Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2017-10-20T14:04:18Z No. of bitstreams: 1 Valdirene da Silva de Sousa.pdf: 1750710 bytes, checksum: 0a8bc028b9358699f47ec528cab9ffba (MD5) / Made available in DSpace on 2017-10-20T14:04:18Z (GMT). No. of bitstreams: 1 Valdirene da Silva de Sousa.pdf: 1750710 bytes, checksum: 0a8bc028b9358699f47ec528cab9ffba (MD5) Previous issue date: 2017-08-16 / The present study has started on the will to investigate the ways teachers on elementary levels, while teaching Math, deal with situations when students make errors when performing tasks. Seeing that, it was built the following question: which characteristics can be identified in the teacher’s speech about their practice in using the error as a resource for learning? It was expected with this question to identify convergences and/or divergences among theoretical instances and teacher’s reports about their practices. The theoretical support was found specially on the ideas of Cury and Brousseau in order to analyze the assumptions teachers have on students errors. In order to focus on teacher’s views we added to the consideration teacher’s backgrounds and the error and obstacles concept, aiming to narrow the ideas adopted although the research. A qualitative approach or research was chosen in which six people took part, all elementary teachers of 4th and 5th grades from private and public schools in the city of São Paulo. The data were collected through semi-structured questionnaires. As a result of this process of reflection, we can infer that teachers use student’s errors in their Math classes in different possibilities inherent to the teaching process. As strategies, the teachers affirmed to use, in most cases, collective corrections in which the students participate by showing their reasoning in carrying out the activities, sometimes going to the blackboard to register them, or participating orally. Teachers have stated that they rarely use individualized correction because there is not enough time to realize this dynamic. All of them ask students to erase the resolution when they miss an activity, and to redo copying from the blackboard. It is expected from this paper to contribute to the thinking of errors students make as part of what they understand of what is taught / O presente estudo teve a intenção de investigar de que forma os professores que ensinam Matemática dos anos finais do Ensino Fundamental I lidam com as situações em que os alunos apresentam erros nas atividades. Diante dessa inquietação, construiu-se o seguinte questionamento: Quais características podem ser identificadas no discurso de professores sobre sua prática na utilização do erro como ferramenta para a aprendizagem? Com essa pergunta esperou-se identificar convergências e/ou divergências entre instâncias teóricas e relatos de professores sobre sua prática. Buscou-se aportes teóricos que permitissem analisar as visões que os professores têm sobre os erros dos alunos, com base, principalmente, nas ideias de Cury e Brousseau. Para tratar a perspectiva dos profissionais, agregamos a essa reflexão, a formação de professores e os conceitos erro e obstáculo, visando delimitar a concepção adotada ao longo da investigação. Desenvolveu-se uma pesquisa de abordagem qualitativa, em que participaram seis sujeitos, professoras de 4º ou 5º ano do Ensino Fundamental I, que lecionam em escolas da rede pública ou privada do município de São Paulo. A coleta de dados foi efetuada por meio de entrevistas semiestruturadas. Como resultado, constata-se que as professoras aproveitam os erros dos alunos em suas aulas de Matemática, utilizando diferentes possibilidades inerentes ao processo de ensino. Como estratégias, elas afirmaram realizar, na maioria das vezes, correções coletivas, em que os alunos participam mostrando seu raciocínio ao realizar as atividades, às vezes indo até a lousa para registrá-los, ou então participando oralmente. As professoras declararam que raramente utilizam a correção individualizada, porque não há tempo suficiente para realizar essa dinâmica. Todas pedem que os alunos apaguem a resolução quando erram alguma atividade, e que refaçam copiando da lousa. Com esse trabalho, espera-se contribuir com uma reflexão sobre a análise dos erros que os alunos cometem nas aulas, como reflexo do que eles compreendem dos conteúdos

What is good mathematics teaching? The answer depends on whom you are asking. Teachers, researchers, policymakers, administrators, and parents usually provide their own view on what they consider is good mathematics teaching and what is not. The purpose of this study was to determine how beginning teachers define good mathematics teaching and what they report as being the most important attributes at the secondary level. This research explored whether there was a relationship between the demographics of the participants and the attributes of good teaching. In addition, factors that influence the understanding of good mathematics teaching were explored. A mixed methodology was used to gather information from the research participants regarding their beliefs and classroom practices of good mathematics teaching. The two research instruments used in this study were the survey questionnaire and a semi-structured interview. Thirty-three respondents who had one to two years of classroom experience comprised the study sample. They had graduated from a school of education in an eastern state and had obtained their teacher certification upon completing their studies. The beginning mathematics teachers selected these four definitions of good teaching as their top choices: 1) have High Expectations that all students are capable of learning; 2) have strong content knowledge (Subject Matter Knowledge); 3) create a Learning Environment that fosters the development of mathematical power; and 4) bring Enthusiasm and excitement to classroom. The three most important attributes in good teaching were: Classroom Management, Motivation, and Strong in Content Knowledge. One interesting finding was the discovery of four groups of beginning teachers and how they were associated with specific attributes of good mathematics teaching according to their demographics. Beginning teachers selected Immediate Classroom Situation, Mathematical Beliefs, Pedagogical Content Knowledge, and Colleagues as the top four factors from the survey analysis that influenced their understanding of good mathematics teaching. The study's results have implications for informing the types of mathematical knowledge required for pre-service teachers that can be incorporated into teacher education programs and define important attributes of good mathematics teaching during practicum.

The release of the Common Core State Standards for Mathematics in 2010 resulted in a new focus on mathematical modeling in United States curricula. Mathematical modeling represents a way of doing and understanding mathematics new to most teachers. The purpose of this study was to determine the conceptions and misconceptions held by teachers about mathematical models and modeling in order to aid in the development of teacher education and professional development programs. The study used a mixed methods approach. Quantitative data were collected through an online survey of a large sample of practicing and prospective secondary teachers of mathematics in the United States. The purpose of this was to gain an understanding of the conceptions held by the general population of United States secondary mathematics teachers. In particular, basic concepts of mathematical models, mathematical modeling, and mathematical modeling in education were analyzed. Qualitative data were obtained from case studies of a small group of mathematics teachers who had enrolled in professional development which had mathematical models or modeling as a focus. The purpose of these case studies was to give an illustrative view of teachers regarding modeling, as well as to gain some understanding of how participating in professional development affects teachers' conceptions. The data showed that US secondary mathematics teachers hold several misconceptions about models and modeling, particularly regarding aspects of the mathematical modeling process. Specifically, the majority of teachers do not understand that the mathematical modeling process always requires making choices and assumptions, and that mathematical modeling situations must come from real-world scenarios. A large minority of teachers have misconceptions about various other characteristics of mathematical models and the mathematical modeling process.

This dissertation is devoted to a history of the New York State elementary school teacher certification requirements, specifically in mathematics, during the nineteenth century. In the last half of nineteenth century, teacher education and uniform certification procedures were beginning to become the norm in the educational systems throughout the United States. The purpose of this study was to determine the ways an individual could obtain a elementary school teachers' certificate in New York State at the genesis of teacher certification examinations and to establish an understanding of what mathematical knowledge was required for elementary school teachers in order to obtain a State Certificate. This study analyzes multiple primary sources, including annual reports from New York State and New York City, along with the State Certificate examinations. The examinations in arithmetic, algebra and geometry are analyzed for the mathematics topic of the questions, the number of steps needed to complete the question and the categorizing of the questions according to Bloom's taxonomy (CBT). Five examinations of each subject, arithmetic, algebra and geometry, are analyzed, from the years 1875 to 1898. The topics addressed on the examinations, the number of steps to complete the questions and the CBT of each question are determined and used to establish the required mathematical knowledge of the elementary school teacher. The analysis of the mathematics examinations for State Certificates showed that the content required for elementary school teachers were comparative to the high school mathematics curriculum at the time. The content on the examinations for mathematics was more than the course of study they were going to be teaching in the elementary schools. They were required to have a broad and sophisticated mathematical education, however, less sophisticated than offered in the University. Also on the mathematics examinations, teachers were not assessed in what is now called pedagogical content knowledge.

As the need for educational reform is increasingly recognized, so too is the need for effective professional development (Guskey, 2000). Historically the evaluation of professional development experiences has been limited to exit surveys, noticeably failing to examine the long-term impact of the effort. This study assessed the impact on the classroom learning environment of a yearlong, job-embedded professional development opportunity for middle-school mathematics teachers. The application of learning environment instruments to the evaluation of professional development is a unique feature of this study. The research employed the Questionnaire on Teacher Interactions (QTI) and a modified version of the What Is Happening In this Class? (WillIC) survey with over 1000 middle-school mathematics students in 57 classrooms in the state of Washington. Both instruments were administered at the beginning and end of the school year. Teacher interviews were conducted with a sample of participants in order to further illuminate the impact of the professional development. Data from the study were examined for changes in the learning environment and to cross-validate the QTI and WIHIC with this specific population. Results indicate that the QTI and WIHIC are valid and reliable with the middle-school population is this study. Statistical analyses of learning environment data indicate that any pretest-posttest changes that were observed are mostly likely too small to be of educational significance. This study contributes to a better general understanding of the impact of this professional development, and its findings could be utilized in the preparation of future professional development opportunities.

Abstract not available

This study addresses the criticism leveled at A.C.T. Mathematics teachers with regard to their failure to use any other method of teaching than chalk-and-talk. By considering the changed needs of society for mathematics and the changed perceptions by society of education, the criticism is placed in context. The importance of spatial ability for mathematics is examined in the context of theories of cognitive abilities and its current under utilization within the classroom. On the basis of the increased need to utilize more talent the study was designed to operationalise in the classroom the constructs of simultaneous and successive synthesis, derived from Luria's model of brain functioning. The question of gender differences in mathematics achievment and spatial ability is addressed. The possible role of the maturation of language in determining differences in the acquistion of ability to form simultaneous synthesis is briefly discussed. The study was designed to utilize and enhance simultaneous synthesis. By demonstrating an Aptitude-Treatment Interaction it was intended to confirm that students, who function at a high level in simultaneous synthesis but at a low level in successive synthesis, would achieve more with experience with spatial activates than in a more traditional chalk-and-talk classroom. Gender differences in achievement were not found. Gender differences in successive/simultaneous profiles were found in accordance with theory predictions. The need for the duration of longer treatment periods is briefly discussed in the context of funding and the appearance of greater efficiency of traditional teaching methods when the students are functioning at the highest level of symbolic thought.

The National Council of Teachers of Mathematics published their vision of active, problem-centered instruction with a goal of conceptual understanding in 1989. Fifteen years after these reforms were proposed the changes are reflected in school policy and elementary mathematics curriculum, but only limited change has actually occurred in classroom instruction. With the belief that the classroom teacher is the key person affecting educational change, this case study examines the journey of five elementary classroom teachers as they transformed their mathematics instruction from traditional to reform-based, with the purpose of identifying the key elements that influenced the changes. This is a multi-case study involving five elementary classroom teachers who have recently been the recipient of the Elementary Presidential Award for Excellence in Teaching Mathematics. All of these teachers began teaching with traditional textbook programs and have changed their teaching to reform-based, problem-centered instruction. Over the course of two one-hour interviews each teacher told the story of his or her changes, explaining the influences, the key resources, the influential people, and the support they received in the process. The cases are individually presented; then all five are examined together in a cross-case analysis using a constructivist theoretical perspective. Three key elements were found to be influential in the teachers' change journeys. First, all five were self-motivated to make changes in their mathematics instruction. They were looking for practices that would give their students both better understanding and positive dispositions. All believed the reform-based instruction met these goals. Second, all five engaged in rich professional discussions about the changes they were making. These discussions were in groups with high levels of trust, in which the teachers freely shared concerns and successes, asked questions, and compared experiences. They were learning communities that supported the teachers' development of pedagogy and knowledge, allowing them to become confident practitioners. Finally, all five teachers were passionate about their teaching. The learning of their students and the improvement of their teaching were the prime considerations in the changes they adopted and the knowledge and skills they developed. / Graduation date: 2004

The study investigated the impact of Thai preservice middle school mathematics teachers' knowledge of subject matter and of students' conceptions of division of rational numbers with respect to their classroom practices in a teaching environment controlled by a required national curriculum. Four preservice teachers were selected with different knowledge of subject matter and of students' conceptions of division of rational numbers: high knowledge of subject matter and high knowledge of students' conceptions, high knowledge of subject matter and low knowledge of students' conceptions, low knowledge of subject matter and high knowledge of students' conceptions, and low knowledge of subject matter and low knowledge of students' conceptions. Each preservice teacher was observed three weeks, each class day during the teaching of units on division of decimals, representing fractions as decimals, and division of fractions. Formal interviews were conducted with each of the four preservice teachers prior to and after teaching each unit. Informal interviews were conducted prior to and after teaching each lesson. Materials used in the normal teaching of the class were collected. Interviews with the preservice teachers' mentors were conducted before and after each unit. The mentors were interviewed daily before or after the instruction. Interviews with supervisors were conducted each time they supervised the preservice teachers. Results showed that all preservice teachers planned and taught division of rational numbers procedurally following an algorithmically-based national curriculum. The preservice teachers with higher subject matter knowledge used multiple examples. They could make up examples when the students asked questions. In contrast, the lower subject matter knowledge preservice teachers rarely created new examples while they were teaching. The high knowledge of students' conceptions preservice teachers used their knowledge of students' conceptions throughout the lessons more often than the low knowledge of students' conceptions preservice teachers. After teaching the lessons, they all gained knowledge of subject matter and of students' conceptions of division of rational numbers. The depth of knowledge of subject matter and of students' conceptions of division of rational numbers is as essential for preservice middle school mathematics teachers' teaching in a nonvoluntary curriculum as it is in a voluntary curriculum. / Graduation date: 2003

This study described how and why a Korean elementary teacher taught mathematics as he did. Specifically, the study sought to describe his beliefs about the teaching and learning of mathematics and relate them to patterns of classroom interaction and norms. An ethnographic inquiry guided the study of one third grade 10-year veteran teacher over three months in Korea. Through participant observation, the researcher observed Teacher Lee's teaching paying special attention to the mathematics lessons for one class of 45 students daily Monday through Saturday. Formal and informal interviews were used to collect data on the teacher and 17 of his students as well as other teachers (such as four teachers in the third grade), two principals, two mothers, and three beginning teachers. In addition to participant observation and interviews, a variety of documents were also collected, including newspapers, articles from journals, test items used in the teacher's classroom, daily worksheets, curriculum guide book, mathematics textbook, the school's newspaper. All videotapes and audiotapes were transcribed for inductive analysis. The analysis generated six major themes of the teacher's beliefs about the teaching and learning of mathematics and how those beliefs impacted the interactions and norms: (a) behave orderly, think freely; (b) teaching mathematics with understanding; (c) manipulative activities and games; (d) discourse-oriented teaching practices; (e) mathematical tasks; and (f) professional development. The teacher's beliefs about the teaching and learning of mathematics were closely related to the interaction patterns and classroom norms. This close relationship implies that identifying interaction patterns and classroom norms may shed light on understanding teachers' beliefs and teaching practices. The teacher's study group activity was a major professional development factor in promoting the consistent relationship. Implications and recommendations included (a) the need for more study of classroom norms and interactions as practical knowledge of teaching mathematics, (b) the need for investigating the effect of a study group to support teacher change, (c) the importance of the relationship between pedagogical content knowledge and teachers' beliefs, and (d) the need for more study of classroom management for teaching mathematics using understanding and discourse as an instructional strategy. / Graduation date: 2000