Integrating algebra and proof in high school: Student work with variables and parameters on the Calendar Sequence.

摘要

Nowadays, the learning and teaching of algebra is a central issue in the mathematics education research agenda (for further readings see e.g., Stacey, 2004; Gutiérrez, 2006). In addition, students' obstacles when learning algebra have been largely documented (Bednarz, 2001; Bednarz & Janvier, 1996; Booth, 1984; Boulton-Lewis, Cooper, Atweh, Pillay, & Wilss, 2001; Demana & Leitzel, 1988; Filloy & Rojano, 1989; Kieran, 1981, 1985, 1989; Kuchemann, 1981; MacGregor, 1996; Steinberg, Sleeman, & Ktorza, 1990). Further, previous research claims that students' performance producing proofs using algebra is poor (Healy & Hoyles, 2000), and that students' opportunities to produce their own conjectures and proofs in school algebra are scarce (Friendlander & Hershkowitz, 1997; Harel & Sowder, 1998). The above sets the stage for the need to conduct more research in these two areas: algebra and proof.;In this study, a group of 9 high school students (9th and 10th graders) participated in fifteen one-hour-long lessons carried out by the author of this dissertation at their school in the Boston area, Massachusetts, United States of America. The overarching research question of the study described in this paper is: what are the consequences of an integrated approach to algebra and proof on students' mathematical knowledge while they work through a didactical sequence (i.e., the "Calendar Sequence")? In particular, the goal of this paper is to report on the challenges that students faced in their work with variables, parameters, and equivalent expressions while engaged in producing and proving conjectures, and how these challenges were overcome. Previous studies on algebra and proof (Barallobres, 2004; Bell, 1993) are scant, nonetheless promising, in regards to students' production of proofs using algebra in an integrated approach. I claim that the results presented in this paper provide promising evidence that an integrated approach towards algebra and proof, such as that implemented in the Calendar Sequence, has a positive impact on students' use of algebra as a tool to prove. The following were the challenges identified that students faced and that were overcome: (a) the use of algebra to prove in contrast with a finite non-exhaustive set of numeric examples; (b) the use of a single variable that could capture all cases at the same time; (c) the number of dependent and independent variables to include in their expressions; (d) how to set up relations among variables and parameters; (e) how to obtain a simpler expression from a more complex algebraic expression; (f) the use of conventions of algebra such as the use of parenthesis; (g) the use of properties such as the use of distributive property.