Visual Mathematics: seeing is understanding
The brain has five pathways for doing mathematics. Two of them are visual. Not as a backup. As architecture.
When you picture the number 8, does anything happen in your mind? Not the digit — something spatial. A sense of its size relative to 10, perhaps. A faint impression of where it sits between other numbers. Maybe two groups of four, arranged somehow.
That image — however faint, however half-formed — is your visual mathematics. You did not lose it when school told you to stop drawing and just do the symbols. It went quiet. It learned to operate in the background. But it never stopped. And the research we are about to look at suggests that recovering it — noticing it, trusting it, giving it room — may be the most direct path back to mathematical confidence that exists.
There is a particular cruelty that happens in mathematics classrooms, so quiet that most people never notice it.
A child draws a picture beside their calculation. Little squares arranged in rows, or dots grouped in clusters, or a line with numbers marked along it. They are thinking visibly — mapping what lives in their mind onto the page.
And then the instruction comes: Stop drawing. Just do the maths.
We mean well when we say this. We believe we are moving the child forward — from the primitive to the sophisticated, from the concrete to the real thing. What we are actually doing is amputating half of their brain.
The five pathways your brain uses
Neuroscientist Vinod Menon at Stanford has spent years mapping what happens in the brain during mathematical thinking. His research reveals something that should change how we teach: when you do mathematics — any mathematics, from simple arithmetic to pure algebra — five distinct neural pathways engage. Among those five, two are visual and spatial.
Not visual as a remedial workaround. Not visual as training wheels for beginners. Visual as architecture — as part of how the human brain is built to understand quantity, relationship, and structure.
One of those pathways — the dorsal visual pathway — is the primary brain region responsible for representing quantity. When you sense that a crowd is larger than you expected, or that a fraction is less than a half, or that two equations are equivalent, that pathway is at work.
Here is the counterintuitive finding: researchers compared brain scans of high-level mathematicians to equally accomplished academics in other fields. The brain activity that distinguished the mathematicians came almost entirely from visual areas. Not just during geometry problems. During algebra. During symbolic calculation. The visual brain is not where mathematics begins and then fades — it is where mathematical thinking lives, at every level of the discipline.
When a child is told to stop drawing and just do the maths, the instruction is not moving them toward the way mathematicians think. It is moving them away from it.
The hand is not a crutch
Here is something most people do not know: the part of the brain that processes finger sensation stays active during arithmetic — in adults who have not counted on their fingers since primary school.
Ilaria Berteletti and James Booth spent years tracking neural activity in children and adults during mathematical tasks. The somatosensory finger area — the brain region that handles sensation in the fingers — lights up during number work even when the hands are perfectly still, flat on the table. The fingers are not where counting begins and then stops. They are where mathematical knowledge first lives in the body, and the neural connections they build remain active for life.
The finding that most changes the picture: the harder the problem, the more intensely the finger area activates. We assume that as mathematics grows more complex, the body steps back and pure thought takes over. The data shows the opposite. Under greater mathematical demand, the brain reaches back toward the body — toward the physical knowing that came first. Under greater demand, the body leans in harder.
Children with stronger finger perception in first grade — the ability to feel which fingers are being touched without looking — show higher mathematical achievement in second grade than children with better test scores. The hand is not a scaffold to be removed once the building stands. It is part of the foundation.
What games can do
In 2008, psychologists Robert Siegler and Geetha Ramani ran a study that should have made headlines everywhere.
They gave preschool children from low-income families a board game to play. Nothing sophisticated — a strip of numbered squares from one to ten, with a spinner to decide how many spaces to move. Four sessions. Fifteen minutes each.
After those four sessions, the number sense gap between children from low-income and middle-income families had closed. Not narrowed — closed. One hour of playing with a linear number line, moving a piece through space and naming numbers aloud, transferred what normally takes years of informal mathematical experience at home.
The game was not about memorisation. It was spatial. The children were physically moving through the number line — building in their bodies the sense of quantity as a landscape, with a smaller and a larger, a left and a right, a position in space. That spatial experience was what their peers from more advantaged homes had been absorbing through play and conversation for years.
Separate research by Park and Brannon confirmed the mechanism: spatial training directly improves mathematical performance — not because spatial reasoning and arithmetic are the same thing, but because they share neural territory. Strengthen the visual-spatial sense, and the mathematical sense grows with it.
The loop that never closes
There is a sequence that mathematics educators call CRA — Concrete, Representational, Abstract — or CPA — Concrete, Pictorial, Abstract — and it describes the natural movement of mathematical understanding.
Hands first. Then eyes. Then symbols.
Most people understand this as a one-way journey: you start with objects, move to diagrams, arrive at equations, and then you are finished with the earlier stages. This is not what the research shows. CRA is a loop — fluid, infinite, and non-hierarchical.
When experts encounter a problem at the edge of their understanding, they fold back. Not down — they reach for the concrete and the visual not because they have regressed, but because genuine understanding always lives there.
Maryam Mirzakhani, the first woman to win the Fields Medal — the highest honour in mathematics — worked by spreading large sheets of paper across the floor and doodling on them, sometimes for hours, until the ideas began to move.
Paul Lockhart described mathematics as "surprisingly tactile and visual — sensory by proxy." He wrote that "most mathematics is done with a friend over a cup of coffee, with a diagram scribbled on a napkin." The Pythagorean theorem, one of the most notated ideas in the history of the subject, "needs to be felt, not just notated."
The abstraction is not the destination.
It is the record of an understanding that was first reached through the hands and the eyes. For the preschooler moving a game piece and for the Fields Medal winner doodling on the floor, the sequence is the same. The subject changes. The movement does not.
If a child is always given the physical object or printed diagram, the brain does not have to do the work of visualisation. Visualising is what we do when we don't have a visual.
The concrete model should support the visual imagination — not replace it. Introduce it, let it do its work, and then occasionally take it away. What the child draws from memory, from the image held in mind, is deeper than anything handed to them on paper.
What the drawing reveals
There is a test that reveals something about the difference between a remembered procedure and genuine understanding.
Ask a child to solve 4 × 6. Most will write 24. Some will answer instantly. This tells you almost nothing about what they know.
Now ask: Can you draw it?
A student who understands multiplication can draw a 4-by-6 array — four rows, six columns, twenty-four squares in a rectangle. They can see why 4 × 6 and 6 × 4 give the same answer: rotate the rectangle and the squares do not change. They can see what it means to multiply by one. They can see why zero produces nothing — because there are no rows, or no columns, and either way there is no rectangle.
Ask another student to compare 5/6 and 2/8 by drawing both fractions.
Their written answer is correct. But the drawing shows one fraction in a rectangle and the other in a circle of a different size. The answer was right. The understanding was absent: fractions can only be compared when they refer to the same-sized whole.
The symbol produced the right output. The drawing revealed that nobody was home.
Ask a student to show 7 × 3 in two ways. They write 7 × 3 = 21 without hesitation. For the drawing, they produce a part-part-whole diagram — the structure used for addition. The multiplication fact is memorised. The meaning of multiplication is not there.
A different context will make this visible, and the student will have no idea why they are suddenly lost.
John SanGiovanni, who has spent years listening to how students think about mathematics, compares visual representations to "photographs from different angles." A correct numerical answer is one photograph — useful, but limited. The drawing is the angle that shows the architecture: what the student genuinely understands, what they are imitating, and exactly where the thread between symbol and meaning has broken.
The visual is not easier than the symbolic. It is harder to fake.
An invitation to look again
The reason a mathematical idea works is not a supplement to understanding it. The reason is the understanding. A symbol without a why is a sound without a meaning — technically correct, functionally hollow.
The child who was told to stop drawing was told to stop asking why. They learned to perform the what instead — to move symbols through procedures without ever inhabiting the space behind them. Somewhere along the way, they stopped believing they could see the mathematics at all.
The doodle-filled notebooks of the best mathematicians in the world all say the same thing: the visual brain did not leave. It was simply told to wait.
The next time your child or your student picks up a pencil to draw something beside the numbers, do not stop it.
Ask instead: "How did you see it?"
Not "what is the answer." Not "is that right?"
The question invites the visual brain to speak — to describe the structure it perceived, the grouping it made, the shape it recognised. And in that description, something arrives that no symbol can carry alone.
The drawing is not an alternative to the mathematics. Cognitive scientists George Lakoff and Mark Johnson have shown that even our most abstract concepts — "knowing," "understanding," "grasping" — are rooted in embodied, visual experience.
We say "I see what you mean" because, neurologically, we do. The abstract math is the shorthand. The drawing is the truth it abbreviates.
Watch what the hand knows that the mind has not yet put into words.
Glossary
Dorsal visual pathway — One of two visual brain pathways engaged during mathematical thinking, identified in research by Vinod Menon at Stanford. The dorsal visual pathway is the primary brain region responsible for representing quantity — it activates not just in geometry, but in algebra and symbolic calculation. Brain scans of high-level mathematicians show this pathway as one of the key regions separating their thinking from non-mathematicians, even when the mathematics is entirely symbolic.
Somatosensory finger area — The region of the brain's somatosensory cortex that processes sensory information from the fingers. Research by Ilaria Berteletti and James Booth has shown this region is active during arithmetic in both children and adults, even when the fingers are completely still. Crucially, the harder the mathematical problem, the more intensely this area activates — the opposite of what intuition predicts. Children with strong finger perception in first grade are better predictors of second-grade mathematical achievement than standard test scores.
Finger perception — The ability to distinguish which fingers are being touched without looking. Research by Berteletti and Booth found that children with stronger finger perception in first grade consistently achieved more in mathematics in second grade than children with higher test scores — suggesting that embodied, physical number sense is a deeper predictor of mathematical capacity than early performance metrics.
Array — A mathematical representation in which objects are arranged in rows and columns. Arrays make multiplication visible: a 4-by-6 array shows not just that 4 × 6 = 24, but why — and why 6 × 4 gives the same result (rotating the rectangle doesn't change the count). They also reveal the commutative property, lay the foundation for understanding area, and provide a concrete path into the distributive property and algebraic reasoning.
CRA sequence — A three-stage model: Concrete (working with physical objects), Representational (drawing or picturing those objects), and Abstract (working with symbols). Research shows this is not a one-way linear sequence but a fluid, infinite loop — even expert mathematicians fold back to concrete models and drawings when encountering new problems. The abstraction is the record of an understanding, not its beginning.