The role of manipulatives in learning to multiply and factor polynomials

• Main Author: Altizer, Carol Jane
• Other Authors: Supervision
• Format: Others
• Language: en
• Published: Virginia Polytechnic Institute and State University 2016
• Subjects: LD5655.V856 1977.A48
• Online Access: http://hdl.handle.net/10919/64736

The purpose of this exploratory research study was to investigate, both theoretically and experimentally, the hypothesis that the manipulation of concrete materials can contribute substantively to the learning of the operation of multiplication of polynomials and its inverse, factorization, in children who are in eighth-grade pre-algebra mathematics classes. The study involved a comparison of the achievement of students who used manipulatives to learn how to multiply and factor polynomials with the achievement of those who did not use manipulatives to learn to operate on the polynomials. The instructional material designed for use by both treatment groups was based on the theory of learning developed by this writer. It was theorized that as students use manipulatives to learn mathematical concepts the actions performed upon the concrete materials would be abstracted or internalized in the mind as operations. The study involved four teachers and 173 students from two middle schools in the Pulaski County School System, Pulaski, Virginia. The means and standard deviations of the students' scores on both the immediate posttest and retention test were compared as well as inferences made from the data using several analyses of covariance. The Orleans-Hanna Algebra Prognosis Test served as the pretest for this study. The F ratios from the analyses of covariance conducted on the immediate posttest scores from Experiment I indicated that (1) using the total population of students, there was no statistical difference in mean scores between the manipulative and nonmanipulative groups (p = 0.385); (2) using only Teacher A's students, there was no statistical difference in mean scores between the manipulative and nonmanipulative groups (p = 0.609); and (3) using only Teacher B's students, there was a marginal difference in mean scores between the manipulative and nonmanipulative groups (p = 0.071), favoring the nonmanipulative group. The F ratios from analyses of covariance conducted on the retention test scores from Experiment I indicated that (1) using the total population of students, there was a statistical difference in mean scores between the manipulative and nonmanipulative groups (p < 0.005), favoring the manipulative groups; (2) using only Teacher A's students, there was a statistical difference in mean scores between the manipulative and nonmanipulative groups (p < 0.009), favoring the manipulative group; (3) using only Teacher B's students there was no statistical difference in mean scores between the manipulative and the nonmanipulative groups (p = 0.241). However, the mean score of Teacher B's manipulative group was higher than the mean score of his nonmanipulative group. The study was replicated (Experiment II) immediately following Experiment I in two classes taught by Teacher D.* The F ratio from an analysis of covariance conducted on the immediate posttest scores indicated that there was no statistical difference in mean scores between the manipulative and nonmanipulative groups (p = 0.762). The F ratio from an analysis of covariance conducted on the retention test scores indicated that there was no statistical difference in mean scores between the manipulative and nonmanipulative groups (p = 0.143). However, the mean score of the manipulative group was higher than the mean score of the nonmanipulative group. In summary, there are implications from these findings that the manipulation of concrete materials by students does aid the learning of the mathematical transformation of multiplication of polynomials and its inverse, factorization. This was especially evident for retention of the operations. These findings support the theory of learning conceptualized for this study. *Teacher C was omitted from the analyses of the data since she taught only a manipulative group. === ED. D.