Fostering Recursive Thinking in Combinatorics through the Use of Manipulatives and Computing Technology

Anne Pieper currently teaches high school geometry at The Westminster Schools in Atlanta, Georgia. She recently graduated fromThe University of Georgia with an M.A. in Mathematics Education. Current trends in the revision of both mathematics curricula and traditional teaching strategies include the incorporation of discrete mathematics into the secondary school curriculum (Dossey, 1990). Serious consideration of this task has occurred as a result of the publication of the NCTM 1991 Year Book (Kenney, 1991), which accumulated perceptions, practical ideas and suggestions of many experienced professionals about discrete mathematics in general and combinatorics in particular. Also, Standard 12 of the Curriculum and Evaluation Standards for Grades 912 emphasizes the importance of this topic: “instruction should emphasize combinatorial reasoning as opposed to the application of analytic formulas for permutations and combinations” (NCTM, 1989, p.179). Combinatorics is one of the oldest branches of discrete mathematics and goes back to the 16th century when games of chance played an important role in the life of a society (Vilenkin, 1971). The need for the theory of such games stimulated the creation of specific counting techniques and mathematical concepts related to the new reallife situations. Further efforts of French scholars Pascal and Fermat in the pursuit of theoretical studies of combinatorial problems laid a foundation for the theory of probability and provided approaches to the development of enumerative combinatorics as the study of methods of counting various combinations of elements of a finite set. The importance of a recursive approach to enumerative combinatorics through using difference equations (recurrence relations) is a common theme (Anderson, 1974; Kenney & Bezuszka, 1993; Olson, 1989). Many counting problems deal with arrangements that involve permutations and combinations. While the study of these concepts provides opportunities for using recursive reasoning as an effective problem-solving means, student learning of combinatorics has often been limited to plugging numbers into formulas for permutations and combinations without developing any conceptual understanding of combinatorial ideas or using recursive reasoning as a means for approaching these ideas. Mathematical visualization provided by computing technology and the use of concrete materials allow for significant improvement in the teaching and learning of basic combinatorial notions. This article describes how employing manipulatives for solving simple combinatorial problems leads to the discovery of recurrence relations for permutations and combinations, so that, in turn, numerical evidence and visual imagery generated by the computer spreadsheet through modeling these relations can enable all students to experience the ease and power of combinatorial reasoning, recursion, and mathematical induction proof. In addition to these teaching ideas, the interplay among geometry, combinatorics, and number theory is discussed. According to Ahlfors et al. (1962), instruction should be a process of “extracting the appropriate concept from a concrete situation.” Combinatorics is an appropriate context for accommodating such instructional philosophy. For instance, what kind of combinatorial reasoning is behind whether one is sorting or selecting books from a shelf? How does one distinguish between combinatorics of a bookshelf and combinatorics of a combination lock or a touch-tone telephone? While these are real-world situations with which students are familiar, they need help in making sense of the mathematics that is behind the reality so that they can extract from it the appropriate concept. Just as the use of dominoes in classroom activities helps to furnish the conceptual basis of combinatorics (Johnson, 1991), colored manipulatives are excellent tools that encourage and support combinatorial reasoning (Spangler, 1991). Because of this, we will formulate realworld situations in terms of colored disks (discussed in this paper as patterned disks) which were found to be extremely appropriate for individual or small group explorations and subsequent student demonstrations using an overhead projector. The paper reflects work done in a lab setting with preservice and inservice secondary teachers enrolled in a contemporary general mathematics course.