Browsing by Subject "representations"
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Item Converting among fractions, decimals, and percents: an exploration of representational usage by middle school teachers(2009-05-15) Muzheve, Michael TapfumaUsing both quantitative and qualitative data collection and analyses techniques, this study examined representations used by sixteen (n = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. The classroom videos used for this study were recorded as part of the Middle School Mathematics Project (MSMP). The study also compared teacher-selected and textbook representations and examined how teachers? use of idiosyncratic representations influenced representational choices on the number test by the teachers? five hundred eighty-one (N = 581) students. In addition to using geometric figures and manipulatives, a majority of the teachers used natural language such as the words nanny, north, neighbor, dog, cowboy, and house to characterize fractions and mathematical procedures or algorithms. Coding of teacher-selected representations showed that verbal representations deviated from textbook representations the most. Some teachers used the words or phrases bigger, smaller, doubling, tripling, breaking-down, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers, such as equations with missing or double equal signs, numbers and operators written as superscripts, and numbers written above and below the equal sign. Although use of idiosyncratic representations by teachers influenced representational choices by students on the number test, no evidence of a relationship between representational forms and degree of correctness of solutions was found. The study did reveal though that teachers? use of idiosyncratic representations can lead to student misconceptions such as thinking that multiplying by a whole number not equal to 1 gives an equivalent fraction. Statistical tests were done to determine if frequency of representation usage by teachers was related to the textbook, highest degree obtained by teacher, certification, number of years spent teaching mathematics, number of years teaching mathematics at grade level, number of hours completed on professional development related to their textbook, and total number of days spent on the Interagency Education Research Initiative (IERI) professional development. The results showed representation usage was related to all the above variables, except the highest degree obtained and the total number of days spent on the IERI professional development.Item The role of instructional representations on students' written representations and achievements(Texas A&M University, 2006-10-30) Sun, YeThis 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.