Effects On Teachers\u27 Mathematics Content Knowledge Of A Professional Learning Community

The purpose of this research was to study the implementation of a professional learning community comprised of one group of third-grade teachers in a Florida elementary school where the emphasis was on research-based practices in the teaching of mathematics. Investigated were the growth of teachers‟ content knowledge in mathematics, specifically in the areas of multiplication and division, and the effects of their professional learning within their classrooms. Specifically this study looked at whether or not the participation of a group of third grade teachers in a professional learning community PLC improved the mathematical content knowledge of the participants of the study. This research design called for the research to be conducted in three phases. In Phase I, the researcher interviewed all participants using a researcher-designed interview guide. A researcher-adapted survey, based upon previously released items Ball (2008) was administered as a pre-test of mathematical content knowledge,. In Phase II of the study, the researcher documented the activities that occurred within a 10-week long professional learning community (PLC) of third-grade teachers. In Phase III of the research, a post-study interview was conducted with each of the participants by an independent observer to elicit participants‟ perceptions and observations based on their participation in the PLC. A post-test of content knowledge was also administered to the participants. Several themes were identified in the research study. These themes led to recommendations for practice and future research. Themes were related to the lack of iv mathematical understanding experienced by some teachers and the lack of professional development specifically related to mathematics, the value of the professional learning community, and the benefits of sharing current research and best practices. During this study, the participants were able to read and share examples of research-based best practices in mathematics, and participants then used this new information and additional mathematics content knowledge in their classrooms in teaching their students

Understanding primes: The role of representation

In this article we investigate how preservice elementary school (K-7) teachers understand the concept of prime numbers. We describe participants' understanding of primes and attempt to detect factors that influence their understanding. Representation of number properties serves as a lens for the analysis of participants' responses. We suggest that an obstacle to the conceptual understanding of primality of numbers is the lack of a transparent representation for a prime number. Key words: Conceptual knowledge; Content knowledge; Number sense; Preservice teacher education; Representations; Teacher education; Teacher knowledge; Whole numbers Prime numbers are often described as building blocks of natural numbers. The term building blocks can be viewed as a metaphorical interpretation of the Fundamental Theorem of Arithmetic, which claims that the prime decomposition of a composite number to prime factors exists and is unique. Although the uniqueness of prime decomposition presents a challenge for many learners, its existence is the property that is usually taken for granted (Zazkis & Campbell, 1996b). However, it is the existence property that is behind the building-blocks metaphor, creating an image of composite numbers being built up multiplicatively from primes. What are the structure and the properties of these building blocks? There are two properties in particular that seem to present a mystery to the learner. One is the existence of infinitely many prime numbers, which entails very large primes. Another is the property that prime numbers are not generated by a simple polynomial function. In fact, mathematicians of different origins have struggled for centuries to discover a prime number generator. A few successes in this area have been recorded in the early 1970s (see for example Gandhi's formula in Ribenboim, 1996), but these developments present considerable mathematical complexity and are beyond the scope of our investigation. Although the understanding of elementary number theory has been the topic of a few recent studies (Campbell & Zazkis, 2002), there has not been any study focusing specifically on prime numbers. On a related matter, Zazkis and Campbell (1996b) investigated preservice teachers' understanding of prime decomposition and concluded that "if the concepts of prime and composite numbers have not been The study reported in this article was supported i

Investigating the relationship between mathematical knowledge for teaching and self-efficacy of pre-service mathematical literacy teachers

Although a good understanding of mathematical content knowledge is essential for effective mathematics teaching, this might not be enough. Mathematical knowledge for teaching (MKT) requires a kind of depth and detail special to teaching, and involves mathematical reasoning as well as thinking from a learners’ perspective. Educational outcomes are also influenced by teachers’ self-efficacy beliefs regarding their ability to teach effectively. This study was an investigation into the relationship between pre-service teachers’ mathematical knowledge for teaching (MKT) and their mathematical self-efficacy with regard to MKT. Participants in the study were 137 BEd (FET) students at Nelson Mandela Metropolitan University, specializing in Mathematical Literacy as teaching subject. The quantitative data used for the study were gathered using a questionnaire on MKT for the topics number concepts and operations. This questionnaire was designed by Deborah Ball’s Michigan research team, to which I added a question on self-efficacy for every item. An analysis of the data gathered from the questionnaire reveals interesting and disturbing trends. The results suggest that, in more than 80% of the cases, respondents were either completely sure their answer was correct, or tended to think their answer was correct, indicating high levels of self-efficacy. Since only about 40% of answers were in reality correct, this indicates that participants believed their answer to be correct, although their interpretation of the mathematical knowledge for teaching involved was incorrect. Hence: they don’t know that they don’t know! The results of this study suggest that there is a need for educators of teachers to help improve prospective mathematical literacy teachers’ mathematical knowledge for teaching. Pre-service teachers should be taught to use cognitive skills that will raise the likelihood of improved learner understanding. For this, robust understanding of the fundamental mathematics involved is needed, as well as high levels of self-efficacy with regard to the teaching of mathematics

Preservice Elementary Teachers\u27 Understandings of Topics in Number Theory

Research suggests that preservice elementary teachers may lack the mathematics understanding necessary to teach mathematics for understanding. The literature has consistently linked student success in mathematics with teacher pedagogical content knowledge (PCK), and recent study suggested a link between teachers’ mathematical content knowledge and student achievement. There are gaps in the literature concerning preservice elementary teachers’ understanding of number theory, and little is known about how they develop number theory PCK or the relationship between their content knowledge and their PCK. The goals of this dissertation were to investigate the nature of mathematics concentration preservice elementary teachers’ content knowledge of number theory, the nature of their potential PCK in number theory, and the relationship between the two. To address these goals, I conducted a qualitative, interpretive case study of undergraduate students enrolled in a number theory course designed for preservice elementary teachers, using an emergent constructivist-based theoretical perspective. I gathered observational, interview, and document data and conducted analysis using constant comparative methods. Many of my findings concerning preservice elementary teachers’ understandings of number theory content pertain to their understandings of greatest common factor (GCF) and least common multiple (LCM). In particular, participants were more comfortable creating LCM story problems than creating GCF story problems, but their understandings of GCF story problems were closely related to the two meanings of division. In contrast to their understanding of story problems, participants were more comfortable with procedures for finding the GCF than with procedures for finding the LCM. In response to my other research questions, evidence suggests that preservice elementary teachers do possess potential PCK in number theory, namely potential knowledge of content and students and potential knowledge of content and teaching, and that they are related and influenced by specialized content knowledge, curricular content knowledge, experiences working with students, and epistemological perspectives. My data also suggest that preservice elementary teachers possess a type of PCK that is not explicitly represented by the literature, which I call general mathematical pedagogy. My findings hold many implications for practice. For example, data suggest a process through which preservice elementary teachers might develop a robust understanding of GCF story problems, which builds on their understandings of division. With this observed development process, instructors can scaffold preservice elementary teachers’ understanding of GCF story problems. My results also imply specific ways in which mathematics teacher educators and mathematicians may help preservice elementary teachers develop PCK in number theory. For example, instructors can pose hypothetical student conjectures and ask preservice elementary teachers to reflect on the knowledge necessary to teach the content, determine the validity of the conjecture, identify the concepts the student does and does not understand, suggest how they might respond to the student, and reflect on how they used their content knowledge to do so

Doctoral students’ use of examples in evaluating and proving conjectures

This paper discusses variation in reasoning strategies among expert mathematicians, with a particular focus on the degree to which they use examples to reason about general conjectures. We first discuss literature on the use of examples in understanding and reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and semantic reasoning strategies relative to a representation system of proof. We then use this conceptualisation as a basis for contrasting the behaviour of two successful mathematics research students whilst they evaluated and proved number theory conjectures. We observe that the students exhibited strikingly different degrees of example use, and argue that previously observed individual differences in reasoning strategies may exist at the expert level. We conclude by discussing implications for pedagogy and for future research

Elementary logic as a tool in proving mathematical statements

Magister Scientiae - MScThe findings of the study indicate that knowledge of logic does help to improve the ability of students to make logical connections (deductions) between and from statements.The results of the study, however, do not indicate that knowledge and understanding of logic translates into improved proving ability of mathematical statements by students.South Afric

Number worlds: Visual and experimental access to elementary number theory concepts.

Recent research demonstrates that many issues related to the structure of natural numbers and the relationship among numbers are not well grasped by students. In this article, we describe a computer-based learning environment called Number Worlds that was designed to support the exploration of elementary number theory concepts by making the essential relationships and patterns more accessible to learners. Based on our research with pre-service elementary school teachers, we show how both the visual representations embedded in the microworld, and the possibilities afforded for experimentation affect learners' understanding and appreciation of basic concepts in elementary number theory. We also discuss the aesthetic and affective dimensions of the research participants' engagement with the learning environment

Specialized Understanding of Mathematics: A Study of Prospective Elementary Teachers

This dissertation study informs the field on how, when and where a specialized understanding of math (SUM) might be developed within a teacher education program by focusing on the three following research questions and related methodology. 1) What are the strengths and weaknesses in prospective elementary teacher’s specialized understanding of mathematics as they enter their mathematics methods course? The Number and Operation and Geometry items from the Content Knowledge for Teaching Mathematics instruments, which have been developed at The University of Michigan’s Learning Mathematics for Teaching Project, were administered to 244 prospective elementary teachers at four universities during the first two weeks of the mathematics methods course. An item analysis sheds light on areas of strengths and weaknesses, and a statistical analysis was conducted to see any relationships between content understanding and quantity and type of content courses. A relationship was found between participants who took specialized content courses and the pretest scores. Another interesting finding was that simply taking more mathematics content courses is not related to higher scores. 2) Does the specialized understanding of mathematics change as they take the mathematics methods course? The CKTM items were administered as a post test during the last two weeks of the methods course and compared with the pre test to look at changes, both as a paired samples t test and an item analysis. Growth in SUM was found between the pretest and posttest. 3) What learning opportunities during the methods course may improve the specialized understanding of mathematics of prospective elementary teachers? Interviews were conducted with mathematics methods instructors who saw significant growth on specific items. The general philosophy of the course, as well as specific learning opportunities that may have helped understanding in the specific items that saw growth were explored, and a framework was created of learning opportunities that may impact understanding of mathematics. The learning opportunities that seem to add to improved SUM include readings, communication, experiencing children’s mathematical thinking, mathematics activities, manipulatives, and field experiences

Integrating Student Led Conferences into an Elementary Classroom

Improving student learning and responsibility using student led conferences was studied. A guide for implementing student led conferences in an elementary school was developed and has been implemented at Camelot Elementary School. The results of the guide support that student led conferences can be utilized by general education teachers to enhance their students ability to develop an understanding of their learning and its processes. Through student led conferences, students learn to effectively set personal academic goals, improve their communication skills and become more active in their learning. The use of student led conferences has also provided evidence that participation and involvement of parents in their child\u27s educational development increases