This article reports on students' learning through conjecturing, by drawing on a semester-long teaching experiment with 6 sixth-grade students. It focuses on 1 of the students, Josh, who developed especially powerful ways of operating over the course of the teaching experiment. Through a fine-grained analysis of Josh's actions, this article integrates Piaget's scheme theory (1950/2001) and Peirce's logic of abduction (1998) into a new theory about conjecturing that explains Josh's learning. Results indicate the power of Josh's operational conjectures in resolving problematic situations and constructing new schemes. Because of the context in which the teaching experiment and Josh's conjecturing occurred, results hold implications for research on fractions and on a particular operation called splitting (Confrey, 1994; Empson, 1999; Sáenz-Ludlow, 1994; Steffe, 2003). The theoretical integration of scheme theory and abduction also holds implications for resolving the learning paradox (Fodor, 1980; Glasersfeld, 2001).
Anderson Norton, Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123; firstname.lastname@example.org