A classroom teaching experiment was conducted in a semester-long undergraduate mathematics content course for elementary education majors. Preservice elementary teachers' development of rational number understanding was documented through the social and psychological perspectives. In addition, social and sociomathematical norms were documented as part of the classroom structure. A hypothetical learning trajectory and instructional sequence were created from a combination of previous research with children and adults. Transcripts from each class session were analyzed to determine the social and sociomathematical norms as well as the classroom mathematical practices. The social norms established included a) explaining and justifying solutions and solution processes, b) making sense of others' explanations and justifications, c) questioning others when misunderstandings occur, and d) helping others. The sociomathematical norms established included determining what constitutes a) an acceptable solution and b) a different solution. The classroom mathematical practices established included ideas related to a) defining fractions, b) defining the whole, c) partitioning, d) unitizing, e) finding equivalent fractions, f) comparing and ordering fractions, g) adding and subtracting fractions, and h) multiplying fractions. The analysis of individual students' contributions included analyzing the transcripts to determine the ways in which individuals participated in the establishment of the practices. Individuals contributed to the practices by a) introducing ideas and b) sustaining ideas. The transcripts and student work samples were analyzed to determine the ways in which the social classroom environment impacted student learning. Student learning was affected when a) ideas were rejected and b) ideas were accepted. As a result of the data analysis, the hypothetical learning trajectory was refined to include four phases of learning instead of five. In addition, the instructional sequence was refined to include more focus on ratios. Two activities, the number line and between activities, were suggested to be deleted because they did not contribute to students' development.

<p> This thesis examines continued fraction expansions of the square root of nonsquare positive integers of periods one to six, and shows their relationships with Newton's method of approximation. It also contains known results concerning continued fractions.</p> / Thesis / Master of Science (MSc)

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No description available.

In the basic understanding of measurement, students are expected to be able to subdivide a given whole into a unit and then change the position of that unit along the entire length of the whole. These basic operations of subdivision and change of position are related to the more formal operations of partitioning and iterating. In the context of fractions, partitioning and iterating play a fundamental role in understanding fractions as measures, where students are expected to partition a whole into an iterable unit. In the context of angle measurement, students are expected to measure angles as a fractional amount of a full rotation or a circle, by partitioning the circle into a unit angle and then iterating that unit angle to find the measure of the given angle. Despite this link between measurement, fractions, and angles, research suggests that there is a disconnect between students' concepts of measurement and geometry concepts, including angle and angle measurement. Therefore, one area of study that might help us understand this disconnection would be to investigate the relationship between students' concepts of measurement and their concepts of angle measurement. This current study documents sixth, seventh, and eighth grade students' measurement schemes for fractions and their quantifications of angularity, and then investigates the relationship between them. This research is guided by the following question: What is the relationship between middle school students' measurement schemes for fractions and their quantifications of angularity? Results indicate that the majority of students involved in this study do not possess a measurement concept of fractions nor a measurement concept of angularity. However, these results demonstrate that there is a relationship between students' measurement schemes for fractions and their quantifications of angularity. It is concluded that students who construct more sophisticated fraction schemes tend to construct more sophisticated quantifications of angularity. / Doctor of Philosophy / Although the concepts of measurement, fractions, and angle measurement are related, research suggests that there is a disconnect between students' concepts of measurement and geometry concepts, including angle and angle measurement. Therefore, one area of study that might help us understand this disconnection would be to investigate the relationship between students' concepts of measurement and their concepts of angle measurement. This current study documents sixth, seventh, and eighth grade students' understandings of measurement, as indicated by their fraction schemes, and angle measurement, as indicated by how they quantify angularity or the openness of an angle. This study then investigates the relationship between them. This research is guided by the following question: What is the relationship between middle school students' measurement schemes for fractions and their quantifications of angularity? Results indicate that the majority of students involved in this study do not possess a measurement concept of fractions nor a measurement concept of angularity. However, these results demonstrate that there is a relationship between students' measurement schemes for fractions and their quantifications of angularity. It is concluded that students who construct more sophisticated fraction schemes tend to construct more sophisticated quantifications of angularity.

no abstract provided by author / Master of Science

Ce travail de thèse s'inscrit dans le cadre de la valorisation des co-produits issus des industries de transformation et de conserve de thon. Il porte sur la mise en oeuvre de l'hydrolyse enzymatique et des procédés de séparation membranaires en vue d'obtenir des composés d'intérêt comme les peptides et les acides aminés. Les techniques utilisées dans ce travail font partie des « technologie propre », visant une dépense énergétique et un investissement modérés. L'hydrolyse enzymatique a été menée pour identifier l'efficacité des enzymes sur la matrice afin de déterminer les conditions optimales d'hydrolyse en utilisant la méthodologie des plans d'expériences avec pour objectif l'enrichissement de la phase soluble en peptides de petite taille dotés d'activités biologiques intéressantes. Le fractionnement par la taille par ultrafiltration et Nanofiltration a été utilisé comme un procédé pour agir sur la distribution de la taille moléculaire de la population peptidique et sur les propriétés d'hydrolysat produit. Tout d'abord, un fractionnement à petite échelle a été réalisé avec des membranes de différents seuils de coupure et de différentes natures (organique et inorganique) afin de sélectionner la meilleure membrane répondant à notre objectif.Ensuite une étude d'optimisation des conditions opératoire a été réalisée pour les deux étapes de fractionnement UF et NF de manière à déterminer les meilleures conditions séparatives en utilisant un hydrolysat commercial. La validation du procédé de fractionnement en utilisant l'hydrolysat des co-produits de thon a été effectuée. Durant cette étude, différents modes de fractionnement utilisant différentes combinaisons d'UF et NFont été testés pour déterminer le meilleur procédé permettant la récupération du maximum de fractions peptidiques intéressantes. L'originalité de ce travail de thèse réside d'une part, dans l'enrichissement de l'hydrolysat des co-produit de thon en composés valorisables comme les acides aminés essentiels ainsi que les peptides dotés de valeur ajoutée et d'autre part, la mise en place d'un procédé de fractionnement pour la récupération des différentes fractions afin de mettre en évidence la conception d'un bioréacteurs enzymatique couplé à la technique membranaire. / This work is performed in the framework of up-grading of tuna by-products generated from processing and conditioning industries. The enzymatic hydrolysis combined with membrane separation processes in order to obtain the fraction of interest peptides and amino acids was studied. The optimal conditions during enzymatic hydrolysis were determined using the methodology of design of experiments in order to enrich the soluble phase in small peptides with interesting biological activities. The fractionation by Ultrafiltration and Nanofiltration following a suitable combination was studied. For this, firstly, a small-scale fractionation was performed with membranes of different cut-off and different natures (organic and inorganic) to select the best membrane processes combination and to optimize the used conditions. Then, a validation study of the fractionation using the hydrolysate of tuna by-products produced during was performed. In this study, different modes of fractionation combination of concentration and diafiltration steps were tested to determine the best method for the recovery of large quantities of interesting peptide fractions. The originality of this PhD work is the enrichment of the tuna by-products hydrolysate with valuable compounds such as essential amino acids and peptides with a high biological activity.

O presente trabalho tem como objetivo trazer uma proposta de aplicação de uma sequência didática para o ensino de frações no fundamental II, utilizando uma narrativa com elementos históricos para motivar os alunos a participarem da sequência. Apresentamos tambem algumas discussões sobre a pertinência do ensino das frações na atualidade, tendo como argumentos de partida alguns trabalhos apresentados por matemáticos, como Peter Hilton e Carlos Roberto Vianna. que discutiram existir a possibilidade da retirada das frações do currículo escolar. Utilizaremos uma abordagem histórica e conceitual das frações, através de duas dimesões temporais, a historiográfica e a dos tempos atuais, mostrando assim sua importância histórica. Baseamo-nos nos diferentes significados das frações: parte-todo, medida, quociente e operador multiplicativo. Por fim, oferecemos uma estratégia para que a aprendizagem de frações aconteça de forma conceitual e significativa. Tal estratégia foi elaborada através da análise teórica e da construção de uma sequência didática (SD), baseada nos princípios das situações didáticas de Guy Brousseau (Teoria da Situação Didática). A sequência foi utilizada em turmas do 6º ano do ensino básico. Buscamos, com isso, criar um ambiente de motivação para aprendizagem da matemática e, ao mesmo tempo, conceder significado aos conhecimentos relacionados às frações. / The present work aims to bring a proposal for the application of a didactic sequence for the teaching of fractions in the fundamental II segment, using a narrative with historical elements to motivate the students to participate in the sequence. We present some discussions about the pertinence of the teaching of fractions in the present time, starting with research presented by mathematicians such as Peter Hilton and Carlos Roberto Vianna, who discussed the possibility of the removal of fractions from the school curriculum. We will use a historical and conceptual approach of the fractions, through two temporal dimensions, the historiographic one and the one of the present times, thus showing its historical importance. We are based on the different meanings of fractions: part-whole, measure, quotient and multiplicative operator. Finally, we offer a strategy for the learning of fractions to happen in a conceptual and meaningful way, being so that such strategy was elaborated through the theoretical analysis and elaboration of a didactic sequence (SD) based on the principles of didactic situations offered by Guy Brousseau (Theory of Didactic Situation). The sequence was used in classes of the 6th grade of the elementary school.. We seek to create a motivational environment for learning mathematics, at the same time giving meaning to the knowledge related to fractions.

O presente trabalho tem como objetivo trazer uma proposta de aplicação de uma sequência didática para o ensino de frações no fundamental II, utilizando uma narrativa com elementos históricos para motivar os alunos a participarem da sequência. Apresentamos tambem algumas discussões sobre a pertinência do ensino das frações na atualidade, tendo como argumentos de partida alguns trabalhos apresentados por matemáticos, como Peter Hilton e Carlos Roberto Vianna. que discutiram existir a possibilidade da retirada das frações do currículo escolar. Utilizaremos uma abordagem histórica e conceitual das frações, através de duas dimesões temporais, a historiográfica e a dos tempos atuais, mostrando assim sua importância histórica. Baseamo-nos nos diferentes significados das frações: parte-todo, medida, quociente e operador multiplicativo. Por fim, oferecemos uma estratégia para que a aprendizagem de frações aconteça de forma conceitual e significativa. Tal estratégia foi elaborada através da análise teórica e da construção de uma sequência didática (SD), baseada nos princípios das situações didáticas de Guy Brousseau (Teoria da Situação Didática). A sequência foi utilizada em turmas do 6º ano do ensino básico. Buscamos, com isso, criar um ambiente de motivação para aprendizagem da matemática e, ao mesmo tempo, conceder significado aos conhecimentos relacionados às frações. / The present work aims to bring a proposal for the application of a didactic sequence for the teaching of fractions in the fundamental II segment, using a narrative with historical elements to motivate the students to participate in the sequence. We present some discussions about the pertinence of the teaching of fractions in the present time, starting with research presented by mathematicians such as Peter Hilton and Carlos Roberto Vianna, who discussed the possibility of the removal of fractions from the school curriculum. We will use a historical and conceptual approach of the fractions, through two temporal dimensions, the historiographic one and the one of the present times, thus showing its historical importance. We are based on the different meanings of fractions: part-whole, measure, quotient and multiplicative operator. Finally, we offer a strategy for the learning of fractions to happen in a conceptual and meaningful way, being so that such strategy was elaborated through the theoretical analysis and elaboration of a didactic sequence (SD) based on the principles of didactic situations offered by Guy Brousseau (Theory of Didactic Situation). The sequence was used in classes of the 6th grade of the elementary school.. We seek to create a motivational environment for learning mathematics, at the same time giving meaning to the knowledge related to fractions.