La présente étude intitulée « utilisation des technologies de l’information et de la communication dans l’enseignement secondaire et développement des compétences des élèves en résolution de problèmes mathématiques au Burkina Faso » est une recherche descriptive de type mixte examinant à la fois des données qualitatives et quantitatives. Elle examine les compétences en résolution de problèmes mathématiques d’élèves du Burkina Faso pour révéler d’éventuelles relations entre celles-ci et l’utilisation des TIC par les élèves ou leur enseignant de mathématiques. L’intérêt de cette recherche est de fournir des informations aussi bien sur la réalité des TIC dans l’enseignement secondaire au Burkina que sur les effets de leur présence dans l’enseignement et l’apprentissage des mathématiques. Les éléments théoriques ayant servi à l’analyse des données sont présentés suivant trois directions : la résolution des problèmes, le développement des compétences, et les relations entre les TIC, le développement de compétences et la résolution de problèmes. Du croisement de ces éléments émergent trois axes pour le développement de la réponse apportée à la préoccupation de l’étude : 1) décrire l’utilisation de l’ordinateur par les élèves du Burkina Faso pour améliorer leur apprentissage des mathématiques ; 2) identifier des rapports éventuels entre l’utilisation de l’ordinateur par les élèves et leurs compétences en résolution de problèmes mathématiques ; 3) identifier des rapports entre les compétences TIC de l’enseignant de mathématiques et les compétences de ses élèves en résolution de problèmes. Les processus de la résolution de problèmes sont présentés selon l’approche gestaltiste qui les fait passer par une illumination et selon l’approche de la théorie de la communication qui les lie au type de problème. La résolution de problèmes mathématiques passe par des étapes caractéristiques qui déterminent la compétence du sujet. Le concept de compétence est présenté selon l’approche de Le Boterf. Les données révèlent que les élèves du Burkina Faso utilisent l’ordinateur selon une logique transmissive en le considérant comme un répétiteur suppléant de l’enseignant. Par la suite, il n’y a pas de différence significative dans les compétences en résolution de problèmes mathématiques entre les élèves utilisant l’ordinateur et ceux qui ne l’utilisent pas. De même, l’étude révèle que les enseignants présentant des compétences TIC n’ont pas des élèves plus compétents en résolution de problèmes mathématiques que ceux de leurs collègues qui n’ont pas de compétences TIC. / This study entitled "Use of information and communication technologies in secondary education and development of students in mathematical problems solving skills in Burkina Faso" is a mixed descriptive research examining both qualitative and quantitative data. It examines the math problem solving skills of students from Burkina Faso to reveal possible relationships between them and the use of ICT by students or teachers in mathematics. The interest of this research is to provide information as well on the reality of ICT in secondary education in Burkina Faso as on the effect of their presence in teaching and learning mathematics. The theoretical elements used for data analysis are presented in three areas: problem solving, skills development and the relationship between ICT, skills development and problem solving. From crossing these areas rose three directions in witch the response to the concern of the study is presented: 1) describe the use of computers by students from Burkina Faso to improve their learning of mathematics; 2) identify possible relationship between the use of computer by students and their mathematical problems solving skills; 3) identify relationships between mathematics teachers ICT skills and their students problems solving skills. The problem solving process is presented according the gestalt theory that takes it through an enlightenment, and according to the communication theory approach that links it to the type of problem to solve. Mathematical problems solving goes through characteristic stages that determine the subject’s abilities. The concept of competence is presented using the approach of Le Boterf. The data show that students from Burkina Faso are using the computer in a transmissive way taking it as a substitute for the teacher. Subsequently, in mathematical problem solving skills there is no meaningful difference between students who use computer and those who do not use it. Similarly, the study found that students whose teacher has ICT skills were not more competent in mathematical problems solving than those whose teacher does not have ICT skills.

Cette recherche, réalisée en milieu scolaire québécois, concerne l’enseignement et l’apprentissage de la géométrie à l’entrée au secondaire. Ce contexte est caractérisé par une géométrie non clairement définie d’un point de vue épistémologique, tant dans le programme d’études du premier cycle que dans les manuels scolaires. Ainsi, nous avons cherché à voir d’une part, l’activité géométrique souhaitée et actualisée par des enseignants incluant les problèmes proposés et, d’autre part, les conceptions d’élèves développées par ces problèmes. À partir de données recueillies auprès de quatre classes, nous avons déterminé cette activité géométrique et répertorié six types de problèmes dont quatre sont dominants ainsi que des conceptions d’élèves. L’activité géométrique en classe a donné lieu à des moments d’hésitation épistémologique, lesquels ne sont pas sans effet dans le développement des conceptions des élèves. / This research was conducted in a Quebec classroom environment. It pertains to the teaching and learning of geometry at the outset of secondary school. This context is characterized by a geometry that is not clearly defined from the epistemological point of view in either the secondary cycle one program or in textbooks. We attempted to find firstly, the desired geometric activity and updated by teachers with the proposed problems and, secondly, students conceptions developed by these problems. Using data collected from four classes, we then determined this geometric activity and identified six types of problems from which four were predominant. We also observed students conceptions. The classroom activity gave birth to moments of epistemological hesitance that may have had a certain effect on the development of the students’conceptions.

L'apprentissage de l’algèbre abstraite semble correspondre, pour les étudiants de niveau universitaire ou collégial, à l'introduction d'une multitude de nouveautés conceptuelles. Afin de mieux comprendre les raisons du taux d'échec important mesuré dans cette discipline, nous avons tenté de dégager les obstacles ou les difficultés rencontrés et nous les avons regroupés en quatre familles. Sur la base d'un exemple tiré d'une séquence d'introduction à l'algèbre abstraite et des productions des étudiants, nous relèverons que, en plus de devoir franchir un cap dans le niveau d'abstraction requis, les étudiants sont, souvent pour la première fois de leur parcours, confrontés à une théorie axiomatique développée comme telle, à des définitions de nature essentielle dont l'emploi va parfois à l'encontre du sens usuel, à l'absence de représentation graphique ainsi qu'à un processus de preuve formelle pour lequel ils n'ont été jusque-là que peu entraînés. / For university or college students, the learning of abstract algebra seems to involve a multitude of conceptual innovations. To better understand the reasons for the high failure rate in abstract algebra courses, we have aimed at identifying the obstacles or difficulties encountered and grouped them into four families. Based on an example from an introductory sequence in abstract algebra, we will show that in addition to having to reach an unprecedented level of abstraction, students, often for the first time in their mathematical instruction, have to face simultaneously an axiomatic theory developed with essential type definitions that seem to go against the usual meaning, a lack of graphical representation as well as a process of formal proof for which they had little to no training.

Master of Science / Department of Psychology / Gary L. Brase / In 2006, the Center for Disease Control and Prevention recommended the revision of state HIV testing laws. With these recommendations, more low-risk individuals are tested regardless of their risk group. However, there is a greater chance of a false positive test result for low-risk individuals than for high-risk individuals. Additionally, previous research found that doctors and HIV counselors in Germany did not accurately communicate the relationship between risk factors and false positive tests (Gigerenzer, Hoffrage, & Ebert, 1998). This study aimed to (1) compare the findings of the 1998 German sample to HIV hotline counselors in the United States in 2011; and (2) to investigate the ability of students to calculate the conditional probability of HIV for a low-risk individual after receiving a positive test, based on idealized transcripts of conversations with HIV hotline counselors. The first study found that HIV hotline counselors use both verbal expressions of risk and percentages to communicate HIV testing statistics. Additionally, 2011 American counselors were more aware of the chance of false positives and false negatives than compared to the 1998 German sample. However, no 2011 American counselors were able to provide an accurate positive predictive value for a low-risk woman. The second study found low performance among students in the calculation of the positive predictive value. Performance was facilitated by a natural frequency format for high numerate individuals. There were different patterns of results for the General Numeracy Scale and the Subjective Numeracy Scale. This would suggest that these two scales might be measuring different constructs. These findings are consistent with the two theories supporting the Frequency Effect, namely the Frequentist Hypothesis and the Nested Sets Hypothesis. Additionally, this research suggests computation of the conditional risk of HIV is facilitated by a natural frequency format. Teaching techniques have been developed and demonstrate long lasting improvement in health related computations. If a few hours of training is all that it takes to communicate these life and death statistics in a manner that is consistent with reasoning, health practitioners and students should be required to have more education in communicating and computing probabilities.

This study used multilevel measurement modeling to examine the differential difficulties of math and reading items for Grade 6 students participating in Ontario’s provincial assessment in 2005-2006, in relation to whether they received a setting accommodation, had a learning disability (LD), and spoke a language in addition to English. Both differences in difficulty between groups of students for all items (impact) and for individual items (differential item functioning) were examined. Students’ language backgrounds (whether they spoke a language in addition to English) were not significantly related to item difficulty. Compared to non-accommodated students with LD, math and reading items were relatively difficult for accommodated students with LD. Moreover, the difference in overall impact on math items was larger than on reading items for accommodated and non-accommodated students with LD. Overall, students without LD and who did not receive a setting accommodation outperformed students with LD and/or who received a setting accommodation as well as accommodated students without LD. It is important to note that, because this was an operational test administration, students were assigned to receive accommodations by their schools based on their individual needs. It is, therefore, not possible to separate the effect of the setting accommodation on item difficulty from the effects of other differences between the accommodated and non-accommodated groups. The differences in math and reading item difficulties between accommodated and non-accommodated students with LD may be due in part to factors such as comorbidity of LD and attention deficit hyperactivity disorder (ADHD) or a possible mismatch between the setting accommodation and the areas of disabilities. Moreover, the results of the present study support the underarousal/optimal stimulation hypothesis instead of the premise of the inhibitory control and attention for the use of setting accommodation. After controlling for the impact across all items of setting accommodation and LD, several math and reading items were found to exhibit differential item functioning (DIF). The possible sources of DIF were (1) math items that were not adherent to specific item-writing rules and (2) reading items targeting different types of comprehension. This study also found that the linguistic features of math items (total words, total sentences, average word length, monosyllabic words for math) and reading items (word frequency, average sentence length, and average words per sentence for reading) were associated with math and reading item difficulties for students with different characteristics. The total sentences and average word length in a math item as well as total words in a reading item significantly predicted the achievement gap between groups. Therefore, the linguistic features should be taken into account when assessments are developed and validated for examinees with varied characteristics.

This action-based, qualitative research project involving 7 early childhood educators working in a well-established preschool child care program examined the influences of job-embedded professional development on children’s experiences of mathematics. Data was collected through observations, journals, conversations, interviews, and surveys, and then analyzed using a grounded theory model. A number of themes emerged, the strongest being those related to teachers’ increased awareness, interpretation, and support of children’s explorations in mathematics during play. This project provides an example of a successful model of teacher development for early childhood educators, and contributes to the growing field of research in mathematics education related to teacher noticing, but at the preschool level. Further, with the introduction of full day kindergarten and the emphasis on play based learning this project provides many rich examples of the mathematics present in children's every day play that can be used in future teacher development.

The first part of the thesis deals with aspects of Loewner theory in several complex variables. First we show that a Loewner chain with minimal regularity assumptions (Df(0,t) of local bounded variation) satisfies an associated Loewner equation. Next we give a way of renormalizing a general Loewner chain so that it corresponds to the same increasing family of domains. To do this we will prove a generalization of the converse of Carathéodory's kernel convergence theorem. Next we address the problem of finding a Loewner chain solution to a given Loewner chain equation. The main result is a complete solution in the case when the infinitesimal generator satisfies Dh(0,t)=A where inf {Re<Az,z>: ||z| =1}> 0. We will see that the existence of a bounded solution depends on the real resonances of A, but there always exists a polynomially bounded solution. Finally we discuss some properties of classes of biholomorphic mappings associated to A-normalized Loewner chains. In particular we give a characterization of the compactness of the class of spirallike mappings in terms of the resonance of A. The second part of the thesis deals with the problem of finding examples of extreme points for some classes of mappings. We see that straightforward generalizations of one dimensional extreme functions give examples of extreme Carathéodory mappings and extreme starlike mappings on the polydisc, but not on the ball. We also find examples of extreme Carathéodory mappings on the ball starting from a known example of extreme Carathéodory function in higher dimensions.

This study used multilevel measurement modeling to examine the differential difficulties of math and reading items for Grade 6 students participating in Ontario’s provincial assessment in 2005-2006, in relation to whether they received a setting accommodation, had a learning disability (LD), and spoke a language in addition to English. Both differences in difficulty between groups of students for all items (impact) and for individual items (differential item functioning) were examined. Students’ language backgrounds (whether they spoke a language in addition to English) were not significantly related to item difficulty. Compared to non-accommodated students with LD, math and reading items were relatively difficult for accommodated students with LD. Moreover, the difference in overall impact on math items was larger than on reading items for accommodated and non-accommodated students with LD. Overall, students without LD and who did not receive a setting accommodation outperformed students with LD and/or who received a setting accommodation as well as accommodated students without LD. It is important to note that, because this was an operational test administration, students were assigned to receive accommodations by their schools based on their individual needs. It is, therefore, not possible to separate the effect of the setting accommodation on item difficulty from the effects of other differences between the accommodated and non-accommodated groups. The differences in math and reading item difficulties between accommodated and non-accommodated students with LD may be due in part to factors such as comorbidity of LD and attention deficit hyperactivity disorder (ADHD) or a possible mismatch between the setting accommodation and the areas of disabilities. Moreover, the results of the present study support the underarousal/optimal stimulation hypothesis instead of the premise of the inhibitory control and attention for the use of setting accommodation. After controlling for the impact across all items of setting accommodation and LD, several math and reading items were found to exhibit differential item functioning (DIF). The possible sources of DIF were (1) math items that were not adherent to specific item-writing rules and (2) reading items targeting different types of comprehension. This study also found that the linguistic features of math items (total words, total sentences, average word length, monosyllabic words for math) and reading items (word frequency, average sentence length, and average words per sentence for reading) were associated with math and reading item difficulties for students with different characteristics. The total sentences and average word length in a math item as well as total words in a reading item significantly predicted the achievement gap between groups. Therefore, the linguistic features should be taken into account when assessments are developed and validated for examinees with varied characteristics.

The first part of the thesis deals with aspects of Loewner theory in several complex variables. First we show that a Loewner chain with minimal regularity assumptions (Df(0,t) of local bounded variation) satisfies an associated Loewner equation. Next we give a way of renormalizing a general Loewner chain so that it corresponds to the same increasing family of domains. To do this we will prove a generalization of the converse of Carathéodory's kernel convergence theorem. Next we address the problem of finding a Loewner chain solution to a given Loewner chain equation. The main result is a complete solution in the case when the infinitesimal generator satisfies Dh(0,t)=A where inf {Re<Az,z>: ||z| =1}> 0. We will see that the existence of a bounded solution depends on the real resonances of A, but there always exists a polynomially bounded solution. Finally we discuss some properties of classes of biholomorphic mappings associated to A-normalized Loewner chains. In particular we give a characterization of the compactness of the class of spirallike mappings in terms of the resonance of A. The second part of the thesis deals with the problem of finding examples of extreme points for some classes of mappings. We see that straightforward generalizations of one dimensional extreme functions give examples of extreme Carathéodory mappings and extreme starlike mappings on the polydisc, but not on the ball. We also find examples of extreme Carathéodory mappings on the ball starting from a known example of extreme Carathéodory function in higher dimensions.

La présente étude intitulée « utilisation des technologies de l’information et de la communication dans l’enseignement secondaire et développement des compétences des élèves en résolution de problèmes mathématiques au Burkina Faso » est une recherche descriptive de type mixte examinant à la fois des données qualitatives et quantitatives. Elle examine les compétences en résolution de problèmes mathématiques d’élèves du Burkina Faso pour révéler d’éventuelles relations entre celles-ci et l’utilisation des TIC par les élèves ou leur enseignant de mathématiques. L’intérêt de cette recherche est de fournir des informations aussi bien sur la réalité des TIC dans l’enseignement secondaire au Burkina que sur les effets de leur présence dans l’enseignement et l’apprentissage des mathématiques. Les éléments théoriques ayant servi à l’analyse des données sont présentés suivant trois directions : la résolution des problèmes, le développement des compétences, et les relations entre les TIC, le développement de compétences et la résolution de problèmes. Du croisement de ces éléments émergent trois axes pour le développement de la réponse apportée à la préoccupation de l’étude : 1) décrire l’utilisation de l’ordinateur par les élèves du Burkina Faso pour améliorer leur apprentissage des mathématiques ; 2) identifier des rapports éventuels entre l’utilisation de l’ordinateur par les élèves et leurs compétences en résolution de problèmes mathématiques ; 3) identifier des rapports entre les compétences TIC de l’enseignant de mathématiques et les compétences de ses élèves en résolution de problèmes. Les processus de la résolution de problèmes sont présentés selon l’approche gestaltiste qui les fait passer par une illumination et selon l’approche de la théorie de la communication qui les lie au type de problème. La résolution de problèmes mathématiques passe par des étapes caractéristiques qui déterminent la compétence du sujet. Le concept de compétence est présenté selon l’approche de Le Boterf. Les données révèlent que les élèves du Burkina Faso utilisent l’ordinateur selon une logique transmissive en le considérant comme un répétiteur suppléant de l’enseignant. Par la suite, il n’y a pas de différence significative dans les compétences en résolution de problèmes mathématiques entre les élèves utilisant l’ordinateur et ceux qui ne l’utilisent pas. De même, l’étude révèle que les enseignants présentant des compétences TIC n’ont pas des élèves plus compétents en résolution de problèmes mathématiques que ceux de leurs collègues qui n’ont pas de compétences TIC. / This study entitled "Use of information and communication technologies in secondary education and development of students in mathematical problems solving skills in Burkina Faso" is a mixed descriptive research examining both qualitative and quantitative data. It examines the math problem solving skills of students from Burkina Faso to reveal possible relationships between them and the use of ICT by students or teachers in mathematics. The interest of this research is to provide information as well on the reality of ICT in secondary education in Burkina Faso as on the effect of their presence in teaching and learning mathematics. The theoretical elements used for data analysis are presented in three areas: problem solving, skills development and the relationship between ICT, skills development and problem solving. From crossing these areas rose three directions in witch the response to the concern of the study is presented: 1) describe the use of computers by students from Burkina Faso to improve their learning of mathematics; 2) identify possible relationship between the use of computer by students and their mathematical problems solving skills; 3) identify relationships between mathematics teachers ICT skills and their students problems solving skills. The problem solving process is presented according the gestalt theory that takes it through an enlightenment, and according to the communication theory approach that links it to the type of problem to solve. Mathematical problems solving goes through characteristic stages that determine the subject’s abilities. The concept of competence is presented using the approach of Le Boterf. The data show that students from Burkina Faso are using the computer in a transmissive way taking it as a substitute for the teacher. Subsequently, in mathematical problem solving skills there is no meaningful difference between students who use computer and those who do not use it. Similarly, the study found that students whose teacher has ICT skills were not more competent in mathematical problems solving than those whose teacher does not have ICT skills.