A case study: The development of Stephanie's algebraic reasoning
This research provides an analysis of the mathematical growth and development of one student, Stephanie, as she worked on early algebra tasks during her eighth-grade year as part of a teaching experiment. Stephanie was among the original participants in a longitudinal study which investigated how students develop mathematical ideas under conditions that fostered independent exploration, reasoning, and justification of ideas (Maher, 2005). A qualitative approach based on the analytical model described by Powell, Francisco, and Maher (2003), was taken in analyzing videotape data from the Robert B. Davis Institute of Learning archive, along with student work. Seven task-based interview sessions were analyzed, spanning a six month period, beginning from November 8, 1995 to April 17, 1996. The research focused on Stephanie's algebraic reasoning; in particular, how she built an understanding of the binomial theorem and related it to Pascal's triangle. Stephanie's representations, her explanations and justifications, and her methods of dealing with obstacles to understanding, were all examined and provided the basis for this research.
The analysis shows that Stephanie built her mathematical understanding through the development of multiple representations of concepts and moved fluidly between and among the representations that she organized into ‘symbolic’ and ‘visual’ representations. Symbolic representations included algebraic expressions, combinatorics notation, and Pascal's triangle while visual representations included drawings, tables, models formed by algebra blocks and other manipulatives, and towers built with unifix cubes. Furthermore, through Stephanie's explanations and justification of her representations and reasoning in general, she invented strategies to convince herself as well as the researchers that she had fulfilled the requirements of the problem task. When dealing with obstacles to her understanding such as lack of information, or calculating obstacles, Stephanie acquired the use of several heuristic methods in order to overcome them. These included the use of substituting in numbers in order to test a conjecture; returning to basic meaning; drawing diagrams; building models; and considering a simpler problem. Throughout the task-based interviews, Stephanie retrieved knowledge from her earlier problem solving and extended this knowledge to build new ideas, while tackling more challenging problems. In particular, Stephanie mapped the coefficients in the binomial expansion to particular rows in Pascal's Triangle; she connected these ideas to her problem solving from earlier work in the elementary grades. The findings are relevant to the timing and method of early algebraic instruction in schools.
Cognition & reasoning