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.

Thesis (doctoral)--Grossherzoglich Hessischen Ludwigs-Universität zu Giessen, 1910. / Vita.

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Typescript (photocopy) / viii, 245 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.) Dept. of Mathematical Physics, University of Adelaide, 1982

Thesis (Ph.D.) Dept. of Mathematical Physics, University of Adelaide, 1982. / Typescript (photocopy).

Thesis (Ph. D.)--School of Mathematics and Statistics, Faculty of Science, University of Sydney, 2007. / Title from title screen (viewed 13 January 2009). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliographical references. Also available in print form.

The matrices of the inequivalent irreducible representations of the space group D[symbol omitted] (Pnma in international symbols) are derived. The allowable representations of the groups of the k-vectors are obtained from certain ray representations of the corresponding point groups. From these allowable representations the irreducible representations of the entire space group are induced. The results are presented in a systematically arranged set of tables. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

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In this thesis, we classify all irreducible admissible mod-p representations ofGL2(Qp)and divide those into 4 exhaustive disjoint categories. This classification was first pioneered by Barthel-Livne (1994-95), and completed by Breuil (2003). This thesis is mainly based on the lecture notes by Herzig (2015).

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