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

Submitted by Fabíola Peixoto Cintra (fabiolacintra17@hotmail.com) on 2018-09-21T22:17:40Z No. of bitstreams: 1 DissertaçãoFabiolaPeixotoCintra (1).pdf: 3861189 bytes, checksum: d6cfcf223f2ed02eb15ba300564fa265 (MD5) / Approved for entry into archive by Cristina Alexandra de Godoy null (cristina@adm.feis.unesp.br) on 2018-09-26T19:53:40Z (GMT) No. of bitstreams: 1 cintra_fp_me_ilha.pdf: 3861189 bytes, checksum: d6cfcf223f2ed02eb15ba300564fa265 (MD5) / Made available in DSpace on 2018-09-26T19:53:40Z (GMT). No. of bitstreams: 1 cintra_fp_me_ilha.pdf: 3861189 bytes, checksum: d6cfcf223f2ed02eb15ba300564fa265 (MD5) Previous issue date: 2018-07-26 / O conhecimento dos professores, juntamente com as suas concepções e crenças, é uma importante variável no processo de ensino e de aprendizagem da Matemática que ocorre em suas aulas, uma vez que o conhecimento do professor é a base para o ensino de um determinado conteúdo. Nesta perspectiva, o objetivo da pesquisa aqui apresentada foi investigar os conhecimentos dos futuros professores de Matemática sobre o conceito de Função e sobre o ensino e a aprendizagem desse conceito. Com isso, tivemos as seguintes questões: i) Como o conceito de função é abordado nos Cursos de Licenciatura em Matemática? ii) Quais os conhecimentos dos alunos do Curso de Licenciatura em Matemática sobre o conceito de função? iii) Quais as implicações dos conhecimentos dos futuros professores de Matemática sobre função para a atividade docente? Para responder a essas questões, tomando como base uma revisão histórica do desenvolvimento do conceito de função e uma revisão bibliográfica sobre o conhecimento dos professores sobre esse conceito, foi aplicado um questionário para alunos do último ano de um curso de Licenciatura em Matemática e foi enviado um questionário para os docentes deste curso. Para a análise dos dados obtidos por meio dos questionários foi adotada a metodologia de Análise de Conteúdo. Os resultados obtidos revelam que, de modo geral, o conceito de função está sendo adequadamente estudado e aprendido pelos alunos, mas ainda existem obstáculos epistemológicos que necessitam ser superados para uma compreensão abrangente do conceito de função, ou seja, há algumas dificuldades de aprendizagem dos alunos em relação ao conceito de função, em virtude da sua variedade de representações, levando aos obstáculos epistemológicos. / Teachers' knowledge, together with their conceptions and beliefs, is an important variable in the teaching and learning process of Mathematics that occurs in their classes, since the knowledge of the teacher is the basis for the teaching of a certain content. In this perspective, the purpose of the research presented here was to investigate the knowledge of the future Mathematics teachers about the concept of Function and about the teaching and learning of this concept. With this, we had the following questions: i) How the concept of function is approached in the Mathematics Degree Courses? ii) What are the students' knowledge of the Degree in Mathematics about the concept of function? iii) What are the implications of the knowledge of the future teachers of Mathematics about function for the teaching activity? For answer these questions, based on a historical review of the development of the concept of function and a bibliographical revision on the teachers' knowledge about this concept, a questionnaire was applied to students of the last year of a degree in Mathematics and a questionnaire was sent to the teachers of this course. For the analysis of the data obtained through the questionnaires, the Content Analysis methodology was adopted. The results show that function concept is be studied and learned by students, but there are still has epistemological obstacles that need to be overcome for a embracing function concept understanding, because of your representations variety, which leads to epistemological obstacles.

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

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-07-13T12:05:08Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Um desconhecido a porta os discursos das professoras da educação infantil sobre o ser o saber e o fazer docente -Silvéria Nascimento Ferreira.pdf: 1333397 bytes, checksum: f978fad92f82eae3c83567eef99d8b01 (MD5) / Made available in DSpace on 2016-07-13T12:05:09Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Um desconhecido a porta os discursos das professoras da educação infantil sobre o ser o saber e o fazer docente -Silvéria Nascimento Ferreira.pdf: 1333397 bytes, checksum: f978fad92f82eae3c83567eef99d8b01 (MD5) Previous issue date: 2015-12-10 / Assistimos com veemência, um reforço discursivo vindo, principalmente de documentos oficiais e pesquisas que configuram novas dimensões sobre o que caracteriza o ser professora na educação infantil, assim como ela deve proceder junto aos seus saberes em seu fazer pedagógico. É comum evidenciarmos discursos sociais que atribuem a essa professora certo despreparo para exercer a função de docência, atenuando a imagem da profissão docente nesse nível de ensino a uma ocupação fácil, exercida mais por amor, do que por saberes e fazeres profissionais. Diante disso, inquieta-nos saber o que o discurso docente enuncia sobre essas questões. Portanto, foi com base nessas premissas que realizamos a pesquisa ora relatada, com o objetivo principal de apreender enunciados atribuídos pelo discurso docente ao que é ser professora na educação infantil e aos seus saberes e fazeres. Nosso estudo realizou-se no município de Major Isidoro – Al e contou com a participação de 10 professoras que atuam na educação infantil. Do ponto de vista teórico dialogamos com autores como Ostetto (2000), Roldão (2005), Tristão (2015), Dubar (2005), Oliveira-Formosinho (2008), Louro (1997), Kramer (2011). O percurso teórico-metodológico foi elaborado com base na abordagem qualitativa de pesquisa e aportes da análise de discurso (FISCHER, 2012). Essa perspectiva escolhida justifica por nos permitir o acesso ao discurso docente, identificando alguns caracterizadores do ser professora na educação infantil, assim como apreendendo elementos constitutivos dos seus saberes e fazeres, de modo que os enunciados atribuídos pelo discurso docente ao que é ser professora na educação infantil e ao seu saberes e fazeres sejam também apreendidos. Para isso, realizaram-se entrevistas semiestruturadas, que permitiram, por meio da análise enunciativa de seus discursos, apreender os enunciados que dizem sobre o ser e os saberes e fazeres da professora na educação infantil. A partir das práticas manifestas pelas professoras nas redes discursivas que atravessam nossa pesquisa, apreendemos no discurso docente dizeres sobre o ser, o saber e o fazer da professora que apontam para uma profissão marcada social, cultural e historicamente pela produção de discursos que suscitam feminilidade, vocacionalidade, afetividade ao ser professora, mobilizando e produzindo saberes e fazeres em sua prática pedagógica, mediante conhecimentos plurais, construindo-se como constituintes de elementos necessários a profissionalidade docente na EI. Nesse processo pudemos perceber que os discursos acerca do ser professora, saber e fazer são heterogêneos, reúne um conjunto de elementos diversos, que se aproxima de uma discursividade que aponta para a afirmação da profissionalidade. Nesse sentido, é possível evidenciar um movimento, sendo realizado pelas professoras, tanto da creche, quanto da pré-escola, em direção à profissionalização docente, sendo apresentados em seus discursos elementos vários desse processo, como o planejamento e a prática pedagógica. / A discursive reinforcement from official documents and research that set new dimensions about what characterizes how the teachers in children education should behave, as well as they should proceed with to their knowledge in their pedagogical practice is watched with vehemence. It is common to identify social discourses that attributes to this professional an unpreparedness to exercise the teaching function, attenuating the image of the teaching profession in this level of education an easy occupation, exercised more by love than by knowledge and practices professional. Thus, restless us know what the teaching speech states on these issues. So, based on these assumptions this work aims to study, from the educational perspective, what is being a teacher in children education, its knowledge and practices. Our study was carried out in the town of Major Isidoro (Alagoas State, Brazil) and it was included the participation of 10 teachers who work in children education. From a theoretical point of view we dialogue with authors as Ostetto (2000), Roldão (2005), Tristão (2015), Dubar (2005), Oliveira-Formosinho (2008), Louro (1997) and Kramer (2011). The theoretical and methodological approach was based on qualitative research and contributions of discourse analysis. That perspective chosen is justified for allowing us access to educational discourse, identifying some characteristics of being a teacher in children education, as well as apprehending the constituent elements of their knowledge and practices, so that the enunciations assigned by the teacher speech to what is being a teacher in children education and their knowledge and practices are also seized. Thus, interviews were partially held, which allowed by means of the enunciation analysis of his discourses, to apprehend the speeches that say about being and the knowledge and practices of the teacher in children education. Starting from the practices expressed by teachers in the discursive networks that run through our research, we learned in teaching speech saying about being, knowing and doing the teacher pointing to a social. Labelled profession, culturally and historically for the production of speeches that give rise femininity, vocational nature, affectivity to being a teacher, mobilizing and producing knowledge and practices in their practice through plural knowledge, constructing themselves as constituent elements necessary for professional teaching in children education. In this process we realize that the speeches about being a teacher, knowing and doing, are heterogeneous, brings together a number of different elements, which sometimes approaches a discourse that points to the affirmation of professionalism. In this sense is possible to show a movement being carried out by the teachers, both day care, as literacy, toward teacher professionalization and presented their speeches in various elements of that process, such as planning and practice teaching.

In the past two decades there has been an increased effort to understand the depth to which mathematics teachers must know their subject to teach it effectively. Researchers have termed this type of knowledge mathematical knowledge for teaching (MKT). Even though recent studies have focused on MKT, the current literature on the subject indicates that this area remains underdeveloped. In an attempt to further refine our conception of MKT this study looked at MKT in Japan. In this thesis I explored and categorized the MKT of three experienced Japanese cooperating teachers (CTs) by looking at the content of their conversations with three Japanese student teachers (STs). I separated the MKT mentioned in these conversations into three categories: knowledge about the students' mathematical knowledge, knowledge about mathematics, and knowledge about school mathematics. I also discussed various implications of this work on the field of MKT.

This study investigated the pedagogical content knowledge supposedly held by two FET mathematics teachers from Mpumalanga Province as they taught quadratic functions in grade 11 classes. The criterion for selecting the two teachers was that they had consistently produced good results (overall pass rate of 80% or more) in the grade 12 mathematics examinations of the National Senior Certificate for the past three years or more and thus, they were classed as effective. The two teachers prepared and taught lessons on quadratic functions in grade 11 whilst they were being observed. The study focused on teacher knowledge base as exemplified in the teachers’ pedagogical content knowledge (PCK). Three elements of PCK were investigated; namely; (i) knowledge of the subject matter; (ii) knowledge of teaching strategies and (iii) knowledge of learners’ conceptions. Qualitative research approach using the case study research method was used to collect qualitative data on the pedagogical content knowledge of the two teachers through lesson observations, lesson plan analysis and interviews. Analysis of the results suggests that the two teachers have adequate subject matter knowledge but have limited knowledge on the aspects of teaching strategies and knowledge of learners’ pre-conceptions and misconceptions on the topics of quadratic functions that they taught. The study recommends that teachers be exposed to workshops that deal specifically with the various topic specific teaching strategies and knowledge of learners’ pre-conception and misconceptions on the topic of quadratic functions. / Dissertation (MEd)--University of Pretoria, 2012. / Science, Mathematics and Technology Education / unrestricted

Syftet med denna studie är att undersöka hur lärare upptäcker elever i behov av särskilt stöd inom matematikundervisningen. Fyra intervjuer har genomförts med två klasslärare respektive två speciallärare, från tre olika skolor. Resultatet indikerar att lärare huvudsakligen upptäcker elever i behov av särskilt stöd genom att se till elever med hög frånvaro och genom diskussion med andra lärare. Gemensamt för alla respondenter är att de påpekar att det är elever som når låga resultat i tester under en längre period som är i behov av det särskilda stödet. Resultatet diskuteras i förhållande till ett ramverk som beskriver lärarkunskaper för matematiklärare. Slutsatsen som dras är att kunskapen att upptäcka elever i behov av särskilt stöd är en viktig lärarkunskap.