Students: Do we need to memorize this?
Parker: Yes.
Many students are worried about memorizing facts in mathematics. Memorization feels so difficult and unnecessary. But, memorization does not have to be so. It is a really well-studied phenomena and there are known methods that work. This note provides some helpful suggestions about how to memorize material in mathematics.
Let’s say that we need to memorize the following fact.
For any $n \in \mathbb{N}$, we have: \[ 1 + 2 + \dots + n = \frac{n(n+1)}{2}. \]
First up, let’s try to understand the thing. Understanding is a personal process. You, and only you, can know if you really understanding something. Can you explain what the formula says? Remember, a formula is a always a series of words that can be said out loud. If someone asked me1 what this means, I would say something like:
If you add up the first $n$ numbers, you get some number. Like $1+2+3+4+5$. In general, adding up the first number $n$ has a specific formula. It is a polynomial, one half of $n$ times $n+1$.
We can always delve deeper in to what something means. One was of delving deeper is creating an explanation. If I had to explain why this formula is true, I would think about the vague idea proof.
Make two copies of the sum: ${\color{red} 1 + 2 + 3}$ and ${\color{blue} 1 + 2 + 3}$.We stack them to form a little grid.
\[ \begin{array}{ccc} {\color{red} \star} & {\color{red} \star} & {\color{red} \star} \\ {\color{blue} \star} & {\color{red} \star} & {\color{red} \star} \\ {\color{blue} \star} & {\color{blue} \star} & {\color{red} \star} \\ {\color{blue} \star} & {\color{blue} \star} & {\color{blue} \star} \\ \end{array} \]
These can be regrouped to form:
\[ ({\color{red}1} + {\color{blue}3}) + ({\color{red}2} + {\color{blue}2}) + ({\color{red}3} + {\color{blue}1}) = 3 \cdot 4 = 12. \] This is double the sum we want, so one copy is:
\[ \displaystyle 1 + 2 + 3 = \frac{1}{2}(12) = 6. \]
Notice that this explanation involved creating an example; we used $n = 3$. What’s another way to explain this? It all depends on your background. You might give an proof using induction. There are fancy proofs using finite differences. You might not even want a proof as your explanation; you might be well served by something like:
If we add up the numbers $1 + 2 + \dots + n$, then we’re going to get a value that’s like half the square $n^2$.
Anything that helps make the fact more memorable is a helpful. There is a story about the Gauss the boy genius solving a problem related to this.
One day, Gauss’s school teacher asked the class: “What’s $1+2+ \dots + 100$?” The teacher probably wanted to relax and read a trashy novel while the kids ground it out. Gauss instantly blurted out: \[ 5050 = \frac{100(100 + 1)}{2}. \] So much for that teacher’s break!
Notice that this story is much more memorable. The little details like a trashy novel and the teacher losing their break make it easier to remember. And yet, the formula is there with $n = 100$.
This is a bit contrived because I’m a mathematics professor. This sort of statement seems totally natural to me. Producing explanations takes hard work. ↩︎
Published: Jan 7, 2026 @ 12:10.
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