Margaret Kinzel

Explicating a Mechanism for Conceptual Learning: Elaborating the Construct of Reflective Abstraction

We articulate and explicate a mechanism for mathematics conceptual learning that can serve as a basis for the design of mathematics lessons. The mechanism, reflec- tion on activity-effect relationships, addresses the learning paradox (Pascual-Leone, 1976), a paradox that derives from careful attention to the construct of assimilation (Piaget, 1970). The mechanism is an elaboration of Piagets (2001) reflective abstrac- tion and is potentially useful for addressing some of the more intractable problems in teaching mathematics. Implications of the mechanism for lesson design are discussed and exemplified.

Moving students through steps of mathematical knowing: An account of the practice of an elementary mathematics teacher in transition☆

Abstract We present an account of a sixth-grade teachers practice as she responds to the challenges of current reform initiatives. We analyzed classroom observations and interviews to understand how the teacher, Ivy, teaches and thinks about teaching mathematics to her students. For Ivy, mathematical meaning is available in particular experiences. She creates these experiences for her students by leading them through a predetermined sequence of steps of mathematical knowing. This account contributed to our postulation of a perspective on mathematics learning that we refer to as perception-based, in which the goal of instruction is to create opportunities for students to perceive, first hand, mathematics that exists as part of an external reality. We examine implications of this account of Ivys practice for mathematics teacher development.