In mathematics, what the expert

sees and does with an image is not what the novice sees, even with the same diagrams. What the teacher

sees is not what the students see. What one student sees is not what their neighbor sees. All of these

differences impact our classroom work with diagrams and visuals.

It is an illusion that mathematical reasoning is done in the brain with language. Standard

presentations of mathematics foster this illusion, but this formal public appearance does not represent

the problem solving, the thinking, the reasoning of many mathematicians. A ballet performance does

not embody the way this performer walks around their home, or the way they practice. Analogously,

what you observe in a mathematics paper or lecture does not embody what a mathematician does while

solving the problem, or when talking with a colleague. It does not match the cognitive processes of the

mathematician, the teacher, or the learner of mathematics.

What students commonly see in a mathematics classroom is also an illusion.

Visual reasoning is not restricted to geometry or spatially represented mathematics.

As an example, combinatorics is very rich in visual patterns and associated reasoning

Even the algebra, and symbolic logic, rely on visual form and appearance to evoke appropriate

steps and comparisons. All fields of mathematics contain processes and properties that afford

visual patterns and visually structured reasoning .

Mathematicians have not developed clear, consistent ways of working with visuals, as we have

with algebra and other symbolic forms. While the larger community has the discipline to agree on

shared definitions and algebraic forms, we continually develop new diagrammatic representations, in

undisciplined ways. This mixes sustainable visuals with good cognitive fit with local eccentricities.

This gap between individual or local practices and shared conventions is an obstacle to effective

sharing and learning.

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I've been through the desert on a horse with no name

It felt good to be out of the rain

In the desert you can remember your name

'Cause there ain't no one for to give you no pain