already seems too broad to say anything of use - but my impression is that China, at least, does promote memorization to a large extent, with less focus on creativity than "the West" (again, I hesitate to talk in such generalities). There are, of course, a great number of excellent mathematicians in China, but assuming my impression is correct, I would posit that their training with memorization did not help them get there. I would like to hear from someone with experience in Asian schools to get their input on this. – Zev Chonoles Nov 26 '11 at 04:57
  • +1 I really like this answer and I especially like the analogy made between learning mathematics and learning a new language. – Amitesh Datta Dec 18 '11 at 02:13
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    +1 "Think of exercises from a book as conversation prompts." I love this line. – Jesse Madnick Feb 28 '12 at 08:15
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There are roughly two kinds of memorization: with sufficient understanding and without. Without enough understanding, our memory tends to fade more quickly than we would like it to. So I would say: try to memorize with understanding as much as we can.

From chat by t.b. about "memorizing" theorems and their proofs:

Always ask: what are the hypotheses used for? How does this assumption enter the argument? What is the crucial point of the proof, what do I need to remember in order to re-prove that result? What is just standard technique, what is new to me?

Well, one could certainly ask for an outline of the proof, I believe. There are a few main ideas that one can try to isolate. I mean these six pages aren't six pages of pure calculation, they certainly are divided in some natural steps. Try to partition the proof in such a way that it looks natural. This takes a few hours to do, but if for some reason you know that this theorem is considered important for your exam, you probably need to really grasp these ideas.

If one proof doesn't help me a lot, I need a different angle of looking at things. If I find a place where things are presented the way I like it, I can then go back and see what the other author emphasizes and thus get a more complete picture.

When we are driven by the deadlines of homework and exams and don't have time to digest the ideas, we usually try to memorize them in a hurry. Courses are sometimes arranged in ways that they contradict their original purposes. That is sad...

Tim
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I think it depends on what your priorities are and what you're trying to achieve. If it is indeed something resembling mastery of maths then my answer would be a resounding no to memorising in the conventional sense being the way to achieve this. If however your requirements are just recollection in the short term then this kind of memorisation is a tried and tested reliable method and perhaps the safest means.

I'd say there are different levels of understanding spanning from just being able to read the words, to following the logic and then to the highest levels where you reconcile with what the mathematician who originally came up with the proof might have been thinking and begin to consider how this could influence your future thoughts on these subjects. If you're under pressure to memorise proofs for a lot of theorems then a way which I think is effective is to break the proof down algorithmically and memorise the algorithm for the proof. Such an algorithm may include steps like "define a such and such set", "sum elements and apply some property". In this way when the time comes to recall it, you can at least do it in a way that still gives you the feel of "doing maths" to some capacity as opposed to just rewriting word for word. It will also help you to develop mathematical technique and you may be able to get through things quicker as there is less to memorise this way.

metamaths
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While committing essential information to memory makes sense, we are far from understanding the optimal use of memorization for learning math beyond the elementary stage, in part because little experimental research has been done. As an analogy to memorization, we can consider distributed practice of problem solving. Experiments have shown that distributed practice works better than mass practice. But even there we don't have enough experiments with varied kinds of problems in real-life contexts with diverse students; and we have little sense of what the optimal spacing and amounts of practice might be. At any rate, if you are going to memorize, it makes sense to distribute your efforts over time. And, to follow up on the perceptive comment above about figuring out what to memorize, it seems reasonable to determine the key concepts and formulas in a given chapter of a textbook, then rank them from most important to least, and finally to apportion time to memorize them accordingly. Note that a "formula" might mean a particularly useful worked problem. It would be very helpful to have an optimal ratio of effort to deploy in memorization versus problem solving, but we don't have any research results to get beyond notional suggestions.

Kenneth J. Dillon
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It is not a waste of time because the fact is when your in a test and it is limited time the faster answers come the better. 9

bob
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Yes to some extend you need to memories.

Example: You must remember what a prime number and a factor is before you can find prime factors of numbers. to remember that you need to memories and it obviously differs from person to person.

benno
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