Learning by Heart is the Surest Way to Not Understand

Why rote learning is counterproductive and how genuine understanding naturally leads to memorization.

October 21, 2024
4 min
By Matazart

And understanding is the surest way to memorize.

If you truly want to progress, excel, and navigate with ease through the mathematical landscape, the most counterproductive way to get there is to memorize as many formulas, properties, and theorems by heart.

Why?

Because memorization or rote learning is the opposite of understanding.

The reason a person learns by heart is that they want to reassure themselves.

They want to feel like they possess what they've learned.

But this possession is extremely fragile and ephemeral.

When you memorize a formula by heart, what do you do:

  • If you have a memory gap?
  • If you can't remember whether the formula has a "+" or a "-"?

But that's not the worst part - by learning by heart, you haven't understood anything.

So, how can you hope to progress in the theory or even apply it?

If you memorize all the formulas of quantum mechanics, do you think you'll understand anything?

Do you think Einstein was so prolific in his creativity and discoveries because he memorized so much by heart?

"Imagination is more important than knowledge" "Any fool can know, the point is to understand"

  • Albert Einstein

The fundamental point is this:

We think we reach understanding by learning by heart, but by learning by heart, we generally get neither, since everything is easily forgotten.

Whereas by truly and deeply understanding, we can easily retrieve everything, and after a certain number of times, things are naturally memorized.

You never learned the way home by heart - it's by doing it repeatedly that you memorized it.

Let's take a mathematical example.

A simple and famous formula:

The one that allows summing integers from 11 to nn:

1+2+3+...+n=n(n+1)21+2+3+...+n=\frac{n(n+1)}{2}

This formula, today, I know it by heart.

But I never deliberately learned it by heart.

When I wanted to apply it, I would rediscover it.

How?

Simply, by re-proving it.

In the beginning, in writing, then mentally, and after rediscovering it many times, the formula easily engraved itself in my brain.

And I didn't invent anything - I just redo the classic proof.

Add this sum to itself, but in the other direction:

1+2+...+(n−1)+n1+2+...+(n-1)+n n+(n−1)+...+2+1n+(n-1)+...+2+1

By doing this, we reveal a symmetry: the sum of aligned terms (top-bottom) is n+1n+1.

Since we have nn terms, 22 times our sum equals n(n+1)n(n+1)

So the sum equals n(n+1)2\frac{n(n+1)}{2} And there's the treasure - if you've truly understood the central idea on which the entire proof rests:

Add the sum to itself in the other direction.

Then, you'll never forget it.

Unlike the formula, which you'll certainly forget.

There's another advantage to understanding the proof rather than memorizing the formula by heart.

If we modify the sum slightly, for example, instead of starting at 1, we start at 5, and instead of summing to nn, we sum to 2n2n.

5+6+...+2n5+6+...+2n

Then someone who only knows the classic formula is lost or must go through many contortions to find the result.

Whereas if you've understood the method, nothing simpler - we again use the central idea of the proof:

Add this sum to itself in the other direction:

5+6+...+(2n−1)+2n5+6+...+(2n-1)+2n 2n+(2n−1)+...+6+52n+(2n-1)+...+6+5

The sum of vertically aligned terms is 2n+52n+5 and we have 2n−42n-4 terms (you also need to know how to count the number of terms).

So 22 times our sum equals (2n+5)(2n−4)(2n+5)(2n-4)

Thus the sum 5+6+...+2n5+6+...+2n equals

(2n+5)(2n−4)2=(n−2)(2n+5)\frac{(2n+5)(2n-4)}{2}=(n-2)(2n+5)

I don't know how people can learn without understanding... they learn otherwise, by heart, or I don't know what...
No wonder after that their knowledge is so fragile!

  • Richard Feynman