Doctor of Philosophy / Department of Mathematics / Andrew Bennett / Engineering majors and Mathematics Education majors are two groups that take the basic, core Mathematics classes. Whereas Engineering majors go on to apply this mathematics to real world situations, Mathematics Education majors apply this mathematics to deeper, abstract mathematics. Senior students from each group were interviewed about “function” and “accumulation” to examine any differences in learning between the two groups that may be tied to the use of mathematics in these different contexts. Variation between individuals was found to be greater than variation between the two groups; however, several differences between the two groups were evident. Among these were higher levels of conceptual understanding in Engineering majors as well as higher levels of confidence and willingness to try problems even when they did not necessarily know how to work them.

Abstract: This is a case study of a mathematics professional learning community. It illustrates the experience of eight Grade 2 teachers as they collaborate to improve their students'understanding of mathematics. In this inquiry, I worked as a participant-observer with the teachers over the course of five months as a witness to their expanding understanding of mathematics and learning. The case study describes two manifestations: the experiences of the teachers as they develop their knowledge of the mathematical learning of young children; and secondly, the teachers' growth as a professional learning community committed to improving the mathematical understanding of their students and of themselves. Collectively, the findings from this study extend other conversations on both professional learning communities and the development of teachers' knowledge about mathematical learning (often called pedagogical content knowledge). This work shows that opportunities for professional learning that are self-directed, context and content specific, within a milieu that is collegial and supportive, enable teachers to bridge the elusive gap between theory and practice. The specific questions addressed are as follows: 1. How does participation in a professional learning community affect teachers‘ iii pedagogical content knowledge and their understanding of students‘ learning of mathematics? 2. How do primary teachers develop an effective mathematics professional learning community? In mathematics, professional development often focuses on the creation of effective lesson design. This study differed in some key ways. Although good lesson design was valued and employed, the stimulus for teacher learning was the observation of the students as they struggled with new complex concepts. From these observations, the teachers became astute at recognizing particular consistencies and inconsistencies in the mathematical learning of the one hundred plus students they each observed within this project. Together, as a professional learning community, the teachers became adept at using external resources such as research and other resource materials to search the reasons and solutions for students‘ difficulty with mathematical concepts. Teachers' cognitive dissonance as they tried new instructional approaches and shared successes and failures with their colleagues provided the foundation for their growth in pedagogical content knowledge.

In a transferable utility context, Choo and Siow (2006) introduced a competitive model of the marriage market with gumbel distributed stochastic part, and derived its equilibrium output, a marriage match- ing function. The marriage matching function defines the gains generated by a marriage between agents of prescribed types in terms of the observed frequency of such marriages within the population, relative to the number of unmarried individuals of the same types. Left open in their work is the issue of existence and uniqueness of equilibrium. We resolve this question in the affirmative, assuming the norm of the gains matrix (viewed as an operator) to be less than two. Our method adapts a strategy called the continuity method,more commonly used to solve elliptic partial differen- tial equations, to the new setting of isolating positive roots of polynomial systems. Finally, the data estimated in [4] falls within the scope of our results.

In a transferable utility context, Choo and Siow (2006) introduced a competitive model of the marriage market with gumbel distributed stochastic part, and derived its equilibrium output, a marriage match- ing function. The marriage matching function defines the gains generated by a marriage between agents of prescribed types in terms of the observed frequency of such marriages within the population, relative to the number of unmarried individuals of the same types. Left open in their work is the issue of existence and uniqueness of equilibrium. We resolve this question in the affirmative, assuming the norm of the gains matrix (viewed as an operator) to be less than two. Our method adapts a strategy called the continuity method,more commonly used to solve elliptic partial differen- tial equations, to the new setting of isolating positive roots of polynomial systems. Finally, the data estimated in [4] falls within the scope of our results.

This research is a qualitative case study examining changes in urban Canadian elementary teachers’ conceptualizations of equity and approaches to pedagogy in their mathematics teaching in relation to their involvement in multiple professional learning contexts. The study focuses on four major professional development (PD) efforts in which five focal teachers participated over a school year. Data sources include researcher observations, field notes, video-recordings of PD sessions and classroom mathematics teaching, as well as a series of one-on-one interviews. Data analysis revealed three main ideas related to equity that were adopted by focal teachers: 1) the importance of developing awareness of students and their communities; 2) teaching strategies to scaffold students’ development of mathematical proficiency; and 3) strategies for structuring student-driven, inquiry-based learning for mathematics. The multiple contexts of professional learning presented contradictory messages. Thus, teachers took up some ideas and left others behind and sometimes took up ideas that served conflicting goals of education. Future studies of teacher PD should focus on the teacher’s perspective and the role of any individual PD within the multiple contexts of professional learning in which teachers participate.

Abstract: This is a case study of a mathematics professional learning community. It illustrates the experience of eight Grade 2 teachers as they collaborate to improve their students'understanding of mathematics. In this inquiry, I worked as a participant-observer with the teachers over the course of five months as a witness to their expanding understanding of mathematics and learning. The case study describes two manifestations: the experiences of the teachers as they develop their knowledge of the mathematical learning of young children; and secondly, the teachers' growth as a professional learning community committed to improving the mathematical understanding of their students and of themselves. Collectively, the findings from this study extend other conversations on both professional learning communities and the development of teachers' knowledge about mathematical learning (often called pedagogical content knowledge). This work shows that opportunities for professional learning that are self-directed, context and content specific, within a milieu that is collegial and supportive, enable teachers to bridge the elusive gap between theory and practice. The specific questions addressed are as follows: 1. How does participation in a professional learning community affect teachers‘ iii pedagogical content knowledge and their understanding of students‘ learning of mathematics? 2. How do primary teachers develop an effective mathematics professional learning community? In mathematics, professional development often focuses on the creation of effective lesson design. This study differed in some key ways. Although good lesson design was valued and employed, the stimulus for teacher learning was the observation of the students as they struggled with new complex concepts. From these observations, the teachers became astute at recognizing particular consistencies and inconsistencies in the mathematical learning of the one hundred plus students they each observed within this project. Together, as a professional learning community, the teachers became adept at using external resources such as research and other resource materials to search the reasons and solutions for students‘ difficulty with mathematical concepts. Teachers' cognitive dissonance as they tried new instructional approaches and shared successes and failures with their colleagues provided the foundation for their growth in pedagogical content knowledge.

This research is a qualitative case study examining changes in urban Canadian elementary teachers’ conceptualizations of equity and approaches to pedagogy in their mathematics teaching in relation to their involvement in multiple professional learning contexts. The study focuses on four major professional development (PD) efforts in which five focal teachers participated over a school year. Data sources include researcher observations, field notes, video-recordings of PD sessions and classroom mathematics teaching, as well as a series of one-on-one interviews. Data analysis revealed three main ideas related to equity that were adopted by focal teachers: 1) the importance of developing awareness of students and their communities; 2) teaching strategies to scaffold students’ development of mathematical proficiency; and 3) strategies for structuring student-driven, inquiry-based learning for mathematics. The multiple contexts of professional learning presented contradictory messages. Thus, teachers took up some ideas and left others behind and sometimes took up ideas that served conflicting goals of education. Future studies of teacher PD should focus on the teacher’s perspective and the role of any individual PD within the multiple contexts of professional learning in which teachers participate.

A qualitative case study was presented in order to explore an inquiry-based learning approach to teaching risk in two different grade 11 mathematics classes in an urban centre in Canada. The first class was in an all-boys independent school (23 boys) and the second class was in a publicly funded religious school (19 girls and 4 boys). The students were given an initial assessment in which they were asked about the safety of nuclear power plants and their knowledge of the Fukushima nuclear power plant accident. Following the initial assessment, the students participated in an activity with the purpose of determining the empirical probability of a nuclear power plant accident based on the authentic data found online. The second activity was then presented in order to determine the impact of a nuclear power plant accident and compare it to a coal power plant accident. The findings provide evidence that the students possess intuitive knowledge that risk of an event should be assessed by both its likelihood and its impact. The study confirms the Levinson et al. (2012) pedagogic model of risk in which individuals’ values and prior experiences together with representations and judgments of probability play a role in the estimation of risk. The study also expands on this model by suggesting that pedagogy of risk should include five components, namely: 1) knowledge, beliefs, and values, 2) judgment of impact, 3) judgment of probability, 4) representations, and 5) estimation of risk. These ii components do not necessarily appear in the instruction or students’ decision making in a chronological order; furthermore, they influence each other. For example, judgments about impact (deciding not to consider accidents with low impact into calculations) may influence the judgments about probability. The implication for mathematics education is that a meaningful instruction about risk should go beyond mathematical representations and reasoning and include other components of the pedagogy of risk. The study also illustrates the importance of reasoning about rational numbers (rates, ratios, and fractions) and their critical interpretation in the pedagogy of risk. Finally, the curricular expectations relevant to the pedagogy of risk from the Ontario secondary curriculum are identified.

A qualitative case study was presented in order to explore an inquiry-based learning approach to teaching risk in two different grade 11 mathematics classes in an urban centre in Canada. The first class was in an all-boys independent school (23 boys) and the second class was in a publicly funded religious school (19 girls and 4 boys). The students were given an initial assessment in which they were asked about the safety of nuclear power plants and their knowledge of the Fukushima nuclear power plant accident. Following the initial assessment, the students participated in an activity with the purpose of determining the empirical probability of a nuclear power plant accident based on the authentic data found online. The second activity was then presented in order to determine the impact of a nuclear power plant accident and compare it to a coal power plant accident. The findings provide evidence that the students possess intuitive knowledge that risk of an event should be assessed by both its likelihood and its impact. The study confirms the Levinson et al. (2012) pedagogic model of risk in which individuals’ values and prior experiences together with representations and judgments of probability play a role in the estimation of risk. The study also expands on this model by suggesting that pedagogy of risk should include five components, namely: 1) knowledge, beliefs, and values, 2) judgment of impact, 3) judgment of probability, 4) representations, and 5) estimation of risk. These ii components do not necessarily appear in the instruction or students’ decision making in a chronological order; furthermore, they influence each other. For example, judgments about impact (deciding not to consider accidents with low impact into calculations) may influence the judgments about probability. The implication for mathematics education is that a meaningful instruction about risk should go beyond mathematical representations and reasoning and include other components of the pedagogy of risk. The study also illustrates the importance of reasoning about rational numbers (rates, ratios, and fractions) and their critical interpretation in the pedagogy of risk. Finally, the curricular expectations relevant to the pedagogy of risk from the Ontario secondary curriculum are identified.

The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.