The role of instructional representations on students' written representations and achievements

Abstract

This research is based on Middle School Mathematics Project (MSMP) funded by the Interagency Educational Research Initiative through a grant to the American Association for the Advancement of Science. Both teacher’s instructional representations and students’ written representations were coded and analyzed to investigate the nature and structure of the representations in teaching fractions, decimals and percents in middle school classrooms in three school districts in Texas. The study further explored the relationship between both the quality and quantity of instructional representations and students’ written representations, and the relationship between students’ written representations and their achievements. This dissertation used a mixed approach utilizing both quantitative and qualitative methods. The data was collected in the first two years of a five-year study. A total of 14 sixth grade mathematics teachers from three school districts in Texas were selected from the MSMP project. Before the actual videotaping procedure, a professional development focusing on multiple representations was held for the teachers. Both pretests and posttests were used to examine the relationship between the structure of students’ written representations and their achievements. The results showed that the both the quantity and quality of teachers’ instructional representations varied a lot. Symbolic representations were the predominant representations in classroom teaching. Structures of instructional representations converge to content sub-constructs rather than format sub-constructs. Here subconstructs include part-whole, measure, quotient, multiplication by one and cross product. Instead, format sub-constructs include real world, manipulatives, pictures, spoken symbolic representations and written symbolic representations, however, connections between these sub-constructs were not statistically significant. Within the three content sub-constructs (part-whole, quotient, and multiplication by one) that revealed by students’ written representations, quotient and multiplication by one significantly predicated the students’ posttest scores. It was also found that, among the three quality criteria (accuracy, comprehensibility and connections) of instructional representations, the comprehensibility score significantly predicated students’ achievement in the posttests.