This circle model (⚪︎ model) I’m proposing should allow us to hold “quantity × count” as a single visual object.
Multiplication has a two-layer structure — a quantity and the number of times it appears — and the ⚪︎ model lets you keep that entire structure in one visual unit.
Let me first show the expression that will appear later.
⚪︎ × 4 (Imagine the circle contains “42.”
I’m not using a special symbol for it.)
=³ (42 + 42 + 42 + 42)
=² (84 + 84)
= 168Isn’t multiplication incredibly hard to grasp at the beginning?
Unlike addition, multiplication forces you to handle a two-layer structure at the same time:
“quantity × number of sets.”
Most people, myself included, probably memorized it as a kind of snapshot rather than understanding the internal process.
For example, when we say:
“2×2 = 4”
many children aren’t actually thinking
(1+1) + (1+1) = 4.
They just remember the pattern “2×2 → 4.”
I certainly learned it that way.
That isn’t always bad, but knowing only the result and not the procedure can cause later problems. And I honestly have no idea how multiplication is currently taught in schools.
So here, I’ll write down the method that I believe is easier to understand.
This is purely a beginner-friendly way of presenting the structure.
How should we understand something like:
7 × 6 = 42 ?
The front number is replicated as many times as the back number.
In other words: “seven, six times.”
If a child understands just that, they can count it out by folding fingers one at a time — increasing 7, six times. I probably did this myself as a kid, though I don’t remember anyone explicitly teaching me that method. (Maybe they did, but I have no memory of it.)
7 7 7 7 7 7Line up six sevens and then add them:
7 → 14 → 21 → 28 → 35 → 42This works fine if you can speak the numbers aloud and you understand two-digit transitions on paper.
But it doesn’t feel elegant.
(Why it feels inelegant — I’ll explain in the appendix at the end.)
This approach is basically for “physical computation specialists.”
If your working memory and finger-counting stamina are strong, you’ll be fine.
If not, it quickly becomes a bottleneck.
So I think multiplication should be taught using a “set” concept instead.
If a student understands the set model, then structural transformations of multiplication —
factoring, regrouping, decomposing — become far easier.
Put simply: it’s better if the mind can see the structure.
Numerical symbols are far too abstract; beginners need something with a concrete mental image.
Copying = high procedural load (many steps)
Sets = structure visible at a glance (most likely)
Imagine placing a round ⚪︎-box in front of you, with a number inside it.
Like this:
⑦ × 6The idea is: the box contains the number.
You can think of it as a toy box.
“A round box with 7 toys inside it — and we have 6 sets of those boxes.”
If someone asks, “What exactly is a set?”
Well, it becomes a pragmatic/linguistic question, but the best I can say is:
“a container holding countable things.”
If someone has a better explanation, I’d genuinely like to hear it.
You might say, “Isn’t copying easier to understand?”
Maybe. For some kids, yes.
But the set concept is probably clearer for most.
That said, some children do understand copying better, and that’s fine.
Knowing both is ideal.
(Division can be visualized using the same ⚪︎-container idea — but I’ll handle that in the next article.)
If we extend the idea a bit, we can do more.
(By using the ⚪︎ symbol, I’m hoping readers will intuitively feel the “movement” or flexibility it’s meant to represent—but whether that works is anyone’s guess.)
Take this example:
⚪︎ × 4
(Imagine the ⚪︎ contains 42. I don’t have a dedicated notation for this yet.)
=³ (42 + 42 + 42 + 42)
=² (84 + 84)
= 168You can teach the idea that the ⚪︎ can be “expanded” into parentheses.
This allows us to see how the child is thinking through the calculation.
Some people may ask, “Does that really matter?”
But it does matter — because the goal is to see whether the student is actually performing a procedure, not just landing on the correct answer by a lucky snapshot.
Did they recall the answer from memory?
Or did they follow a structure?
This makes it easy to see where a child got stuck, and it’s far more satisfying for them than simply being told their answer is wrong.
It’s not that results don’t matter; they do.
But learning is a different domain.
There is a meaningful distinction between:
• getting the answer after understanding the procedure, and
• getting the answer without understanding any procedure at all.
That difference matters.
That’s all for now.
Next, I’ll write about division, which will also use the same ⚪︎-model.
・Supplement section
Here are a few supplemental notes.
Why the step-by-step method isn’t very elegant
7 → 14 → 21 → 28 → 35 → 42
Each step consumes working memory, which means performance varies wildly from person to person.
Humans tend to struggle with more than three consecutive operations — the cognitive load spikes fast.
Card games often limit hand size to around five for the same reason: managing more becomes difficult.
I’m not sure what the average elementary-school working memory capacity is, but it’s almost certainly weaker than an adult’s.
“Set” vs “Copy” — they aren’t actually the same
A set means:
You assume from the beginning that there are “six boxes, each containing seven toys.”
It treats the quantity as a unit and repeats the unit as a group.A copy means:
You explicitly replicate “the box containing seven toys” six times.
It treats the world as “amount × number of copies” but requires procedural repetition.
They look similar, but they are cognitively different.Copy (repeated addition)
Treats the world as one-dimensional
A linear operation: “7 → 14 → 21 → …”
High working-memory load
Easy to make mistakes
Still important for understanding procedures
Essentially an extension of addition
Correct, but cognitively heavy
Set (equal groups)
Treats the world as two-dimensional
“Amount in one set × number of sets”
Visually: ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ (seven six times)
The structure becomes visible instantly
Strong for reverse operations (division, factorization)
Intuitive for children
Provides a one-shot view of the structure
Directly connects to division and factorization
So why use a circle (⚪︎) instead of a square?
It’s mostly intuition, but I think a circle works better.
⚪︎ = a “mass,” something that can be stretched or reshaped
◻︎ = a “box,” something that feels rigid and harder to deform
People may visualize differently, but a circle makes it easier to imagine “stretching” into parentheses.
That morphing process reduces cognitive load:
you can imagine taking the quantity inside the circle and splitting it into parts as the boundary stretches.
A circle behaves like a single malleable chunk.
You can stretch it, split it, compress it, recombine it — all without breaking the mental model.
The more freely the container can transform, the easier it becomes to treat multiplication, division, and factorization as the same underlying operation.
Whether you call it a container, a lump, or a bundle doesn’t matter.
What matters is that the child can grasp it intuitively.
If another shape works better for them, that’s perfectly fine.
A small detour: the strange case of “倍” (“times as much”)
We casually use phrases like “three times as much,” but the concept is oddly opaque if you stop and think about it.
If 4 “three-times” becomes 12, I’m not running a procedure to compute it.
I simply know that it becomes 12.
It’s a stored result, not a reconstructed process.
Whether that’s good or bad isn’t the point.
It’s better if one can reconstruct the process when needed.
“倍” is essentially a snapshot concept for most people —
stored results without explicit procedures —
and in that sense it behaves just like multiplication.
Maybe we outgrow the need for the process.
Or maybe we never learned the process in the first place.
Do you remember?
(Though kids today might understand “2× speed” simply because of YouTube…)
(Small aside: sometimes the shape of a symbol carries its own intuitive weight.
If any symbol in this piece gave you a small “oh, that feels like something” moment,
that effect was intentional.)
・I opened a related question on Hacker News:
If you want to share how you learned arithmetic,
or just read how others learned, you can find it here.
・Diary
It took me longer than I expected.
Multiplication runs deeper than it looks.
I talk about the ⚪︎ model, but Δ also becomes ⚪︎.
Spin a Δ and it looks like a circle, doesn’t it?
That’s the self-différance engine.
Strictly speaking, you’re supposed to sense Achilles, the tortoise, and the dragon before reaching it.
But once you get past that ridge, the self-différance engine is waiting.
At that point Achilles turns into either Heracles or Siegfried.
And at that point you turn into a dragon.
Just imagery, of course—imagery is always the important part.
Even the West Coast is obsessed with “visualizing” things these days, right?
Kidding.
I’m heading to my part-time job.
By the way, doesn’t Δ look a bit like a slice of pizza?
Pretty sure I said this before.
Derrida’s différance even sounds like delta if you squint.