Doctor of Philosophy / Curriculum and Instruction Programs / David S. Allen / Margaret G. Shroyer / Ongoing, effective professional development is viewed as an essential mechanism for eliciting change in teachers’ knowledge and practice in support of enacting the vision of NCTM’s Principles and Standards of School Mathematics. This case study of the Infinite Mathematics Project, a Title IIB MSP professional development initiative, seeks to provide a qualitative examination of the characteristics and strategies used in the project and their impact on teacher learning and practice. The project embodied many features and strategies of effective professional development such as: mathematics content focus; sustained over time; reform activities (e.g., lesson study, teacher collaboration); active learning opportunities (e.g., implementing an action plan; developing differentiated instruction activities for a mathematics classroom); coherence with NCTM and state standards; and collective participation by IHE facilitators and participant K-12 teachers from partner districts. The findings reveal teachers gained both content knowledge (knowledge about mathematics, substantive knowledge of mathematics, pedagogical content knowledge, and curricular knowledge) and pedagogical knowledge (knowledge about strategies for differentiating instruction in a mathematics classroom, for supporting students’ reading in the content area, for fostering the development of number sense, for implementing standards-based teaching, and for critically analyzing teaching). The study also provides some evidence that the project had an impact on teaching practice. In addition, an implication of the study suggests the positive impact of Title IIB MSP partnership requirements.

Master of Science / Mathematics / Andrew G. Bennett / This study utilizes eye-tracking technology as a tool to measure college algebra students’ mathematical noticing as defined by Lobato and colleagues (2012). Research in many disciplines has used eye-tracking technology to investigate the differences in visual attention under the assumption that eye movements reflect a person’s moment-to-moment cognitive processes. Motivated by the work done by Madsen and colleagues (2012) who found visual differences between those who correctly and incorrectly solve introductory college physics problems, we used eye-tracking to observe the visual attention difference between correct and incorrect solvers of college algebra problems. More specifically, we consider students’ visual attention when presented tabular representations of linear functions. We found that in several of the problems analyzed, those who answered the problem correctly spend more time looking at relevant table values of the problem while those who answered the problem incorrectly spend more time looking at irrelevant table labels x, y, y = f(x) of the problem in comparison to the correct solvers. More significantly, we found a noteworthy group of students, who did not move beyond table labels, using these labels solely to solve the problem. Future analyses need to be done to expand on the differences between eye patterns rather than just focusing on dwell time in the relevant and irrelevant areas of a table.

Doctor of Philosophy / Curriculum and Instruction Programs / Andrew G. Bennett / On-line homework is becoming a larger part of mathematics classrooms each year. Thus, ways to maximize the effectiveness of on-line homework for both students and teachers must be investigated. This study sought to provide one possible answer to this aim, by requiring students to achieve at least 50% for any on-line homework assignment in order to receive credit. Research shows that students respond well to reasonably set high expectations, and coupling this with one of the primary advantages of on-line homework, the ability to rework assignments, provided the basis for this study. Data for this experimental study was collected from the spring semester of 2008 until the fall semester of 2009, and included student exam scores, the number of on-line assignments above and below the 50% threshold, and the number of times students accessed help features of the on-line homework system when given the ability to do so. Analysis at both the whole-class and cluster levels attempted to discern the effectiveness of the intervention. Results indicated that significantly fewer students settled for on-line homework scores less than 50% in the experimental semesters where the 50% requirement was in place than in the control semesters in which the requirement was absent. Certain clusters of students seemed to benefit even more than others from this higher expectation, leading to the possibility of differentiated instruction or differentiated interventions in the future. In addition to fewer sub-par on-line homework scores, students also demonstrated other positive traits, such as accessing the on-line help links more within the experimental semesters.

Doctor of Philosophy / Department of Curriculum and Instruction / Michael F. Perl / This exploratory correlational study seeks to answer the question of whether a relationship exists between student average test score gains on state exams and teachers’ rating of values on the Schwartz Values Survey. Eighty-seven randomly selected Kansas teachers of math and/or reading, grades four through eight, participated. Student test score gains were paired with teachers and averaged. The results of these backward stepwise entries of multiple regressions using SPSS software are reported. Significant relationships with large effect sizes are reported for teacher values and student test score gains in reading and math. Models of teacher values are found that account for thirty-two percent of the average student test score gains in reading and for forty-three percent of the average student test score gains in mathematics. The significant model of values with the greatest adjusted relationship with reading test score gains is described as the Relational Teacher Value Type. The valuing of True Friendship (close supportive friends) and the valuing of Sense of Belonging (feeling that others care about me) proved to be the most powerful indicators of student reading score gains within this type. The significant model of values with the greatest adjusted relationship with mathematics test score gains is described as the Well-Being Teacher Value Type. The valuing of Healthy (not being sick physically or mentally), the valuing of Reciprocation of Favors (avoidance of indebtedness), and Self Respect (belief in one’s own worth) proved to be the most powerful indicators of student mathematics test score gains within this type. The significant value items within each of the above types’ models are discussed regarding possible reasons for their relationships to student test score gains. A value that is found significant for both reading and mathematics teachers in accounting for student test score gains is Moderate (avoiding extremes of feeling and action). Of the teachers in the study that taught mathematics and reading, their students’ mathematics score gains did not correlate in a statistically significant way with their students’ reading score gains, suggesting that a teacher’s ability to teach math has little to do with a teacher’s ability to teach reading.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / The goal of this work is to provide instructors with detailed information about their classes at each assignment during the term. The information is both on an individual level and at the aggregate level. We used the large number of grades, which are available online these days, along with data-mining techniques to build our models. This enabled us to profile each student so that we might individualize our approach. From these profiles, we began to investigate what can be done in order to get students to do better, or at least be less frustrated. Regardless, the interactions with our undergraduates will improve as our knowledge about them increases. We start with a categorization of Studio College Algebra students into groups, or clusters, at some point in time during the semester. In our case, we used the grouping just after the first exam, as described by Dr. Rachel Manspeaker in her PhD. dissertation. From this we built a naive Bayesian model which extends these student clusters from one point in the semester, to a classification at every assignment, attendance score, and exam in the course. A hidden Markov model was then constructed with the transition probabilities being derived from the Bayesian model. With this HMM, we were able to compute the most likely path that students take through the various categories over the semester. We observed that a majority of students settle into a group within the first two weeks of the term.

The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets.\\ Dynamics of simple parabolic germs. Let $F$ be a germ with a simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The dynamics of a germ $f$ can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution \[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where $\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show that $\tilde$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J.~\'Ecalle and S.~Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of \'Ecalle's construction. \\ Computability of Julia sets. Informally, a compact subset of the complex plane is called \emph if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let $f$ be a rational function of degree $d\ge 2$. Assume that for each critical point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain either a critical point or a parabolic periodic point of $f$. Then the Julia set $J_f$ is computable in a polynomial time.

This study investigated secondary school students’ errors and misconceptions in algebra with a view to expose the nature and origin of those errors and to make suggestions for classroom teaching. The study used a mixed method research design. An algebra test which was pilot-tested for its validity and reliability was given to a sample of grade 11 students in an urban secondary school in Ontario. The test contained questions from four main areas of algebra: variables, algebraic expressions, equations, and word problems. A rubric containing the observed errors was prepared for each conceptual area. Two weeks after the test, six students were interviewed to identify their misconceptions and their reasoning. In the interview process, students were asked to explain their thinking while they were doing the same problems again. Some prompting questions were asked to facilitate this process and to clarify more about students’ claims. The results indicated a number of error categories under each area. Some errors emanated from misconceptions. Under variables, the main reason for misconceptions was the lack of understanding of the basic concept of the variable in different contexts. The abstract structure of algebraic expressions posed many problems to students such as understanding or manipulating them according to accepted rules, procedures, or algorithms. Inadequate understanding of the uses of the equal sign and its properties when it is used in an equation was a major problem that hindered solving equations correctly. The main difficulty in word problems was translating them from natural language to algebraic language. Students used guessing or trial and error methods extensively in solving word problems. Some other difficulties for students which are non-algebraic in nature were also found in this study. Some of these features were: unstable conceptual models, haphazard reasoning, lack of arithmetic skills, lack or non-use of metacognitive skills, and test anxiety. Having the correct conceptual (why), procedural (how), declarative (what), and conditional knowledge (when) based on the stage of the problem solving process will allow students to avoid many errors and misconceptions. Conducting individual interviews in classroom situations is important not only to identify errors and misconceptions but also to recognize individual differences.

This study investigated secondary school students’ errors and misconceptions in algebra with a view to expose the nature and origin of those errors and to make suggestions for classroom teaching. The study used a mixed method research design. An algebra test which was pilot-tested for its validity and reliability was given to a sample of grade 11 students in an urban secondary school in Ontario. The test contained questions from four main areas of algebra: variables, algebraic expressions, equations, and word problems. A rubric containing the observed errors was prepared for each conceptual area. Two weeks after the test, six students were interviewed to identify their misconceptions and their reasoning. In the interview process, students were asked to explain their thinking while they were doing the same problems again. Some prompting questions were asked to facilitate this process and to clarify more about students’ claims. The results indicated a number of error categories under each area. Some errors emanated from misconceptions. Under variables, the main reason for misconceptions was the lack of understanding of the basic concept of the variable in different contexts. The abstract structure of algebraic expressions posed many problems to students such as understanding or manipulating them according to accepted rules, procedures, or algorithms. Inadequate understanding of the uses of the equal sign and its properties when it is used in an equation was a major problem that hindered solving equations correctly. The main difficulty in word problems was translating them from natural language to algebraic language. Students used guessing or trial and error methods extensively in solving word problems. Some other difficulties for students which are non-algebraic in nature were also found in this study. Some of these features were: unstable conceptual models, haphazard reasoning, lack of arithmetic skills, lack or non-use of metacognitive skills, and test anxiety. Having the correct conceptual (why), procedural (how), declarative (what), and conditional knowledge (when) based on the stage of the problem solving process will allow students to avoid many errors and misconceptions. Conducting individual interviews in classroom situations is important not only to identify errors and misconceptions but also to recognize individual differences.

This study investigates the teaching practices that three Grade 9 Applied Mathematics teachers use to increase student engagement and enhance student learning. Specifically, the study examines the factors within social and academic domains that teachers used to increase student engagement. Qualitative data were collected in the form of teacher interviews, classroom observations and teacher journals. The evidence from the study shows that all three teachers were cognizant of attributes of their early adolescent learners as the teachers sought to increase student engagement in their Grade 9 Applied Mathematics classes. Six major findings as suggested by the case studies can be summarized as follows: (1) developing student self-confidence is a critical component of increasing student engagement for early adolescent learners; (2) teachers may focus on one domain more than the other as a result of their personal comfort with that domain; (3) domains for student engagement and the factors found within these domains are not independent; (4) the Ontario Ministry of Education’s TIPS4RM resource is an effective way to increase student engagement; (5) technology is also an effective and relevant way to increase student engagement; and (6) the use of a framework for student achievement may support teachers efforts to increase student engagement. Implications from this study suggest that teachers should consider a variety of factors to increase student engagement in the Grade 9 Applied Mathematics class. Teachers can consider characteristics of their early adolescent learners, and factors for social and academic engagement. Teachers will favour approaches that parallel their personality and values and efforts in one factor may support another factor of student engagement. Suggestions for areas of further research are included at the end of the study.

This study examined how an elementary teacher fostered student mathematical understanding and the strategies that she used to help students learn mathematical concepts. A case study of a Grade 1 teacher is described based on qualitative data from interviews and classroom observation sessions using a peer coaching model. The evidence from the study suggests that this teacher benefited from professional development opportunities to gain deeper insights regarding her teaching practices. There were five major findings: (1) enthusiasm for improving her practices was necessary to successfully meet her goals; (2) this teacher’s role in the classroom was important to facilitate the construction of knowledge; (3) the classroom was an environment where her students felt safe; (4) a variety of tasks and strategies that students of varied abilities, interests and aptitudes can enjoy were used; and (5) multiple representations (including the use of manipulatives) were used to scaffold the construction of knowledge.