This report explores several geometric representations of quadratic equations and their solutions. Topics discussed include applications of geometry relating to solving quadratic equations using graphs and constructions as well as deriving compatible pairs of equations from Pythagorean triples. A brief discussion on the inclusion of advanced graphing methods and constructions into a secondary mathematics class is also included. / text

Magister Educationis - MEd / This study analysed the modelling competencies of grade 11 learners and also explored the degree to which the learners’ competency in setting up a mathematical model inhibits the development of an acceptable solution for word problems. The research data comprised 30 learners drawn from a secondary school in the Western Cape Province, South Africa. Data was collected via a task-based activity response sheet containing five word problems linked to either one of the following concepts: rectangle, two-digit number, average speed and petrol price. Learners’ responses were graded into four categories viz: correct, partially correct, incorrect and no response. Thereafter, the modelling competency framework was used to diagnose the modelling competencies of the sampled learners.

The aim of the thesis is to find out students' reasoning of chosen word problem by using quadratic equation. The work focuses on specific errors end problems reported by students and evaluated by Newman's method of Error Causes for Written Mathematical Tasks. The theoretical work was based on analyse of current mathematical textbooks and comparison with several international pedagogical studies and thesis with similar specialization. Furthermore, a detailed description of methodology and my own research are described in practical part of the thesis. Principle of study was to chose group of the students from two different high schools ( one well know grammar school and one business high school in Prague) and record the process of reasoning the word problem by camera. The concluison is dedicated to proper analyses of mistakes and problems during the student's reasonings. Powered by TCPDF (www.tcpdf.org)

The aim of this study was to investigate the effects of using interactive whiteboard and computer technology on graduate language and mathematics student&rsquo / performance in quadratic equation functions graphics. Two groups of language and mathematics graduate students were selected for the study / one for experimental group and the other for control group. The experimental group consists of 32 students, and control group consists of 33 students. The control group is thought by traditional learning whereas the experimental group is thought the same topic by interactive whiteboard and computer technology. Graphic Achievement Test (GAT), Attitudes toward Technology Scale (ATTS) and Attitude toward Mathematics Scale (ATMS) and interviews were used as a data collection instruments. GAT applied as a pre-test, post-test and delayed post-test on both of the groups. However the attitudes scales applied only experimental group before and after the treatment.

M.Ed. / This dissertation is about the instructional approaches used by teachers to teach Grade 11 quadratic equations, errors learners made and misconceptions they possessed. The main topics that I had focused on were solving quadratic equations by factoring, completing a square and using quadratic formula. The intention was to observe teachers’ teaching approaches in quadratic equations and diagnosed types of errors learners displayed and misconceptions they possessed in quadratic equations. Literature review had served as a secondary source of information that was relied upon for the research study. Sources such as scholarly books, government documents, dissertations, professional journals and electronic resources were used to gather the information pertinent to the research topic. Review was also done on how teachers teach quadratic equations, learners’ learning of quadratic equations and teachers teach and learners learn mathematics. This study is action research under qualitative research paradigm in which the information collected was analyzed through thick description and not statistically. Pre-test, self and post test evaluation methods are discussed of quadratic equations by factorization, completing a square and using quadratic formula. Learners were tested on factoring, completing a square and using quadratic formula. In addition to the learners’ class exercises and home work, these scripts were also analysed for errors and misconceptions. Collected data is presented that helped to address errors and misconceptions learners displayed in solving quadratic equations and teachers’ teaching methods and approaches. Data was collected from schools in the neighborhood and the school I was attached as a mathematics teacher. In all schools, five teachers’ three lessons were observed which added up to a total of fifteen. All five teachers were interviewed as well as five learners in each school. Interviews were analyzed by comparing what their teaching approaches with the types of learners’ errors and misconceptions. In classroom observations, Indicator Evaluation Form adopted from Luneta (2006) was used to collect data as well as analyzing it. Questionnaires were prepared for both teachers and learners for interviews.

<p>Utilizing action research as the methodology, this study was developed with the ultimate goal of describing and reflecting on my implementation of one aspect of the <i>Common Core State Standards for Mathematics (</i><i>CCSSM)</i> in an algebra classroom. This implementation focused on the Problem-Solving Standard of Mathematical Practice (SMP) as described in <i>CCSSM </i>(Making sense of problems and persevere in solving them). The research question that guided my work was the following: How is the <i>Common Core State Standards for Mathematics </i>(<i>CCSSM</i>) Problem-Solving Mathematical Standard enacted in an algebra class while using a <i>Standards-</i>based curriculum to teach a quadratics unit?</p> <p>I explored this by focusing on the following sub-questions:</p> <ul> <li>Q1. What opportunities to enact the components of the Problem-Solving Mathematical Standard are provided by the written curriculum? </li> <li>Q2. In what way does the teacher’s implementation of the quadratics unit diminish or enhance the opportunities to enact the components of the Problem-Solving Mathematical Standard provided by the written curriculum? </li> <li>Q3. In what ways does the teacher’s enactment of problem-solving opportunities change over the course of the unit? </li> </ul> <p>Reviewing the literature related to the relevant learning theories (sociocultural theory, the situated perspective, and communities of practice), I outlined the history of <i>CCSSM, </i>National Council of Teachers of Mathematics <i>(</i>NCTM), National Research Council (NRC), and the <i>No Child Left Behind Act of 2001</i>. Exploring the details of <i>CCSSM</i>’s Standards of Mathematical Content (SMCs) and Standards of Mathematical Practice (SMPs), I discussed problem solving, the Problem Solving Components (PSCs) listed in the Problem-Solving SMP of <i>CCSSM</i>, teaching through problem solving, and <i>Standards-</i>based curricula, such as <i>College Preparatory Mathematics (CPM)</i> which is the algebra curricula I chose for this study. </p> <p>There are many definitions of the construct problem solving. <i>CCSSM </i>describes this construct in unique ways specifically related to student engagement. The challenge for teachers is to not only make sense of <i>CCSSM</i>’s definition of problem solving and its components, but also to enact it in the classroom so that mathematical understanding is enhanced. For this reason, studies revealing how classroom teachers implemented <i>CCSSM</i>, especially in terms of problem solving, are necessary. </p> <p>The Critical Theoretic/Action Research Paradigm is often utilized by researchers trying to improve their own practice; thus, I opted for an action research methodology because it could be conducted by the practitioner. These methods of data collection and analysis were employed in order to capture the nature of changes made in the classroom involving my teaching practice. I chose action research because this study met the key tenets of research in action, namely, a collaborative partnership concurrent with action, and a problem-solving approach. </p> <p>While I knew how I wanted to change my classroom teaching style, implementing the change was harder than anticipated. From the onset, I never thought of myself as an absolute classroom authority, because I always maintained a relaxed classroom atmosphere where students were made to feel comfortable. However, this study showed me that students did view my presence as the authority and looked to me for correct answers, for approval, and/or for reassurance that they were on the right track. My own insecurities of not knowing how to respond to students in a way to get them to interact more with their group and stop looking to me for answers, while not being comfortable forcing students to talk in front of their peers, complicated this study. While it was easy to anticipate how I would handle situations in the classroom, it was hard to change in the moment. </p> <p>The research revealed the following salient findings: while the written curriculum contained numerous opportunities for students to engage with the Focal PSCs, the teacher plays a crucial role in enacting the written curriculum. Through the teacher’s enactment of this curriculum, opportunities for students to engage with the Focal<i> </i>PSCs can be taken away, enacted as written, or enhanced all by the teacher. Additionally, change was gradual and difficult due to the complexities of teaching. Reflection and constant adapting are crucial when it comes to changing my practice. </p> As a classroom teacher, I value the importance of the changes that need to be made in the classroom to align with <i>CCSSM</i>. I feel that by being both a teacher and a researcher, my work can bridge the gap between research and classroom practice.

In school algebra, using different methods including factorization to solve quadratic equations is one common teaching and learning topic at upper secondary school level. This study is about analyzing the algebra content related to solving quadratic equations and the method of factorization as presented in Swedish mathematics textbooks with subject matter content knowledge (CK) and pedagogical content knowledge (PCK) as analytical tools. Mathematics textbooks as educational resources and artefacts are widely used in classroom teaching and learning. What is presented in a textbook is often taught by teachers in the classroom. Similarly, what is missing from the textbook may not be presented by the teacher. The study is based on an assumption that pedagogical content knowledge is embedded in the subject content presented in textbooks. Textbooks contain both subject content knowledge and pedagogical content knowledge. The primary aim of the study is to explore what pedagogical content knowledge regarding solving quadratic equations that is embedded in mathematics textbooks. The secondary aim is to analyze the algebra content related to solving quadratic equations from the perspective of mathematics as a discipline in relation to algebra history. It is about what one can find in the textbook rather than how the textbook is used in the classroom. The study concerns a teaching perspective and is intended to contribute to the understanding of the conditions of teaching solving quadratic equations. The theoretical framework is based on Shulman’s concept pedagogical content knowledge and Mishra and Koehler’s concept content knowledge. The general theoretical perspective is based on Wartofsky’s artifact theory. The empirical material used in this study includes twelve mathematics textbooks in the mathematics B course at Swedish upper secondary schools. The study contains four rounds of analyses. The results of the first three rounds have set up a basis for a deep analysis of one selected textbook. The results show that the analyzed Swedish mathematics textbooks reflect the Swedish mathematics syllabus of algebra. It is found that the algebra content related to solving quadratic equations is similar in every investigated textbook. There is an accumulative relationship among all the algebra content with a final goal of presenting how to solve quadratic equations by quadratic formula, which implies that classroom teaching may focus on quadratic formula. Factorization method is presented for solving simple quadratic equations but not the general-formed quadratic equations. The study finds that the presentation of the algebra content related to quadratic equations in the selected textbook is organized by four geometrical models that can be traced back to the history of algebra. These four geometrical models are applied for illustrating algebra rules and construct an overall embedded teaching trajectory with five sub-trajectories. The historically related pedagogy and application of mathematics in both real world and pure mathematics contexts are the pedagogical content knowledge related to quadratic equations.

Orientador: José Mario Martínez Pérez / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T05:08:42Z (GMT). No. of bitstreams: 1 FloresCallisaya_Hector_D.pdf: 2324904 bytes, checksum: e15e7624ccad0fdf64ce3c4d8095c20a (MD5) Previous issue date: 2012 / Resumo: Neste trabalho, serão propostos modelos matemáticos para problemas de empacotamento não reticulado de esferas em regiões limitadas por quadráticas no plano e no espaço. Uma técnica para construir representações ou parametrizações será introduzida, mediante a qual será possível encontrar um sistema de desigualdades que determinam o empacotamento de um número fixo de esferas. Desta forma, resolvemos o problema de empacotamento de esferas através de uma sequência de sistemas de desigualdades. Finalmente, para obter resultados eficientes, minimizaremos a função de sobreposição, usando o método do Lagrangiano Aumentado / Abstract: In this work, we will propose mathematical models for not latticed packing of spheres problems in regions bounded by quadratic in the plane and in the space. A technique to construct representations or parameterizations will be introduced, by which it will be possible to find a system of inequalities which determine the packing of a fixed number of spheres. Thus, we solve the problem of packing spheres through a sequence of systems of inequalities. Finally, to obtain effective results, we will minimize the overlay function using the Augmented Lagrangian Method / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada