Cognitive Theories of Development and Instruction: Algebraic Reasoning in The Middle Grades

摘要

Cognitive science rests a great deal on the belief that research on learning and reasoning has direct implications for classroom instruction. In particular, cognitive theories attempt to specify how children develop increasingly complex reasoning abilities, and how learning environments can facilitate this process. Indeed, some prominent examples support this view (e.g., Carpenter et al., 1988). However, in our theory-based cognitive research on classroom teaching and learning (e.g., Kalchman et al, 2001; Nathan & Koedinger, 2000), it is evident that, as with any design science (Glaser, 1976; Simon, 1969) there are substantial gaps between the dictates that can be expected from cognitive theory and what is needed to implement learning environments in practice. It is here that we feel our research programs can make solid contributions to theoretically-guided cognitive research in educational settings. Both authors situate their work in middle school mathematics classrooms and study the effects of theoretically based classroom interventions (derived from laboratory studies and theoretical models) to investigate students' thinking as they make the transition from relatively concrete arithmetic thinking to more abstract algebraic reasoning. In this work, we discuss the many of faces of the meaning of "development" as it relates to students' learning and reasoning about mathematics, the role of curriculum, as well as teacher cognition and practices.