Students' conjectural operations
Norton, Anderson Hassell
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In taking a scheme theoretic approach to studying mathematical conjectures, this paper examines the conjectural operations that underlie conjecturing activity. It provides answers to the following two questions: How might conjectural operations engender accommodations in schemes within a cognitive system, and how do students develop a conjectural disposition? In answering these questions, the paper reports on the conjecturing activity of four sixth-grade students, working in the context of fractions problems and using software called TIMA:Bars. The researcher also served as the students’ teacher during two teaching experiments, one with each of two pairs of students. The teaching experiments were conducted twice per week for one school semester. Findings include that conjectural operations, themselves, sometimes serve as functional accommodations in schemes. At least one kind of conjecturing operation, generalizing assimilation, can modify the trigger of a scheme and raise student awareness about new constraints to students’ ways of operating. Abduction was another kind of conjectural operation and often reorganized existing operations into new patterns for operating. Conjectural operations occurred among all four students and served in actualizing their zones of potential construction. Not all conjecturing activity was constructive. Several affective and environmental factors contributed to differences between students’ success in constructing new ways of operating through conjecturing activity. The role of the teacher in designing appropriate tasks and interpreting students’ actions appropriately was particularly important. Differences between the students’ initial levels of development were a factor in the particular schemes that they constructed, but such differences did not appear to determine the ways in which conjectural operations were used, nor the constructiveness of their use.