| Combinatorics is the mathematics of counting a finite number of elements over a finite number of operations (Hart, 1992). As these problems are finite and the elements can be common physical objects, these problems would appear to lend themselves to images and conceptualizations that rely on this concrete aspect. However, as Eizenberg and Zaslavsky (2004) showed, the introduction of a formula can interfere with the students' conceptualizations and can even result in problems that were easily solved beforehand to become much more difficult for the student to solve. This study looks at how a group of students at a large Midwestern public university think about a series of combinatorics problems which involve several different physical objects as problem elements as well as a standard symbolic representation of the problems. The study was conducted as a multiple case study with students with a high ability in combinatorics as one bounded unit and students with low ability as a second bounded unit. Students were selected from a mathematics course with the problems discussed as a part of the curriculum and will be assigned to the high or low ability groups based on preliminary interviews, self-assessment and work with a basic combinatorics problem. Participants then returned for two problem-solving sessions which were videotaped for further analysis and a final interview session for clarification of the collected data. An analysis of the results concludes with observations on the teaching and learning of permutations and combinations, but individually as well as comparatively, as well as suggestions for alternative approaches or perspectives on the instruction of this topic. |
