No description available.

Doctor of Philosophy / Curriculum and Instruction Programs / David S. Allen / This study examined pedagogical content knowledge (PCK) and Mathematics knowledge for teaching (MKT) from a perspective that blends existing definitions, questions, and methodologies into a unique method of collecting and analyzing data. Many studies on MKT have been done using qualitative methodology, usually with the researcher interviewing or testing individual teachers and analyzing that data using qualitative methods. A smaller number of studies have attempted to measure MKT using a quantitative approach, often times involving paper pencil tests with multiple choice and some open ended questions. Current research is also heavily weighted towards pre-service elementary teachers in the area of MKT for division by fractions (Depaepe et al, 2013). Although it may be true many pre-service elementary teachers have difficulties with division by fractions (Li & Kulm, 2008; Ma, 1999; Tirosh, 2000), we do not know if these problems persist for in-service elementary teachers or with secondary teachers at any stage. This study used a survey created by the researcher as a synthesis of existing questions from research on MKT with regards to division by fractions. This survey was delivered through an online format. The qualitative data in this research was then coded into quantitative data using a rubric developed by the researcher producing MKT scores that could be analyzed using statistical methods and generalized to a larger population. This study examined the Mathematical Knowledge for Teaching held by elementary, middle, and secondary in-service teachers in the domain of division by fractions. In particular this research asked if there was a relationship between the educational background, training, and experience of teachers and their MKT. Multiple analysis including independent t-tests, independent one-way ANOVAs, and Multiple regression analyses revealed that Middle school teachers and teaching at a middle school were significant predictors of increased MKT scores when compared with other groups of teachers. The type of teaching license, the type of degree held, total experience, and grade level experience were all found to have no significant relationship to MKT scores.

This study investigated how student teachers used their mathematical knowledge for teaching and pedagogical knowledge to design and modify mathematical tasks. It also examined the relationship between teacher knowledge and the cognitive demands of a task. The study relied heavily on the framework in Hill, Ball, and Shilling (2008), which describes the different domains of knowledge in mathematical knowledge for teaching, and the framework on the cognitive demands of mathematical tasks in Stein, Smith, Henningsen, and Silver (2000). Results of the study indicated that the student teachers used their common content knowledge when they lacked sufficient knowledge in other domains, especially specialized content knowledge, to perform a particular job of teaching. There was often a decrease in the cognitive demands of a task when it was modified by the student teachers. These drops were often associated with a lack of specialized content knowledge.

This thesis concerns teaching in mathematics teacher education and is based on the implementation of a learning study at teacher training. The overall purpose was to investigate in what way teacher training could facilitate and improve student teachers’ Mathematical Knowledge for Teaching (MKT). In the learning study design, MKT was conceptualized as an object of learning with a meta-character, which meant that it was applicable to and transferable between different content areas of mathematics. This made it possible to vary the mathematical content between lessons but to keep the object of learning constant. Four critical features of the object of learning were found, giving insight in some of the problems related to teacher education. Student teachers had to be able to formulate proper aims for a lesson and to give detailed descriptions of elements of MKT for coherence in their MKT to occur. A focus on student teachers’ role as mathematics teachers had to be established and finally, sufficient mathematical knowledge was found to be a prerequisite for their MKT to develop. The study shows that enactment of these critical features improved the teaching by the teacher educators, which in its turn improved the student teachers’ learning with regard to MKT. The study also indicates that the prescribed design is worth considering for future collaborative efforts of improving teaching where other objects of learning with a similar meta-character are involved.

Knowledge of curriculum is a significant component of mathematical knowledge for teaching. However, clearly understanding knowledge of curriculum requires further refinement and substantial research. This study consists of three papers that aim to explore prospective and practicing middle school teachers’ Knowledge of Curriculum for Teaching Simple Algebraic Equations (KCTE). The first paper reviews trends in and the evolution of standards and policies and synthesizes significant findings of research on mathematics curriculum and Knowledge of Curriculum for Mathematics Teaching (KCMT). Through this synthesis, the paper examines policy changes and research relevant to mathematics curriculum and KCMT and anticipates future research approaches and topics that show promise. Building on the context provided by the first paper, the following two papers investigate KCTE from the perspectives of prospective and practicing middle school mathematics teachers. For the second paper, data was collected from a convenience sample of 58 prospective middle school mathematics teachers and a subsample of six participants. The findings of this study identify patterns of key mathematical topics in the teaching sequence of simple algebraic equations, compare the participants’ sequences with experts’, reveal participants’ orientations toward KCTE, draw connections between participants’ KCTE and their knowledge of content and teaching, and establish relationships between participants’ KCTE and their knowledge of content and students. Four middle school mathematics teachers participated in the third study. The results indicate that state-level intended curriculum is the most prevailing component of participants’ KCTE. Furthermore, from a vertical view of curriculum, participants’ awareness of their students’ lack of basic mathematical knowledge impacted their KCTE. The paper also identifies the role of the state-level intended curriculum in participants’ KCTE, alternative approaches to curriculum implementation that participants used to respond to the multiple intelligences of their students, and the participants’ lack of lateral curriculum knowledge in KCTE. Together, these three papers offer a closer look at KCMT with a focus on simple algebraic equations. This research broadens our understanding of prospective and practicing middle school teachers’ KCMT and discusses implications for professional development.

In the Teaching Principle (National Council of Teachers of Mathematics [NCTM], 2000), it explained that development and utilization of pedagogical content knowledge required teachers to continually increase their knowledges of mathematics content and pedagogy. This study researched the amalgamation of multi-faceted factors and inter-relatedness of pedagogical content knowledge (PCK), content knowledge for teaching mathematics (CKTM), and mentoring support perceptions throughout elementary and middle level student teachers’ and interns’ initial full-time teaching experiences. In the first article 13 elementary and seven middle grade student teachers’ are examined based on differences between pedagogical content knowledge and content knowledge for teaching mathematics. Standardized difference scores were calculated and compared using multivariate contrasts on certification level. Results showed statistically significant differences (p < .01) on all three CKTM domains but no statistically significant differences were found on any of the five Classroom Observation and Assessment for Teachers (COPAT) domains. Both groups had the highest mean difference in the CKTM number/concept domain. COPAT results showed middle grade level pre-service teachers primarily had all positive mean differences, in comparison to the elementary level pre-service teachers, which had all negative mean differences. In the second article the mathematics mentoring support perceptions of 11 first year teachers who participated in a year-long urban internship program were examined. Semi-structured interviews revealed that district and grade level campus mentors provided the most mathematics instruction and pedagogically-based support to both groups of interns. Middle school level interns relied more on their team of mathematics teachers and elementary level interns received more mathematical content support from their district mentor than did middle level interns. Pedagogical support was greatest in the areas of lesson design and implementation of classroom management strategies. In the third article 14 elementary and six middle level student teachers were observed and interviewed on general and content-specific pedagogical skills and perceptions. Results indicated both groups of student teachers perceived themselves as most competent in having lesson plans ready, routines evident, and utilizing studentcentered instruction. Conversely, both groups felt least competent in getting students on task quickly, using a variety of teaching strategies, using critical thinking skills, and handling inappropriate behavior effectively.

This study examined teachers‘ knowledge for teaching algebra, with a particular focus on teaching the concept of function and quadratic relations in China and the United States. An embedded mixed methods design was adapted, a design in which the main data set consists of written answers to a questionnaire, while the supportive data set is comprised of the written answers to open-ended questions and follow-up interviews. A structural equation model was adopted to analyze the status and structure of teacher knowledge for teaching algebra in China and the U.S. A qualitative analysis of the answers to the open-ended questions and follow-up interviews is aimed to further illustrate and interpret the quantitative findings. Three hundred and seventy six Chinese and 115 U.S. prospective middle and high school mathematics teachers participated in this survey. Based on an extensively quantitative and qualitative data analysis, the following conclusions were made. First, the Chinese participants demonstrated a stronger knowledge for teaching algebra when compared with their U.S. counterparts. Second, the structure of knowledge for teaching algebra of the Chinese participants is much more interconnected than that of their U.S. counterparts. Third, the Chinese participants showed flexibility in choosing appropriate perspectives of function concept and in selecting multiple representations in contrast to their U.S. counterparts. Fourth, this flexibility is found to be closely related to school math and teaching math. Finally, the number of college math and math education courses taken impacts teachers‘ knowledge for teaching algebra. The findings of this study hold several implications for mathematics teacher preparation in general and studies on mathematics teachers‘ knowledge in particular. Theoretically, the complexity of understanding and measuring mathematics teachers‘ knowledge for teaching was examined and discussed. This study also enriches the understanding of mathematics teachers‘ knowledge for teaching at middle and high schools in China and the United States. Specifically, the Chinese practice of developing teachers‘ basic knowledge, skills, and flexibility provides an alternative for U.S. mathematics teacher educators to reflect on their practice. Practically, what we can learn from this study to improve mathematics teacher preparation in China and the U.S. is discussed. Finally, the limitations of this study are discussed and further studies are suggested.

A longstanding problem in mathematics education has been to determine the knowledge that teachers need in order to teach mathematics effectively. It is generally agreed that teachers need a more advanced knowledge of the mathematical content that they are teaching. That is, teachers must know more about the content that they are teaching than their students and also know more than simply how to "do the math" at a particular grade level. At the same time, research does not clearly indicate what advanced mathematical knowledge (AMK) is useful in teaching or how it can be developed and identified in teachers. In particular, the potential AMK that is useful for teaching is too vast to be enumerated and may involve a great deal of tacit knowledge, which might be difficult to detect through observations of practice alone. In the last decade, researchers have identified that teaching practice entails a specialized knowledge of mathematics but the role of advanced mathematical knowledge in teaching practice remains unclear. However, the construct of horizon content knowledge (HCK) has emerged in the literature as a promising tool for characterizing AMK as it relates specifically to teaching practice. I propose an operationalization of HCK and then use that as a lens for analyzing the knowledge resources that a fourth and fifth grade teacher draw on in their encounters with the mathematical horizon. The analysis identifies what factors contribute to teachers' encounters with the horizon, characterizes the knowledge resources, or HCK, that teachers draw on to make sense of mathematics they engage with during their horizon encounters, and explores how HCK affords and constrains teachers' ability to navigate mathematical territory. My findings suggest that experienced teachers' HCK includes a situated, professional teaching knowledge that, while sometimes non-mathematical in nature, informs their understanding of mathematical content and teaching decisions. This professional teaching knowledge guides how teachers use and generate mathematical structures that sometimes align with established mathematical structures and in other cases do not. These findings have implications regarding the way in which the development of AMK is approached relative to teacher education, ongoing professional development, and curriculum design.

abstract: Teachers must recognize the knowledge they possess as appropriate to employ in the process of achieving their goals and objectives in the context of practice. Such recognition is subject to a host of cognitive and affective processes that have thus far not been a central focus of research on teacher knowledge in mathematics education. To address this need, this dissertation study examined the role of a secondary mathematics teacher’s image of instructional constraints on his enacted subject matter knowledge. I collected data in three phases. First, I conducted a series of task-based clinical interviews that allowed me to construct a model of David’s mathematical knowledge of sine and cosine functions. Second, I conducted pre-lesson interviews, collected journal entries, and examined David’s instruction to characterize the mathematical knowledge he utilized in the context of designing and implementing lessons. Third, I conducted a series of semi-structured clinical interviews to identify the circumstances David appraised as constraints on his practice and to ascertain the role of these constraints on the quality of David’s enacted subject matter knowledge. My analysis revealed that although David possessed many productive ways of understanding that allowed him to engage students in meaningful learning experiences, I observed discrepancies between and within David’s mathematical knowledge and his enacted mathematical knowledge. These discrepancies were not occasioned by David’s active compensation for the circumstances and events he appraised as instructional constraints, but instead resulted from David possessing multiple schemes for particular ideas related to trigonometric functions, as well as from his unawareness of the mental actions and operations that comprised these often powerful but uncoordinated cognitive schemes. This lack of conscious awareness made David ill-equipped to define his instructional goals in terms of the mental activity in which he intended his students to engage, which further conditioned the circumstances and events he appraised as constraints on his practice. David’s image of instructional constraints therefore did not affect his enacted subject matter knowledge. Rather, characteristics of David’s subject matter knowledge, namely his uncoordinated cognitive schemes and his unawareness of the mental actions and operations that comprise them, affected his image of instructional constraints. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2015

abstract: In this mixed-methods study, I examined the relationship between professional development based on the Common Core State Standards for Mathematics and teacher knowledge, classroom practice, and student learning. Participants were randomly assigned to experimental and control groups. The 50-hour professional development treatment was administered to the treatment group during one semester, and then a follow-up replication treatment was administered to the control group during the subsequent semester. Results revealed significant differences in teacher knowledge as a result of the treatment using two instruments. The Learning Mathematics for Teaching scales were used to detect changes in mathematical knowledge for teaching, and an online sorting task was used to detect changes in teachers' knowledge of their standards. Results also indicated differences in classroom practice between pairs of matched teachers selected to participate in classroom observations and interviews. No statistical difference was detected between the groups' student assessment scores using the district's benchmark assessment system. This efficacy study contributes to the literature in two ways. First, it provides an evidence base for a professional development model designed to promote effective implementation of the Common Core State Standards for Mathematics. Second, it addresses ways to impact and measure teachers' knowledge of curriculum in addition to their mathematical content knowledge. The treatment was designed to focus on knowledge of curriculum, but it also successfully impacted teachers' specialized content knowledge, knowledge of content and students, and knowledge of content and teaching. / Dissertation/Thesis / Ph.D. Curriculum and Instruction 2013