This study examined teachers' learning and understanding of mathematical representation through the Middle School Mathematics Project (MSMP) professional development, investigated teachers' use of mathematics representations in teaching fractions and algebraic functions, and addressed patterns of teachers' changes in learning and using representation corresponding to Teachers' Zone of Proximal Development (TZPD). Using a qualitative research design, data were collected over a 2-year period, from eleven participating 6th and 7th grade mathematics teachers from four school districts in Texas in a research-designed professional development workshop that focused on helping teachers understand and use of mathematical representations. Teachers were given two questionnaires and had lessons videotaped before and after the workshop, a survey before the workshop, and learning and discussion videotapes during the workshop. In addition, ten teachers were interviewed to find out the patterns of their changes in learning and using mathematics representations. The results show that all teachers have levels of TZPD which can move to a higher level with the help of capable others. Teachers' knowledge growth is measurable and follows a sequential order of TZPD. Teachers will make transitions once they grasp the specific content and strategies in mathematics representation. The patterns of teacher change depend on their learning and use of mathematics representations and their beliefs about them. This study advocates teachers using mathematics representations as a tool in making connections between concrete and abstract understanding. Teachers should understand and be able to develop multiple representations to facilitate students' conceptual understanding without relying on any one particular representation. They must focus on the conceptual developmental transformation from one representation to another. They should also understand their students' appropriate development levels in mathematical representations. The findings suggest that TZPD can be used as an approach in professional development to design programs for effecting teacher changes. Professional developers should provide teachers with opportunities to interact with peers and reflect on their teaching. More importantly, teachers' differences in beliefs and backgrounds must be considered when designing professional development. In addition, professional development should focus on roles and strategies of representations, with ongoing and sustained support for teachers as they integrate representation strategies into their daily teaching.

A new continuous one-dimensional sedimentation model incorporating a new continuous flocculation model that considers aggregation and fragmentation processes was derived and tested. Additionally, a new procedure to model sediment particle size distribution (PSD) was derived. Basic to this development were three different parametric models: Jaky, Fredlund and the Gamma probability distribution (GPD) were chosen to fit three different glass micro-spheres PSDs having average particle sizes of 7, 25 and 35 microns. The GPD provided the best fit with the least parameters. The bimodal GPD was used to fit ten sediment samples with excellent results (< 5% average error). A continuous flocculation model was derived using the method of moments for solving the continuous Smoluchowski coagulation equation with fragmentation. The initial sediment PSD was modeled using a bimodal GPD. This new flocculation model resulted in a new general moments’ equation that considers aggregation and fragmentation processes, which is represented by a system of ordinary differential equations. The model was calibrated using a genetic algorithm with initial and flocculated PSDs of four sediment samples and four anionic polyacrylamides flocculants. The results show excellent correlation between predicted and observed values (R2 > 0.9878). A new continuous one-dimensional sedimentation model that resulted in a scalar hyperbolic conservation law was derived from the well-known Kynch kinematic sedimentation model. The model was calibrated using column tests results with glass micro-spheres particles. Two different glass microspheres particle size distributions (PSDs) were used with average diameters of 7 and 37 microns. Excellent values of coefficient of determination (R2 > 0.89, except for one test replicate) were obtained for both the small and large glass micro-spheres PSDs. These results suggest that the proposed sedimentation model can be expanded to model the sedimentation process inside a sediment pond.

Made available in DSpace on 2016-04-27T16:58:43Z (GMT). No. of bitstreams: 1 Sergio Antonio Wielewski.pdf: 5622287 bytes, checksum: f8af86ba1d6cfaf4c528be38a00df366 (MD5) Previous issue date: 2008-09-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This doctoral thesis contains theoretical discussions as well as results of an empirical study. The general starting point has been the thesis that our mathematical thinking is largely ruled by certain dualities or complementarities of which that between the representational and instrumental aspects of concepts is best known. Ernst Cassirer, presents in his famous book Substanzbegriff und Funktionsbegriff (Substance and Function) of 1910 the general thesis that the historical development of science could be described as a transition from merely referential Aristotelian concepts to operative concepts or functions. The very same duality has been discussed widely in mathematics education starting from the work of Richard Skemp. Our first goal has consequently been to find connections between Cassirer and Skemp. The discussion of these connections and differences leads then in a second part of the thesis to a presentation of the results of an empirical case study with fourteen participants. These had been confronted with a number of problem situations and their problem solving activities have afterwards been analyzed in terms of the aforementioned complementarity between relational and operative thinking / Nesta tese estão apresentados resultados de investigação teórica e empíricos. O alvo da pesquisa é identificação de características e análise das reflexões relativas a dualidades inerentes ao pensamento matemático. Tomou-se como pressuposto que o conhecimento de dualidades do pensamento matemático, e o como se utilizar desse conhecimento, se num sentido de complementaridade, seja relevante para o processo de ensino e aprendizagem da Matemática. A referência inicial do estudo foi a obra de Ernst Cassirer, Substance and Function (1910). Nessa obra é apresentado o desenvolvimento histórico da teoria do conceito de Aristóteles ao século XIX, isto é, desenvolvimento esse que vai das propriedades de substância à noção de função. Cassirer,como neo-kantiano, dá forte ênfase aos aspectos operativos e instrumentais do conceito. Na continuidade do estudo é destacado a fundamental importância de um conceito teórico ser compreendido nos termos de uma dualidade, em seus aspectos operativos e referencial. O trabalho didático de Richard Skemp é outro que explora dualidade semelhante. Trata -se da dualidade de aprender e de compreender, que Skemp chama de compreensão instrumental e relacional. Nossa investigação centra-se então na busca de conexão entre as concepções de Cassirer e Skemp. Para tal levamos em conta aspectos educacionais, reflexões filosóficas e pedagógicas, postura profissional do educador, exemplos de situações a-didáticas e didáticas. Esses aspectos, reflexões e exemplos nortearam a exploração empírica desta tese. Esta exploração teve o caráter de uma pesquisa qualitativa, tendo sido desenvolvidas atividades didáticas. O objetivo dessas atividades era avaliar a utilização pelos sujeitos do pensamente relacional e do pensamento instrumental

Made available in DSpace on 2016-04-27T16:57:16Z (GMT). No. of bitstreams: 1 Aspectos do Pens Matematico na RP-Segurity printing.pdf: 2060132 bytes, checksum: c0711be8b2de82245dd639a3d704a6be (MD5) Previous issue date: 2005-11-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This doctorate thesis aims to identify the characteristics and the dimensions of mathematical thinking in experimental and theoretical terms which may be useful to teachers with respect to teaching processes, development of mathematical ideas and the delineation of learning contexts. Our study began with a detailed analysis of the work of Krutetskii (1968). This book is very rich in theoretical examples and reflections. It is, however, a completely psychological work and provided few indications of the more general mathematic knowledge and thinking. For this reason we added detailed information about the work of other authors such as Gowers, Poincaré, Boutroux, Otte and Kurz, and this added other dimensions which assisted our understanding of the nature of mathematics. These authors were concerned with the problems of cognitive styles, cultural and historical differences, differences that are the results of mathematics itself and distinctive ways of representing mathematics. The experimental dimensions consisted of analysis of data obtained from qualitative research with students whereby one was taken from the literature (Krutetskii) and the other an exploratory survey which we carried out for the purposes of this thesis. Krutetskii carried out an experimental investigation involving 201 Russian students with different mathematic abilities, attending elementary school. These students were presented with a number of different series of mathematic problems and their mathematic abilities were observed during the problem solving process. In our survey we carried out case studies exploring mathematic problem solving involving 13 students from the Federal University of Mato Grosso with 9 students from the Mathematics/Education course and 4 students from the Computer Sciences Course. The exploratory survey was organized into 3 phases. The first was the completion of a questionnaire with subjective questions about Mathematics and preferred ways of thinking and dealing with this subject. The second phase was reserved for the solution of 13 varied mathematical problems. The final phase was the completion of another questionnaire with subjective questions which sought to obtain information about the experiences of the students when solving the problems set. With our exploratory survey we were able to document and verify several parameters and characteristics of mathematical thinking which were described in the theoretical chapters as well as being able identify the problems themselves and the experience of solving them also influenced mathematical thinking. As a general result we concluded that mathematical thinking must be considered in the light of different parameters since this can help to characterize more complete mathematical thinking / A presente Tese de Doutorado pretende indicar características e dimensões do pensamento matemático, em termos teóricos e experimentais, que podem ser úteis aos professores no que se refere aos processos de ensino, ao desenvolvimento de idéias matemáticas e ao delineamento de contextos de aprendizagem. Nosso estudo começou com uma análise detalhada do trabalho de Krutetskii (1968). Esse livro é muito rico em exemplos e reflexões teóricas. No entanto, é um trabalho completamente psicológico e forneceu poucas indicações a respeito dos aspectos mais gerais do conhecimento matemático e do pensamento matemático. Por esse motivo, adicionamos informações detalhadas sobre o trabalho de outros autores como Gowers, Poincaré, Boutroux, Otte e Kurz que acrescentaram outras dimensões que auxiliaram a nossa compreensão da natureza da Matemática. Esses autores se preocuparam com problemas de estilos cognitivos, de diferenças culturais e históricas, de diferenças que são resultados das várias áreas da própria Matemática e distintas formas de representação na Matemática. As dimensões experimentais consistiram na análise de dados obtidos em pesquisas qualitativas com estudantes, sendo uma da literatura (Krutetskii) e outra uma pesquisa exploratória realizada por nós para a presente Tese. Krutetskii realizou uma investigação experimental envolvendo 201 estudantes russos do Ensino Fundamental, com diferentes habilidades matemáticas. A esses estudantes foram propostas diversas séries de problemas matemáticos, em que foram observadas suas habilidades matemáticas durante o processo de resolução. Na nossa pesquisa, realizamos estudos de caso exploratório na resolução de problemas matemáticos envolvendo 13 estudantes da Universidade Federal de Mato Grosso, sendo 09 do Curso de Licenciatura Plena em Matemática e 04 do Curso de Ciências da Computação. A pesquisa exploratória foi organizada em três momentos. O primeiro foi destinado a responder um questionário com perguntas subjetivas acerca da Matemática e de preferências na forma de pensar e de lidar com a mesma. O segundo momento foi reservado para a resolução de 13 problemas matemáticos variados. E o último momento foi destinado para responder a outro questionário com perguntas subjetivas que procurava obter informações sobre a experiência dos estudantes na atividade de resolução dos problemas propostos. Com a nossa pesquisa exploratória pudemos documentar e verificar vários parâmetros e características do pensamento matemático que foram descritos nos capítulos teóricos, bem como identificar que os próprios problemas e as experiências com a resolução dos mesmos também influenciam o pensamento matemático. Como resultado geral, concluímos que o pensamento matemático deve ser considerado sob diferentes parâmetros, pois eles podem auxiliar na caracterização mais completa do pensamento matemático

Les rapports d’activité de l’Établissement Français du Sang (EFS) font état d’une demande croissante de produits sanguins labiles (PSL) tels les concentrés globules rouges (CGR), les plaquettes, et le plasma. Afin d’assurer la demande vitale en PSL, il est primordial d’optimiser la logistique liée aux activités de collecte du sang et de ses composants. Pour faire face à cette situation, l’EFS Auvergne-Loire mène une réflexion dans le but d’utiliser de manière plus efficiente les dispositifs de collecte en sites fixes et mobiles pour améliorer (i) la qualité de service rendue au donneur, et (ii) l’efficience de l’utilisation des ressources humaines. Dans ce contexte nous avons développé dans cette thèse des outils opérationnels pour (i) la modélisation des dispositifs de collecte, (ii) la régulation des flux de donneurs, et (iii) la planification de collectes mobiles.La méthode d'analyse des dispositifs de collecte est basée sur des techniques de simulation à événements discrets. Une modélisation préalable des flux de donneurs dans les systèmes de collecte en sites fixes et mobiles à l’aide de réseaux de Petri a été proposée. Pour la régulation de flux de donneurs, notamment pour la planification optimale des rendez-vous des donneurs et la planification de la capacité dans les systèmes de collecte au site fixe, deux approches ont été abordées: (a) Construction d'un algorithme basée sur techniques d'optimisation stochastique via simulation ; (b) Programmation mathématique: Modèle de programmation en nombres entiers non-linéaire (MINLP) basée sur réseaux de files d'attente et représentation et évaluation des systèmes à événements discrets à travers de programmation mathématique. Pour la planification de collectes mobiles. Deux types de modèles ont été développés : (a) Au niveau tactique : Modèles de programmation en nombres entiers linéaire (MIP) pour planifier les semaines de collectes pour chaque ensemble disponible sur un horizon de temps pour garantir l'autosuffisance à niveau régional des CGR. (b) Au niveau opérationnel : Modèle de programmation en nombres entiers linéaire (MIP) pour l’organisation du travail des équipes en charge de la collecte. / Activity reports of the French Blood Establishment (EFS) indicate a growing demand for Labile Blood Products (LBP) as red blood cells (RBC), platelets and plasma. To ensure the vital demand of labile blood products (LBP), it’s essential to optimize the logistics related with the collection of blood components. To deal with this situation, the EFS Auvergne-Loire carry out a reflection in order to use more efficiently the collection devices in fixed and mobile sites, to improve the quality of service offered to the donor and the efficiency of human resources. In this context we have developed in this thesis operational tools for (i) modeling of blood collection devices (ii) The regulation of flows donors (iii) Planning of bloodmobile collections.The method analysis of collection devices is based on techniques of discrete event simulation. A preliminary modeling of donors’ flow in fixed and mobile collection systems using Petri nets was conducted. For the regulation of flow of donors, i.e. the optimal capacity planning and appointment scheduling of blood collections, two approaches were considered: (a) Simulation based-optimization.(b) Mathematical Programming: Mixed integer nonlinear programming (MINLP) based on queuing networks and mathematical programming representation of discrete event systems. For planning of bloodmobile collections. Two models have been developed: (a) At the tactical level: Mixed integer linear programming (MIP) to determine the weeks in which the mobile collection must be organized in order to ensure the regional self-sufficiency of RBC. (b) At the operational level: Mixed integer linear programming (MIP) for the planning of human resources in charge of blood collections.