1

This question is about efficiency of studying habits in math.

Clearly, rote learning everything without trying to develop an understanding of concepts in a math education is a bad idea, and this doesn't need an explanation.

On the other hand, rejecting rote learning completely is obviously highly inefficient, since there are things that cannot be understood, but merely memorized (e.g. the fact that $\partial$ is the symbol for partial derivatives).

My question is about what the optimal role is for rote memorization in math. What are best practices that are generally accepted regarding rote memorization for optimal learning speed in math?

  • Should you rote memorize definitions before trying to understand more intricate aspects of their meaning? Or should you study examples and theorems so that you will eventually remember the definition "naturally" by understanding the deeper meaning?

  • Should you memorize key theorems as facts without understanding why they're true, and try to get an overview of the theory first before studying the deeper nature why they're true? Or should you ruthlessly try to understand the proof of every important theorem, and not give up until you've understood it so well that you no longer need to rote-memorize it because you can just "see it"?

user56834
  • 12,925
  • On 15 November 2012 I wrote, among other things, "Since when was math class about memorization? Math has always involved the least amount of memorization of any subject I can think of ..." – Dave L. Renfro Apr 29 '18 at 16:28
  • Are you assuming everyone learns maths the same way? or that that most people learn maths the same way and that losing the few that top and tail the distribution is the way to go. How do know in advance that there aren't two major cohorts in the class you are about to teach, and this varies from class to class, and which ever way you proceed you are about to lose a large percentage of them or at least severely limit their interest and maximum attainment in the subject? – James Arathoon Apr 29 '18 at 16:46
  • 1
    @JamesArathoon, I am not a teacher but a selftaught student never enrolled in a math degree. So I am trying to get the type of advice I would have asked a professor if I knew any. – user56834 Apr 29 '18 at 16:59
  • The link to my 15 November 2012 post in my 29 April 2018 comment is now dead, and I have been unable to find it using the WayBack Machine. However, I happen to have a copy of that post, so because I think it is especially relevant to the present question, here is the entire post (math expressions formatted for posting here): “One glance at her safety glasses, and one might think 7th grader (continued) – Dave L. Renfro Dec 07 '23 at 19:09
  • Anne-Wesley Taylor is busy in science class. But this lesson on water quality is actually a new, interactive way to understand a sometimes confusing subject: Algebra.” I haven't listened to the video and don't really have comment about the teaching method (getting students engaged is good, assuming it's in a way that leads to appropriate learning), but I did want to complain about something I see way too often (and have complained about before), which is the tendency of advocates of “the latest new thing” to misrepresent the past. (continued) – Dave L. Renfro Dec 07 '23 at 19:10
  • The first sentence of the article follows: “When many of us were in school, math class was about word problems and memorization.” Since when was math class about memorization? Math has always involved the least amount of memorization of any subject I can think of, with the possible exception of P.E. classes. And, now that I think about it, I took a number of multiple choice tests on volleyball rules and other sports rules in my high school P.E. class, tests whose preparation for involved nothing but memorization. As someone who always had great difficulty with memorization (continued) – Dave L. Renfro Dec 07 '23 at 19:11
  • (I had to transfer to another undergraduate university due to Foreign language requirements, I got a 60 (under 70 was an F) on my 3rd quarter 9th grade English report card because I was making 30s to 50s on the spelling tests our class began taking that quarter, I almost failed a supposedly easy classics elective because I couldn't remember the various painting and sculpture and architecture styles we needed to distinguish on tests, etc.), I'm EXTREMELY AWARE of the amount of memorization in various subjects. (continued) – Dave L. Renfro Dec 07 '23 at 19:11
  • Sure, I often forgot things in math too (e.g. is the derivative of $u/v$ equal to $(u'v - uv')/v^2$ or $(uv' - u'v)/v^2),$ but almost always you can “see the complete picture” by filling in the missing parts by using some alternate method. For example, in the case of the quotient rule, see which of the two possibilities work for the case of $u/v = 1/x$ (whose result you know by using the power rule). – Dave L. Renfro Dec 07 '23 at 19:11

3 Answers3

3

I feel that rote learning could be used as a step in the process of understanding mathematics.

Much of mathematics is attempting to obtain an answer for a given information with given information. For example, if you have a right angled triangle with the shorter sides being 4cm and 3cm long, how do you determine the size of the larger triangle.

For a given question in which one does not know how to obtain said answer, they could start with rote learning the steps involved in obtaining the answer. For example, squaring the size of the other two sides, adding them together, and then obtaining the square root, as per Pythagoras' theorem. By rote learning it, they could then begin applying it to similar questions to become more familiar with it.

If one wanted to truly understand Pythagoras' theorem, they would need to know a proof or multiple proofs behind it. Not everyone would do this; many people would know it well enough to be able to apply it without it (what I call partial memorisation). The proof(s) could in themselves also be rote learned until one understands them fully.

This has practical implications. If for an exam you are unable to understand a topic, you could just rote learn the steps (and perhaps write them on a note sheet if you're allowed to bring one into the exam) and understand when to apply them e.g. for this question with this given information.

Definitions are a similar issue to the application of the knowledge. You could start by rote learning them and, in the process, become more familiar with how to apply them to given questions. I see rote learning definitions, study examples, and theorems as a process towards learning and understanding them.

MBorg
  • 211
2

I’m the last person on earth who would willingly memorize something. I love mathematics because you can do so much without any memorization, but I hit a wall once I did actual mathematics for the first time in Real Analysis and Abstract Algebra. Don’t get me wrong, I didn’t have to memorize formulas or anything like that. The techniques, on the other hand, didn’t really hit me in the right time and even if it did, I’d miss out a crucial conditions of the theorem. This happened because all of this was way too new for me. Never did proof-based mathematics rigorously. And not only that, different books would have different conditions for the same theorem. So when applying, I had to make sure I’m in the right context.

I ended up somewhat memorizing the extremely new techniques that I had never encountered before. However, it was not a complete rote memorization, I visualized the logical flow of the proofs — kind of like visualizing an algorithm. I’m a visual learner and so it’s easier for me to remember pictures. Sometimes, I just sit down and find for myself the motivation behind the results, for e.g., an easy one is Lagrange’s Theorem in Algebra which was outright obvious when you drew some disjoint sets.

However, as time goes on, the familiarity of the process of mathematics really reduces the amount of memorization required. Now it’s much easier to remember a statement of a theorem and the logical flow of the proof once it has been learned. So, my advice would be to memorize smartly. Memorize the process and the flow but not the way it’s written word by word. I never did rote memorization and never will, hopefully. If I really need to remember a difficult formula, then I just practice a bunch and that’s all it takes.

Foon
  • 137
1

The only case where rote memorization can help a little, is if you are preparing for an exam.

If your goal is to learn, you should focus on understanding. Which of course will imply reading, practicing, noticing you didn't understand something, going back to read, practice again, etc. By the time you practiced enough to understand, you will have memorized many useful things.

Martin Argerami
  • 205,756
  • 1
    Yes, but… I think that there may be cases where memorization precedes understanding. For the definition of a group, for instance, I think that if you have it memorized, you can then turn it over and over in your mind and see how the parts fit together, till the understanding hits you. After the understanding, the memorization can be allowed to lapse. This is sure: when I’m teaching abstract algebra, and a student can’t reproduce the definition, I know without doubt that the student isn’t understanding. – Lubin Apr 29 '18 at 20:56
  • My argument is that when you learn the definition of group, you should immediately do exercises showing that certain sets and operations are groups, and certain other are not. After this, and more practice, normal human beings remember the definition without having to look it up. – Martin Argerami Apr 29 '18 at 21:03
  • Yes indeed. But if it’s memorized, you can do many of those exercises while walking in the woods. – Lubin Apr 29 '18 at 21:14
  • @Lubin, the disagreement you and Martin are having, these are precisely the kind of arguments I'm having with myself in my head. Which is most efficient? And how far should you go with memorization? – user56834 Apr 30 '18 at 04:52
  • You just have to do what you find works best for you. @MartinArgerami and I agree that the goal is understanding, and that memorization is at best a tool toward that end. – Lubin Apr 30 '18 at 14:47