6. Model choice

Experts hold multiple models and select the best model for a problem. Novices do not!

——————————

Think back to your models of the integers; if the only way you conceived the integers was in a one-to-one correspondence with pieces of fruit, you would struggle to do problems (such as working with negative numbers) that require different models.

For example:

“Imagine a wire-frame cube resting on a tabletop with the front face directly in front of you and perpendicular to your line of sight. Imagine the long diagonal that goes from the bottom, front, left-hand comer to the top, back, right-hand one. Now imagine that the cube is reoriented so that this diagonal is vertical and the cube is resting on one comer. Place one fingertip about a foot above a tabletop and let this mark the position of the top corner on the diagonal. The corner on which the cube is resting is on the tabletop, vertically below your fingertip. With your other hand point to the spatial locations of the other corners of the cube.”

Hinton, G. (1979). Some Demonstrations of the Effects of Structural Descriptions in Mental Imagery. Cognitive Science, 3, 231–250.

Geoffrey Hinton performed this experiment on over 20 people, only one of whom gave the correct answer.

When presented with this problem most people show the locations of 4 additional vertices, forgetting that the cube has 8 vertices. It appears that rotating all the parts of a cube is too difficult.

In Hinton’s study, the person who got the answer correct rotated both the axes and the cube, a model that allowed the problem to be solved easily.

Weak students tend to have a single model for an item that they inflexibly try to use in all situations.