What is the equivalent of musical ear training with regards to studying mathematics

Question

When one aspires to be a professional musician, it is made clear that ear training is a very valuable skill that must be cultivated on a daily basis. The student is advised to put in the time and effort required to hone this skill.

I’ve read many great answers on math.SE which explained how to learn math, and I’m doing my best to apply the advice given in those answers.

I was wondering if there is some sort of recommended daily practice regimen that should be pursued, such as (this might be a bad example) knowing several calculus formulae pages by heart?

Any advice would be most welcomed :)

Answer 1

I don't think there is a direct equivalent, but one similar thing in mathematics is mathematical intuition. It's the "feeling" of what you need to do to solve a given task, and the "feeling" whether the answer to some question is yes or no.

For example, if I ask you "Is $10!>2^{10}$", you will probably have no problem answering "yes", and if I ask you to prove it, you will probably write down something like $$10! = 1\cdot 2\cdot 4\cdot\cdot 3\cdot 5\cdots 10> 1\cdot 2\cdot 4\cdot 2\cdots 2 = 2^{10}$$

But why would one prove this in this way? Well, mostly because this is the easiest (or one of the easiest) ways to prove such a thing. And how do you know that you should answer yes and not no? Well, you have a rough idea about how fast $n!$ increases, and it feels much faster than $2^n$, so your intuition may say yes, it's larger.

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There are two main ways to increase one's mathematical intuition.

You could just be born with it. You could be one of the lucky $0.000001%$ that is born with such a remarkable brain that his mathematical intuition is almost frightening. Ramanujan is a nice example of that.

The other method is practice. A whole lot of practice. Practice to the point when you think you have formulas sticking out of your ears. If you solve $100$ equations, then the $101$st will be trvial.

Answer 2

There are all sorts of fields of endeavor, each with their own training regimen. I'm not certain that you can expect analogies. For instance, animators are told that everyone has 10,000 bad frames of animation in them, and their goal should be to get those out as soon as possible, so drawing a 20-frame walking cycle first thing each day is a good practice. (There are variations on this, of course.) But mathematicians don't say "Hey, maybe I should do 10 integration problems each morning, in hopes of making all possible mistakes so that I won't make those in the future."

I find that answering a stackExchange question each morning helps solidify my understanding of some concepts, and hones my ability to explain them, both of which seem useful. But I'm not sure I'd recommend that to everyone.

Answer 3

My two cents.Watch for tendencies,watch how changes are recorded in curves and formats to re-generate them as patterns. If a pattern is clear, then you have a victory over a class. Capturing relationships among variables .. I think is in the heart of math... Even recognizing something which does not change while change is the order everywhere ... that makes for a great result.. like the Gauss Egregium Theorem.

Jargon is essential noise, but its pattern is music. It involves separating wheat from the chaff all the time.