A Non-Visual Counterexample in Elementary Geometry

Abstract

Future math teachers, whether at the primary or secondary level, should enter their profession with a solid foundation in geometry. There are times when, in teaching elementary topics to such students, we have an opportunity to introduce more profound ideas at the same time. An example of this occurs in the context of area and perimeter. Every student needs to understand that there is no dependence whatsoever between a change in perimeter and a change in area-one can go up or down while the other remains constant. (Some theoretical background on this topic may be found in Courant and Robbins [1, Ch. VII], and in a more general discussion leading to the general isoperimetric problem and its dual in Jacobs [2, pp. 307-328].) In my experience, many students do not comprehend this as it contradicts their erroneous intuition that an increase in one of these quantities leads to an increase in the other. An exchange similar to the following takes place between me (Instructor) and my students nearly every year.