| This study examines eighth grade students' work in solving an open-ended activity of finding surface area and volume of various models of stacked Cuisenaire rods. This research is part of a longitudinal study of a focus group of students, conducted at Rutgers University, of the development of students' mathematical ideas and ways of reasoning from grades 1 through present. The focus of this research is the investigation of early algebraic learning, a critical dimension of mathematics education today. This study examines students' use of generalizations, verbal and symbolic, and the methods, procedures and representations used by students for creating generalizations.;The methodology centered around viewing of archived video tapes, student work and student transcripts that were analyzed according to the model for studying the development of mathematical thinking described by Powell, Francisco and Maher (2003).;The study was guided by seven research questions: (1) What representations do students use to express their ideas? (2) How, if at all, do the representations support students' solutions (3) How, if at all, do the representations support students' building of a general solution? (4) How were the problems solved by the students (5) What evidence, if any, is there of students generalizing their solutions? (6) What interventions, if any, are made by the researcher and (7) How, if at all, do researcher interventions influence student reasoning?;The findings for questions 1, 2 and 3 suggest that students used a variety of representations in solving the tasks, connected the physical model of the rods and the symbolic notation of their formulas, connected the spoken language with the symbolic notations, made connections within representations, identified components of a previous formula, which was altered to solve a subsequent task, and expressed generalized methods. The main findings for research question 4 suggested that students searched for patterns and built on previous formulas that they had developed. The main finding for question 5 is that students were able to express a generalized method. The main finding for research questions 6 and 7 suggest that researcher intervention of questioning influenced students' creation of a formula. |
