The Use of Research Problems in High School Geometry

Abstract

Despite curricular changes associated with “the new math”, many high school mathematics students do not realize that the body of mathematical knowledge is constantly growing. That is, they regard mathematics as an essentially “dead” science, while being at least vaguely aware of research progress in physics, chemistry, and biology. In order to help them appreciate the lively nature of mathematics, and thus to quicken their interest in the subject, they should be told about recent mathematical discoveries and about some of the as-yet-unsolved problems that occupy the attention of mathematical researchers. This is not feasible at the high school level in most areas of mathematics, but it is feasible in Euclidean and analytic geometry. By way of illustration, some suitable problems are presented below. The discussions are not complete, but should suffice to convey the flavor of each problem. References to more detailed discussions are also included. All but one of the problems appear in two films that I have recently completed [16], and additional references concerning them can be found in the viewer’s manuals for those films.