An analysis of the relationship between two alternate procedures for the utilization of teaching aids and Piaget's developmental theory during the initial introduction of selected fifth grade mathematical topics



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The problem of this study was to investigate the effects upon initial achievement and retention of selected fifth grade mathematics operations attributed to the sequence of introductory experiences and the Piagetian Developmental Level of the students. Two types of introductory experiences were compared, namely, a concrete-semi-concrete-abstract sequence and a semi-concrete-abstract sequence. All students were classified as presently in either Level III or Level IV of Piagetian Developmental Stages. The study also investigated the effects of the introductory sequences upon attitudes toward mathematics. The purposes of the study were: (1) to determine if there was a significant difference between initial achievement or retention of fifth grade students as a result of the sequence of introductory experiences, (2) to determine if there was a significant difference between initial achievement or retention as a result of the Piagetian Level of Development, (3) to determine if there was a significant difference between initial achievement or retention caused by an interaction of the introductory sequence and the Piagetian level, (4) to determine if there was significant change in attitude towards mathematics on the part of fifth grade students caused by the introductory sequence of experiences, and (5) to determine if there was a significant difference in the attitude changes of fifth grade students as a result of the introductory sequence of experiences. The initial achievement and retention levels of the students were measured by a set of three author constructed instruments. The sum of the scores of the first two instruments constituted the initial achievement score for each student and the third instrument administered four weeks after the conclusion of the instructional phase of the study provided the retention score. The reliability and validity of all three instruments were obtained through a pilot administration program prior to the actual study. The Fellows Attitudinal Device was used to survey pre- and post-attitudes towards mathematics. The study was based on a sample of ninety-nine fifth grade students selected by random procedures from the Lafayette Parish School System during the 1974 Spring Semester. The ninety-nine students comprised four heterogeneously grouped self-contained classes. Two classes were instructed through the utilization of concrete-semi-concrete-abstract experiences and the other two with semi-concrete and abstract experiences only. All students were classified according to present Piagetian Level at the start of the experiment. All classes were instructed for twenty-seven consecutive school days. All instruction was conducted by the principal investigator. The two topics selected were multiplication and division of fractions. It was the first formal exposure to the two topics for all students. The statistical technique used for analyzing the initial achievement and retention data included Analysis of Variance and Covariance Tables generated through the use of Multiple Regression Analysis. The Analysis of Covariance used Mathematics Achievement as the covariate. The statistical technique used for analyzing the attitude data included the NcNemar Test for Significance of Changes and a Chi-square Proportion Division Test. The hypotheses were either rejected or accepted at the .05 level of significance. Findings included the following: 1. Level TV students secure significantly higher achievement and retention scores, and when mathematics achievement was introduced as a covariate, retention differences remained significant. 2. Both teaching procedures affected attitudes equally and both resulted in significant changes toward more positive attitudes for the sample students. Additional analysis of the test scores produced strong evidence supporting the following: 1. In the area of comprehension, a teaching sequence utilizing concrete aids results in higher retention results. 2. In the area of computation, interaction occurs between a teaching procedure utilizing concrete aids and Level III students. Recommendations included the following: 1. Upper elementary teachers should determine the levels of Piagetian development of their students and utilize the information for grouping purposes. 2. Upper elementary teachers should develop meaningful teaching approaches utilizing the appropriate instructional aids for all mathematics concepts presented.