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Probing students’ thinking when introduced to formal combinatorics theory in grade 12 Perrin, Thomas

Abstract

This research explores students’ mathematical thinking when introduced to formal combinatorics theory. It identifies how students understand formal theory and modify their mathematical thinking and resolution strategies after having been introduced to combinatorics. This research is situated in the study of a Mathematics 12 class for the duration of a teaching unit on combinatorics, and of two groups of two students that solved specific combinatorial problems outside of class hours. Data includes videotapes of the classes and group sessions, copies of students’ work and tests, students’ answers to meta-cognitive questions and field notes. I describe how students solved a specific combinatorial problem - the pathway problem - arguing that this description exemplifies how students shifted from resolution strategies based on counting and the use of different techniques such as drawings, graphs, lists, trees, amongst others, to the sole use of the taught algorithm. I argue that this shift followed both the emphasis given to the use of formulae during instruction and the students’ lack of proficiency in the use of counting techniques. The latter is described in detail and points to the fact that students lacked practice and were not systematic. Results from this study suggest that the shift from using counting techniques to using formulae was common throughout the unit. In particular, it was the case with the permutation and combination formulae. Nevertheless, in the case of permutations, some students still used repeated multiplication instead of the formulae. Students were confused as to which formula to choose between the permutation and combination formula. I illustrate how students saw combinations only as permutations without order and did not understand the impact of the division in the combination formulae. Students’ understanding was limited and they had no other way to solve the problem than to apply a formula they did not understood. Following these findings, I suggest teachers should not overlook the instruction of counting techniques and should make connections between these and the formulae, for instance in showing various methods for resolving problems using both methods. I also recommend teaching combinations by emphasizing the role of division in the formula and in computations when solving problems without using the formula.

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