When Should Students Memorize Math Facts?

It's helpful to loosely understand what something means before memorizing it, but this does not have to be a rigorous derivation.

On one hand, I wouldn’t ask a student to memorize times tables until they understand that multiplication is repeated addition and are able to calculate times tables using that definition.

But on the other hand, I would not say that students always have to derive things before memorizing them. For instance, I would agree with the usual approach of having students memorize derivative rules first and then later derive them from scratch.

In general, I would say that a student should loosely understand what something means before memorizing it.

This does not have to be a derivation / full computation – it’s just that in arithmetic, full computation turns out to be the most straightforward way to understand things.

In calculus, if you want to loosely explain what the derivative means, then derivations / full computations are not as straightforward, and instead it’s more straightforward to just explain that the derivative represents the instantaneous rate of change, or the slope of the tangent line to the curve. With that information, it makes sense that, e.g., the derivative of $x^2$ should get larger and larger as x increases, and that the derivative of $\sin(x)$ oscillates up and down.

That’s the main thing I’m trying to get at – the reason why it’s helpful to “loosely understand what something means” before memorizing it is that it makes the memorization process a lot easier.

And I would say that the goal should always be to approach that baseline understanding in the most quick / easy / straightforward / simple way, whatever it may be.