abstract: Researchers have described two fundamental conceptualizations for division, known as partitive and quotitive division. Partitive division is the conceptualization of a÷b as the amount of something per copy such that b copies of this amount yield the amount a. Quotitive division is the conceptualization of a÷b as the number of copies of the amount b that yield the amount a. Researchers have identified many cognitive obstacles that have inhibited the development of robust meanings for division involving non-whole values, while other researchers have commented on the challenges related to such development. Regarding division with fractions, much research has been devoted to quotitive conceptualizations of division, or on symbolic manipulation of variables. Research and curricular activities have largely avoided the study and development of partitive conceptualizations involving fractions, as well as their connection to the invert-and-multiply algorithm. In this dissertation study, I investigated six middle school mathematics teachers’ meanings related to partitive conceptualizations of division over the positive rational numbers. I also investigated the impact of an intervention that I designed with the intent of advancing one of these teachers’ meanings. My findings suggested that the primary cognitive obstacles were difficulties with maintaining multiple levels of units, weak quantitative meanings for fractional multipliers, and an unawareness of (and confusion due to) the two quantitative conceptualizations of division. As a product of this study, I developed a framework for characterizing robust meanings for division, indicated directions for future research, and shared implications for curriculum and instruction. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2019

This thesis is essentially about the theory of the transition to socialism and aims at comprehending the Soviet experience of attempting to re-mould the relations of production and the forces of production mainly at the level of Soviet labour processes. The main finding is that the Soviet Union was rich in debate in the 20s but short in actually pioneering new labour processes in the years of construction and accelerated industrialization. As a by-product of this larger project, the thesis has begun to demarcate the division of labour from technology by constructing models on the basis of their shared characteristics of efficiency, output, cost, control and welfare. It is argued- that an attempt to 'abolish' the fragmentation of tasks arising from the technical . division of labour would require a redirection of the physical organization of technology itself. The thesis emphasizes the importance not only the re-arranging of macro-societal-level social relations but also micro production unit level changes. The socialist project ought to include simultaneous interventions at the micro and macro levels of reality. It is suggested that despotic control at both the society and production-unit levels would hardly expand the emancipatory possibilities for labour. Nor democratization at one or the other level alone whilst keeping despotic control will do. Only simultaneous democratizations at the enterprise and society levels will make the socialist development attractive. The thesis concludes by stressing the need to reinstate democracy in relation to the organization of the labour process pointing out the responsibility of the Gorbachov leadership in relation to the democratization of Soviet society and enterprises.

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published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

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