Children's construction of the multiplicative principle in combinatorial reasoning
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The multiplicative principle, one of the fundamental principles of counting, is crucial in the learning of combinatorics. The purpose of this study was to investigate operations, schemes, and conceptual constructs that contributed to the construction of the multiplicative principle. By means of clinical interviews, two pairs of seventh-grade students were selected based on three principal factors: units-coordinating activities, levels of the units-coordination, and combinatorial reasoning. The three factors were determined as being quite similar within each pair and distinct between the pairs. After the clinical interviews, each of the pairs participated in the constructivist teaching experiment for approximately five months during two school semesters where the researcher served as the students’ teacher. The findings revealed that only the pair of students who were capable of taking three levels of units as given prior to operating constructed the multiplicative principle. In addition to their levels of units-coordination as being necessary toward the construction of the multiplicative principle, the students needed to establish at least five basic operations and five schemes. The operations include counting, making selections, pairing, distinguishing, and monitoring. And, the schemes consist of an enactive pairing scheme; an iterating pairing scheme; a multiplicative pairing scheme; an iterating tripling scheme: independent sequenced composite units; and an iterating tripling scheme: dependent sequenced composite units. Besides these operations and schemes, the students needed to produce two kinds of reasoning called recursive reasoning as well as progressive recursive reasoning in their construction of the multiplicative principle.