The reification of the solution of linear equations in algebra.

Abstract

This study investigated the use of manipulative materials and symbolic representations in teaching the concept of variable to twenty-six English speaking middle school (grade 7) beginning algebra students. It examined the effect of manipulating Tac-Tiles (a set of tiles which are designed to represent mathematical ideas and the symbolic representations of those ideas) on (a) learning how to solve equations, (b) learning how to solve simple word problems requiring the use of equations, and (c) describing the concept of a variable. The study used four groups. One treatment group was taught to solve linear equations with integer solutions using st and ard symbolic notation and traditional instruction. The second treatment group was taught using Tac-Tiles. The third treatment group used Tac-Tiles and was also asked to develop a symbolic notation for the operations carried out. The fourth group received treatment using a balance bar as a manipulative material. The total instruction time for all groups remained the same. The method of collecting data was to use an adaptive test in which subjects were given questions (ordered in difficulty from easy to hard) on a task outlined above until three errors were encountered, at which time questions on the next task were administered. The process continued the next day and began two questions prior to the first error on that task. Several problems with the use of the adaptive test complicated the analysis of data and deserve attention in the future. Data from the daily tests was analysed using two-way analyses of variance with repeated measures, followed by paired comparisons where indicated. Three-way analyses were used to examine the data which was reorganized for the post-hoc hypotheses. The data from the final exam were examined using a one-way analysis of variance. The subjects in the Tac-Tile Symbolic group performed better (p $<$.05) than the other groups on the task of solving linear equations both on the daily tests and on the final exam. When the groups were reorganized into Tac-Tile and non-Tac-Tile clusters, the Tac-Tile cluster performed better (p $<$.05) than the non-Tac-Tile cluster on the task of solving linear equations on the final exam. Analyses of data on the tasks of word problems and describing variable found no significant differences due to treatment.