Forskning om de särskilda matematikkunskaper som lärare behöver för att undervisa om proportionella samband är begränsad. Därav är syftet med denna studie att bidra med insikter om pedagogiska ämneskunskaper inom proportionella samband. Det görs med utgångspunkt i två frågeställningar, som utgår från två kategorier av pedagogiska ämneskunskaper i matematik som Ball et al. (2008) beskrivit: (1) Vilka kunskaper om proportionalitet och elever visar lärare? (2) Vilka kunskaper om proportionalitet och undervisning visar lärare? Studien har genomförts med hjälp av kvalitativa semistrukturerade intervjuer med fyra lärare som arbetar i grundskolan och undervisar i matematik. Resultatet visar på flera exempel på vad som är kunskap inom de båda kategorierna. Bland annat beskrivs olika representationsformer och hur de kan användas för progression av undervisning som exempel på kunskaper om proportionalitet och undervisning. Sådana kunskaper är av stor betydelse att lärare besitter eftersom förståelse för proportionella samband framhålls som viktigt inom flera områden i matematiken. / Research about the special kind of mathematical knowledge that teachers need to teach proportional relationships is limited. Therefore, the aim of this study is to contribute with insights into pedagogical content knowledge about proportionality. To study this, two questions were posed: (1) What can knowledge of proportionality and students be? (2) What can knowledge of proportionality and teaching be? The two questions are based on two different categories of pedagogical content knowledge in mathematics that Ball et al. (2008) have described. This study has been conducted through qualitative semi-structured interviews with four teachers who work with students in the ages of 10-13 and teaches mathematics.  The result of the study shows several examples of pedagogical content knowledge in both categories that has been studied. For example, different representations and how they can be used to progress the teaching are described as examples of knowledge of proportionality and teaching. It is of great importance that teachers possess such knowledge considering understanding of proportional relationships has been described as a key to many areas of mathematics.

Thesis (MEd (Curriculum Studies)--Stellenbosch University, 2010. / ENGLISH ABSTRACT: The aim of this research is three-fold. Firstly I aimed to show the difficulty of the concept of proportional reasoning through empirical research. Several researchers have shown the degree of difficulty learners experience with proportional reasoning and have even indicated that many university students (and adults) do not have sound proportional reasoning skills. Piaget’s controversial developmental levels classify proportional reasoning as a higher order thinking skill in his highest level of development, formal operational thought, and claims that most people do not reach this level. The difficulty of proportional reasoning and the fact that it is a skill needed within all Learning Outcomes of Mathematical Literacy creates a predicament in terms of the difficulty of the subject in general. Is it then fair to classify Mathematical Literacy as an inferior subject in the way it has been done over the last few years if it is a subject that requires learners to operate at such a high level of thought through proportional reasoning? Secondly, I would like to confirm with the use of a baseline assessment that learners entering Grade 10 Mathematical Literacy have poor proportional reasoning skills and have emotional barriers to Mathematics and therefore Mathematical Literacy. The research will be done in three private schools located in the West Coast District of the Western Cape in South Africa. If learners in these educationally ideal environments demonstrate poor proportional reasoning skills even though they were privileged enough to have all the possible support since their formative years, then results from overcrowded government schools may be expected to be even worse. The learners in Mathematical Literacy classes often lack motivation, interest and enthusiasm when it comes to doing mathematics. Through the baseline assessment I confirm this and also suggest classroom norms and values that will help these learners to become involved in classroom activities and educational discourse. Thirdly and finally this research will focus on the design of activities that will aim to build on learners’ prior knowledge and further develop their proportional reasoning skills. I argue that activities to develop proportional reasoning should take equivalence of fractions as basis to work from. The activities will aim to help learners to set up questions in such a way that they can solve it with techniques with which they are familiar. Interconnectivity will form a vital part to this investigation. Not only do I indicate the interconnectivity between concepts in the Mathematical Literacy Learning Outcomes of the National Curriculum Statement, but I would like to make these links clear to learners when working through the proposed activities. Making links between concepts is seen as a higher order thinking skill and is part of meta-cognition which involves reflection on thoughts and processes. In short, this research can be summarised as the design of activities (with proposed activities) that aims to develop proportional reasoning by making connections between concepts and requires of learners to be active participants in their own learning. / AFRIKAANSE OPSOMMING: Die doel van hierdie navorsing is drieledig. Eerstens will ek die probleme met die konsep van proporsionele denke uitlig deur eksperimentele ontwerp navorsing. Verskeie navorsers verwys na die moeilikheidsgraad van probleme wat leerders ondervind met proporsionele denke. Sommige van hierdie navorsers het ook bevind dat verskeie universiteitstudente (en ander volwassenes) nie oor die vaardigheid van proporsionele denke beskik nie. Piaget se kontroversiële ontwikkelingsvlakke klassifiseer proporsionele denke as ‘n hoër orde denkvaardigheid in sy hoogste vlak van ontwikkeling, formele operasionele denke, en noem dat meeste mense nooit hierdie vlak bereik nie. Die hoë moeilikheidsgraad van proporsionele denke en die feit dat dit ‘n vaardigheid is wat binne al die Leeruitkomste van Wiskundige Geletterdheid benodig word veroorsaak ‘n dilemma as mens dit vergelyk met die moeilikheidsgraad van die vak oor die algemeen. Tweedens wil ek met behulp van ‘n grondfase assessering bewys dat leerders wat Graad 10 Wiskunde Geletterdheid betree swak proporsionele denkvaardighede het, gepaardgaande met emosionele weerstand teenoor Wiskunde en Wiskunde Geletterdheid. Die navorsing sal gedoen word in drie privaatskole in die Weskus distrik van die Wes-Kaap van Suid-Afrika. Indien leerders in hierdie ideale opvoedkundige omstandighede swak proporsionele denkvaardighede ten toon stel, ten spyte van die feit dat hulle bevoorreg was om sedert hulle vormingsjare alle moontlike opvoedkundige ondersteuning te geniet, dan kan verwag word dat resultate komende van oorvol staatskole selfs swakker mag wees. By leerders in Wiskunde Geletterdheid klasse kan daar gereeld ‘n gebrek aan motivering, belangstelling en entoesiasme ten opsigte van Wiskunde bespeur word. Deur gebruik van die grondfase assessering wil ek hierdie stelling bewys en ook voorstelle maak vir klaskamernorme en waardes wat sal help om die leerders meer betrokke te maak by klaskameraktiwiteite en opvoedkundige gesprekke.

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2016-05-12T14:21:23Z No. of bitstreams: 1 PDF - Edvanilson Santos de Oliveira.pdf: 3909149 bytes, checksum: a0062756c15dd26b2f383e858a5cd279 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2016-07-21T20:45:31Z (GMT) No. of bitstreams: 1 PDF - Edvanilson Santos de Oliveira.pdf: 3909149 bytes, checksum: a0062756c15dd26b2f383e858a5cd279 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2016-07-21T20:45:42Z (GMT) No. of bitstreams: 1 PDF - Edvanilson Santos de Oliveira.pdf: 3909149 bytes, checksum: a0062756c15dd26b2f383e858a5cd279 (MD5) / Made available in DSpace on 2016-07-21T20:45:42Z (GMT). No. of bitstreams: 1 PDF - Edvanilson Santos de Oliveira.pdf: 3909149 bytes, checksum: a0062756c15dd26b2f383e858a5cd279 (MD5) Previous issue date: 2015-12-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Our research work aimed to investigate the use of Robotics in Mathematics Education as technology capable of contributing in the development of the proportional reasoning achieved by students in Elementary Years, revealing itself as a new field that traces the national panorama. Besides new technologies being implanted in schools, Robotics is a pedagogic instrument still little broadcasted in Brazil, especially in the northeast region. The experiences and investigations involving Educational Robotics in the teaching of Mathematics are scarce. Our research work involved the first years of Robotics introduction in the Mathematics Education in a public school located in Campina Grande, Paraiba. For such we elaborated a theoretical contribution based on Educational Robotics (ER), characteristics and conceptual aspects of proportional reasoning, and the Theory of Relation with Knowledge. With this, we presented a didactic proposal developed from a collaborative work with teachers and Mathematics undergraduate students who made part of a bigger project, OBEDUC/CAPES in UFMS, UEPB and UFFAL institutions. The field work was performed with 8th grade students of elementary years. For our investigation we explored the way students relate themselves with the ER activities that explores the development of proportional reasoning, considering identity, epistemic and social dimensions; and in which way these relations can mobilize the potential of learning. We analyzed the students register, our subjects, through questionnaires, essays, videos, audios and resolution activities with robots. We also observed and interviewed them. From our research results we can assert that Educational Robotics, when worked with an adequate didactic proposal, can promote the development of proportional reasoning in a broader way, providing meaningful changes in the classroom. / Nossa pesquisa teve como objetivo investigar o uso da Robótica no âmbito da Educação Matemática como tecnologia capaz de contribuir no desenvolvimento do raciocínio proporcional por alunos do Ensino Fundamental, revelando-se como novo campo que delineia o panorama nacional. Apesar da inserção de novas tecnologias na escola, a Robótica constitui- se de um instrumento pedagógico ainda pouco difundido no Brasil, em especial na região nordeste. Escassas são as experiências e investigações envolvendo Robótica Educacional no ensino da Matemática. Nossa pesquisa envolveu-se nos primeiros anos de introdução da Robótica no contexto da Educação Matemática em uma escola pública localizada na cidade de Campina Grande, Paraíba. Para tanto, elaboramos como aporte teórico Robótica Educacional (RE), características e aspectos conceituais do raciocínio proporcional e a Teoria da Relação com Saber. Neste caminhar, apresentamos uma proposta didática desenvolvida a partir de um trabalho colaborativo com professores e alunos de graduação em Matemática, participes de um projeto maior, em rede, OBEDUC/CAPES, entre as instituições UFMS, UEPB e UFAL. A pesquisa de campo foi realizada com alunos do 8º ano do Ensino Fundamental. Para nossa investigação exploramos como se dá a relação de alunos do 8º ano do Ensino Fundamental com a RE em atividades que buscam explorar o desenvolvimento do raciocínio proporcional, considerando as dimensões identitária, epistêmica e social; e de que maneira estas relações podem mobilizar o potencial de aprendizagem. Analisamos o registro dos alunos, nossos sujeitos, a partir de questionários, redação, vídeos, áudios e resolução de atividades com robôs, além de os observarmos e entrevistarmos. A partir dos resultados de nossa pesquisa, podemos afirmar que a Robótica Educacional, aliada a uma proposta didática adequada, pode vir a promover o desenvolvimento do raciocínio proporcional de forma ampla, propiciando mudanças significativas na sala de aula.

Instructional coaching has been a professional learning opportunity that many school districts have employed to support teacher practice. Pairing instructional coaching with on-going workshops is a relatively new approach to professional development. Participants for this study include fourteen middle school teachers that teach either mathematics or collaborate with special needs students. This study examines the effect that pairing instructional coaching with on-going workshops (with a primary focus on proportional reasoning) has on participants’ content knowledge and their perceptions of coaching. Drawing on Wenger’s community of practice theory and post-modern theory of power, this study employs mixed-methods design. Pre- and post-tests for proportional reasoning were administered to analyze the extent to which content knowledge changed over the course of the study. Pre- and post-interviews were conducted with each participant to determine any misconceptions each had on proportional reasoning and their perceptions of coaching (before and after the study’s instructional coaching). Grounded theory and thematic analysis was employed on the pre-and post-interviews to examine the role that power played in the participants’ perceptions of effective coaching attributes. Results suggest that (a) instructional coaching coupled with on-going professional workshops can change content knowledge in participants; (b) perceptions of coaching can change as the result of experiencing a coaching relationship and (c) power dynamics in the coaching experience determine the extent to which participants see the effectiveness of coaching as a professional development activity.

The main purpose of this study was to investigate the effect of creative drama based instruction on seventh grade students&rsquo / achievement in ratio and proportion concepts and their attitudes toward mathematics. Another purpose of this study was to investigate students&rsquo / self-reported views related to creative drama based instruction. The study was conducted in a public school in K&ouml / rfez-Kocaeli with a total of 58 seventh grade students, lasting 12 lesson hours (three weeks). Thirty of the participants received Creative Drama Based Instruction (CDBI), and twenty-eight received Traditional Instruction (TI). The data were collected through Ratio and Proportion Achievement Test (RPAT), Mathematics Attitude Scale (MAS), and interviews. The RPAT and MAS were administered as both pretest and posttest. In addition, interviews were conducted with the ten randomly selected students. The quantitative analyses were carried out by using One-Way Analysis of Covariance (ANCOVA) with covariate preRPAT and dependent variable postRPAT at the significance level 0.05. Moreover, independent samples t-test was performed on gain scores of MAS. The results of the study indicated that there was a statistically significant mean difference between the students who received creative drama based instruction and traditional instruction in terms of achievement in ratio and proportion concepts and in terms of gain scores of attitudes toward mathematics, in favor of CDBI. Furthermore, according to the interview responses of the experimental group students, significantly better performance of the experimental group students was attributable to the potential of the creative drama based instruction to provide actively involvement, work with friends and collaboratively and providing selfawareness.

Este trabalho, realizado no âmbito do Curso de Mestrado Profissional em Ensino de Ciências e Matemática, do Programa de Pós-Graduação em Ensino de Ciências e Matemática da Faculdade de Ciências Integradas do Pontal da Universidade Federal de Uberlândia, visa apresentar uma sequência didática para favorecer o desenvolvimento do raciocínio proporcional tendo como suporte teórico a Teoria dos Campos Conceituais (TCC) de Gerard Vergnaud. São objetivos específicos: (a) analisar uma sequência didática, organizada na forma de situações-problema visando favorecer o estabelecimento das relações de covariação e de invariância de grandezas, necessárias para conceituar razão e proporção e (b) analisar o desempenho e as estratégias utilizadas pelos alunos para resolver situações- problema envolvendo o raciocínio proporcional, ao longo da aplicação da proposta didática. O trabalho tem apoio metodológico na Engenharia Didática e foi desenvolvido junto a aproximadamente 26 alunos do 6º ano do Ensino Fundamental de uma escola da cidade de Ituiutaba/MG, no período regular de aulas. A sequência teve seis etapas, na primeira foi aplicada uma avaliação tipo lápis e papel e as etapas seguintes foram constituídas por situações-problema aplicadas com mediação da professora e também por avaliações, sendo que estes instrumentos foram elaborados com base na literatura existente sobre o tema raciocínio proporcional. Os dados foram analisados quantitativamente por meio da estatística descritiva e qualitativamente quando foram organizadas categorias de análise. Na primeira etapa, houve mais dificuldade nos problemas de comparação que nos de valor omisso e as estratégias multiplicativas utilizadas pelos alunos indicaram alguma inferência e predição na compreensão de que as grandezas envolvidas nos problemas variavam em conjunto. Ao longo da aplicação da sequência, verificou-se que os alunos passaram a identificar as grandezas proporcionais envolvidas nas situações e a maioria deles conseguiu justificar as repostas por meio da relação de covariação, valendo-se da simbologia adequada. Considera-se que a opção metodológica de oferecer situações diversificadas antes da apresentação formal desse conteúdo (que acontece a partir do sétimo ano do Ensino Fundamental) contribui para desenvolver o raciocínio proporcional dos alunos. Espera-se que as análises e discussões teóricas realizadas nesse trabalho possam contribuir para a prática do professor de matemática. / This work, carried out during the Masters Course for teaching Science and Mathematics, within the postgraduate program in teaching Science and Mathematics at the Pontal College of Integrated Sciences of the Federal University of Uberlândia. It a aims to introduce a didactic sequence to encourage the development of proportional reasoning with the theoretical support the Theory of Conceptual Fields (TCC) by Gerard Vergnaud. The specific objectives are: (a) to analyze a didactic sequence, arranged in the form of problem situations in order to promote the establishment of relations of covariance and invariance of quantities, required to conceptualize reason and proportion and (b) examine the performance and the strategies used by the students to solve problem situations involving proportional reasoning, through the application of didactic proposal. The work has methodological support in Teaching Engineering and was developed together with approximately 26 students of the sixth grade of an elementary school in the city of Ituiutaba, Minas Gerais, during regular class periods. The sequence had six stages, the first was a pencil and paper type evaluation and review the following stages were composed of problem situations applied with mediation of a teacher as well as evaluations, being that these instruments have been drawn up on the basis of the existing literature on the subject of proportional reasoning. Data was analyzed through quantitative and qualitative descriptive statistics when the categories of analysis were set up. In the first step, there was more difficulty with the problems of comparison than those of omissive value and the multiplicative strategies used by the students indicated some inference and prediction on the understanding that the values involved in the problems varied together. Throughout the implementation of the result, it was found that the students have come to identify the proportional quantities involved in situations and most of them managed to justify the answers through the relationship of covariance, using the appropriate symbol. The methodological option of offering diverse situations before the formal presentation of this content (which happens from the seventh grade of primary school) helps to develop proportional reasoning of students. It is expected that the analysis and theoretical discussions undertaken in this work could contribute to the work of a Maths teacher. / Dissertação (Mestrado)

<p>  </p> <p>Radian angle measure is central to learning trigonometry, with researchers providing evidence that a coherent understanding of radian contributes to a coherent understanding of trigonometric and inverse trigonometric functions. However, there are few opportunities for students to engage with curricular situations that involve radian angle measure. The purpose of this dissertation is to explore and provide insights into preservice mathematics teachers’ (PMTs’) conceptions of radian angle measure using three curricular situations. The first chapter reviews the relevant literature, which reported that PMTs’ conceptions of radian angle measure involve angles measured in terms of π, in relation to degrees, and in relation to the unit circle. In chapter two, I explored PMTs’ conceptions of radian angle measure using textbook representations. Seven PMTs participated in a think-aloud semi-structured interviews, where they defined radian angle measure from six textbook diagrams of radian, including a diagram of the unit circle. In chapter three, building on literature that reported that PMTs’ conceptions of radian angle measure involve relating radian to degrees, I explored how PMTs conceptualize this relationship. Five PMTs participated in semi-structured interviews, where they described radian angle measure given the angle measure in degrees. In chapter four, I explored the PMTs’ conceptions of radian angle measure given a novel context. Four PMTs participated in semi-structured virtual interviews, where they engaged with a digital activity that involves radian angle measure in the context of light reflection. Some of the dissertation’s findings align with previous research, where PMTs’ conceptualized radian angle measure in relation to the unit circle. However, this dissertation provides empirical evidence of why the PMTs refer to the unit circle. The PMTs acknowledged knowing the unit circle from memorization, but also explained that the purpose for using the unit circle is efficiency. At the same time, the PMTs acknowledged limitations in the unit circle and in their conceptions of it. Overall findings from the dissertation demonstrate the complexity of PMTs’ conceptions of radian angle measure. The PMTs’ conceptions were reported as concept definitions, ways of thinking, and spatial ways of thinking. The PMTs demonstrated flexibility with reasoning about radian angle measure using foundational conceptions in learning higher mathematics topics (e.g., proportional reasoning concepts, spatial ways of thinking). By positioning the PMTs as knowers and thinkers with valuable insights to provide, I was able to uncover and report a collection of conceptions that were demonstrated by PMTs when a curricular situation involved radian angle measure. The findings from this dissertation extend existing research that explored conceptions of angle measure and radian angle measure by reporting PMTs’ conceptions of radian angle measure given three different curricular situations. While there is still much that needs to be investigated about complexities in PMTs’ conceptions of radian angle measure, this dissertation represents one step toward providing insights about those complexities. </p>

Underachievement in Mathematics hangs over South African Mathematics learners like a dark cloud. TIMSS studies over the past decade have confirmed that South African learners‟ results (Grades 8 and 9 in 2011) remained at a low ebb, denying them the opportunity to compete and excel globally in the field of Mathematics. It is against this backdrop that the researcher investigated the meaningful understanding of the important yet challenging algebraic concept of Proportion. The theoretical as well as the empirical underpinnings of the fundamental idea of Proportion are highlighted. The meaningful learning of Algebra was explored and physical, effective and cognitive factors affecting meaningful learning of Algebra, views on Mathematics and learning theories were examined. The research narrowed down to the meaningful understanding of Proportion, misconceptions, and facilitation in developing Proportional reasoning. This study was embedded in an interpretive paradigm and the research design was qualitative in nature. The qualitative data was collected via task sheets and interviews. The sample informing the central phenomenon in the study consisted of a heterogeneous group of learners and comprised a kaleidoscope of nationalities, both genders, a variety of home languages, differing socio-economic statuses and varying cognitive abilities. The findings cannot be generalised. Triangulation of the literature review, the analysis of task sheets and interviews revealed that overall the participants have a meaningful understanding of the Proportion concept. However, a variety of misconceptions were observed in certain cases. Finally, recommendations are made to address the meaningful learning of Proportion and its associated misconceptions. It is hoped that teachers read and act on the recommendations as it is the powerful mind and purposeful teaching of the teacher that can make a difference in uplifting the standard of Mathematics in South African classrooms! / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Underachievement in Mathematics hangs over South African Mathematics learners like a dark cloud. TIMSS studies over the past decade have confirmed that South African learners‟ results (Grades 8 and 9 in 2011) remained at a low ebb, denying them the opportunity to compete and excel globally in the field of Mathematics. It is against this backdrop that the researcher investigated the meaningful understanding of the important yet challenging algebraic concept of Proportion. The theoretical as well as the empirical underpinnings of the fundamental idea of Proportion are highlighted. The meaningful learning of Algebra was explored and physical, effective and cognitive factors affecting meaningful learning of Algebra, views on Mathematics and learning theories were examined. The research narrowed down to the meaningful understanding of Proportion, misconceptions, and facilitation in developing Proportional reasoning. This study was embedded in an interpretive paradigm and the research design was qualitative in nature. The qualitative data was collected via task sheets and interviews. The sample informing the central phenomenon in the study consisted of a heterogeneous group of learners and comprised a kaleidoscope of nationalities, both genders, a variety of home languages, differing socio-economic statuses and varying cognitive abilities. The findings cannot be generalised. Triangulation of the literature review, the analysis of task sheets and interviews revealed that overall the participants have a meaningful understanding of the Proportion concept. However, a variety of misconceptions were observed in certain cases. Finally, recommendations are made to address the meaningful learning of Proportion and its associated misconceptions. It is hoped that teachers read and act on the recommendations as it is the powerful mind and purposeful teaching of the teacher that can make a difference in uplifting the standard of Mathematics in South African classrooms! / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

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