### Perceptions

Yesterday a hardworking but unsuccessful student comes to my office hours to ask for help on a homework problem. We have spent the past few days working with the Euler Characteristic: a result that states that the number of vertices (corners), faces, and edges of a polyhedron are related by the equation V - E + F = 2. I have brought models of polyhedra to class, and I have had the students count vertices, faces, and edges; I have modeled these calculations with my large, demonstration-sized shapes; we have spent a lot of time in class on this. The student was working on a problem which asked students to verify that this equation holds for a box ("like an unopened tissue box" the problem suggested).

What I expected the student to do was to either find or imagine a box* and count the vertices, faces, and edges then plug these numbers into the equation. My student struggled with the problem.

There was no illustration of the box in the problem, so my student drew a picture much like this one:

My student then went to work counting vertices, faces, and edges, counting 10 vertices and 16 edges, as follows:

(These are my recreations of the diagrams -- not the student's original work. For non-math people: the right answers is that a box has 8 vertices, 6 faces, and 12 edges.)

When my student translated the idea of a box to a 2D representation on the page, it was as if a certain aspect of its "boxness" had vanished. My student did not perceive the vertical lines representing the sides of the box as passing either in front of or behind the horizontal lines defining the base and the top: these were now

This is the second time that I've had such a revelation about a student's perception of a drawing like this one. The first time was when I was working with an autistic student** who viewed pictures like the first one as a rectangle surrounded by four trapezoids and two triangles: totally flat.

And now I know: I really have no clue what it is that my students see when I draw pictures on the board.

*Or, better yet, to remember that we did this calculation several times for a cube and realize that a cube is a special case of a box.

**Bonus points to the first person who guesses why my autistic student was late to class

What I expected the student to do was to either find or imagine a box* and count the vertices, faces, and edges then plug these numbers into the equation. My student struggled with the problem.

There was no illustration of the box in the problem, so my student drew a picture much like this one:

My student then went to work counting vertices, faces, and edges, counting 10 vertices and 16 edges, as follows:

(These are my recreations of the diagrams -- not the student's original work. For non-math people: the right answers is that a box has 8 vertices, 6 faces, and 12 edges.)

When my student translated the idea of a box to a 2D representation on the page, it was as if a certain aspect of its "boxness" had vanished. My student did not perceive the vertical lines representing the sides of the box as passing either in front of or behind the horizontal lines defining the base and the top: these were now

*lines that crossed*. My student knew that this picture was a standard way of drawing a box on a sheet of paper but did not seem to perceive this as an attempt to represent a 3D object by showing perspective.This is the second time that I've had such a revelation about a student's perception of a drawing like this one. The first time was when I was working with an autistic student** who viewed pictures like the first one as a rectangle surrounded by four trapezoids and two triangles: totally flat.

And now I know: I really have no clue what it is that my students see when I draw pictures on the board.

*Or, better yet, to remember that we did this calculation several times for a cube and realize that a cube is a special case of a box.

**Bonus points to the first person who guesses why my autistic student was late to class

*every single day*.