Imagery and Mathematics Learning Grayson H. Wheatley, Ph.D.

There is compelling evidence that imagery plays a significant role in mathematical reasoning. For example, a young child may add 7 + 5 by mentally "moving" 1 from the 7 to the 5 to form 6 + 6, a known double. Or a child might determine how many one inch cubes there are in a rectangular solid 3" by 3" by 4" by visualizing the solid as composed of three layers. Whether working in a numerical or geometric context, when students are engaged in meaningful mathematics rather than rote computation, it is quite likely they will be using some form of imagery. There is also evidence that imaging plays an essential role in many mathematician's activity. Logic alone does not account for mathematical reasoning. It has been reported that mathematicians who feel they have a deep understanding have constructed some abstract image which makes the knowledge into a whole - an abstract image.

When doing mathematics, the nature of the images formed depends on prior mental constructions, intentions, and the situation under which the image is constructed. For example, a child might form an image of 'triangle' as formed by a horizontal base and a point above the base. If this image of a triangle is the child's only image of a triangle, then their concept of triangle is quite limited. A child has a richer concept of triangle when they can transform their image of triangle flexibly.

What an individual constructs depends on their mental schemes. The line drawing shown was presented briefly to a class and they were then asked to draw what they saw. This figure was described as two squares, a small square and two trapezoids, a hallway, a skylight and a pyramid with the top cut off. Some individuals constructed an image of regions and others of joined segments, some two-dimensional and some-three dimensional interpretations. Even though the same figure was presented in the same manner to individuals, the nature of the images constructed varied greatly.

As an example of imaging in problem solving, consider the Long Table Problem, which we have used with fifth grade students.

The Long Table

Tiffany is arranging tables for a party. She has 12 square tables which she wants to put together to make one long table. Each of the small tables seats one person or side. How many people can be seated?

Most students successful in solving this problem elaborated their image of one table to twelve, often drawing out the twelve tables, and proceeded to count how many seats were available. One student made no marks on paper but explained he had a mental picture of the twelve tables in a row and he could count the number of seats available. This student had powerful mental imagery. The individual differences in imaging among children is striking.

A fifth grade girl's solution to a mathematics problem shows the importance of imaging. The problem was "On a Rubik's Cube (3x3x3), how many small cubes have exactly two faces showing?" While at first she answered 24 (four on each of six faces), once she looked at a Rubik's Cube she quickly revised her answer and confidently said 12. The nature of the image constructed was crucial in her solution. Her first image of a cube as composed of six faces is a frequently reported image. The second is more sophisticated and, in this case, more useful in solving the question posed.

Meaningful mathematics learning is usually imaged-based. While there may be certain forms of mathematical reasoning which do not use imagery, most mathematical activity has a spatial component. If school mathematics is procedural, students may fail to develop their capacity to form necessary images of mathematical patterns and relationships. It is well documented that students who reason from images tend to be powerful mathematics students. When students are encouraged to develop mental images and use those images in mathematics, they show surprising growth. All students can learn to use images effectively. Thus every mathematics teacher or parent should make improving spatial sense a priority.