Does the phrase "If you don't use it, you lose it" apply to mathematics?

Question

I'm asking this because I ran into the following particular situation:

I took some calc courses over 2013, where I learned, amongst other things, to integrate some pretty nasty functions, and this semester, I started taking a differential equations course. To my surprise, I realized I had forgotten most I had learnt in terms of integration, and had to relearn most of the stuff on my own.

How do you guys manage to keep things you won't use for a year+ but you'll need later on in your head?

Answer 1

(In my experience)

Yes, you start to forget things pretty quickly if you don't practice at all. However, you don't forget them permanently--ideas and concepts usually stay, and all of the material will be much easier to re-learn later if you've learned it once.

Answer 2

I left academics 20 years ago and likely have forgotten more than I retained. And so, yes, there is something to be said for "use it or lose it.

But even then, I always tried to understand and retain the underlying principles and not just try to memorize "certain" details and results from applying those principles.

So, how does one filter "certain" details? It is completely a judgment call.

Is it useful to memorize, for example, the indefinite integral of $x\,e^x$

$$\int x\,e^x\,dx=e^x(x-1)+C$$

or better just to understand and recall that the principle for integrating by parts and the product rule for differentiation so that

$$uv'=(uv)'-u'v\implies \int u\,dv=(uv)-\int v\,du$$

I don't find it useful to memorize the former. But, in my humble opinion, the differentiation product rule ought to be "memorized," although I suggest that it would be even better yet to understand how to prove the rule using a basic $\delta-\epsilon$ set up.

So, the details one chooses to "memorize" are quite discretionary and one must be judicious in selecting how to separate the foundational concepts - the building blocks, so to speak - from the myriad of "results" that can be derived therefrom.

There really is no "one-shoe-fits-all" advice. In the end, just use your best judgment and ask yourself, "Is it better to memorize a result, or recall the tools and derive the result myself?"