Although the Technological Pedagogical Content Knowledge (TPACK) framework has shown a lot of promise as a theoretical perspective, researchers find it difficult to use it in particular environments because the requirements of the framework change in specific contexts. The purpose of this study was to explore and produce more flexible ways of using the TPACK for inservice mathematics secondary teachers. Three such teachers at an urban public school were observed in their classrooms and interviewed about their experiences of teaching mathematics and integrating computer technology in their day-to-day activities. Each participant had over 10 years experience in teaching mathematics in secondary schools in Ontario, and expertise in using computers in mathematics curriculum. The research questions were: 1) How do secondary school mathematics teachers describe their ways of integrating technology? 2) What difficulties do teachers have when they try to integrate technology into mathematics classrooms? The findings from the first research question show that teachers displayed a high degree of integration of technology. Their activities were very clearly designed, conferring clear roles to the use of integrating computer technology in mathematics classes. Teachers had specific approaches to integrate computer technology: a) to allow students opportunities to learn and experiment with their mathematical knowledge; b) to help them pass the content to the students in the process of teaching mathematics; and c) to assess and evaluate students’ work, and give them feedback. The findings from the second research question reveal that teachers had difficulties in purchasing and maintaining the computer equipment. They had some difficulties in trying to integrate new technologies as these required time, preparation, and dedication. In addition, teachers had some difficulties in making students use computers in a significant way. The implication for teacher education is that inservice teachers should have opportunities to update their computer and pedagogical skills, a long term perspective in integrating technology in mathematics education, and professional and technical support from teaching colleagues and administrators. Finally, the integration of computer technology in mathematics requires more intensive teamwork and collaboration between teachers, technical support staff, and administrators.

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.

Doctor of Philosophy / Curriculum and Instruction / David S. Allen / This study addresses how the flipped method of classroom instruction differs from traditional classroom instruction when comparing student achievement measures in middle and high school mathematics classrooms. The flipped classroom is defined by the Flipped Learning Network (2014) as an instructional method that moves direct instruction outside of the classroom in order to make room in the classroom for a more interactive learning environment where students can actively engage in the content. The flipped classroom strategy theoretically allows teachers the time to develop mathematical ideas and the ability to facilitate that development. For the Common Core State Standards initiative to be effective, teachers need to engage students in new learning experiences that support college and career readiness. By implementing a technology based instructional approach, like the flipped classroom strategy, teachers are able to blend twenty-first century skills with the development of the essential habits of mind of mathematically proficient students (Brunsell & Horejsi, 2013). This study seeks to understand how the flipped method of classroom instruction can lead to improved student achievement in mathematics courses and improve student perceptions about math in order to encourage course consumption in the future (Zollman, 2011). A modified explanatory sequential mixed methods design was used, and it involved collecting quantitative data and then explaining the quantitative results with in-depth qualitative data. In the quantitative phases of the study, NWEA Mathematics MAP Assessment data were collected from middle school students and course common final assessment scores were collected from middle school and high school students in a large Midwestern suburban school district to determine how student math achievement was impacted for students in a flipped classroom as compared to a traditionally instructed classroom. The frequency of active learning incidents was also collected during classroom observations. The qualitative phase was conducted as a follow up to the quantitative results to help explain the quantitative results. In this exploratory follow-up, student and teacher perceptions of mathematics achievement as a result of the flipped classroom approach to instruction with middle and high school math students and how those perceptions might be different than those of students and teachers in traditionally taught classrooms along with descriptions of observable active learning incidents in the school district were explored.

Doctor of Philosophy / Curriculum and Instruction Programs / Jennifer M. Bay-Williams / This study examined end-of-the-year seventh grade students’ rational number knowledge using comparison tasks and rational number subconstruct tasks. Comparison tasks included: comparing two decimals, comparing two fractions and comparing a fraction and a decimal. The subconstructs of rational number addressed in this research include: part-whole, measure, quotient, operator, and ratio. Between eighty-six and one-hundred-one students were assessed using a written instrument divided into three sections. Nine students were interviewed following the written instrument to probe for further understanding. Students were classified by error patterns using decimal comparison tasks. Students were initially to be classified into four groups according to the error pattern: whole number rule (WNR), zero rule (ZR), fraction rule (FR) or apparent expert (AE). However, two new patterns emerged: ignore zero rule (IZR) and money rule (MR). Students’ knowledge of the subconstructs of rational numbers was analyzed for the students as a whole, but also analyzed by classification to look for patterns within small groups of students and by individual students to create a thick, rich description of what students know about rational numbers. Students classified as WNR struggled across almost all of the tasks. ZR students performed in many ways similar to WNR but in other ways performed better. FR and MR students had more success across all tasks compared to WNR and ZR. On average apparent experts performed significantly better than those students classified by errors. However, further analysis revealed hidden misconceptions and deficiencies for a number of apparent experts. Results point to the need to make teachers more aware of the misconceptions and deficiencies because in many ways errors reflect the school experiences of students.

Doctor of Philosophy / Curriculum and Instruction Programs / Margaret G. Shroyer / While there has been much quantitative research done in the area of attitudes and self-efficacy beliefs, this study sought hear the voices of the middle school child. Therefore, this qualitative study investigated the attitudes toward mathematics and mathematics self-efficacy beliefs of middle school students in one middle school in western Kansas. The conceptual framework for this study is supported by the research of Albert Bandura on Social Cognitive Theory. This study used a naturalistic inquiry approach and data were collected from multiple sources, including short-answer questionnaires, classroom observations, and one-on-one interviews. Coded data were examined for patterns, themes, and relationships. Middle school students in this study exhibited positive, negative, and variable attitudes toward mathematics, and both positive and negative mathematics self-efficacy beliefs. Students attribute their high mathematics self-efficacy beliefs to the teacher or the high grades they receive on daily assignments, as well as the scores they receive on state and local assessments. Conversely, middle school students have low mathematics self-efficacy beliefs when they feel unsuccessful or distressed, and they attribute those beliefs to the low grades they receive on daily assignments and assessments, as well as the distress of not understanding the mathematics. Middle school students told their mathematical stories of the change in attitudes toward mathematics and mathematics self-efficacy beliefs, and attributed positive changes to the mathematics teacher. Negative changes in attitudes toward mathematics and mathematics self-efficacy beliefs were attributed to the amount of homework expected at the middle school level, as well as the lack of hands-on activities. The influence of the teacher, grades, and hands-on activities impact middle school students’ attitudes toward mathematics and mathematics self-efficacy beliefs. There is a relationship between attitudes toward mathematics and mathematics self-efficacy beliefs. Low mathematics self-efficacy beliefs and poor attitudes toward mathematics are related since low mathematics self-efficacy beliefs and poor attitudes toward mathematics are highly connected. Conversely, high mathematics self-efficacy beliefs and good attitudes toward mathematics are highly related. Middle school students’ experiences impact both mathematics self-efficacy beliefs and attitudes toward mathematics. Students’ mathematics self-efficacy beliefs impact their attitudes toward mathematics.

Doctor of Philosophy / Curriculum and Instruction Programs / Gail Shroyer / David Allen / This study was designed to gather input from early career elementary teachers with the goal of finding ways to improve elementary mathematics methods courses. Multiple areas were explored including the degree to which respondents’ elementary mathematics methods course focused on the NCTM Process Standards, the teachers’ current standards-based teaching practices, the degree to which various pedagogical strategies from mathematics methods courses prepared preservice teachers for the classroom, and early career teachers’ suggestions for improving methods courses. Both qualitative and quantitative methodologies were used in this survey study as questions were of both closed and open format. Data from closed-response questions were used to determine the frequency, central tendencies and variability in standards-based preparation and teaching practices of the early career teachers. Open-ended responses were analyzed to determine patterns and categories relating to the support of, or suggestions for improving, elementary mathematics methods courses. Though teachers did not report a wide variation in the incorporation of the NCTM Process Standards in their teaching practices, some differences were worth noting. Problem Solving appeared to be the most used with the least variability in its frequency of use. Reasoning, in general, appeared to be used the least frequently and with the most variability. Some aspects of Communication, Connections and Representation were widely used and some were used less frequently. From a choice of eight methods teaching practices, ‘Observing in actual classrooms or working with individual students’ and ‘Planning and teaching in actual classrooms’ were considered by early career teachers to be the most beneficial aspects of methods courses.

L’un des buts de l’apprentissage des mathématiques est le développement du raisonnement et celui-ci participe à la compréhension des mathématiques. Très liée au raisonnement, la notion de preuve est aussi fondamentale à l’apprentissage des mathématiques, car elle permet d’établir la validité d’arguments mathématiques et de conférer un sens à différents concepts à travers l’explication de l’organisation logique du travail effectué. Toutefois, malgré l’importance accordée au développement de différents types de raisonnements, plusieurs élèves éprouvent des difficultés lorsqu’ils sont appelés à concevoir ou à évaluer des preuves. Dans le cadre de cette recherche, nous avons étudié l’impact de l’utilisation d’un forum électronique sur le développement d’habiletés de validation algébrique ainsi que sur le développement d’habiletés en lien avec l’évaluation de preuves en algèbre chez des élèves de 13 et 14 ans du Nouveau-Brunswick et du Québec. Les résultats laissent supposer que l’utilisation du forum électronique encourage le passage des preuves pragmatiques aux preuves intellectuelles, en plus de favoriser une utilisation adéquate des règles du débat mathématique. / One of the goals of learning mathematics is the development of reasoning, because it is essential to understand mathematics. Closely related to reasoning, the notion of proof is also fundamental in the learning of mathematics, because it allows students to establish the validity of mathematical arguments and put a sense on various concepts through logical explanation of their work. However, in spite of the importance placed on the development of the capacity to reason mathematically, several students are confronted with difficulties during the development or the evaluation of proofs. This study examined the impact of the use of a discussion forum on the development of algebraic validation skills as well as on the development of skills linked with the evaluation of the proof process in algebra with 13 and 14 year old students from New Brunswick and Quebec (Canada). The results lead us to believe that the use of the electronic forum encourages the passage from pragmatic proofs to intellectual proofs. It also seems to facilitate an adequate use of the rules of the mathematical debate.

Ancrée dans le domaine de la didactique des mathématiques, notre thèse cible le « travail de l’erreur » effectué par trois enseignants dans leur première année de carrière. Libérés des contraintes associées au système de formation initiale, ces sujets assument pleinement leur nouveau rôle au sein de la classe ordinaire. Ils se chargent, entre autres, de l’enseignement de l’arithmétique et, plus précisément, de la division euclidienne. Parmi leurs responsabilités se trouvent le repérage et l’intervention sur les procédures erronées. Le « travail de l’erreur » constitue l’expression spécifique désignant cette double tâche (Portugais 1995). À partir d’un dispositif de recherche combinant les méthodes d’observation et d’entrevue, nous documentons des séances d’enseignement afin de dégager les situations où nos maîtres du primaire identifient des erreurs dans les procédures algorithmiques des élèves et déploient, subséquemment, des stratégies d’intervention. Nous montrons comment ces deux activités sont coordonnées en décrivant les choix, décisions et actions mises en œuvre par nos sujets. Il nous est alors possible d’exposer l’organisation de la conduite de ces jeunes enseignants en fonction du traitement effectif de l’erreur arithmétique. En prenant appui sur la théorie de champs conceptuels (Vergnaud 1991), nous révélons l’implicite des connaissances mobilisées par nos sujets et mettons en relief les mécanismes cognitifs qui sous-tendent cette activité professionnelle. Nous pouvons ainsi témoigner, du moins en partie, du travail de conceptualisation réalisé in situ. Ce travail analytique permet de proposer l’existence d’un schème du travail de l’erreur chez ces maîtres débutants, mais aussi de spécifier sa nature et son fonctionnement. En explorant le versant cognitif de l’activité enseignante, notre thèse aborde une nouvelle perspective associée au thème du repérage et de l’intervention sur l’erreur de calcul de divisions en colonne. / Rooted in the Didactic of Mathematics’ field, this thesis looks into the practice of three new teachers. Free from the constraints and pressures associated with the teacher training context, these subjects take on a new role in their first year on the job. Among all of the affiliated responsibilities are those related to the teaching of arithmetic and, more specifically, the long division algorithm. “Le travail de l’erreur” is the expression used by Portugais (1995) to design error management by the teacher. It includes the diagnosis of errors and the strategies unfurled to help the pupil remedy his/her mistakes. This double task is the object of this study. Combining research methods of observation and interview, we aim to describe error management in a regular elementary class setting. We delineate situations where our subjects identify and intervene on errors made by students during the arithmetic procedure. We examine how these two activities are coordinated, by documenting the novice teachers choices, decisions and actions. We focus on the organization of their conduct throughout the different situations. A theoretical framework based on the “conceptual field theory” or “théorie des champs conceptuels” (Vergnaud, 1991) enables us to reveal the implicit teaching knowledge comprised in the behavior adopted by the young professionals. This analysis reveals the dynamics of piagetian assimilation/ accommodation mechanisms. It also gives us evidence of the conceptualizing process underlying teacher conduct. We utilize the concept of “scheme” to better understand this cognitive activity. We propose the existence of “le schème du travail de l’erreur” and aim to specify his nature and function. This allows us to describe the conceptual structure, which shapes and organizes the novice’s ability to manage errors in their pupil’s calculation of written divisions.

La thèse porte sur l’analyse qualitative de situations didactiques intégrées au programme de prévention au préscolaire Fluppy. Conçu pour la prévention de la violence et du décrochage scolaire (Tremblay et al., 1992, Tremblay et al., 1995), ce programme s’est enrichi depuis une dizaine d’années de différentes composantes d’intervention, dont une sur l’enseignement du français et des mathématiques. Ce programme, relevant aujourd’hui d’une approche multimodale, a fait l’objet d’une évaluation d’impact en 2002-2004 (Capuano et al., 2010). Le devis quasi-expérimental n’a cependant pas permis de procéder à une analyse appropriée au cadre méthodologique, l’ingénierie didactique (Artigue, 1990), sur lequel se fondent les situations didactiques en mathématiques. La thèse procède donc à la validation interne des trois séquences numériques, issues de la composante mathématique, telles qu’expérimentées dans deux classes du préscolaire en 2011-2012. La première séquence vise au développement des connaissances sur la désignation de quantités. La deuxième sur la comparaison numérique et, la troisième, sur la composition additive des nombres. Les analyses mettent en évidence : 1) certains décalages entre la proposition didactique et la réalisation effective des situations; 2) l’évolution des connaissances numériques des élèves; 3) les forces et les limites de l’analyse a priori. L’interprétation des résultats ouvre sur un enrichissement de l’analyse a priori des situations didactiques ainsi que sur de nouvelles considérations relatives aux processus de dévolution et d’institutionnalisation dans le cadre de l’appropriation de situations didactiques par des enseignants du préscolaire. / The thesis focuses on the qualitative analysis of didactic situations incorporated in the prevention program, Fluppy, intended for preschool children. Originally designed for the prevention of violence and school dropout (Tremblay et al., 1992; Tremblay et al.,1995), over the last decade, this program has abundantly been enhanced of different intervention components, including French and mathematics teaching. This program, which is now part of a multimodal approach, has been the subject of an impact assessment in 2002-2004 (Capuano et al., 2010). The quasi-experimental instrument, has however failed to conduct a proper analysis of the methodological framework, the didactical engineering (Artigue, 1990); basis of the theory of didactical situations in mathematics. The thesis undertakes the internal validation of three numeric sequences –from the mathematical component– such as they were tested in two preschool classes in 2011-2012. The first sequence studies the development of C-knowledge regarding the designation of quantities. The second one tackles the numerical comparison, and the third one studies the additive composition of numbers. Analyzes reveal: 1) some discrepancies between the didactical proposal and the actual situations, 2) the development of students’ c-knowledge, and 3) the strengths and limitations of the a priori analysis. The interpretation of the results broadens the a priori analysis of didactical situations as well as arises new considerations on the devolution and institutionalization phenomena within the framework of preschool teachers’ appropriation of didactical situations.

This study followed three elementary-school teachers as they engaged in online discussions about inquiry-based mathematics teaching, and wrote and tested inquiry lessons for their own classrooms. In an inquiry lesson, students bring their own knowledge to open-ended problem situations, and build on that knowledge as they try out solutions and share their ideas with others. Evidence from the study suggests that teachers may turn to inquiry as an antidote to the way they learned about mathematics as schoolchildren, and that participating in an online community is a way for teachers to gain new mathematical and pedagogical knowledge and to change their conceptual understanding of inquiry-based teaching. The study results indicate that online professional development can help teachers improve their practice, but that care must be taken to build social ties within the group, and to structure tasks in a way that encourages collaboration and constructive criticism.