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 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.

Doctor of Philosophy / Curriculum and Instruction Programs / Jennifer M. Bay-Williams / Graphing calculators have been used in education since 1986, but there is no consensus as to how, or if, they should be used. The National Council of Teachers of Mathematics and the National Research Council promote their use, and ample research supports the positive benefits of their use, but not all teachers share this view. Also, rural schools face obstacles that may hinder them from implementing technology. The purpose of this study is to determine how graphing calculators are used in mathematics instruction of high schools in western Kansas, a rural region of the state. In addition to exploring the introduction level of graphing calculators, the frequency of their use, and classes in which they are used, this study also investigated the beliefs of high school mathematics teachers as related to teaching mathematics and the use of graphing calculators. Data were collected through surveys, interviews, and observations of classroom teaching. Results indicate that graphing calculators are allowed or required in almost all of the high schools of this region, and almost all teachers have had some experience using them in their classrooms. Student access to graphing calculators depends more on the level of mathematics taken in high school than on the high school attended; graphing calculator calculators are allowed or required more often in higher-level classes than in lower-level classes. Teachers believe that graphing calculators enhance student learning because of the visual representation that the calculators provide, but their teaching styles have not changed much because of graphing calculators. Teachers use graphing calculators as an extension of their existing teaching style. In addition, nearly all of the teachers who were observed and classified as non-rule-based based on their survey utilized primarily rule-based teaching methods.

Doctor of Philosophy / Department of Curriculum and Instruction / Jennifer M. Bay-Williams / The purpose of this study was to assess the impact of student attitudes in a college algebra mathematics classroom when lessons are primarily composed of standards-based pedagogy. National reports advocate for a change in teaching K-12. Nowhere is this more needed than in community colleges where students are taught in traditional formats and rarely challenged to make connections between mathematics and their personal experiences. A thorough review of the literature shows the need for mathematics reform at every level, including the college mathematics classroom. There are several national reports, Principles and Standards for School Mathematics, Adding it Up, How People Learn, and Undergraduate Programs and Courses in the Mathematical Sciences, that have been published to address the need to change mathematics teaching and learning. They are advocates for the implementation of standards-based instruction into the mathematics classroom. This study focused on students’ perceptions about the nature of mathematics and learning mathematics, specifically, does such a learning environment impact students’ perceptions of being a student of mathematics in the areas of confidence, anxiety, enjoyment, and motivation, and relevance of mathematics in personal and professional experiences. Over the course of one semester, two sections of college algebra students participated in the study. By using both qualitative and quantitative data collection methods, the study was able to see if there was an impact in student attitudes toward mathematics. The standards-based pedagogy used in this study was cooperative learning, problem solving, discourse, and the graphing calculator. Changes in attitude were determined by attitudinal surveys, student questionnaires, observations, and focus groups. College algebra students had a statistically significant change in their enjoyment of mathematics. Although the other attitudes, confidence, motivation, and value did not have a statistically significant change, the qualitative data indicates a change in these attitudes did occur. This study identified that cooperative learning, problem-solving, discourse, and graphing calculators increased student confidence in doing mathematics because they felt more competent in working problems on exams. Students also found the class enjoyable, anxiety was reduced as students became more familiar with the instructional strategies, and students recognized the value of mathematics for job skills and personal business.

Doctor of Philosophy / Department of Educational Leadership / Sarah Jane Fishback / This study explored the effects of personal epistemological beliefs about mathematics and beliefs about the ability to do well in mathematics on achievement in a college-level, developmental mathematics class. The influences of gender, age, and ethnicity on these beliefs as they relate to mathematics achievement were also explored. The Mathematics Belief Scales (MBS) was adapted from the Indiana Mathematics Belief Scales and Self-Description Questionnaire III to measure beliefs about the time it takes to solve mathematics problems, the importance of conceptual understanding in mathematics, the procedural emphasis in mathematics, the usefulness of mathematics, and self-concept about mathematics. MBS was administered to 159 participants enrolled in Intermediate Algebra over two semesters at an urban, state-supported mid-western university and two small private mid-western universities. Responses to the surveys and scores on the final exams for the Intermediate Algebra courses were analyzed using descriptive statistics, the Pearson product-moment correlations, analysis of variance techniques, and hierarchical regression analysis. Results indicated that students generally held nonavailing beliefs about mathematics and mathematics self-concept. Students typically believed that mathematical problems should be solved within ten minutes. Students generally did not believe that math problems can be solved with logic and reason instead of learned math rules. Over 40% of the students did not believe that mathematics beyond basic mathematics was useful to everyday life. Students were also generally not confident in their ability to solve mathematics problems. Additionally, men’s self-concept was significantly higher than women’s self-concept. Adult learners’ self-concept was also significantly higher than traditional age students’ self-concept. Hierarchical regression analyses revealed that the importance of understanding mathematical concepts positively influenced final exam scores for men more so than women and self-concept positively influenced final exam scores for women more so than men. These results indicate a need for academic experiences at the college-level that will challenge students’ current belief system and provide an environment that is supportive and conducive to building individual self-confidence.

Doctor of Philosophy / Curriculum and Instruction Programs / Margaret G. Shroyer / This study was designed to investigate teaching practices of mathematics teachers in rural high schools in Kansas in the context of the NCTM Principles and Standards. National reports advocate for change in the mathematics classroom while state assessments force teachers to focus on test scores. This study investigated the extent to which teachers whose students experienced repeated success on state assessments integrated the NCTM Process and Content Standards into the mathematics classroom. Those data were then compared with the teaching practices in schools whose students repeatedly did poorly on state assessments. This two-phase study used both quantitative and qualitative data from four main sources: survey, interview, observation, and collection of artifacts. Phase I surveyed all mathematics teachers in high performing and low performing rural high schools throughout the state of Kansas. Data collected in Phase I were used to examine differences and similarities in teaching practices of teachers from high and low performing schools. During Phase II qualitative data were collected and analyzed to further explore any existing patterns among high performing and low performing schools. Results from teachers in high and low performing schools were compared and contrasted to determine if there were differences between the teaching practices that were demonstrated by each group of teachers. Results of surveys, interviews, observations, and artifacts revealed teachers in high performing schools used a variety of different representations to teach and assess a topic while those teachers from low performing schools used one or two representations. Students from high performing schools had more frequent opportunities to communicate with the teacher to gain additional assistance in learning the mathematics content. Teachers in high performing schools also used formal assessment strategies as part of the learning process more consistently than their counterparts from low performing schools. Results from interviews, observations, and artifacts reveal that teachers in high and low performing schools implement teaching practices aligned with the algebra content standards in a very similar manner.

Doctor of Philosophy / Curriculum and Instruction Programs / Diane McGrath / The goal of this study was to explore the use of interdisciplinary PBL projects for teaching mathematical concepts according to NCTM (2000) goals for mathematics instruction. This study sought to answer the question: what are the teaching issues and evidence of student learning of mathematical concepts over a series of three interdisciplinary PBL projects involving STELLA™ modeling software which are designed to engage students, integrate technology, and provide a context for learning mathematics based on the 5 NCTM (2000) goals? HyperStudio™ was used as a communication tool with which students built artifacts of understanding. This study was a naturalistic case study employing videotaped observations, interviews, student-peer reviews and student generated artifacts of learning as data sources. Data were categorized into two variable clusters: Teaching and Learning. Implementation issues for three computer-based PBL simulations are discussed. Themes that emerged from analysis of the data are grouped into teaching themes and learning themes. Themes relating to teaching include the struggle to form a community of learners, relevancy of the simulations to middle school students, need for group-worthy projects, helping students balance creativity and content, lesson adaptation, and critical review and student reflection on constructive feedback. Findings of the study suggest the students were able to meet a majority of the expected content goals. Themes relating to learning include the struggle to find a balance between creativity and content, ownership and control, engagement with the simulations, and students’ ability to think and express themselves mathematically. Recommendations are made for teachers who wish to implement PBL, simulations, and similar teaching strategies and for researchers who are studying similar learning environments.

Doctor of Philosophy / Department of Secondary Education / Jacqueline D. Spears / Moses & Cobb (2001) argue that algebra is a “civil right” and assert that limited algebraic understanding has an unfavorable impact on African American students’ entry into post-secondary education. Gay (2000) outlines six pedagogical methods, known as culturally responsive teaching (CRT), which emphasize the importance of teachers creating learning environments that relate to the personal experiences and cultural perspectives of minority students. The National Council of Teachers of Mathematics (NCTM) prescribes five process standards (communication, problem solving, connections, representation, and reasoning and proof) and the Equity Principle (includes setting high expectations, responding to the needs of culturally and linguistically diverse students, and providing support) for effective mathematics instruction. CRT, the NCTM Process Standards, and the NCTM Equity Principle served as the conceptual framework for this mixed-method study. Thirty-four teachers from two elementary and two middle schools in one school district in the Midwest responded to The Powell Teaching Mathematics Index (PTMI), a five-option Likert survey that explored teachers’ current “use” and “desire” to use CRT methods, NCTM process standards, NCTM Equity Principle, and teachers’ personal efficacy in learning and teaching mathematics in general and in algebra. Results from the PTMI revealed that teachers had a “desire” to use CRT in mathematics with AA students (M=4.41, SD=0.70); and although there was more variance among respondents, teachers also reported a “desire” to use process standards in algebra with AA students (M=3.94, SD=1.03). One bivariate correlation revealed a relationship between “use” of process standards in general and “efficacy” (r =0.681, p[less than or equal to]0.01). Eight volunteer teachers participated in a professional development workshop on CRT and integrated one of the six pedagogical methods into their classrooms for one month. Teachers reported “strengths” from the implementation phase as: increased student engagement, transition from teacher-directed to student-directed learning and an increase in student confidence in mathematics. Implementation “strains” were reported as: a time consuming process, difficulty in providing individual attention and an increase in classroom noise level. Findings have implications for teacher education programs, local school district and teacher networks.

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.