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

The aim of the study was to investigate what influence the use of computers using MS Excel and RJS Graph software has on grade 11 Eritrean students' understanding of functions in the learning of mathematics. An empirical investigation using quantitative and qualitative research methods was carried out. A pre-test (task 1) and a post-test (task 2), a questionnaire and an interview schedule were used to collect data. Two randomly selected sample groups (i.e. experimental and control groups) of students were involved in the study. The experimental group learned the concepts of functions, particularly quadratic functions using computers. The control group learned the same concepts through the traditional paper-pencil method. The results indicated that the use of computers has a positive impact on students' understanding of functions as reflected in their achievement, problem-solving skills, motivation, attitude and the classroom environment. / Educational Studies / M. Ed. (Math Education)

The objective of this multiple case study was to examine how three pairs of high school students from a northern Vermont high school approached quadratic functions through traditional and multiple representation tasks. Four research questions were examined: 1) How do students think about the quadratic function as they work on a series of tasks? 2) What mathematical strategies do students employ when they work on a series of tasks related to the quadratic function? 3) How does the type of task, traditional versus multiple representation, impact students' understanding of the quadratic function? 4) What kinds of knowledge (procedural or conceptual) do students utilize when completing a series of tasks about the quadratic function? Qualitative research methods that utilized think-aloud protocols while students were engaged in four tasks pertaining to the quadratic function were employed in this study. Results suggested that students tend to think about isolated parts of the problem when solving quadratic problems. Early on in their learning about quadratics, students primarily relied on procedural strategies such as think-alouds, gestures, algebraic formulas, converting equation forms, process of elimination, dissecting problems, backtracking, and drawing pictures. In addition, students preferred the standard form to the vertex form when solving quadratics and often confused the y-intercept of the standard form with the y-coordinate of the vertex when the function was in vertex form. Results also indicated that students preferred to algebraically solve a problem versus tabular or graphical strategies. By exploring how students approach the quadratic function through their own voices, this study offers some insight into the conceptions and strategies that students use for solving problems that involve the quadratic function as well as possibilities for how quadratics may be taught in high school.

<p>Algebraic equations and functions play an important role in various mathematical topics, including algebra, trigonometry, linear programming and calculus. Accordingly, various documents, such as the most recent Swedish curriculum (Lpf 94) for upper secondary school and the course syllabi in mathematics, specify what the students should learn in Mathematics Course B. They should be able to solve quadratic equations and apply this knowledge in solving problems, explain the properties of a function, as well as be able to set up, interpret and use some nonlinear functions as models for real processes. To implement these recommendations, it is crucial to understand the students’ way of experiencing quadratic equations and functions, and describe the meaning these have for the students in relation to the possibility they have to their experience of them.</p><p>The aim of this thesis is to analyse, understand and explain the relation between the handled and learned content, which consists of second-degree equations and quadratic functions, in classroom practice. This means that content is the research object and not the teacher’s conceptions or knowledge of, or about this content. This restriction implies that the handled and learned contents are central in this study and will be analysed from different perspectives.</p><p>The study includes two teachers and 45 students in two different classes. The data consist of video-recordings of lessons, individual sessions, interviews and the teachers’/researcher’s review of the individual sessions. The students’ tests also constituted an important part of the data collection.</p><p>When analysing the data, concepts relating to variation theory have been used as analytical tools. Data have been analysed in respect of the teachers’ focus on the lesson content, which aspects are ignored and which patterns of dimensions of variations are constituted when the contents are handled by the teachers in the classroom. Also, data have been analysed in respect of the students’ focus when they solve different exercises in a test situation. It can be shown that the meaning of parameters, the unknown quantity in an equation and the function’s argument change several times when the teacher presents the content in the classroom and when the students solve different exercises. It can also be shown that the teachers and the students develop complicated patterns of variation during the lessons and that the ways in which the teachers open up dimensions of variation play an important role in the learning process. The results indicate that there is a convergent variation leading the students to improve their learning. By focusing on some aspects of the objects of learning and create convergent variations, it is possible for the students to understand the difference between various interpretations of these aspects and thereafter focus on the interpretation that fits in a certain context. Furthermore, this variation leads the students to make generalisations in each object of learning (equations and functions) and between these objects of learning. These generalisations remain over time, despite working with new objects of learning. An important result in this study is that the implicit or explicit arguments of a function can make it possible to discern an equation from a function despite the fact that they are constituted by the same algebraic expression.</p>

Algebraic equations and functions play an important role in various mathematical topics, including algebra, trigonometry, linear programming and calculus. Accordingly, various documents, such as the most recent Swedish curriculum (Lpf 94) for upper secondary school and the course syllabi in mathematics, specify what the students should learn in Mathematics Course B. They should be able to solve quadratic equations and apply this knowledge in solving problems, explain the properties of a function, as well as be able to set up, interpret and use some nonlinear functions as models for real processes. To implement these recommendations, it is crucial to understand the students’ way of experiencing quadratic equations and functions, and describe the meaning these have for the students in relation to the possibility they have to their experience of them. The aim of this thesis is to analyse, understand and explain the relation between the handled and learned content, which consists of second-degree equations and quadratic functions, in classroom practice. This means that content is the research object and not the teacher’s conceptions or knowledge of, or about this content. This restriction implies that the handled and learned contents are central in this study and will be analysed from different perspectives. The study includes two teachers and 45 students in two different classes. The data consist of video-recordings of lessons, individual sessions, interviews and the teachers’/researcher’s review of the individual sessions. The students’ tests also constituted an important part of the data collection. When analysing the data, concepts relating to variation theory have been used as analytical tools. Data have been analysed in respect of the teachers’ focus on the lesson content, which aspects are ignored and which patterns of dimensions of variations are constituted when the contents are handled by the teachers in the classroom. Also, data have been analysed in respect of the students’ focus when they solve different exercises in a test situation. It can be shown that the meaning of parameters, the unknown quantity in an equation and the function’s argument change several times when the teacher presents the content in the classroom and when the students solve different exercises. It can also be shown that the teachers and the students develop complicated patterns of variation during the lessons and that the ways in which the teachers open up dimensions of variation play an important role in the learning process. The results indicate that there is a convergent variation leading the students to improve their learning. By focusing on some aspects of the objects of learning and create convergent variations, it is possible for the students to understand the difference between various interpretations of these aspects and thereafter focus on the interpretation that fits in a certain context. Furthermore, this variation leads the students to make generalisations in each object of learning (equations and functions) and between these objects of learning. These generalisations remain over time, despite working with new objects of learning. An important result in this study is that the implicit or explicit arguments of a function can make it possible to discern an equation from a function despite the fact that they are constituted by the same algebraic expression.

The aim of the study was to investigate what influence the use of computers using MS Excel and RJS Graph software has on grade 11 Eritrean students' understanding of functions in the learning of mathematics. An empirical investigation using quantitative and qualitative research methods was carried out. A pre-test (task 1) and a post-test (task 2), a questionnaire and an interview schedule were used to collect data. Two randomly selected sample groups (i.e. experimental and control groups) of students were involved in the study. The experimental group learned the concepts of functions, particularly quadratic functions using computers. The control group learned the same concepts through the traditional paper-pencil method. The results indicated that the use of computers has a positive impact on students' understanding of functions as reflected in their achievement, problem-solving skills, motivation, attitude and the classroom environment. / Educational Studies / M. Ed. (Math Education)

The aim of the diploma theses is to find out if the study in a form of an e-learning course is more beneficial to the student than the isolated study text. To find the benefits, the research among nineth grade students from two different schools was done. A study text called Functions for the nineth grade was prepared together with a collection of solved and unsolved problems. Based on this text an e-learning course called Functions for the nineth grade was prepared. The e-learning course was placed on the website linked http://funkcepro9r.maweb.eu/. For the communication of the students and the tutor of the course, for the completion of the compulsory tasks and for the carrying out the study agenda, the modul iTřída on the web link itrida.dumy.cz was selected. Both forms of the study materials were tested by the nineth grade students on two selected schools. The efectivity of both of the forms was tested by the written test. The evaluation of the both forms was done by the electronic questionnaire. According to the results of the written test, the students which used the isolated study text were more succesful than the students studying the e-learning course. The results of the questionnaire verified that the study text was more acceptable for the students. The e-learning course did not suit the...

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This piece of work has as its main objective, verifying the possibility of insertion in high-school, of the intuitive ideas of Differential and Integral Calculus applied to the study of quadratic functions. Intuitive ideas of limits of a function, average variation rate, instantaneous variation and even the calculus of areas below the graph of positive functions, limited by the abscissa axis and by vertical lines, or even among positive functions in a determined interval of domain of those, e.g. are fairly simple concepts, which can be inserted in high-school. In order to facilitate the understanding of those ideas, coupled to the study of functions, one can make use of a computational resource such as Geogebra, software used as a learning tool on the activities suggested in the present piece of work. The activities proposed here are intended to first-year students of high-school, ally to the study of quadratic functions, because of the need to restrict the theme on this occasion; however the proposed material may be adapted and applied to other functions studied in this series. Is to highlight the importance of introduction of those ideas in high-school, in a way that stimulates the construction of more solid knowledge about the behavior of functions and many other related concepts, such as consequences and the very construction of numerical conjuncts, especially real numbers. This way the student can expand, for example, his views on the construction of graphs from the idea of continuity of a function which may be approached by simple propositions involving the limits of a function and its behavior, insofar as we take values from its domain each time bigger or smaller. It s believed that, this way, on the long range, the students who entered in college education in the subjects of calculus, will have better conditions to understand the necessary concepts and, thus, the rate of failure in those subjects and other related, may not be so high. We will see in the course of this work, that the cause to that high rate of failure may be related to a faulty forming of intuitive ideas of calculus in high-school. The following piece of work presents a proposal of activities about the teaching of those topics with the aid of the software Geogebra and the analyses of the results of the applying of those activities on an experimental class of first-year high-school students. It s been verified that it s possible opening the horizons in the sphere of learning and teaching of mathematics in high-school, for the intuitive ideas of calculus, making use of several tools, such as the use of adequate technologies and that it s even possible providing the students with new methodologies of teaching that can favor the learning of those and other mathematical concepts. / Este trabalho tem como objetivo principal verificar a possibilidade da inserção, no Ensino Médio, das ideias intuitivas do Cálculo Diferencial e Integral. Ideias intuitivas de limites de uma função, de taxa de variação média, variação instantânea e o cálculo de áreas abaixo do gráfico de funções positivas, limitadas pelo eixo das abscissas e por retas verticais, ou até mesmo entre funções positivas em um intervalo determinado pelo domínio das mesmas, por exemplo. São conceitos razoavelmente simples, que podem ser introduzidos no ensino médio. Para facilitar o entendimento dessas ideias, aliado ao estudo de funções, pode-se fazer o uso de um recurso computacional como o Geogebra, software utilizado como ferramenta de apoio a aprendizagem nas atividades sugeridas nesse trabalho. As atividades aqui propostas destinam-se a alunos do primeiro ano do ensino médio, aliado ao estudo de funções quadráticas. Pela necessidade de restrição do tema nessa ocasião, o material proposto pode ser adaptado e aplicado às demais funções estudadas nessa série. Destaca-se a importância da introdução dessas ideias no Ensino Médio, de modo a estimular a construção de conhecimentos mais sólidos sobre o comportamento de funções e muitos outros conceitos relacionados, tais como sequências e a própria construção dos conjuntos numéricos, especialmente os números reais. Dessa forma, o estudante pode ampliar sua visão sobre a construção de gráficos a partir da ideia de continuidade de uma função a qual pode ser abordada por problemas simples envolvendo os limites de uma função e seu comportamento, na medida em que tomamos valores de seu domínio cada vez maiores ou menores. Acredita-se que, assim, a longo prazo, os alunos que ingressarem no Ensino Superior nas disciplinas de Cálculo terão condições melhores de compreender os conceitos necessários e, então, os índices de não aprovação nessas disciplinas e outras relacionadas, poderão deixar de ser tão altos. Veremos no decorrer desse trabalho que a causa para esse índice elevado pode estar relacionada com uma formação deficiente das ideias intuitivas de cálculo no Ensino Médio. O trabalho a seguir apresenta uma proposta de atividades sobre o ensino desses tópicos com auxílio do software Geogebra e a análise dos resultados da aplicação dessas atividades a uma turma experimental de alunos do 1o ano do Ensino Médio. Verificou-se que é possível abrir os horizontes no âmbito do ensino e aprendizagem de Matemática no Ensino Médio, com as ideias intuitivas de Cálculo, fazendo o uso de ferramentas diversas, como a utilização de tecnologias apropriadas, e que assim, pode-se inclusive proporcionar aos estudantes novas técnicas de ensino que favoreçam a aprendizagem desses e demais conceitos matemáticos.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-23T17:39:24Z No. of bitstreams: 1 DissFMGD.pdf: 5584383 bytes, checksum: a1c7f3be1865a53b851640fd3b5643c9 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:41:50Z (GMT) No. of bitstreams: 1 DissFMGD.pdf: 5584383 bytes, checksum: a1c7f3be1865a53b851640fd3b5643c9 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:41:55Z (GMT) No. of bitstreams: 1 DissFMGD.pdf: 5584383 bytes, checksum: a1c7f3be1865a53b851640fd3b5643c9 (MD5) / Made available in DSpace on 2016-09-26T20:42:01Z (GMT). No. of bitstreams: 1 DissFMGD.pdf: 5584383 bytes, checksum: a1c7f3be1865a53b851640fd3b5643c9 (MD5) Previous issue date: 2015-08-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / This work presents a didactic proposal for the classes of affine and quadratic functions in elementary school. From our experiences in the teaching profession, we noted that the study of this content requires experimentation and contextualized activities. The study of affine and quadratic functions in basic education is of great importance for abstraction capacity development, solving practical problems and helps to get skills of comparing results, recognizing the characteristics of these functions. Our work is not intended to define affine and quadratic function, but to develop the perception ability of the results that leads to graphs and charts building and can be confirmed using the GeoGebra software. This activity was implemented in two classes of the ninth year of elementary school at Colégio Batista Brasileiro in Bauru. To do this we need four classes of 100 minutes and 1 class of 50 minutes. This is a didactic sequence that requires the use of computers and of the GeoGebra software. / Este trabalho apresenta uma proposta didática para as aulas de funções afim e quadrática no ensino fundamental. A partir de nossas experiências no exercício da docência observamos que o estudo desse conteúdo necessita de atividades de experimentação e contextualização. O estudo de funções afim e quadrática no ensino fundamental é de grande importância para o desenvolvimento da capacidade de abstração, resolução de problemas práticos do cotidiano e ajuda a adquirir habilidades de comparar resultados, reconhecendo as características dessas funções. Nosso trabalho não tem como objetivo definir função afim e quadrática, mas desenvolver a capacidade de percepção dos resultados que levam a construção de gráficos e tabelas, podendo ser confirmados utilizando o software GeoGebra. Esta atividade foi aplicada em duas turmas do nono ano do ensino fundamental de uma escola particular chamada Colégio Batista Brasileiro em Bauru. Para isso foram utilizadas 4 aulas de 100 minutos e 1 aula de 50 minutos. Trata-se de uma sequência didática que requer o uso de computadores e do software GeoGebra.

Made available in DSpace on 2015-03-03T15:36:11Z (GMT). No. of bitstreams: 1 ClesioRB_DISSERT.pdf: 808792 bytes, checksum: 2d1aa128ad7360c7d557a02b2b246fb8 (MD5) Previous issue date: 2013-05-03 / Universidade Estadual do Rio Grande do Norte / Across the centuries, Mathematics - exact science as it is - has become a determining role in the life of man, which forms to use suprir needs of their daily lives. With this trajectory, is characterized the importance of science as an instrument of recovery not only conteudstica, but also a mathematician to know that leads the apprentice to be a dynamic process of learning ecient, able to find solutions to their real problems. However, it is necessary to understand that mathematical knowledge today requires a new view of those who deal directly with the teaching-learning process, as it is for them - Teachers of Mathematics - desmistificarem the version that mathematics, worked in the classroom, causes difficulties for the understanding of students. On this view, we tried to find this work a methodology that helps students better understand the Quadratic functions and its applications in daily life. Making use of knowledge Ethnomathematics, contextualizing the problems relating to the content and at the same time handling the software GeoGebra, aiming a better view of the behavior of graphs of functions cited / Atravessando os s?culos, a Matem?tica - como ci?ncia exata que ? - vem assumindo um papel determinante na vida do homem, que dela faz uso para suprir necessidades de sua vida di?ria. Com essa trajet?ria, fica caracterizada a import?ncia dessa ci?ncia como instrumento de valoriza??o n?o s? conteud?stica, mas tamb?m de um saber matem?tico que conduz o ser aprendiz a um processo din?mico de aprendizagem e - ciente, capaz de buscar solu??es para seus problemas reais. No entanto, necess?rio se faz compreender que o conhecimento matem?tico hoje requer um novo olhar daqueles que diretamente lidam com o processo ensino-aprendizagem, pois cabe a eles - professores de Matem?tica - desmistificarem a vers?o de que a Matem?tica, trabalhada na sala de aula, traz dificuldades para a compreens?o do alunado. Sobre esse prisma, buscou-se com este trabalho encontrar uma metodologia que auxilie os alunos a entenderem melhor as Fun??es Quadr?ticas e suas aplica??es no cotidiano. Fazendo-se uso dos conhecimentos etnomatem?ticos, contextualizando os problemas referentes ao conte?do e, ao mesmo tempo, manuseando o software GeoGebra, objetivando uma melhor visualiza??o do comportamento dos gr?ficos das citadas fun??es.