Games, Geometry, and Teaching.

We believe that games can be used ef fectively in high school geometry both to teach content and to structure prac tice of problem-solving skills. In part, our belief stems from our research on the use of games in a wide range of situations (Bright, Harvey, and Wheeler 1985). Also, we recog nize that games can help teach problem solving (NCTM 1980). By the time you finish reading this article, we hope we will have convinced you to share our belief and to try out our games. One way to assess the nature of problem solving that can occur during game playing is through an analysis of the taxonomic level of the game. The taxonomic level is de fined in terms of Bloom's taxonomy (1956), whose categories are knowledge, compre hension, application, analysis, synthesis, and evaluation. Knowledge and comprehen sion activities form a critical base of knowl edge for problem solving. Application ac tivities (e.g., solving "standard'' word prob lems) can be viewed as a transition between that base of knowledge and dealing with more complex situations. Analysis, syn thesis, and evaluation activities require students to probe deeply into content to identify essential concepts or relationships and then to reconfigure a problem's ele ments to solve that problem. Most mathematics can be taught at sev eral taxonomic levels. For example, the classification of quadrilaterals might be taught through writing definitions (perhaps a comprehension-level activity) or through grouping one set of pictures of quadrilat erals in many ways?for example, accord ing to the commonalities of relationships among their sides, angles, or diagonals (per haps an analysis-level activity). During the second activity, students could group pic tures at random, but to succeed, they would need to operate at a high taxonomic level. The taxonomic level that is needed for success in an activity seems to be an impor tant determinant of the quality of learning attributable to that activity. In the study of polygons, students might be asked to list the properties of a polygon (knowledge or perhaps comprehension level) or to identify the properties satisfied by one set of poly gons but not by another set (analysis level). Students interact with the content during both activities, but the second activity gives them an opportunity to learn at a higher taxonomic level. An activity de signed to teach at a high taxonomic level may also offer opportunities to practice that content at lower taxonomic levels, but the reverse seems rarely true. We think that this broad range of opportunities avail able in high-taxonomic-level activities im proves instruction. Because students often have uneven backgrounds in geometry, it is especially important that the content of a geometry game be carefully selected and that its taxo nomic level be known. An analysis of the taxonomic level of a game helps teachers to know both what situations might arise during play of a game and how to help stu dents get past any hurdles they may experi ence with the content. In addition, careful analysis by the teacher will help to match the game with those students who are most likely to benefit from playing it. Appropriately chosen games can accom plish a sizable number of desirable instruc tional goals in geometry instruction. Games provide situations in which students can visualize and construct geometric figures, see applications of general results, and apply logical reasoning skills. Games also