What is the “optimal” use of rote learning in studying (pure) math?

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"?

• 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
• @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

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 of 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.

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.

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.

• 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 4:52
• 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