| The purpose of this study was to determine the effects of the use of Dienes' variability principles implemented with manipulatives on middle school students' ability to perform the algebraic processes of (1) simplifying algebraic expressions, (2) solving linear equations, (3) multiplying algebraic expressions, primarily binomials, and (4) identifying multiple representations of a linear function. The differences in the effects that are related to factors including ability level, achievement in prior mathematics courses, age, and gender were studied. Interviews with selected students were also conducted.;The subjects, 53 sixth graders who completed only the two units on algebraic expressions and equations, and 53 seventh graders who did all four topics of the study, were aged 11 and 12. The standardized test results from school records were used to measure the mathematics ability of each student prior to the experiment, and to classify each subject as being below-average, low, medium or high in mathematics ability.;The investigator identified two “perceptual variates” and a different number of “mathematical variates” for each topic. “Perceptual variates” simply represented the materials used in the experiment, while “mathematical variates” described the many irrelevant attributes whose variations do not change the general mathematical concept. A preliminary unit on integers was also implemented with all students so they could form mental representations they used later when studying the chosen units. An investigator-designed test administered after each topic, had questions classified each as belonging to a certain difficulty level determined by the number of mathematical variates identified in the question.;Results showed that the application of Dienes' variability principles for all four topics was successful for almost all students. The study found no gender-related differences in either age group. In both groups however, there were significant differences favoring the high achievement groups, and none between the medium and low groups. While age-related differences favored the 12-year-olds in the equation-solving topic, 11-year-olds performed almost as well as 12-year-olds in the algebraic expression topic. In general, performance on all topics was described as satisfying, considering that, questions students solved in the study, were usually well beyond middle school expectations. |
