Evaluation of an innovative strategy for teaching systems of linear equations in terms of classroom environment, attitudes and conceptual development

Abstract

This study, which was conducted among middle-school students in California, focused on the effectiveness of using innovative strategies for enhancing the classroom environment, students' attitudes, and conceptual development. Six hundred and sixty-one (661) students from 22 classrooms in four inner city schools completed the modified actual forms of the Constructivist Learning Environment Survey (CLES), the What Is Happening In this Class? (WIHIC) questionnaire, and the Test Of Mathematics Related Attitudes (TOMRA). The data were analyzed for the CLES, WIHIC, and TOMRA to check their factor structure, reliability, discriminant validity, and the ability to distinguish between different classes and groups. In terms of the validity of the CLES, WIHIC, and TOMRA when used with middle-school students in California, the factor analysis results attest to the sound factor structure of each questionnaire. The results for each CLES, WIHIC, and TOMRA scale for the alpha reliability and discriminant validity for two units of analysis (individual and class mean) compare favorably with the results for other well-established classroom environment instruments. A one-way analysis of variance (ANOVA) was also calculated for each scale of the CLES and WIHIC to investigate its ability to differentiate between the perceptions of students in different classrooms. The ANOVA results suggest that students perceived the learning environments of different mathematics classrooms differently on CLES and WIHIC scales. In general, the results provided evidence of the validity of these instruments in describing psychosocial factors in the learning environments of middle-school mathematics classrooms in California. The effectiveness of the innovative strategy was evaluated in terms of classroom environment and attitudes, as well as achievement, among a subgroup of 101 students.Effect sizes and t-tests for paired sample were used to determine changes in classroom environment perceptions, attitudes, and achievement for experimental and control groups. Pretest-posttest differences were statistically significant (p<0.05) for: the CLES scale of Shared Control for the experimental group, the TOMRA scale of Normality of Mathematicians for both the control and the experimental groups, the TOMRA scale of Enjoyment of Mathematics for the experimental group, and the achievement measure for both groups. Also ANCOVA was calculated to determine if differential pretest-posttest changes were experienced by the experimental and control groups in classroom environment perceptions, attitudes, and achievement. The results suggest that there were a statistically significant differential changes for Task Orientation, Normality of Mathematicians, Enjoyment of Mathematics, and achievement between the experimental and control groups. In each case, the experimental group experienced larger pretest-posttest changes than the control group. Overall, a comparison of the pretest-posttest changes for an experimental group, which experienced the innovative strategy, with those for a control group, supported the efficacy of the innovative teaching methods in terms of learning environment perceptions, attitudes to mathematics, and mathematics concept development. The results of simple correlation and multiple correlation analyses of outcome-environment associations for two units of analysis clearly indicated that there is an association between the learning environment and students’ attitudes and mathematics achievement for this group of middle-school mathematics students.In particular, there is a positive and statistically significant correlation between: Normality of Mathematicians and Student Negotiation, Involvement, and Task Orientation with the individual as the unit of analysis; Enjoyment of Mathematics and all three CLES and three WIHIC scales with the student as a unit of analysis, and for the four scales of Personal Relevance, Shared Control, Involvement, and Task Orientation with the class mean as the unit of analysis. The multiple correlations between the group of three CLES and three WIHIC scales and each of the two TOMRA scales are statistically significant for the individual as a unit of analysis. Overall, the study revealed positive and statistically significant associations between the classroom learning environment and students’ attitudes to mathematics. A two-way MANOVA with repeated measures on one factor was utilized to investigate gender differences in terms of students’ perceptions of classroom environment and attitudes to mathematics, as well as mathematics achievement. A statistically significant but small difference was found between the genders for Student Negotiation and Task Orientation. Female students perceived their mathematics classrooms somewhat more positively than did the male students. There was no statistically significant difference between the genders on achievement and students’ attitudes to mathematics. Qualitative information, gathered through audiotaped interviews, students’ journal, and analysis of students’ work, was used to clarify students’ opinions about the new approach, classroom environment perceptions, attitudes, and conceptual development.These qualitative information-gathering tools were utilized to obtain a more in-depth understanding of the learning environments (Tobin, Kahle, & Fraser, 1990) and the results of my study (Punch, 1998), as well as insights into students’ perceptions (Spinner & Fraser, 2005). The responses from the students’ interviews and students’ reflective journals from the group that experienced the innovative methods generally suggested that introducing Cramer’s rule as a method for solving systems of linear equations in the middle school can be beneficial and therefore might be considered for inclusion in the middle-school Algebra 1 curriculum more widely in California. Using only quantitative data would not have provided the richness that was derived from using mixed methods (Johnson & Onwuegbuzie, 2004). Therefore, qualitative data obtained from students who experienced the innovative method generally supported the quantitative findings concerning the effectiveness of this method for teaching and learning systems of linear equations.